Geometry and Topology of Manifolds: Surfaces and Beyond 1470461323, 9781470461324

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GRADUATE STUDIES I N M AT H E M AT I C S

208

Geometry and Topology of Manifolds Surfaces and Beyond Vicente Muñoz Ángel González-Prieto Juan Ángel Rojo

Geometry and Topology of Manifolds Surfaces and Beyond

GRADUATE STUDIES I N M AT H E M AT I C S

208

Geometry and Topology of Manifolds Surfaces and Beyond

Vicente Muñoz Ángel González-Prieto Juan Ángel Rojo

Editorial Board of Graduate Studies in Mathematics EDITORIAL COMMITTEE Gigliola Staffilani (Chair) Marco Gualtieri

Bjorn Poonen

Jeff A. Viaclovsky

Editorial Committee of the Real Sociedad Matem´ atica Espa˜ nola Pedro J. Pa´ ul, Director EDITORIAL COMMITTEE Luis J. Al´ıas Alberto Elduque Manual Maestre Ana Mar´ıa Mancho

Andrei Martinez-Finkelshtein Rosa Mir´o-Roig Luz Roncal Mar´ıa Dolores Ugarte

2010 Mathematics Subject Classification. Primary 57-02, 53-02, 55-02, 30Fxx, 53Cxx.

The photographs on the back cover were supplied by the authors.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-208

Library of Congress Cataloging-in-Publication Data Names: Mu˜ noz, V. (Vicente), 1971– author. ´ Title: Geometry and topology of manifolds : surfaces and beyond / Vicente Mu˜ noz, Angel ´ Gonz´ alez-Prieto, Juan Angel Rojo. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Graduate studies in mathematics, 1065-7339 ; 208 | Includes bibliographical references and index. Identifiers: LCCN 2020019160 | ISBN 9781470461324 (softcover) | ISBN 9781470461621 (ebook) Subjects: LCSH: Manifolds (Mathematics) | Geometry, Differential. | Algebraic topology. | AMS: Manifolds and cell complexes – Research exposition (monographs, survey articles). | Differential geometry – Research exposition (monographs, survey articles). | Algebraic topology – Research exposition (monographs, survey articles). | Functions of a complex variable – Riemann surfaces. | Differential geometry – Global differential geometry Classification: LCC QA613 .M86 2020 | DDC 514/.34–dc23 LC record available at https://lccn.loc.gov/2020019160

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established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

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Contents

Preface Chapter 1.

vii Topological surfaces

§1.1. Topological and smooth manifolds

1 1

§1.2. PL structures

12

§1.3. Planar representations of surfaces

28

§1.4. Orientability

34

§1.5. Classification of compact surfaces

41

Problems

47

References and extra material

49

Chapter 2.

Algebraic topology

53

§2.1. Homotopy theory

53

§2.2. Covers

69

§2.3. Singular homology

84

§2.4. Simplicial homology

94

§2.5. De Rham cohomology

104

§2.6. Poincaré duality

120

Problems

131

References and extra material

133

Chapter 3.

Riemannian geometry

137

§3.1. Riemannian metrics. Curvature. Geodesics

137

§3.2. Riemannian surfaces and the Gauss-Bonnet theorem

160

§3.3. Isotropic, symmetric, and homogeneous manifolds

181

Problems

192 v

vi

Contents

References and extra material Chapter 4.

Constant curvature

194 197

§4.1. Positive constant curvature

197

§4.2. Vanishing constant curvature

217

§4.3. Negative constant curvature

232

§4.4. Classical geometries

258

Problems

260

References and extra material

262

Chapter 5.

Complex geometry

265

§5.1. Complex manifolds

265

§5.2. Kähler manifolds

284

§5.3. Complex curves

300

§5.4. Classification of complex curves

315

§5.5. Elliptic curves

323

Problems

337

References and extra material

340

Chapter 6.

Global analysis

343

§6.1. Conformal structures

343

§6.2. Hodge theory

350

§6.3. Metrics of constant curvature

362

§6.4. The curvature flow

374

Problems

386

References and extra material

388

Subject Index

391

Symbols

403

Preface

The present book arises from notes of a master’s course that the first author delivered from the academic course 2011/12 until 2017/18 for the Master’s Degree in Mathematics at Universidad Complutense de Madrid. The prerequisites are basic courses in linear algebra, elementary topology and algebraic topology, differential geometry, complex analysis, and partial differential equations. This is a somewhat non-standard course on differential topology, that is, its main focus is the geometry and topology of manifolds, trying to touch on many of its ramifications. Being of an introductory nature, it cannot aspire to include all material on the interplay of geometry and topology. On the other hand, we consider it important in a text of this type to give complete proofs of some of the important landmarks in the development of the area. For this reason, we have decided to introduce the different aspects of the theory of manifolds in arbitrary dimension, but then at every chapter we move to give important results and full proofs for the case of dimension 2, that is, surfaces. The title of the book arises from this consideration. The content of the book is distributed into six chapters, following the philosophy of going from the soft to the hard. In Chapter 1, we start with the topological side of the course, introducing manifolds as natural objects to study (natural from the geometrical and from the physical points of view) and setting up the core problem of the classification of manifolds. Smooth manifolds are introduced, where a differentiable structure on a manifold is the way to make sense of the concept of differentiation, which is needed to pose (and solve) physical problems on a manifold. The chapter contains complete proofs of two important results in the case of surfaces: the existence of triangulations and the classification of compact surfaces. Chapter 2 introduces the theory of algebraic topology. This is the area of mathematics that constructs algebraic invariants to study the topology of spaces. For manifolds, topology is the way to go from the local to the global. By definition, a manifold is a space that locally looks like a Euclidean space (which is the model of the space in which we live). So locally there is no information on a manifold, and the global

vii

viii

Preface

information is encoded in the algebraic topology invariants: homotopy and homology groups. We introduce de Rham cohomology by its paramount importance, where the differentiable structure of a smooth manifold is used to get implications in the algebraic topology of the manifold, giving a link between geometry and topology. We delve in Chapter 3 into Riemannian geometry. Both from the geometric and from the physical points of view, it is natural that a smooth manifold can have more structure. For example, this could be a way to measure the lengths, angles, or whatever other concept of geometric significance can be computed locally (at a point). These give rise to different types of geometric structures. We focus on Riemannian metrics, as these were historically the first ones to be studied, and the most thoroughly considered in differential geometry, but by no means the only interesting ones. A metric allows us to define curvature, which helps in understanding how much a manifold is intrinsically bended. We introduce the theory in arbitrary dimensions and focus on the case of surfaces. The chapter includes the proof of the very important Gauss-Bonnet theorem for surfaces, which links the curvature of a surface to its global topology. We include a discussion of orbifolds (which are like manifolds but with singular points) and their Riemannian structures, where the curvature appears concentrated. This point of view is of interest in itself. To deepen on the relation that takes us from the local to the global, it is necessary to focus on manifolds which look the same from one point to another (homogeneous, isotropic). From the physical point of view, according to the Einstein conception of the universe, isotropic spaces arise as cosmological models in which the physical mechanisms that rule the natural phenomena are the the same at every point of the space, and with no preferred directions. From the geometric point of view, these are the spaces where we can move figures from one point to another. It is natural that they are at the heart of the origin of geometry, first with the Euclidean geometry and later with the non-Euclidean geometries. Chapter 4 focuses on the specific study of the three possibilities for isotropic surfaces, i.e., constant curvature surfaces. We will analyse the case of positive curvature which gives rise to spherical and projective geometry, the case of zero curvature which is Euclidean geometry, and the case of negative curvature which produces hyperbolic geometry. Our main focus is on surfaces, so we give a classification of compact surfaces with these types of geometries. The classification is thoroughly given for the case of tori (zero curvature), and an introduction is given to Teichmüller theory for the important topic of hyperbolic metrics on compact surfaces. Chapter 5 analyses a very important enrichment that a manifold may admit, namely that of a complex manifold, in which the tangent space has a natural complex structure. This upgrade has deep implications from the point of view of differential geometry that echoes in the global topology of the manifold. For instance, this complex setting induces a new enhanced algebraic structure on de Rham cohomology, relating it with another cohomology intrinsically tied to the complex structure called Dolbeault cohomology. But this enhancement also has important consequences from from the point of view of algebra, building a bridge between complex manifolds and (smooth) projective varieties, that is the zero locus of complex polynomials. These varieties are the main objects of study in algebraic geometry. Following the general philosophy of the book, we particularize to the case of surfaces with complex structures, that is complex

Preface

ix

curves. These structures are equivalent to conformal structures (Riemannian metrics up to a variable dilation factor) thanks to the uniformization theorem. Strikingly, the classification of complex curves parallels that of surfaces with constant curvature. A very important result that we fully prove in the chapter is the degree-genus formula. It says that the topology of a complex curve (its genus) is given by the degree of the planar model (a polynomial in the complex plane which defines the complex curve). The chapter ends with the classification of elliptic curves, which parallels that of tori given in Chapter 5. The final chapter, Chapter 6, moves into the links of geometry and the theory of partial differential equations, consisting of the study of differential equations on smooth manifolds. The more relevant equations have their origin in geometrical questions or are of physical significance. This area is commonly known as global analysis, since its main features are the implications of the global nature of geometrical spaces on the properties of differential equations. The local aspect, the study of differential equations on open sets of the Euclidean space, are studied in mathematical analysis. We review the theory of harmonic forms and its interplay with de Rham cohomology, which is an elliptic problem on a compact manifold. The chapter gives a proof of the existence of metrics of constant curvature on a conformal structure of a compact surface. Our proof in the positive curvature case hinges on a nice trick using orbifolds. In particular, we give a complete proof of the uniformization theorem with analytical techniques. We end up with a brief introduction to the Ricci flow, a current topic which has produced very strong results in geometrization. This manuscript can be used to deliver a course at postgraduate level. As such, this book may serve as a reference for a first course that explores the interface between differential topology and algebraic topology. With this objective, the course should be focused on the material of Chapter 1, with special attention to the classification of compact triangulated surfaces, Chapter 2, especially singular homology and its interplay with de Rham cohomology and Chapter 3, reaching the celebrated Gauss-Bonet theorem as an interplay between topology and geometry. However, the main aim of this book is to serve as basic reference for a postgraduate course, at the level of a master’s course or an advanced PhD course depending on the background of the targeted audience. In this way, in a half-year course, most of the material of Chapters 1, 2, 3, 4, and 5 may be covered (perhaps omitting the review of known topics). With a view towards a course presenting a more analytic perspective, the material of Chapter 2 can be reduced (especially higher homotopy groups, Seifert-van Kampen theorem, and simplicial homology) and the final part of Chapter 5 (degree-genus formula and elliptic curves) may be replaced with the contents of Chapter 6. Following this aspiration as a textbook, each chapter has been complemented with an extensive collection of problems, ranging from easier to very difficult ones. Some of them complete results appearing inthe text, and others give hints to profound ramifications that have not been treated. Each chapter ends with a list of topics for further study and with a number of references. The topics have been chosen so that they can be

x

Preface

proposed to students, who will write small dissertations, as the first author has successfully done during the years that he has delivered the master’s course. The references are divided into basic reading (i.e., texts where the content of the chapter is treated in full), bibliography for the topics for further study, and references which go beyond of the content of the chapter or that have been mentioned in the text. Most of the topics of the book can be found in other texts with more specific aims. Some of the material has been written in a review form (such as homology theory, Riemannian geometry, or differential equations on manifolds). We hope that this book serves as motivation for learning all these aspects by reading deeper treatises which cover the different theories at large. Our aim is to focus in the interconnections between all these aspects. Modern geometry (from the mid-twentieth century) has seen the most important advances produced on the interaction with algebra, physics, or analysis. Vicente Muñoz Ángel González-Prieto Juan Ángel Rojo

Chapter 1

Topological surfaces

Topology is the area of mathematics which studies topological spaces, i.e., spaces where continuity can be defined. But topological spaces can be very weird, and at some point, one may wonder whether spaces with strange topologies serve any purpose apart from being helpful to understand the theory. Natural spaces are those attached to geometrical problems (that is, about figures that can be drawn and manipulated in our space) or physical problems (that is, particles and systems moving according to some physical laws). In these spaces, we can naturally put coordinates to locate the figures or the particles (at least in a local region of the space), as happens for the surface of the Earth. This leads to the notion of manifold as the most important object to study in topology. In this chapter we introduce manifolds, and we focus on those of dimension 2 (i.e., surfaces), providing a proof of the classification theorem for compact surfaces.

1.1. Topological and smooth manifolds In order to fix notation within this book, we denote by ℝ𝑛 the 𝑛-dimensional Euclidean space, with the usual topology. The open ball of radius 𝑟 > 0 centered at 𝑝 ∈ ℝ𝑛 will be denoted 𝐵𝑟𝑛 (𝑝) = {𝑥 ∈ ℝ𝑛 | ||𝑥 − 𝑝|| < 𝑟}. We will shorten 𝐵 𝑛 = 𝐵1𝑛 (0), the standard open ball. The closed ball, or disc, will be denoted 𝐵𝑟̄𝑛 (𝑝) = {𝑥 ∈ ℝ𝑛 | ||𝑥 − 𝑝|| ≤ 𝑟}, and the standard disc is 𝐷𝑛 = 𝐵1̄𝑛 (0). The sphere will be 𝑆𝑟𝑛−1 (𝑝) = {𝑥 ∈ ℝ𝑛 | ||𝑥 − 𝑝|| = 𝑟}, and the standard sphere will be shortened as 𝑆 𝑛−1 = 𝑆𝑛−1 (0). Finally, when the 1 underlying dimension is clear from the context, we will omit it from the notation. Definition 1.1. A topological manifold 𝑀 (or just a manifold) is a (non-empty) Hausdorff and second countable topological space which is locally homeomorphic to Euclidean spaces, i.e., for every point 𝑝 ∈ 𝑀 there exists an open neighbourhood 𝑈 = 𝑈 𝑝 ⊂ 𝑀 of 𝑝 and a homeomorphism 𝜑 ∶ 𝑈 → 𝜑(𝑈) ⊂ ℝ𝑛 onto an open subset of some ℝ𝑛 . We call (𝑈, 𝜑) a chart of 𝑀. For 𝑞 ∈ 𝑈 we call 𝜑(𝑞) = (𝑥1 (𝑞), . . . , 𝑥𝑛 (𝑞)) the coordinates of 𝑞, and the functions 𝑥𝑖 ∶ 𝑈 → ℝ are called the coordinate functions. If we 1

2

1. Topological surfaces

'

M

'(U ) ⊂ Rn U

take 𝜖 > 0 small so that 𝐵𝜖𝑛 (𝜑(𝑝)) ⊂ 𝜑(𝑈), and 𝑉 = 𝜑−1 (𝐵𝜖𝑛 (𝜑(𝑝))), we have a chart 1 (𝑉, 𝜑) whose image is a ball. Translating and expanding with 𝜏(𝑥) = 𝜖 (𝑥 − 𝜑(𝑝)), we ′ ′ ′ have a chart (𝑉, 𝜑 = 𝜏 ∘ 𝜑) such that 𝜑 (𝑉) = 𝐵1 (0) and 𝜑 (𝑝) = 0. Finally, using the 𝑥 homeomorphism 𝐹 ∶ 𝐵1𝑛 (0) → ℝ𝑛 , 𝐹(𝑥) = 1−||𝑥|| , we get a chart (𝑉, 𝜙 = 𝐹 ∘ 𝜑′ ) whose image is the whole of ℝ𝑛 . To have this freedom of ranges of charts is useful in what follows. A collection of charts 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )} such that ⋃ 𝑈𝛼 = 𝑀 is called an atlas. The number 𝑛 in Definition 1.1 is called the dimension of 𝑀 at 𝑝. The fact that 𝑛 does not depend on the chart 𝜑 follows from the theorem of invariance of dimension. Theorem 1.2 (Invariance of dimension). Let 𝑈 ⊂ ℝ𝑛 and 𝑉 ⊂ ℝ𝑚 be open subsets such that there is a homeomorphism 𝑓 ∶ 𝑈 → 𝑉. Then 𝑛 = 𝑚. Proof. As the main focus of the course is that of surfaces (i.e., when 𝑛 = 2), we will prove the result for 𝑛 = 2, where it can be proved by means of the fundamental group (see section 2.1.3). For 𝑛 > 2, it can be proved using homology groups, which we introduce in section 2.3 (see Exercise 2.20). For 𝑛 = 1, it is proved by using arguments of connectivity (Exercise 1.2). So we assume also that 𝑚 ≥ 2. Let 𝑝 ∈ 𝑈 and 𝑓(𝑝) ∈ 𝑉. Take 𝜖 > 0 so that there is a ball 𝐵𝜖𝑚 (𝑓(𝑝)) ⊂ 𝑉. Take 𝛿 > 0 such that 𝐵𝛿̄𝑛 (𝑝) ⊂ 𝑓−1 (𝐵𝜖𝑚 (𝑓(𝑝))). There is a radial retraction 𝑟 ∶ 𝑓−1 (𝐵𝜖𝑚 (𝑓(𝑝))) − 𝑛−1 {𝑝} → 𝑆 𝑛−1 = 𝑆 1 . Therefore there is a surjection on the fundamental 𝛿 (𝑝) ≅ 𝑆 −1 𝑚 groups 𝜋1 (𝑓 (𝐵𝜖 (𝑓(𝑝))) − {𝑝}) → 𝜋1 (𝑆 1 ) ≅ ℤ. This implies that 𝑓−1 (𝐵𝜖𝑚 (𝑓(𝑝))) − {𝑝} is not simply connected. Clearly 𝑓 ∶ 𝑓−1 (𝐵𝜖𝑚 (𝑓(𝑝))) − {𝑝} → 𝐵𝜖𝑚 (𝑓(𝑝)) − {𝑓(𝑝)} is a homeomorphism, so 𝐵𝜖𝑚 (𝑓(𝑝))−{𝑓(𝑝)} is not simply connected. The space 𝐵𝜖𝑚 (𝑓(𝑝)) − {𝑓(𝑝)} is of the homotopy type of 𝑆 𝑚−1 = 𝑆 𝑚−1 (0), and for 𝑚 > 2, 𝑆 𝑚−1 is simply 1 connected. This is a contradiction, hence 𝑚 = 2. □ The fact that the number 𝑛 is independent of the chart readily yields that 𝑛 is constant on each connected component of 𝑀, so the dimension is a well defined number for each connected topological manifold. We say that 𝑀 is an 𝑛-manifold if 𝑀 is a manifold with all its connected components of dimension 𝑛. We also write 𝑛 = dim 𝑀. Remark 1.3. It is customary to call a topological 1-manifold a (topological) curve. Analogously, a topological 2-manifold is called a (topological) surface. 1.1.1. Categories. A large part of modern geometry consists of studying the geometric structures that a manifold can have. Each geometric structure allows one to use efficiently some mathematical tools (such as algebra, analysis, combinatorics) in the

1.1. Topological and smooth manifolds

3

study of manifolds. The final aim is to solve the classification problem, i.e., to give a complete list of all different manifolds and geometric structures. The correct setting for phrasing this problem is category theory. Let us pause to introduce this. Definition 1.4. A category 𝒞 consists of a collection of objects, denoted Obj(𝒞), and for every pair of objects 𝑋, 𝑌 ∈ Obj(𝒞), a collection of arrows or morphisms, denoted Mor𝒞 (𝑋, 𝑌 ), satisfy the following. • For object 𝑋 there exists an arrow denoted 1𝑋 ∈ Mor𝒞 (𝑋, 𝑋) (or Id if the object is clear from the context), called identity. • For objects 𝑋, 𝑌 , 𝑍 we have a binary operation called composition, Mor𝒞 (𝑋, 𝑌 ) × Mor𝒞 (𝑌 , 𝑍) → Mor𝒞 (𝑋, 𝑍),

(𝑓, 𝑔) ↦ 𝑔 ∘ 𝑓,

such that (1) 𝑓 ∘ 1𝑋 = 𝑓 = 1𝑌 ∘ 𝑓, 𝑓 ∈ Mor𝒞 (𝑋, 𝑌 ). (2) ℎ ∘ (𝑔 ∘ 𝑓) = (ℎ ∘ 𝑔) ∘ 𝑓 for all 𝑓 ∈ Mor𝒞 (𝑋, 𝑌 ), 𝑔 ∈ Mor𝒞 (𝑌 , 𝑍), ℎ ∈ Mor𝒞 (𝑍, 𝑊). 𝑓

It is common to denote by 𝑓 ∶ 𝑋 → 𝑌 or 𝑋 ⟶ 𝑌 , or even 𝑋 → 𝑌 , an arrow 𝑓 ∈ Mor𝒞 (𝑋, 𝑌 ). If 𝑓, 𝑔 can be composed, we say that they are composable. Example 1.5. • The category 𝐒𝐞𝐭 whose objects are sets and whose arrows are maps between sets. The composition is the usual composition of maps. • The category 𝐓𝐨𝐩 whose objects are topological spaces and whose morphisms are continuous maps. The category 𝐓𝐨𝐩∗ whose objects are pointed topological spaces, that is pairs (𝑋, 𝑝) with 𝑋 a topological space and 𝑝 ∈ 𝑋. A morphism 𝑓 ∶ (𝑋, 𝑝) → (𝑌 , 𝑞) is a continuous map 𝑓 ∶ 𝑋 → 𝑌 with 𝑓(𝑝) = 𝑞. • The category 𝐓𝐌𝐚𝐧 of topological manifolds, whose arrows are continuous maps. Recall that manifolds may be disconnected and may have components of different dimensions. In general, it is useful to specify the dimension and the compactness, so we will deal with the categories 𝐓𝐌𝐚𝐧𝑛 and 𝐓𝐌𝐚𝐧𝑛𝑐 whose objects are topological 𝑛-manifolds and compact topological 𝑛-manifolds, respectively. • Given a field 𝐤, the category 𝐕𝐞𝐜𝐭𝐤 whose objects are 𝐤-vector spaces, and whose arrows are 𝐤-linear maps. • The category 𝐆𝐫𝐨𝐮𝐩, whose objects are groups and whose arrows are homomorphisms of groups. The category 𝐀𝐛𝐞𝐥, whose objects are Abelian groups and arrows homomorphisms of groups. Here 𝐀𝐛𝐞𝐥 is a subcategory of 𝐆𝐫𝐨𝐮𝐩, that is, its objects and morphisms are contained in those of 𝐆𝐫𝐨𝐮𝐩. • For a topological space we define the category Π1 (𝑋) called the fundamental groupoid of 𝑋. The objects of this category are the points of 𝑋. The arrows between two points 𝑥, 𝑦 ∈ 𝑋 are the paths Ω𝑥,𝑦 (𝑋) = {𝛾 ∶ [0, 1] → 𝑋 | 𝛾 continuous, 𝛾(0) = 𝑥, 𝛾(1) = 𝑦},

4

1. Topological surfaces

modulo the equivalence relation of being homotopic relative to {𝑥, 𝑦} (Definition 2.1). The composition is the usual juxtaposition of equivalence classes of paths modulo homotopy Ω𝑥,𝑦 (𝑋) × Ω𝑦,𝑧 (𝑋) → Ω𝑥,𝑧 (𝑋). The identity 1𝑥 is the constant map with image {𝑥}. This gives an example of arrows not being functions, and composition not being composition of functions. ˆ • For a group (𝐺, ∘) we define the groupoid associated to 𝐺 as the category 𝐺 ˆ such that Obj(𝐺) = {⋆} and Mor𝐺ˆ (⋆, ⋆) = 𝐺. For two morphisms 𝑔1 and 𝑔2 , their composition is 𝑔1 ∘ 𝑔2 = 𝑔1 ⋅ 𝑔2 , i.e., their product as elements of 𝐺. Definition 1.6. Given a category 𝒞, we say that an arrow 𝑓 ∶ 𝑋 → 𝑌 is invertible if there exists 𝑔 ∶ 𝑌 → 𝑋 so that 𝑓 ∘ 𝑔 = 1𝑌 and 𝑔 ∘ 𝑓 = 1𝑋 . We write 𝑔 = 𝑓−1 and say that 𝑔 is the inverse of 𝑓. It is an exercise to see that the inverse is unique whenever it exists (cf. Exercise 1.1). It is also easy to see that if 𝑓 and ℎ are invertible and its composition makes sense, then the composition 𝑓 ∘ ℎ is also invertible and (𝑓 ∘ ℎ)−1 = ℎ−1 ∘ 𝑓−1 . If 𝑓 ∶ 𝑋 → 𝑌 is invertible, we say that 𝑋 and 𝑌 are isomorphic in the category 𝒞, and we write 𝑋 ≅𝒞 𝑌 (or 𝑋 ≅ 𝑌 , if there is no risk of misunderstanding). This defines an equivalence relation in Obj(𝒞). Note however that Obj(𝒞) may not be a set (with Zermelo-Fraenkel axioms), so we must be careful when it comes to talking of the “equivalence classes” of objects of 𝒞 up to isomorphism. Example 1.7. • In 𝐒𝐞𝐭 the invertible maps are the bijective maps between sets 𝑓 ∶ 𝑋 → 𝑌 . Two sets are isomorphic if and only if they have the same cardinal. • In 𝐓𝐌𝐚𝐧𝑛 and 𝐓𝐌𝐚𝐧𝑛𝑐 the invertible maps are the homeomorphisms, and two manifolds are isomorphic when they are homeomorphic. Observe that, in 𝐓𝐌𝐚𝐧𝑛𝑐 a continuous bijective map is automatically a homeomorphism (Proposition 1.28). • In 𝐕𝐞𝐜𝐭𝑘 the invertible maps are the bijective linear maps. Two objects are isomorphic when their dimensions have the same cardinal. • In Π1 (𝑋) all arrows are invertible. Indeed it is enough to define 𝛾−1 (𝑡) = 𝛾(1 − 𝑡) for any arrow 𝛾. Two objects are isomorphic if and only if there exists an arrow between them, that is they are in the same path connected component of 𝑋. In general, a groupoid is a category whose morphisms are always ˆ of Example 1.5 is also invertible. So Π1 (𝑋) is a groupoid, and the category 𝐺 a groupoid. With this language, solving the classification problem in a given category 𝒞 consists of giving a list 𝕃𝒞 which has exactly one element of each isomorphism class in 𝒞. In other words, we find 𝕃𝒞 ⊂ Obj(𝒞) such that: (1) if 𝑋, 𝑌 ∈ 𝕃𝒞 , then 𝑋 ≅ 𝑌 if and only if 𝑋 = 𝑌 . (2) for each 𝑋 ∈ Obj(𝒞) there exists an 𝑌 ∈ 𝕃𝒞 such that 𝑋 ≅ 𝑌 .

1.1. Topological and smooth manifolds

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Example 1.8. Examples of lists are 𝕃𝐒𝐞𝐭 = {cardinals} and 𝕃𝐕𝐞𝐜𝐭𝐤 = {𝐤𝛼 | 𝛼 ∈ {cardinals}}. For manifolds, we usually restrict the classification lists to connected manifolds, fol𝑐𝑜 lowing the notation in (1.2). We have 𝕃𝐓𝐌𝐚𝐧0 = {⋆}, where ⋆ is the set with one point (called singleton), whereas 𝕃𝐓𝐌𝐚𝐧0 = ℕ ∪ {∞}, where 𝑘 ∈ ℕ indicates the set of 𝑘 points and ∞ the set of countable many points (with the discrete topology). Also 𝑐𝑜 𝑐𝑜 𝕃𝐓𝐌𝐚𝐧1 = {𝑆 1 , ℝ}, 𝕃𝐓𝐌𝐚𝐧1 = {𝑆 1 } (cf. Exercise 1.3). 𝑐

Maps between categories are defined by the concept of functor, which will be useful throughout this course. Definition 1.9. (1) A covariant functor 𝐹 ∶ 𝒞1 → 𝒞2 between two categories 𝒞1 and 𝒞2 consists of a map 𝐹 ∶ Obj(𝒞1 ) → Obj(𝒞2 ) and for all 𝑋, 𝑌 ∈ Obj(𝒞1 ), a map 𝐹 ∶ Mor𝒞1 (𝑋, 𝑌 ) → Mor𝒞2 (𝐹(𝑋), 𝐹(𝑌 )). It is usually denoted 𝑓∗ = 𝐹(𝑓). These satisfy the following. • 𝐹(𝑓 ∘ 𝑔) = 𝐹(𝑓) ∘ 𝐹(𝑔), for composable morphisms 𝑓, 𝑔. In other words, (𝑓 ∘ 𝑔)∗ = 𝑓∗ ∘ 𝑔∗ . • It satisfies 𝐹(1𝑋 ) = 1𝐹(𝑋) , for all 𝑋. (2) A contravariant functor 𝐹 ∶ 𝒞1 → 𝒞2 between two categories 𝒞1 and 𝒞2 consists of a map 𝐹 ∶ Obj(𝒞1 ) → Obj(𝒞2 ) and for all 𝑋, 𝑌 ∈ Obj(𝒞1 ), a map 𝐹 ∶ Mor𝒞1 (𝑋, 𝑌 ) → Mor𝒞2 (𝐹(𝑌 ), 𝐹(𝑋)). It is usually denoted 𝑓∗ = 𝐹(𝑓). These satisfy the following. • It is reverse compatible with the compositions, i.e., 𝐹(𝑓∘𝑔) = 𝐹(𝑔)∘𝐹(𝑓). In other words, (𝑓 ∘ 𝑔)∗ = 𝑔∗ ∘ 𝑓∗ . • It satisfies 𝐹(1𝑋 ) = 1𝐹(𝑋) , for all 𝑋. Example 1.10. The fundamental group is a covariant functor 𝜋1 ∶ 𝐓𝐨𝐩∗ → 𝐆𝐫𝐨𝐮𝐩, (𝑋, 𝑥) → 𝜋1 (𝑋, 𝑥) which maps each 𝑓 ∶ (𝑋, 𝑥) → (𝑌 , 𝑦) to 𝑓∗ ∶ 𝜋1 (𝑋, 𝑥) → 𝜋1 (𝑌 , 𝑦), the induced map on loops (see section 2.1). A functor 𝐹 ∶ 𝒞1 → 𝒞2 preserves isomorphisms: (1.1)

𝑋 ≅𝒞1 𝑌 ⟹ 𝐹(𝑋) ≅𝒞2 𝐹(𝑌 ).

This is an important property with an easy proof. Let us check in the case that 𝐹 is covariant. Let 𝑓 ∶ 𝑋 → 𝑌 , 𝑔 ∶ 𝑌 → 𝑋 with 𝑓 ∘ 𝑔 = 1𝑌 , 𝑔 ∘ 𝑓 = 1𝑋 . Apply 𝐹 to get 𝐹(𝑓) ∘ 𝐹(𝑔) = 𝐹(𝑓 ∘ 𝑔) = 𝐹(1𝑌 ) = 1𝐹(𝑌 ) and 𝐹(𝑔) ∘ 𝐹(𝑓) = 𝐹(𝑔 ∘ 𝑓) = 𝐹(1𝑋 ) = 1𝐹(𝑋) . Therefore 𝐹(𝑓) ∶ 𝐹(𝑋) → 𝐹(𝑌 ) is an isomorphism, with inverse 𝐹(𝑔). Note that this can be written as 𝐹(𝑓−1 ) = 𝐹(𝑓)−1 . The property (1.1) is usually used as 𝐹(𝑋) ≇𝒞2 𝐹(𝑌 ) ⟹ 𝑋 ≇𝒞1 𝑌 , to distinguish objects in 𝒞1 . As a consequence, 𝐹 induces a map between the classifying lists of the categories, 𝐹 ∶ 𝕃𝒞1 → 𝕃𝒞2 , which can serve to help in classification problems, as will be clear in later chapters.

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1.1.2. Smooth structures. In this section we introduce an extra structure into topological manifolds, the so-called smooth structure, giving rise to differentiable manifolds. This extra piece of data will allow us to transfer concepts from mathematical analysis, such as differentiable functions and differential equations, to manifolds. This opens the door to new and powerful techniques for tackling classification problems unavailable with raw topological structure. These techniques will appear continuously within the following chapters with a particularly prominent role in section 2.5 and Chapter 5. For this purpose, we need a further concept, sheaves on topological spaces. These are defined as follows. Definition 1.11. For a topological space 𝑋 we define the category 𝐎𝐩𝐞𝐧(𝑋) whose objects are the open sets of 𝑋. For any open sets 𝑈, 𝑉 of 𝑋, Mor(𝑈, 𝑉) = {𝑖 ∶ 𝑈 → 𝑉} is the natural inclusion if 𝑈 ⊂ 𝑉, and Mor(𝑈, 𝑉) = ∅ otherwise. Definition 1.12. Let 𝑋 be a topological space. A presheaf on 𝑋 is a contravariant functor 𝐹 ∶ 𝐎𝐩𝐞𝐧(𝑋) → 𝐒𝐞𝐭. To be more specific, for each open set 𝑈 ⊂ 𝑋, we have a set 𝐹(𝑈). It is customary to call the elements 𝑠 ∈ 𝐹(𝑈) the sections on 𝑈. This satisfies the following. • If 𝑉 ⊂ 𝑈 and 𝑖𝑉 ,𝑈 denotes the inclusion, we have a map 𝑟𝑈,𝑉 = 𝐹(𝑖𝑉 ,𝑈 ) ∶ 𝐹(𝑈) → 𝐹(𝑉) called restriction. It is usually denoted 𝑠|𝑉 = 𝑟𝑈,𝑉 (𝑠). • If 𝑊 ⊂ 𝑉 ⊂ 𝑈, then 𝑟𝑉 ,𝑊 ∘ 𝑟𝑈,𝑉 = 𝑟𝑈,𝑊 . In other words, (𝑠|𝑉 )|𝑊 = 𝑠|𝑊 , for 𝑠 ∈ 𝐹(𝑈). • For the identity 𝑈 = 𝑈, we have 𝑟𝑈,𝑈 = 1𝐹(𝑈) . The most interesting presheaves arise from functions. With this in mind we have the following definition. Definition 1.13. Let 𝑋 be a topological space. A sheaf on 𝑋 is a presheaf 𝐹∶𝐎𝐩𝐞𝐧(𝑋) → 𝐒𝐞𝐭 that satisfies two additional properties. Suppose that 𝑈𝛼 ⊂ 𝑋 is a collection of open sets, and denote 𝑈 = ⋃ 𝑈𝛼 . Then, (1) Local property. Given sections 𝑠1 , 𝑠2 ∈ 𝐹(𝑈), if 𝑠1 |𝑈𝛼 = 𝑠2 |𝑈𝛼 for all 𝛼, then 𝑠1 = 𝑠 2 . (2) Gluing property. Given sections 𝑠𝛼 ∈ 𝐹(𝑈𝛼 ), if 𝑠𝛼 |𝑈𝛼 ∩𝑈𝛽 = 𝑠𝛽 |𝑈𝛼 ∩𝑈𝛽 for all 𝛼, 𝛽, it follows that there exists 𝑠 ∈ 𝐹(𝑈) such that 𝑠|𝑈𝛼 = 𝑠𝛼 for all 𝛼. Example 1.14. • The most basic example is the sheaf 𝐶 0 of continuous functions on a space 𝑋, i.e., 𝐶 0 (𝑈) = {𝑓 ∶ 𝑈 → ℝ continuous}. The restrictions are the restriction of functions. • A subsheaf of 𝐹 is a sheaf 𝐹 ′ such that 𝐹 ′ (𝑈) ⊂ 𝐹(𝑈) for all 𝑈, and the restriction maps are the same.

1.1. Topological and smooth manifolds

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• For any topological space, we have the sheaf 𝐶 0 (−, 𝑌 ) of continuous maps with values in 𝑌 . In particular, if 𝑌 has the discrete topology 𝐶 0 (−, 𝑌 ) is the sheaf of locally constant functions with values in 𝑌 . • Consider the presheaf given by assigning to 𝑈 the injective functions 𝑈 → ℝ. It is clearly a presheaf with the restrictions of functions. However, it is easy to verify that it is not a sheaf because condition (2) of Definition 1.13 fails in general. Definition 1.15. A smooth (or differentiable) manifold is a topological manifold 𝑀 0 endowed with a subsheaf S ⊂ 𝐶𝑀 called the sheaf of smooth or differentiable functions. The sheaf S must satisfy that for every 𝑝 ∈ 𝑀 there exists a chart 𝜑 ∶ 𝑈 = 𝑈 𝑝 → 𝜑(𝑈) ⊂ ℝ𝑛 which induces a bijection 𝜑∗ ∶ S(𝑈) → 𝐶 ∞ (𝜑(𝑈)), 𝑠 ↦ 𝑠 ∘ 𝜑−1 , being 𝐶 ∞ (𝜑(𝑈)) the set of 𝐶 ∞ functions in the classical sense for open sets of ℝ𝑛 . The inverse of this bijection is given by 𝜑∗ ∶ 𝐶 ∞ (𝜑(𝑈)) → S(𝑈), 𝑓 ↦ 𝑓 ∘ 𝜑. When a topological manifold 𝑀 can be given a sheaf S as above, one says that 𝑀 admits a smooth structure (𝑀, S). Theorem 1.16. Let 𝑀 be a topological manifold. There exists a sheaf of differentiable functions S ⊂ 𝐶 0 if and only if we can find an atlas 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )} on 𝑀 such that the changes of coordinates (also called changes of charts) 𝜑𝛼 ∘ 𝜑−1 𝛽 ∶ 𝜑 𝛽 (𝑈 𝛽 ∩ 𝑈𝛼 ) → 𝜑𝛼 (𝑈 𝛽 ∩ 𝑈𝛼 ) are 𝐶 ∞ diffeomorphisms between open subsets of ℝ𝑛 , for all 𝛼, 𝛽. The cor∞ respondence is given by saying that 𝑓 ∈ S(𝑈) if and only if 𝑓 ∘ 𝜑−1 𝛼 ∈ 𝐶 (𝜑 𝛼 (𝑈 ∩ 𝑈𝛼 )) for all 𝛼. Proof. Suppose that S exists. For each 𝑝 ∈ 𝑀 there exists a chart (𝑈 𝑝 , 𝜑𝑝 ) so that 𝜑𝑝∗ induces a bijection between 𝐶 ∞ (𝜑𝑝 (𝑈 𝑝 )) and S(𝑈 𝑝 ). We can cover 𝑀 = ⋃𝑝∈𝑀 𝑈 𝑝 with such open sets. We claim that this is the required atlas with 𝐶 ∞ change of coordinates. Indeed, let (𝑈𝛼 , 𝜑𝛼 ), (𝑈 𝛽 , 𝜑𝛽 ) be two open sets of the cover and its respective charts, and suppose they have non-empty intersection. Consider the function 𝑦 𝑖 ∶ 𝜑𝛽 (𝑈𝛼 ∩ 𝑈 𝛽 ) → ℝ, (𝑦1 , . . . , 𝑦𝑛 ) ↦ 𝑦 𝑖 , which is 𝐶 ∞ . Therefore 𝑦 𝑖 ∘ 𝜑𝛽 ∈ S(𝑈𝛼 ∩ 𝑈 𝛽 ), which implies that ∞ 𝑦 𝑖 ∘ 𝜑𝛽 ∘ 𝜑−1 𝛼 ∶ 𝜑 𝛼 (𝑈𝛼 ∩ 𝑈 𝛽 ) → ℝ is 𝐶 . Since this is true for 𝑖 = 1, . . . , 𝑛, it follows −1 ∞ ∞ that 𝜑𝛽 ∘ 𝜑𝛼 is 𝐶 . An analogous argument shows that 𝜑𝛼 ∘ 𝜑−1 𝛽 is 𝐶 . Now suppose we have an atlas {(𝑈𝛼 , 𝜑𝛼 )} with 𝐶 ∞ changes of coordinates, and we want to define the sheaf S. We put S(𝑈𝛼 ) = 𝜑∗𝛼 (𝐶 ∞ (𝜑𝛼 (𝑈𝛼 ))). If 𝑈 is an open set contained in some 𝑈𝛼 , we also put S(𝑈) = 𝜑∗𝛼 (𝐶 ∞ (𝜑𝛼 (𝑈))). This does not depend on the choice of 𝛼, because the changes of coordinates are smooth. So we have succeeded in defining S in sufficiently small open sets. For general 𝑈 ⊂ 𝑋 we define S(𝑈) = {𝑓 ∈ 𝐶 0 (𝑈) | 𝑓|𝑈∩𝑈𝛼 ∈ S(𝑈 ∩ 𝑈𝛼 ), for all 𝛼}. It is easy to verify that S satisfies all the properties of sheaves because 𝐶 ∞ functions do. □ Remark 1.17. The last proposition opens the door for an equivalent definition of smooth manifold. Namely, a smooth manifold is a topological manifold 𝑋 endowed with a maximal smooth atlas, which is an atlas 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )} such that the following hold. (1) For every non-empty intersection 𝑈𝛼 ∪ 𝑈 𝛽 ≠ ∅, the change of coordinates 𝜑𝛽 ∘ 𝜑−1 𝛼 ∶ 𝜑 𝛼 (𝑈𝛼 ∩ 𝑈 𝛽 ) → 𝜑 𝛽 (𝑈𝛼 ∩ 𝑈 𝛽 )

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is a 𝐶 ∞ diffeomorphism between open sets of ℝ𝑛 . Recall that, by the inverse function theorem, a 𝐶 ∞ diffeomorphism is the same as a bijection between open sets of ℝ𝑛 whose differential at every point is an invertible linear map. (2) The atlas 𝒜 is maximal, in the following sense. If there exists a homeomorphism 𝜑 ∶ 𝑈 → 𝑉 between open sets 𝑈 ⊂ 𝑋 and 𝑉 ⊂ ℝ𝑛 , such that for every non-empty intersection 𝑈 ∩ 𝑈𝛼 ≠ ∅, the map 𝜑 ∘ 𝜑−1 𝛼 ∶ 𝜑 𝛼 (𝑈 ∩ 𝑈𝛼 ) → 𝜑(𝑈 ∩ 𝑈𝛼 ) ⊂ 𝑉 is a diffeomorphism between open sets of Euclidean spaces, then it follows that (𝑈, 𝜑) ∈ 𝒜. Note that any atlas 𝒜′ of 𝑀 has its own maximal atlas 𝒜, which is obtained from 𝒜 by adding to it all the homeomorphisms 𝜑 as in condition (2) above. Therefore, a smooth manifold is completely determined by a topological manifold 𝑀 and a (not necessarily maximal) smooth atlas on it, since this automatically yields a maximal smooth atlas. ′

If two different smooth atlases 𝒜1 and 𝒜2 on a topological manifold 𝑀 have the same maximal atlas, then they define the same smooth structure on 𝑀, i.e., the sheaf of differentiable functions is the same for the two atlases. Now we define the category 𝐃𝐌𝐚𝐧𝑛 of differentiable 𝑛-manifolds. The objects are pairs (𝑀, S𝑀 ) of smooth 𝑛-manifolds and their sheaf of differentiable functions. Recall that by our definitions, these have all connected components of dimension 𝑛. The arrows are continuous functions 𝑓 ∶ 𝑀 → 𝑁 such that for each open set 𝑉 ⊂ 𝑁 we have 𝑓∗ (S𝑁 (𝑉)) ⊂ S𝑀 (𝑓−1 (𝑉)). In other words, if 𝑔 ∶ 𝑉 → ℝ is smooth, then 𝑔 ∘ 𝑓 ∶ 𝑓−1 (𝑉) → ℝ is smooth. Such functions 𝑓 are called differentiable (or smooth) maps, and we write 𝑓 ∈ 𝐶 ∞ (𝑀, 𝑁). It is easy to check that 𝑓 ∈ 𝐶 ∞ (𝑀, 𝑁) if and only ∞ if 𝜓𝛽 ∘ 𝑓 ∘ 𝜑−1 𝛼 is a 𝐶 map between open subsets of Euclidean spaces, where (𝑈𝛼 , 𝜑 𝛼 ) is a smooth chart of 𝑀 and (𝑉 𝛽 , 𝜓𝛽 ) is a smooth chart of 𝑁. The isomorphisms in the category 𝐃𝐌𝐚𝐧𝑛 are called diffeomorphisms. Note that if we endow the same topological manifold 𝑀 with two different smooth atlases 𝒜1 and 𝒜2 , then the two atlases define the same smooth structure if and only if Id ∶ (𝑋, 𝒜1 ) → (𝑋, 𝒜2 ) is an isomorphism in the category 𝐃𝐌𝐚𝐧𝑛 . We also consider the subcategory 𝐃𝐌𝐚𝐧𝑛𝑐 of compact differentiable 𝑛-manifolds. And we denote the category 𝐃𝐌𝐚𝐧 of all smooth manifolds, which may have components of different dimensions. Remark 1.18. It is important to note the difference between being isomorphic and being equal in the category 𝐃𝐌𝐚𝐧𝑛 . For example, the real line ℝ can be endowed, as a topological manifold, with different smooth structures, which nevertheless are isomorphic in 𝐃𝐌𝐚𝐧1 . For instance, we have the standard smooth structure in ℝ given by the chart Id ∶ ℝ → ℝ, 𝑡 ↦ 𝑡. Another smooth structure is given by the chart 𝜙 ∶ ℝ → ℝ, 𝑡 ↦ 𝑡3 . These two smooth structures are not the same, since 𝜙 does not belong to the atlas (ℝ, Id). Also, Id ∶ ℝ → ℝ is not even differentiable for (ℝ, 𝜙), since Id ∘𝜙−1 ∶ ℝ → ℝ is not 𝐶 ∞ at 0 ∈ ℝ. On the other hand we do have a diffeomorphism 𝐹 ∶ (ℝ, 𝜙) → (ℝ, Id), 𝐹(𝑥) = 𝑥1/3 which expressed in charts is just the identity map ℝ → ℝ. This proves that (ℝ, 𝜙) ≅ (ℝ, Id) in 𝐃𝐌𝐚𝐧1 .

1.1. Topological and smooth manifolds

9

Remark 1.19. Recall that an embedding between two topological spaces, 𝑓 ∶ 𝑋 → 𝑌 , is a continuous map which is a homeomorphism onto the image 𝑓(𝑋) ⊂ 𝑌 (with subspace topology). If 𝑀, 𝑁 are (topological) manifolds and 𝑁 ⊂ 𝑀, we will say that 𝑁 is a submanifold if the inclusion map 𝑖 ∶ 𝑁 → 𝑀 is a smooth map that is an embedding. Moreover, if 𝑀 and 𝑁 are also smooth, we can also consider a weaker notion called an immersion. This is a smooth map 𝑓 ∶ 𝑁 → 𝑀 such that its differential (see Remark 3.5) 𝑑𝑝 𝑓 ∶ 𝑇𝑝 𝑁 → 𝑇 𝑓(𝑝) 𝑀 is a linear monomorphism for all 𝑝 ∈ 𝑁. This implies that 𝑓 is locally invertible onto its image, but it may happen that globally it is not injective or it is not open onto its image for subspace topology on 𝑓(𝑁) (see [Boo]). Remark 1.20. There is a notion of 𝐶 𝑟 structure, 𝑟 ≥ 1. This is given by an atlas in which the changes of charts are given by 𝐶 𝑟 differentiable maps. A 𝐶 𝑟 manifold is a manifold with a 𝐶 𝑟 structure. It is a fact that a manifold with a 𝐶 𝑟 structure admits a compatible 𝐶 𝑠 atlas for any 𝑠 ≥ 𝑟, including 𝑠 = ∞ (see [Hir]). There is also the notion of analytic manifold, in which the changes of charts are given by (real) analytic maps, usually denoted 𝐶 𝜔 . It is a deep theorem of Whitney that a compact smooth manifold admits a compatible real analytic structure [Wh1]. Remark 1.21. One of the main concerns of differential geometry is the classification problem. Suppose that 𝒞 is a subcategory of locally connected spaces of 𝐓𝐨𝐩, and we want to study 𝕃𝒞 . Without loss of information, we can restrict our attention to (1.2)

𝑐𝑜

𝕃𝒞 = {𝑋 ∈ 𝕃𝒞 | 𝑋 is connected} . 𝑐𝑜

From this we recover 𝕃𝒞 by taking disjoint unions of elements of 𝕃𝒞 . Observe that, if 𝒞 is only composed of compact topological spaces, we take finite disjoint unions up to reordering. Remark 1.22. It is interesting to study the forgetful functor 𝐃𝐌𝐚𝐧𝑛 → 𝐓𝐌𝐚𝐧𝑛 which ignores the smooth structure. It is important to study whether this functor induces a bijection or not, 𝑐𝑜

(1.3)

𝑐𝑜

𝕃𝐃𝐌𝐚𝐧𝑛 → 𝕃𝐓𝐌𝐚𝐧𝑛 . 𝑐

𝑐

Note that (1.3) is surjective if and only if every topological 𝑛-manifold admits some smooth structure. It is injective if and only if every topological manifold admits at most one smooth structure up to diffeomorphism. Said otherwise, if two smooth 𝑛manifolds are homeomorphic, then they are diffeomorphic. 𝑐𝑜

(a) In dimension 0, 𝕃𝐃𝐌𝐚𝐧0 = {⋆} (see Example 1.8). 𝑐

𝑐𝑜 𝕃𝐃𝐌𝐚𝐧1 𝑐

𝑐𝑜

(b) In dimension 1, = {𝑆 1 } and 𝕃𝐃𝐌𝐚𝐧1 = {𝑆 1 , ℝ} (see Exercise 1.4). The forgetful map (1.3) is therefore bijective. (c) In dimension 2 we will give a complete classification of compact surfaces in Chapters 1 and 2 (Theorem 2.29), and we will see that all of them admit a 𝑐𝑜 𝑐𝑜 unique smooth structure (Theorem 6.56). Therefore 𝕃𝐃𝐌𝐚𝐧2 → 𝕃𝐓𝐌𝐚𝐧2 is 𝑐 𝑐 bijective. 𝑐𝑜

𝑐𝑜

(d) In dimension 3, a result of Moise proves that 𝕃𝐃𝐌𝐚𝐧3 → 𝕃𝐓𝐌𝐚𝐧3 is bijective 𝑐 𝑐 [Moi]. However, the classification of compact 3-manifolds has not been completed.

10

1. Topological surfaces

𝑐𝑜

𝑐𝑜

(e) In dimensions 𝑛 ≥ 5, the forgetful map 𝕃𝐃𝐌𝐚𝐧𝑛 → 𝕃𝐓𝐌𝐚𝐧𝑛 is neither injective 𝑐 𝑐 nor surjective. This failure is controlled by homotopy theoretic invariants, so it is known for a topological 𝑛-manifold how many smooth structures it admits (in particular, there are finitely many for a given topological compact 𝑛-manifold). The most remarkable achievement was the discovery by Milnor [Mi1] of a differentiable manifold homeomorphic to 𝑆 7 but not diffeomorphic to the standard round sphere 𝑆 7 ⊂ ℝ8 . This gave rise to a collection of examples of exotic differentiable structures on spheres 𝑆 𝑛 , which happen in some dimensions 𝑛 ≥ 7. 𝑐𝑜

𝑐𝑜

(f) In dimension 4, the forgetful map 𝕃𝐃𝐌𝐚𝐧4 → 𝕃𝐓𝐌𝐚𝐧4 is neither injective nor 𝑐 𝑐 surjective. The classification of compact topological simply connected 4-manifolds was given by Freedman [Fre]. Also strong restrictions for a 4-manifold to admit a smooth structure have been determined, initially by Donaldson [Do1], although they are not of homotopy theoretic type and depend on tools of global analysis (i.e., partial differential equations on manifolds). About the failure of injectivity, this is known by means of invariants of the smooth structure (Donaldson and Seiberg-Witten invariants), and there are instances of infinitely many (countable) smooth structures on a given compact connected topological 4-manifold. (g) The non-compact case is by far more involved. The following result is remarkable: ℝ𝑛 admits a unique smooth structure for 𝑛 ≠ 4, and ℝ4 admits a non-countable number of non-diffeomorphic smooth structures [Sco]. Remark 1.23. In algebraic geometry, the main objects of study are the algebraic varieties. It is remarkable that we can give a definition in large analogy with that of smooth manifold. For the general theory we refer to [Har] or to Chapter 5. An (abstract) algebraic variety over the complex numbers is a topological space 𝑋, endowed with a alg sheaf 𝒪𝑋 , called the sheaf of regular functions or algebraic functions, such that every point 𝑝 ∈ 𝑋 has an open neighbourhood 𝑈 𝑝 and a homeomorphism 𝜑 ∶ 𝑈 𝑝 → 𝑉(𝐼) = {𝑥 ∈ ℂ𝑛 |𝐹(𝑥) = 0, for all 𝐹 ∈ 𝐼} ⊂ ℂ𝑛 , where 𝐼 = 𝐼(𝐹1 , . . . , 𝐹 𝑑 ) is the ideal generated by some polynomials with complex coefficients 𝐹𝑖 (𝑥1 , . . . , 𝑥𝑛 ) ∈ ℂ[𝑥1 , . . . , 𝑥𝑛 ]. Moreover, alg for a function 𝑓 ∶ 𝑈 𝑝 → ℂ, we require that 𝑓 ∈ 𝒪𝑋 (𝑈 𝑝 ) if and only if the function −1 𝑓 ∘ 𝜑 ∶ 𝑉(𝐼) → ℂ is a well defined rational function, i.e., 𝑝 𝑓 ∘ 𝜑−1 ∈ { || 𝑝, 𝑞 ∈ ℂ[𝑥1 , . . . , 𝑥𝑛 ], 𝑞 never vanishes on 𝑉(𝐼)}. 𝑞 In other words, in the category of algebraic varieties, we want to be able to define the sheaf of rational functions on 𝑋. We have the following alternative characterization. An algebraic variety 𝑋 is a topological space with an atlas {(𝑈𝛼 , 𝜑𝛼 )} with charts 𝜑𝛼 ∶ 𝑈𝛼 → 𝑉(𝐼𝛼 ) = 𝜑𝛼 (𝑈𝛼 ) such that for each non-empty intersection 𝑈𝛼 ∩ 𝑈 𝛽 ≠ ∅, the change of charts 𝜑𝛼𝛽 = 𝜑𝛽 ∘ 𝜑−1 𝛼 ∶ 𝜑 𝛼 (𝑈𝛼 ∩ 𝑈 𝛽 ) → 𝜑 𝛽 (𝑈𝛼 ∩ 𝑈 𝛽 ) can be expressed as the restriction of a map whose components are rational functions, i.e., 𝑝 𝑝 𝜑𝛼𝛽 (𝑥1 , . . . , 𝑥𝑛 ) = ( 1 (𝑥1 , . . . , 𝑥𝑛 ), . . . , 𝑛 (𝑥1 , . . . , 𝑥𝑛 )), 𝑞1 𝑞𝑛 being 𝑝 𝑖 , 𝑞𝑖 ∈ ℂ[𝑥1 , . . . , 𝑥𝑛 ] polynomials.

1.1. Topological and smooth manifolds

11

A couple of comments are relevant: The varieties 𝑋 = 𝑉(𝐼) are called affine varieties (these are algebraic subvarieties of ℂ𝑛 , see Remark 5.47). These are not open subsets of ℂ𝑛 , and the dimension of 𝑋 is not 𝑛 in general. The second remark is that the topology used in algebraic geometry for 𝑋 is highly non-Hausdorff. It is called the Zariski topology, and its closed subsets are given by the algebraic subvarieties of 𝑋. 1.1.3. Manifolds with boundary. Now we come back to topological manifolds and introduce the concept of manifold with boundary. Definition 1.24. A manifold with boundary 𝑀 is a topological space which is Hausdorff, second countable, and such that each 𝑝 ∈ 𝑀 has a neighbourhood 𝑈 = 𝑈 𝑝 which is homeomorphic to an open set of ℝ𝑛+ = {(𝑥1 , . . . , 𝑥𝑛 ) ∈ ℝ𝑛 |𝑥1 ≥ 0} via 𝜑 ∶ 𝑈 → 𝜑(𝑈) ⊂ ℝ𝑛+ . '

M

'(U ) ⊂ Rn+

U

It is easy to see that in the previous definition, one can reduce the open set 𝑈 𝑝 so that 𝜑(𝑈) is either a ball 𝐵𝑟 (0) or a semiball 𝐵𝑟+ (0) = 𝐵𝑟 (0) ∩ ℝ𝑛+ , with 𝜑(𝑝) = 0. In the first case the point 𝑝 is called an interior point, and we write 𝑝 ∈ Int(𝑀); in the second case 𝑝 is called a boundary point,1 and we write 𝑝 ∈ 𝜕𝑀. We have the following. Proposition 1.25. The definition of being an interior and boundary point does not depend on the chosen chart 𝜑. Proof. We shall prove this for our most relevant case 𝑛 = 2, where we use the fundamental group. The general case can be treated with the aid of homology groups (cf. Exercise 2.21). Suppose that there exists a point 𝑝 ∈ 𝑀 that has neighbourhoods 𝑈 𝑝 , 𝑉 𝑝 with homeomorphisms 𝑓 ∶ 𝑈 → 𝐵𝜖 (0) and 𝑔 ∶ 𝑉 → 𝐵𝜂+ (0) with 𝑓(𝑝) = 0, 𝑔(𝑝) = 0. Take 0 < 𝛿 < 𝜂 such that 𝑔−1 (𝐵𝛿+ (0)) ⊂ 𝑈 ∩ 𝑉, and take 0 < 𝜈 < 𝜖 such that 𝐵𝜈̄ (0) ⊂ 𝑓(𝑔−1 (𝐵𝛿+ (0))). Then 𝑓 ∘𝑔−1 ∶ 𝐵𝛿+ (0)−{0} → 𝑓(𝑔−1 (𝐵𝛿+ (0)))−{0} is a homeomorphism. The first space is contractible. The second space is contained in 𝐵𝜖 (0) − {0} and retracts radially to 𝑆𝜈 (0) = 𝜕𝐵𝜈̄ (0) ≅ 𝑆 1 . Therefore it must be that 𝜋1 (𝐵𝛿+ (0) − {0}) = {1} surjects to 𝜋1 (𝑆𝜈 (0)) = ℤ, which is impossible. □ We conclude that 𝑀 = Int(𝑀) ⨆ 𝜕𝑀 as a disjoint union of sets. Since Int(𝑀) is clearly open, 𝜕𝑀 is closed. (A comment in terminology: Technically, a topological 1 We use the notation 𝜕𝑀 for the boundary of 𝑀, whereas the notation 𝜕𝐴 is also used for the topological boundary of a set 𝐴 ⊂ 𝑋 of a topological space. We hope this causes no confusion as the meaning should be clear from the context in each case.

12

1. Topological surfaces

manifold 𝑀 is also a manifold with boundary such that 𝜕𝑀 = ∅. We sometimes stress this as saying that 𝑀 is a manifold without boundary.) As in the case of manifolds, a manifold with boundary is a union of connected components and each of them has a well defined dimension. An 𝑛-manifold with boundary is a manifold with boundary (or without!) such that all connected components have dimension 𝑛. If 𝑀 is an 𝑛-manifold with boundary, then Int(𝑀) is an 𝑛-manifold without boundary. Moreover, 𝜕𝑀 is a manifold of dimension 𝑛 − 1 without boundary (maybe not connected). To see this, take 𝑝 ∈ 𝜕𝑀 and a chart 𝜑 ∶ 𝑈 𝑝 → 𝐵𝜖+ (0), 𝜑(𝑝) = 0. Then (𝜕𝑀) ∩ 𝑈 = 𝜑−1 (𝐵𝜖+ (0) ∩ {𝑥1 = 0}), and 𝜑 ∶ (𝜕𝑀) ∩ 𝑈 → 𝐵𝜖+ (0) ∩ {𝑥1 = 0} = 𝐵𝜖𝑛−1 (0) is an (𝑛 − 1)-dimensional chart. We can also define the concept of smooth manifold with boundary. To do this, first we define what it means to be a 𝐶 ∞ function 𝑓 ∶ 𝐵𝜀+ (0) → ℝ. This simply means that 𝑓 has an extension 𝐹 ∈ 𝐶 ∞ to some open set of ℝ𝑛 containing 𝐵𝜀+ (0). A map 𝜑 ∶ 𝐵𝜀+ (0) → 𝐵𝑟+ (0) is 𝐶 ∞ if all its coordinate functions are 𝐶 ∞ , and it is a 𝐶 ∞ diffeomorphism if it is a homeomorphism that is 𝐶 ∞ and its inverse is also 𝐶 ∞ . Having done this, we say that a manifold with boundary 𝑀 is smooth if it has an atlas whose change of coordinates are 𝐶 ∞ maps between open sets of ℝ𝑛+ . In analogy with the case without boundary, this definition is equivalent to defining a sheaf S of differentiable functions on 𝑀. If a manifold with boundary 𝑀 is smooth, then its boundary 𝜕𝑀 is also smooth, and it has a canonical smooth structure inherited from that of 𝑀 by restricting the charts to 𝜕ℝ𝑛+ = {𝑥1 = 0}. The categories of 𝑛-manifolds with boundary, compact 𝑛-manifolds with boundary, and the smooth counterparts are 𝐓𝐌𝐚𝐧𝑛𝜕 , 𝐓𝐌𝐚𝐧𝑛𝜕,𝑐 , 𝐃𝐌𝐚𝐧𝑛𝜕 , and 𝐃𝐌𝐚𝐧𝑛𝜕,𝑐 , respectively. There are corresponding classification problems (cf. Exercise 1.7).

1.2. PL structures 1.2.1. Quotient topologies. When we have a topological space, any equivalence relation on it yields a natural topology on the quotient space, as follows. Definition 1.26. Let 𝑋 be a topological space, and suppose that we have an equivalence relation ∼ on 𝑋. Let 𝜋 ∶ 𝑋 → 𝑋 ⋆ = 𝑋/ ∼ be the map passing to the quotient. The quotient topology on 𝑋 ⋆ is the topology where 𝑈 ⋆ ⊂ 𝑋 ⋆ is open if and only if 𝜋−1 (𝑈 ⋆ ) ⊂ 𝑋 is open. Accordingly with the notation above, open and closed subsets of 𝑋 ⋆ will be denoted with the superscript ⋆. Given an element 𝑥 ∈ 𝑋, its equivalence class under ∼ will be denoted by [𝑥]. It is easy to see that the open subsets of 𝑋 ⋆ are the images by 𝜋 of the open sets of 𝑋 which are the union of equivalence classes. These open sets are called saturated for the equivalence relation ∼ . Since it is always uncomfortable (and usually ineffective) to work with equivalence classes of objects, it is useful to have criteria for knowing whether a space is homeomorphic to a quotient 𝑋/∼ . For this purpose the following is defined. Definition 1.27. Let 𝑋 and 𝑌 be topological spaces. A quotient map 𝑞 ∶ 𝑋 → 𝑌 is a continuous, surjective map, such that if we define the equivalence relation ∼ on 𝑋 by

1.2. PL structures

13

𝑥 ∼ 𝑥′ if and only if 𝑞(𝑥) = 𝑞(𝑥′ ), then the natural map 𝑞 ̄ that makes the following diagram commutative, 𝑞

/𝑌 }> } } 𝜋 }}  }} 𝑞̄ 𝑋⋆ 𝑋

≅

where 𝑋 ⋆ = 𝑋/∼ is the quotient space, is a homeomorphism 𝑞 ̄ ∶ 𝑋 ⋆ ⟶ 𝑌 . Obviously the quotient map 𝑞 ∶ 𝑋 → 𝑋 ⋆ = 𝑋/∼ satisfies the above definition for any equivalence relation ∼ , by the sheer definition of the quotient topology in 𝑋 ⋆ . Proposition 1.28. Let 𝑞 ∶ 𝑋 → 𝑌 be a continuous, surjective map between topological spaces 𝑋, 𝑌 . The following are true: (1) If 𝑞 is an open map, then it is a quotient map. (2) If 𝑞 is a closed map, then it is a quotient map. (3) If 𝑋 is compact and 𝑌 is Hausdorff, then 𝑞 is a quotient map. Proof. Let ∼ be the equivalence relation 𝑥 ∼ 𝑥′ if and only if 𝑞(𝑥) = 𝑞(𝑥′ ), let 𝜋 ∶ 𝑋 → 𝑋 ⋆ = 𝑋/∼ be the quotient map given by ∼ , and let 𝑞 ̄ ∶ 𝑋 ⋆ → 𝑌 be the induced map. To see (1) and (2) it is enough to note that for an open (resp., closed) set 𝐴⋆ ⊂ 𝑋 ⋆ , we have that 𝑞(𝐴 ̄ ⋆ ) = 𝑞(𝜋−1 (𝐴⋆ )) is an open (resp., closed) subset of 𝑌 since 𝑞 is an open (resp., closed) map and 𝜋 is continuous. We conclude that 𝑞 ̄ is continuous, bijective, and open (resp., closed), and therefore is a homeomorphism and 𝑞 is a quotient map. Let us see (3) by proving that 𝑞 is a closed map. If 𝐹 ⊂ 𝑋 is closed, then it is compact. Therefore 𝑞(𝐹) ⊂ 𝑌 is compact, and then it is closed since 𝑌 is Hausdorff. □ Definition 1.29. Let 𝑋 be a topological space, and let 𝑍 ⊂ 𝑋 be a subspace. We denote 𝑋/𝑍 the space obtained by collapsing 𝑍, that is, the quotient 𝑋/ ∼ , where ∼ is defined by 𝑧 ∼ 𝑧′ , for all 𝑧, 𝑧′ ∈ 𝑍. Example 1.30. Let us prove that the closed disc 𝐷𝑛 collapsing 𝜕𝐷𝑛 , 𝐷𝑛 /𝜕𝐷𝑛 , is homeomorphic to the sphere 𝑆 𝑛 . The equivalence relation in 𝐷𝑛 is 𝑥 ∼ 𝑥′ ⇔ 𝑥, 𝑥′ ∈ 𝜕𝐷𝑛 . First define the map 𝑥 = (𝑥1 , . . . , 𝑥𝑛 ) ↦ (𝑥1 , . . . , 𝑥𝑛 , 2||𝑥||2 −1), which puts the points of the disc on a paraboloid. Now we expand the meridians (that is, the 𝑛 first coordinates (𝑥1 , . . . , 𝑥𝑛 )) by a factor of 𝜆 while leaving the height (the last coordinate) fixed, so as to make them lie in 𝑆 𝑛 . Substituting in the equation of 𝑆 𝑛 , we get ||𝜆𝑥||2 +(2||𝑥||2 −1)2 = 1, which yields 𝜆 = 2√1 − ||𝑥||2 . We obtain the map 𝑞 ∶ 𝐷𝑛 → 𝑆 𝑛 , 𝑞(𝑥1 , . . . , 𝑥𝑛 ) = (2√1 − ||𝑥||2 𝑥1 , . . . , 2√1 − ||𝑥||2 𝑥𝑛 , 2||𝑥||2 − 1) . The map 𝑞 is continuous. Moreover, 𝑞 maps 𝜕𝐷𝑛 = {||𝑥|| = 1} to the north pole 𝑁 = (0, . . . , 0, 1), and it is a bijection between Int(𝐷𝑛 ) and 𝑆 𝑛 − {𝑁}, so 𝑞 induces the desired equivalence relation. Finally, since 𝐷𝑛 is compact and 𝑆 𝑛 is Hausdorff, Proposition 1.28(3) yields that the quotient 𝐷𝑛 /∼ is homeomorphic to the sphere 𝑆 𝑛 , as required.

14

1. Topological surfaces

Example 1.31. The (real) projective space ℝ𝑃 𝑛 is defined as the quotient space (ℝ𝑛+1 − {0})/ℝ∗ , where ℝ∗ = ℝ − {0} acts by 𝜆 ⋅ (𝑥0 , . . . , 𝑥𝑛 ) = (𝜆𝑥0 , . . . , 𝜆𝑥𝑛 ), with its quotient topology. The equivalence relation is thus (𝑥0 , . . . , 𝑥𝑛 ) ∼ (𝜆𝑥0 , . . . , 𝜆𝑥𝑛 ), for 𝜆 ∈ ℝ∗ . The quotient is denoted as 𝜋 ∶ ℝ𝑛+1 − {0} → ℝ𝑃 𝑛 and points in the quotient as [𝑥0 , . . . , 𝑥𝑛 ] = 𝜋(𝑥0 , . . . , 𝑥𝑛 ). • ℝ𝑃𝑛 is compact. Take 𝑆 𝑛 ⊂ ℝ𝑛+1 − {0}. As any vector 𝑣 ∼ 𝑣/||𝑣||, we have that 𝜋 ∶ 𝑆 𝑛 → ℝ𝑃 𝑛 is surjective, and since 𝑆 𝑛 is compact, so is ℝ𝑃𝑛 . By the same argument, ℝ𝑃 𝑛 is connected. • The projective space is also Hausdorff. If [𝑣] ≠ [𝑤], take representants 𝑣, 𝑤 ∈ 𝑆 𝑛 . Then 𝑣 ≠ ±𝑤, so we can take neighbourhoods 𝑈 𝑣 , 𝑉 𝑤 such that 𝑈∩𝑉 = ∅ and 𝑈 ∩ (−𝑉) = ∅. So 𝑈 ⋆ = 𝜋(𝑈 ∪ (−𝑈)) and 𝑉 ⋆ = 𝜋(𝑉 ∪ (−𝑉)) are disjoint neighbourhoods of [𝑣], [𝑤], respectively. By Proposition 1.28, 𝜋 is a quotient map. Therefore ℝ𝑃 𝑛 ≅ 𝑆 𝑛 /∼ , where 𝑣 ∼ −𝑣. • ℝ𝑃𝑛 is a topological 𝑛-manifold. Let 𝜋 ∶ 𝑆 𝑛 → ℝ𝑃 𝑛 be the quotient map. Take 𝑝 = 𝜋(𝑥) ∈ ℝ𝑃 𝑛 . Consider an open set 𝑈 = 𝑈 𝑥 ⊂ 𝑆 𝑛 , and a chart 𝜑 ∶ 𝑈 → 𝐵𝜖 (0) ⊂ ℝ𝑛 , small enough so that 𝑈 ∩ (−𝑈) = ∅. Then 𝜋|𝑈 ∶ 𝑈 → 𝑈 ⋆ = 𝜋(𝑈) = 𝜋(𝑈 ∪(−𝑈)) is a homeomorphism. Therefore 𝜑̄ = 𝜑∘(𝜋|𝑈 )−1 ∶ 𝑈 ⋆ → 𝐵𝜖 (0) is a chart for ℝ𝑃 𝑛 . • Affine charts. When we remove a hyperplane of ℝ𝑃𝑛 , we have an affine space; e.g., take 𝐻 = {𝑥0 = 0} ⊂ ℝ𝑃 𝑛 , and accordingly in the previous item we can take 𝑈 = {𝑥0 > 0} ⊂ 𝑆 𝑛 . As a chart, we consider 𝜑 ∶ 𝑈 → ℝ𝑛 , 𝜑(𝑥0 , 𝑥1 , . . . , 𝑥𝑛 ) = (𝑥1 /𝑥0 , . . . , 𝑥𝑛 /𝑥0 ). This gives a chart for 𝑈 ⋆ = ℝ𝑃 𝑛 − 𝐻, ̄ 0 , 𝑥1 , . . . , 𝑥𝑛 ]) = (𝑥1 /𝑥0 , . . . , 𝑥𝑛 /𝑥0 ). This is the same as the defined as 𝜑([𝑥 affine structure for 𝑈 ⋆ , given as 𝜑̄ ∶ 𝑈 ⋆ → ℝ𝑛 . • ℝ𝑃𝑛 is a smooth manifold. The affine charts give a smooth atlas. Let us see this. For each 𝑖 = 0, 1, . . . , 𝑛, take the open set 𝑈 𝑖 = {[𝑥0 , . . . , 𝑥𝑛 ]| 𝑥𝑖 ≠ 0} ⊂ ℝ𝑃𝑛 . Define the chart 𝜑𝑖 ∶ 𝑈 𝑖 → ℝ𝑛 , 𝜑𝑖 ([𝑥0 , . . . , 𝑥𝑛 ]) = (

𝑥0 𝑥 𝑥 ˆ , ... , 𝑖 , ... , 𝑛 ) , 𝑥𝑖 𝑥𝑖 𝑥𝑖

where the symbol ˆ means that the entry has been deleted. The inverse is 𝜑−1 𝑖 (𝑦 1 , . . . , 𝑦 𝑛 ) = [𝑦 1 , . . . , 𝑦 𝑖 , 1, 𝑦 𝑖+1 , . . . , 𝑦 𝑛 ]. The changes of charts are given, for 𝑖 < 𝑗 as 𝜑𝑖 ∘ 𝜑𝑗−1 (𝑦1 , . . . , 𝑦𝑛 ) = (

𝑦𝑗 1 𝑦𝑗+1 ˆ 𝑦1 𝑦 𝑦 , ... , 𝑖 , ... , , , , ... , 𝑛 ) , 𝑦𝑖 𝑦𝑖 𝑦𝑖 𝑦𝑖 𝑦𝑖 𝑦𝑖

defined for (𝑦1 , . . . , 𝑦𝑛 ), 𝑦 𝑖 ≠ 0. This is a diffeomorphism onto its image, so 𝒜 = {(𝑈 𝑖 , 𝜑𝑖 )|0 ≤ 𝑖 ≤ 𝑛} is a smooth atlas. The following topological construction is very useful when it comes to constructing new topological spaces from old ones. Definition 1.32. Let 𝑋, 𝑌 be topological spaces, let 𝐴 ⊂ 𝑋, and let 𝑓 ∶ 𝐴 → 𝑌 be a continuous map. We define the topological space 𝑋 ∪𝑓 𝑌 = (𝑋 ⊔ 𝑌 )/∼,

1.2. PL structures

15

where the identifications are for each 𝑎 ∈ 𝐴, 𝑎 ∼ 𝑓(𝑎). We endow it with the quotient topology. Example 1.33. (1) Wedge (pointed union) of spaces. Consider (𝑋, 𝑝), (𝑌 , 𝑝′ ) ∈ Obj(𝐓𝐨𝐩∗ ). We define the wedge 𝑋 ∨𝑌 as 𝑋 ∪𝑓 𝑌 , where 𝐴 = {𝑝}, 𝑓(𝑝) = 𝑝′ . Let [𝑝] = 𝜋(𝑝) = 𝜋(𝑝′ ), where 𝜋 ∶ 𝑋 ⊔ 𝑌 → 𝑋 ∨ 𝑌 is the quotient map. Then (𝑋 ∨ 𝑌 , [𝑝]) ∈ ′ Obj(𝐓𝐨𝐩∗ ). The neighbourhoods of [𝑝] in 𝑋 ∨ 𝑌 are given as 𝜋(𝑈 𝑝 ⊔ 𝑉 𝑝 ), ′ where 𝑈 𝑝 and 𝑉 𝑝 are neighbourhoods of 𝑝 and 𝑝′ , respectively. (2) Attaching 𝑛-cells. Given a topological space 𝑋 and 𝑓 ∶ 𝜕𝐷𝑛 → 𝑋 a continuous map, we define 𝑋 ′ = 𝑋 ∪𝑓 𝐷𝑛 , and we say that 𝑋 ′ is obtained from 𝑋 by attaching an 𝑛-cell. (3) Mapping torus. Let 𝑓 ∶ 𝑋 → 𝑋 be a homeomorphism, We define the mapping torus as 𝑇 𝑓 = 𝑋 × [0, 1]/∼ , where (𝑥, 0) ∼ (𝑓(𝑥), 1), so we have the quotient map 𝑞 ∶ 𝑋 × [0, 1] → 𝑇 𝑓 . If 𝑋 is a topological 𝑛-manifold, then 𝑇 𝑓 is a topological (𝑛 + 1)-manifold. Also if 𝑋 is an 𝑛-manifold with boundary, 𝑇 𝑓 is a (𝑛 + 1)-manifold with boundary (Exercise 1.12). In general the manifold 𝑇 𝑓 depends crucially on 𝑓. When the map 𝑓 ∶ 𝑋 → 𝑋 is the identity, we get 𝑇Id = 𝑋 × 𝑆1 . For instance, if 𝑋 = [0, 1] is a segment, 𝑇Id = [0, 1] × 𝑆 1 is the cylinder. When 𝑋 = [0, 1] and 𝑓(𝑡) = 1 − 𝑡, the manifold 𝑇 𝑓 is called the Möbius band, denoted Mob (see Figure 1.1). Id

f

Figure 1.1. Cylinder and Möbius band.

(4) For 𝑋 = 𝑆 1 a circle, 𝑇Id = 𝑆 1 × 𝑆 1 is the torus (see Figure 1.2). If we consider the map 𝑓 ∶ 𝑆 1 → 𝑆 1 , 𝑓(𝑥, 𝑦) = (𝑥, −𝑦), we have the Klein bottle2 Kl as the mapping torus of 𝑓. Equivalently, Kl = 𝑆 1 × [0, 1]/ ∼ , where (𝑒2𝜋i𝑡 , 0) ∼ (𝑒2𝜋i(1−𝑡) , 1). This surface cannot be depicted in ℝ3 without selfintersections because the identification 𝑓 forces one of the boundaries to be inserted inside the “tube” to glue with the other boundary with the right orientation. This is due to the fact that Kl is non-orientable (Example 1.75) and non-orientable compact surfaces without boundary cannot be embedded in ℝ3 (Exercise 2.36). 2 Anecdotally, the Klein bottle was first described by Klein in 1882. It was given the name Kleinsche Fläche (“Klein surface” in German), and then misinterpreted as Kleinsche Flasche (“Klein bottle”), which ultimately led to the adoption of this name. Probably the picture of Figure 1.2 helped for this.

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f

Id

Figure 1.2. Torus and Klein bottle.

1.2.2. Connected sum. Let 𝑀1 and 𝑀2 be two topological connected 𝑛-manifolds. Choose points 𝑝 𝑖 ∈ 𝑀𝑖 and neighbourhoods 𝑈 𝑝𝑖 = 𝑈 𝑖 ⊂ 𝑀𝑖 with homeomorphisms 𝑜 𝜑𝑖 ∶ 𝑈 𝑖 → 𝐵2 (0). Set 𝑊 𝑖 = 𝜑−1 𝑖 (𝐵1 (0)), so that 𝑀𝑖 = 𝑀𝑖 − 𝑊 𝑖 are manifolds with boundary, being the boundary homeomorphic to the (𝑛 − 1)-sphere via 𝜑𝑖 ∶ 𝜕𝑀𝑖𝑜 → 𝑆 1 (0) = 𝑆 𝑛−1 . 𝑜 𝑜 𝑜 Therefore 𝑓 = 𝜑−1 2 ∘ 𝜑 1 ∶ 𝜕𝑀1 → 𝜕𝑀2 ⊂ 𝑀2 can be used as a gluing map to form the connected sum (see Figure 1.3),

𝑀1 #𝑓 𝑀2 = 𝑀1𝑜 ∪𝑓 𝑀2𝑜 .

Figure 1.3. Connected sum.

It is easy to see that 𝑀 = 𝑀1 #𝑓 𝑀2 is a connected topological 𝑛-manifold. To see the existence of charts for 𝑀, consider the quotient map 𝜋 ∶ 𝑀1𝑜 ⊔ 𝑀2𝑜 → 𝑀, and let 𝐶 = 𝜋(𝜕𝑀1𝑜 ) = 𝜋(𝜕𝑀2𝑜 ) be the “core” of the connected sum. If 𝑝 ∈ 𝑀 − 𝐶, then 𝑝 lies in the interior of 𝑀1𝑜 or of 𝑀2𝑜 . In either case, a chart inside it serves as a chart for 𝑀. If 𝑝 ∈ 𝐶, then 𝑝 = 𝜋(𝑝1 ) = 𝜋(𝑝2 ), with 𝑝𝑗 ∈ 𝜕𝑀𝑗𝑜 , 𝑗 = 1, 2, and 𝑓(𝑝1 ) = 𝑝2 . Pick 𝑝

neighbourhoods 𝑉𝑗 𝑗 ⊂ 𝑈 𝑗 ∩ 𝑀𝑗𝑜 , 𝑗 = 1, 2, homeomorphic to semiballs of ℝ𝑛+ , and such that 𝑓(𝜕𝑀1𝑜 ∩ 𝑉1 ) = 𝜕𝑀2𝑜 ∩ 𝑉2 . Therefore 𝑉1 ∪𝑓 𝑉2 = 𝜙(𝑉1 ⊔ 𝑉2 ) is a neighbourhood of 𝑝 and it is homeomorphic to a ball in ℝ𝑛 . The following concept is important in order to understand the connected sum. Two homeomorphisms 𝑓0 , 𝑓1 ∶ 𝑋 → 𝑌 are called isotopic if they are homotopic via a homotopy 𝐻 ∶ 𝑋 × [0, 1] → 𝑌 with the maps 𝑓𝑠 ∶ 𝑋 → 𝑌 , 𝑓𝑠 (𝑥) = 𝐻(𝑥, 𝑠), 𝑠 ∈ [0, 1], ≅

being homeomorphisms. The set of homeomorphisms 𝑋 ⟶ 𝑌 under the equivalence relation of being isotopic is called the mapping class set of (𝑋, 𝑌 ), and it is denoted MCS(𝑋, 𝑌 ). When 𝑋 = 𝑌 it is called the mapping class group and is denoted MCG(𝑋). In the latter case, it is a group under composition. Lemma 1.34.

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(1) For intervals [𝑎, 𝑏] and [𝑐, 𝑑], MCS([𝑎, 𝑏], [𝑐, 𝑑]) = {𝑙1 , 𝑙2 }, where 𝑙1 and 𝑙2 are the two affine homeomorphisms between the intervals, one of them increasing and the other decreasing. (2) For the circle 𝑆 1 , MCG(𝑆 1 ) = {Id, 𝑟} where 𝑟(𝑥, 𝑦) = (𝑥, −𝑦). In other words, any homeomorphism of 𝑆 1 is isotopic either to the identity or a reflection. Proof. Any homeomorphism 𝑓 ∶ [𝑎, 𝑏] → [𝑐, 𝑑] is either an increasing or a decreasing map. Two increasing maps 𝑓0 , 𝑓1 are isotopic via 𝐻𝑠 = 𝑠𝑓0 + (1 − 𝑠)𝑓1 which is also increasing and therefore a homeomorphism. Analogously, two decreasing maps are isotopic. Finally, a decreasing and an increasing map are not isotopic because their restrictions to {𝑎, 𝑏} are different. A homeomorphism 𝑓 ∶ 𝑆 1 → 𝑆 1 is equivalent to a homeomorphism of the interval ̃ 𝑓 ∶ [0, 2𝜋] ⟶ [𝜃0 , 𝜃0 + 2𝜋] via ̃ ̃ 𝑓(cos(𝑡), sin(𝑡)) = (cos(𝑓(𝑡)), sin(𝑓(𝑡))). If 𝑓 ̃ is increasing, then 𝑓 is isotopic to the identity via ̃ + (1 − 𝑠)𝑡), sin(𝑠𝑓(𝑡) ̃ + (1 − 𝑠)𝑡)). 𝐻(𝑡, 𝑠) = (cos(𝑠𝑓(𝑡) Note that 𝐻(𝑡, 𝑠) are homeomorphisms for 𝑠 fixed because the linear interpolation of increasing maps is increasing. If 𝑓 ̃ is decreasing, then 𝑓 is isotopic to the reflection 𝑟 via ̃ + (1 − 𝑠)(2𝜋 − 𝑡)), sin(𝑠𝑓(𝑡) ̃ + (1 − 𝑠)(2𝜋 − 𝑡))). 𝐻(𝑡, 𝑠) = (cos(𝑠𝑓(𝑡) □ It is also true that MCG(𝑆𝑛 ) = {Id, 𝑟}, for 𝑛 ≥ 2, where 𝑟 is a reflection on a hyperplane, although this is harder to prove. Theorem 1.35. The connected sum 𝑀1 #𝑓 𝑀2 does not depend on the points 𝑝1 , 𝑝2 , up to homeomorphism. With respect to the chosen charts, suppose that we have a fixed chart 𝜑2 𝑜 𝑜 𝑜 ′ −1 ′ ′ and two charts 𝜑1 , 𝜑′1 . Let 𝑓 = 𝜑−1 2 ∘ 𝜑 1 ∶ 𝜕𝑀1 → 𝜕𝑀2 and 𝑓 = 𝜑 2 ∘ 𝜑 1 ∶ 𝜕(𝑀1 ) → 𝑜 𝑜 𝑜 ′ 𝜕𝑀2 . There exists a homeomorphism 𝜙 ∶ 𝜕𝑀1 → 𝜕(𝑀1 ) such that if 𝑓 is isotopic to 𝑓′ ∘𝜙, then the connected sums with the chart 𝜑1 and with the chart 𝜑′1 are homeomorphic. Proof. This is a deep result that requires hard techniques. We shall sketch the case 𝑛 = 2, where we use Lemma 1.34. Let 𝑝1 , 𝑝1′ ∈ 𝑀1 be two different points, and let 𝑝2 ∈ 𝑀2 . By Exercise 1.14, there is a homeomorphism 𝜙 ∶ 𝑀1 → 𝑀1 with 𝜙(𝑝1 ) = 𝑝1′ . Thus doing the connected sum with the charts 𝜑1 , 𝜑2 (centered at the points 𝑝1 , 𝑝2 ) gives a surface homeomorphic to the surface obtained by doing a connected sum with the charts 𝜑1 ∘ 𝜙−1 , 𝜑2 (centered at the points 𝑝1′ , 𝑝2 ). This proves the invariance with respect to the choice of points. Now suppose that we have two charts 𝜑1 ∶ 𝑈 𝑝1 → 𝐵2 (0), 𝜑′1 ∶ 𝑉 𝑝1 → 𝐵2 (0) both centered at 𝑝1 ∈ 𝑀1 , and a chart 𝜑2 ∶ 𝑊 𝑝2 → 𝐵2 (0) centered at 𝑝2 ∈ 𝑀2 . Take a small ′ −1 ̄ 2 ̄ ball 𝐵𝜖 (0) such that 𝜑−1 1 (𝐵𝜖 (0)) ⊂ 𝑈 ∩ 𝑉. The image 𝐷 = 𝜑 1 (𝜑 1 (𝐵𝜖 (0))) ⊂ 𝐵2 (0) ⊂ ℝ is a closed disc. We use the following result, whose proof can be found in [Cai]. Theorem 1.36 (Jordan-Schönflies). If 𝐶 ⊂ ℝ2 is a Jordan curve (a subspace homeomorphic to 𝑆 1 ), then there is a homeomorphism ℝ2 → ℝ2 that sends 𝐶 to 𝑆 1 = 𝜕𝐵1 (0) ⊂ ℝ2 .

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With Theorem 1.36, we can prove that 𝐵2̄ (0) − Int(𝐷) is homeomorphic to the annulus 𝐵2̄ (0) − 𝐵1 (0). Certainly, take two non-intersecting paths joining points of 𝜕𝐵2 (0) to 𝜕𝐷. This divides the region into two parts bounded by a Jordan arc. So these two regions are homeomorphic to discs, and their union is homeomorphic to the annulus. This can be arranged to be the identity on 𝜕𝐵2 (0). There are homeomorphisms −1 ′ −1 𝑀1 − 𝜑−1 1 (𝐵2 (0)) ≅ 𝑀1 − 𝜑 1 (𝐵𝜖 (0)) ≅ 𝑀1 − (𝜑 1 ) (Int 𝐷) = (𝑀1 − (𝜑′1 )−1 (𝐵2 (0))) ∪ (𝜑′1 )−1 (𝐵2̄ (0) − Int(𝐷))

≅ (𝑀1 − (𝜑′1 )−1 (𝐵2 (0))) ∪ (𝜑′1 )−1 (𝐵2̄ (0) − 𝐵1 (0)) = 𝑀1 − (𝜑′1 )−1 (𝐵1 (0)) ≅ 𝑀1 − (𝜑′1 )−1 (𝐵2 (0)). Let 𝜙 ∶ 𝑀1𝑜 → (𝑀1𝑜 )′ be the induced homeomorphism by this identification. Taking the union with 𝑀2 − 𝜑−1 2 (𝐵1 (0)), we have a homeomorphism between the connected sum with the chart 𝜑1 and with the chart 𝜑′1 ∘ 𝜙. To see that the gluing 𝑓 is only important up to isotopy, fix homeomorphisms 𝑆 1 ≅ and 𝑆 1 ≅ 𝜕𝑀2𝑜 (for instance, through the charts 𝜑1 and 𝜑2 ). Now we can see 𝑓 as a map 𝑓 ∶ 𝑆 1 → 𝑆 1 . Suppose that 𝑓0 , 𝑓1 are two isotopic homeomorphisms, with 𝐻 ∶ 𝑆 1 × [0, 1] → 𝑆 1 their isotopy. There is a homeomorphism 𝑀1 − 𝜑−1 1 (𝐵1 (0)) ≅ 1 1 ̄ (𝑀1 − 𝜑−1 (𝐵 (0))) ∪ (𝑆 × [0, 1]), where the map ℎ ∶ 𝐵 (0) − 𝐵 (0) → 𝑆 × [0, 1] is 2 𝑓0 2 1 1 𝜕𝑀1𝑜

𝑥

defined as ℎ(𝑥) = (𝐻 ( ||𝑥|| , 2 − ||𝑥||) , 2 − ||𝑥||). Hence −1 (𝑀1 − 𝜑−1 1 (𝐵1 (0))) ∪𝑓1 (𝑀2 − 𝜑 2 (𝐵2 (0))) −1 1 ≅ (𝑀1 − 𝜑−1 1 (𝐵2 (0))) ∪𝑓0 (𝑆 × [0, 1]) ∪𝑓1 (𝑀2 − 𝜑 2 (𝐵2 (0))) −1 ≅ (𝑀1 − 𝜑−1 1 (𝐵2 (0))) ∪𝑓0 (𝑀2 − 𝜑 2 (𝐵1 (0))),

doing the same trick on the 𝑀2 side. At this point the result follows from the obvious homeomorphisms that allow us to change the radii of the chosen holed balls. □ Remark 1.37. 𝑛−1 • Take 𝜓 = 𝜑′1 ∘ 𝜙 ∘ 𝜑−1 → 𝑆 𝑛−1 so that gluing with the chart 𝜑′1 ∘ 𝜙 1 ∶ 𝑆 is the same as gluing with the chart 𝜓 ∘ 𝜑1 . Observe that 𝑓 is isotopic to 𝑓′ if and only if 𝜓 ∶ 𝑆 𝑛−1 → 𝑆 𝑛−1 is isotopic to the identity. In this way, we can reduce the indeterminacy to the mapping class group of the standard sphere. However, there are two isotopy classes of homeomorphisms 𝑆 𝑛−1 → 𝑆 𝑛−1 , so Theorem 1.35 implies that there are at most two possible connected sums.

• The indeterminacy in Theorem 1.35 can be sometimes reduced. Let 𝑋#Id 𝑌 and 𝑋#𝑟 𝑌 denote the two connected sums (with 𝜓 the identity, and with 𝜓 the antipodal map). Then if 𝑋 is non-orientable, 𝑋#Id 𝑌 ≅ 𝑋#𝑟 𝑌 (cf. Exercise 1.26). If 𝑋, 𝑌 are orientable and there exists an orientation reversing homeomorphism 𝑓 ∶ 𝑋 → 𝑋, then 𝑋#Id 𝑌 ≅ 𝑓(𝑋)#𝑟 𝑌 = 𝑋#𝑟 𝑌 (see section 1.4 for the notion of orientability). This happens in dimension 2 (Remark 1.81), so it is customary to denote just 𝑋#𝑌 . The first dimension where this does not occur is 4 (see Exercise 5.19).

1.2. PL structures

19

Remark 1.38. The connected sum operation can be performed on connected 𝑛-manifolds with boundary. One only has to take care that the points chosen for removing the balls should be interior. Note that 𝜕(𝑋#𝑌 ) = (𝜕𝑋) ⊔ (𝜕𝑌 ). Also, the connected sum can be performed on non-connected 𝑛-manifolds by choosing a connected component on each of the summands, but it will depend on the choice. Now we discuss the connected sum in the smooth category 𝐃𝐌𝐚𝐧𝑛 . Let 𝑀1 , 𝑀2 be two smooth connected 𝑛-manifolds and pick interior points 𝑝 𝑖 ∈ 𝑀𝑖 . Choose 𝐶 ∞ 𝑝 𝑝 charts 𝜑1 ∶ 𝑈1 = 𝑈1 1 → 𝐵2 (0) and 𝜑2 ∶ 𝑈2 = 𝑈2 2 → 𝐵2 (0). Put also ̄ 𝐴𝑖 = 𝜑−1 𝑖 (𝐵1+𝜀 (0) − 𝐵1/(1+𝜀) (0)), ̄ 𝜀 > 0 small, which is an open subset of 𝑀𝑖 , and 𝑀𝑖′ = 𝑀𝑖 − 𝜑−1 𝑖 (𝐵1/(1+𝜀) (0)) for 𝑖 = 1, 2, 𝑜 𝑜 which is an open 𝑛-manifold. The diffeomorphism 𝑓 = 𝜑−1 ∘ 𝜑 1 ∶ 𝜕𝑀1 → 𝜕𝑀2 can 2 be extended to a diffeomorphism 𝐹 = 𝜑−1 2 ∘ 𝜑 1 ∶ 𝐴1 → 𝐴2 . Note that 𝑓 is the identity when expressed in the charts. Definition 1.39. We define the smooth connected sum as 𝑀1 #𝑓 𝑀2 = 𝑀1′ ∪𝐹 𝑀2′ . Since 𝐹 ∶ 𝐴1 → 𝐴2 is a diffeomorphism between open subsets 𝐴𝑖 ⊂ 𝑀𝑖′ , 𝑖 = 1, 2, it is easy to see that 𝑀1 #𝑓 𝑀2 has a canonical smooth structure inherited from the structures of 𝑀1 and 𝑀2 . Indeed, the atlas for 𝑀1 #𝑓 𝑀2 is obtained from an atlas of 𝑀1′ and 𝑀2′ , where the changes of coordinates involve the diffeomorphism 𝐹 for open sets contained in 𝐴1 or 𝐴2 . The manifold 𝑀 is topologically the same as the topological connected sum 𝑀1 #𝑓 𝑀2 . This is seen as follows. The inclusion 𝚤 ∶ 𝑀1𝑜 ⊔ 𝑀2𝑜 → 𝑀1′ ⊔ 𝑀2′ is compatible with the equivalence relations, so it induces a continuous map 𝐹 ∶ (𝑀1𝑜 ⊔ 𝑀2𝑜 )/ ∼ → (𝑀1′ ⊔ 𝑀2′ )/ ∼ . This is clearly injective, and it is easily seen to be surjective since the 𝑜 −1 sets 𝜋(𝑀1′ − 𝑀1𝑜 ) = 𝜋(𝜑−1 1 (𝐵1+𝜀 (0) − 𝐵1 (0))) = 𝜋(𝜑 2 (𝐵1 (0) − 𝐵1/(1+𝜀) (0))) ⊂ 𝜋(𝑀2 ) 𝑜 𝑜 ′ and 𝜋(𝑀2 − 𝑀2 ) ⊂ 𝜋(𝑀1 ). It is also seen to be open, since a basis of open sets has been constructed for the charts of (𝑀1𝑜 ⊔ 𝑀2𝑜 )/∼ , and its image under 𝐹 is open. In analogy with the continuous case, we say that two diffeomorphisms 𝑓0 , 𝑓1 ∶ 𝑋 → 𝑌 between smooth manifolds are diffeotopic if there exists a differentiable isotopy 𝐻 ∶ 𝑋 × [0, 1] → 𝑌 between 𝑓0 , 𝑓1 , where all maps 𝑓𝑠 ∶ 𝑋 → 𝑌 , 𝑓𝑠 (𝑥) = 𝐻(𝑥, 𝑠), are diffeomorphisms. Theorem 1.40. The smooth connected sum 𝑀1 #𝑓 𝑀2 does not depend on the points 𝑝1 , 𝑝2 , up to diffeomorphism. With respect to the chosen charts, suppose that we have 𝑜 𝑜 a fixed chart 𝜑2 and two charts 𝜑1 , 𝜑′1 . Let 𝑓 = 𝜑−1 2 ∘ 𝜑 1 ∶ 𝜕𝑀1 → 𝜕𝑀2 and let 𝑜 ′ 𝑜 𝑜 −1 ′ ′ 𝑓 = 𝜑2 ∘ 𝜑1 ∶ 𝜕(𝑀1 ) → 𝜕𝑀2 . There exists a diffeomorphism 𝜙 ∶ 𝜕𝑀1 → 𝜕(𝑀1𝑜 )′ such that if 𝑓 is diffeotopic to 𝑓′ ∘ 𝜙, then the smooth connected sums with the chart 𝜑1 and with the chart 𝜑′1 are diffeomorphic. Proof. This result (not being completely trivial) is simpler than that of Theorem 1.35. The case 𝑛 = 2 appears in Exercise 1.15. □ Remark 1.41. As in Remark 1.37, the indeterminacy on 𝑓 is controlled by a diffeomorphism 𝜓 ∶ 𝑆 𝑛−1 → 𝑆 𝑛−1 . As the charts are defined on balls, 𝜓 is the restriction of

20

1. Topological surfaces

a diffeomorphism between open balls. Up to diffeotopy, there are only two choices for this type of diffeomorphism, given by the identity and the antipodal map. There is a notion of twisted connected sum given by using a diffeomorphism 𝜓 ∶ 𝑆 𝑛−1 → 𝑆 𝑛−1 which cannot be extended to the ball. In this case we may obtain many non-diffeomorphic twisted connected sums 𝑀1 #𝑓 𝑀2 . For instance, the Milnor spheres (cf. Remark 1.22(e)) can be obtained by gluing two discs 𝐷7 of dimension 7 via an exotic diffeomorfism of the sphere 𝑆 6 = 𝜕𝐷7 . This feature is in sharp contrast with the case of the topological connected sum. 1.2.3. Piecewise linear manifolds. In order to better understand the differences between the world of topological manifolds and that of smooth manifolds, it is convenient to introduce an intermediate kind of structure on manifolds, the so-called PL structures (PL stands for piecewise linear). This is a structure of a combinatorial nature and provides useful algebraic tools for topological computations. The strategy consists of putting triangulations on manifolds. We need first a technical definition. Definition 1.42. Let 𝑋 be a topological space, and let 𝒞 = {𝐶𝑖 | 𝑖 ∈ 𝐼} be a covering of 𝑋 (this means that 𝐶𝑖 ⊂ 𝑋 and ⋃𝑖∈𝐼 𝐶𝑖 = 𝑋). We say that 𝑋 has a coherent topology with respect to the cover 𝒞 if the natural inclusion map 𝑞∶

⨆

𝐶𝑖 ⟶ 𝑋

𝑖∈𝐼

is a quotient map. In other words, 𝑈 ⊂ 𝑋 is open if and only if 𝑈 ∩ 𝐶𝑖 ⊂ 𝐶𝑖 is open as a subset of 𝐶𝑖 for each 𝑖 ∈ 𝐼. Example 1.43. (1) The topology of 𝑌 = ⨆𝑖∈𝐼 𝐶𝑖 is the disjoint union topology, that is, 𝑈 ⊂ 𝑌 is open if 𝑈 = ⨆𝑖∈𝐼 𝑈 𝑖 , with 𝑈 𝑖 ⊂ 𝐶𝑖 open for all 𝑖 ∈ 𝐼. (2) If all the 𝐶𝑖 are open subsets of 𝑋, then it is clear that 𝑋 has a coherent topology with respect to 𝒞. Indeed, if 𝐴 ∩ 𝐶𝑖 ⊂ 𝐶𝑖 is open as a subset of 𝐶𝑖 , since 𝐶𝑖 ⊂ 𝑋 is open, then 𝐴 ∩ 𝐶𝑖 ⊂ 𝑋 also is open as a subset of 𝑋, so 𝐴 = ⋃𝑖∈𝐼 (𝐴 ∩ 𝐶𝑖 ) is open in 𝑋. (3) If the 𝐶𝑖 are compact, |𝐼| < ∞, and 𝑋 is Hausdorff, then 𝑋 has a coherent topology with respect to the 𝐶𝑖 . Here we denote by |𝐼| the cardinal of 𝐼. To see it, simply note that 𝑞 ∶ ⨆𝑖∈𝐼 𝐶𝑖 → 𝑋 is continuous and surjective between a compact and a Hausdorff space, so it is a quotient map. Recall that a cover 𝒞 = {𝐶𝑖 } is locally finite if for every 𝑥 ∈ 𝑋 there exists an open neighbourhood 𝑈 𝑥 of 𝑥 in 𝑋 such that 𝑈 𝑥 intersects only a finite number of the sets 𝐶𝑖 . Lemma 1.44. Let 𝒞 = {𝐶𝑖 } be a covering of a topological space 𝑋. If 𝒞 is a closed and locally finite covering, then the topology of 𝑋 is coherent with respect to 𝒞. Proof. Let 𝐴 ⊂ 𝑋 such that 𝐴 ∩ 𝐶𝑖 ⊂ 𝐶𝑖 is open for all 𝑖 ∈ 𝐼. We want to see that 𝐴 is open in 𝑋. For this, take 𝑥 ∈ 𝐴, and let us find a suitable open neighbourhood of 𝑥 contained in 𝐴. Since 𝒞 is locally finite, there exists an open neighbourhood 𝑈 = 𝑈 𝑥 of 𝑥 in 𝑋 such that 𝑈 only intersects a finite number of the 𝐶𝑖 ’s. Suppose that 𝑈 only

1.2. PL structures

21

intersects 𝐶𝑖1 , . . . , 𝐶𝑖𝑚 . Among these, 𝑥 will only be contained in some of them, say that 𝑥 ∈ 𝐶𝑖1 ∩ ⋯ ∩ 𝐶𝑖𝑠 and 𝑥 ∉ 𝐸 = 𝐶𝑖𝑠+1 ∪ ⋯ ∪ 𝐶𝑖𝑚 . Put 𝑈 ′ = 𝑈 − 𝐸, which is also an open neighbourhood of 𝑥, since 𝐸 is closed and 𝑥 ∉ 𝐸. On the other hand, by hypothesis, 𝐴 ∩ 𝐶𝑖 = 𝑉 𝑖 ∩ 𝐶𝑖 for some open set 𝑉 𝑖 of 𝑋. Now put 𝑉 ̃ = 𝑈 ′ ∩𝑉 𝑖1 ⋯∩𝑉 𝑖𝑠 which clearly contains 𝑥 and is open in 𝑋. We claim that 𝑉 ̃ is contained in 𝐴. To see it, it suffices to see that 𝑉 ̃ ∩ 𝐶𝑖 ⊂ 𝐴 for all 𝑖 ∈ 𝐼, since 𝑉 ̃ = ⋃𝑖∈𝐼 (𝑉 ̃ ∩ 𝐶𝑖 ). By definition of 𝑈 ′ it is clear that for 𝑖 ∉ {𝑖1 , . . . , 𝑖𝑠 }, 𝑉 ̃ ∩ 𝐶𝑖 = ∅ ⊂ 𝐴, and if 𝑖 ∈ {𝑖1 , . . . , 𝑖𝑠 }, then 𝑉 ̃ ∩ 𝐶𝑖 ⊂ 𝑉 𝑖 ∩ 𝐶𝑖 = 𝐴 ∩ 𝐶𝑖 ⊂ 𝐴. We conclude that 𝐴 is open in 𝑋, and therefore the topology of 𝑋 is coherent with 𝒞. □ Remark 1.45. One of the consequences of the fact that a topological space 𝑋 has a coherent topology with respect to a cover are the so-called gluing lemmas. Indeed, the following are equivalent: (1) 𝑋 has a coherent topology with respect to the cover {𝐶𝑖 |𝑖 ∈ 𝐼}. (2) For each topological space 𝑌 and any map 𝑓 ∶ 𝑋 → 𝑌 , 𝑓 is continuous if and only if all the restrictions 𝑓|𝐶𝑖 ∶ 𝐶𝑖 → 𝑌 are continuous, for all 𝑖 ∈ 𝐼. A PL structure is based on the concept of a triangulation of a topological manifold. This consists of expressing an 𝑛-manifold as a union of 𝑛-polyhedra which patches nicely. A (convex) 𝑘-polyhedron 𝑃𝑘 is the convex hull of finitely many points in ℝ𝑘 with non-empty interior. An 𝑙-face of 𝑃𝑘 is a subset 𝑃 𝑙 ⊂ 𝜕𝑃 𝑘 which is the intersection of 𝑃𝑘 with a hyperplane (not passing through the interior of 𝑃𝑘 ) and which is affinely isomorphic to an 𝑙-polyhedron (that is, it has dimension 𝑙). Definition 1.46. A triangulation of a topological space 𝑋, denoted 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 )| 0 ≤ 𝑘 ≤ 𝑛, 𝛼 ∈ Λ}, consists of the following. (1) A family of polyhedra 𝑃𝛼𝑘 with dimensions varying between 0 and 𝑛. (2) For each polyhedron, a continuous map 𝑓𝛼 ∶ 𝑃𝛼𝑘 → 𝑋. Its image 𝐶𝛼𝑘 = 𝑓𝛼 (𝑃𝛼𝑘 ) is called a 𝑘-cell. (3) The maps 𝑓𝛼 ∶ Int 𝑃𝛼𝑘 → 𝑓𝛼 (Int 𝑃𝛼𝑘 ) ⊂ 𝑋 are homeomorphisms. (4) The sets 𝐶𝛼𝑘 ⊂ 𝑋 form a locally finite covering of 𝑋 by closed sets. (5) The images 𝑓𝛼 (Int 𝑃𝛼𝑘 ) ⊂ 𝑋 form a disjoint covering of 𝑋. (6) For each 𝑙-face 𝑃 𝑙 ⊂ 𝜕𝑃𝛼𝑘 there exists a map in the triangulation 𝑓𝛽 ∶ 𝑃𝛽𝑙 → 𝑋 such that the following diagram commutes 𝑃𝛽𝑙 𝐿

 𝜕𝑃𝛼𝑘

𝑓𝛽

/𝑋 ?

𝑓𝛼

where 𝐿 ∶ 𝑃𝛽𝑙 → 𝑃 𝑙 ⊂ 𝑃𝛼𝑘 is an affine isomorphism followed by the inclusion. In general, 0-cells are called vertices, 1-cells are called edges, and 2-cells are called faces. The dimension of 𝑋 is 𝑛 = dim 𝑋, the maximum of the dimensions of the polyhedra of the triangulation.

22

1. Topological surfaces

Remark 1.47. • The topology of 𝑋 is the coherent topology with respect to the cells {𝐶𝛼𝑘 } by Lemma 1.44. • Condition (6) is a patching condition. If two polyhedra 𝑃𝛼𝑘 and 𝑃𝛽𝑡 have intersecting images, then the intersection is formed by 𝑙-faces 𝐶𝛾𝑙 , where 𝑖1 ∶ 𝑃𝛾𝑙 ↪ 𝜕𝑃𝛼𝑘 and 𝑖2 ∶ 𝑃𝛾𝑙 ↪ 𝜕𝑃𝛽𝑡 , and moreover 𝑓𝛼 ∘ 𝑖1 = 𝑓𝛽 ∘ 𝑖2 = 𝑓𝛾 ∶ 𝑃𝛾𝑙 → 𝐶𝛾𝑙 . • We allow for 𝑓𝛼 to send two (or more) 𝑙-faces 𝑃 𝑙 ⊂ 𝜕𝑃𝛼𝑘 to the same 𝑙-cell 𝐶𝛾𝑙 ⊂ 𝑋 of the triangulation. A stronger definition of triangulation can be made with the following requirements. (1) All 𝑓𝛼 ∶ 𝑃𝛼𝑘 → 𝐶𝛼𝑘 are homeomorphisms. (2) The 𝑃𝛼𝑘 are 𝑘-simplices. A 𝑘-simplex is the convex hull of 𝑘 + 1 affinely independent points in ℝ𝑘 (3) Any two of the 𝑘-cells cannot share all their vertices. We call this a regular triangulation. A triangulation can always be subdivided (by subdividing each polyhedron into smaller ones) to achieve this (see Remark 1.49). • Given a regular triangulation, take S to be the set of vertices. Any 𝑘-cell is determined by a finite collection 𝑣 0 , . . . , 𝑣 𝑘 ∈ S, so the triangulation is given by a collection 𝜎 of finite subsets of S. Here 𝜎 satisfies that if 𝐴 ∈ 𝜎 and 𝐵 ⊂ 𝐴, then 𝐵 ∈ 𝜎. If 𝑋 is compact, then 𝐿 = |S|−1 is finite. Take a map 𝑓 ∶ S → {𝑤 0 , . . . , 𝑤 𝐿 } ⊂ ℝ𝐿 , where 𝑤 0 , . . . , 𝑤 𝐿 are affinely independent points. For any 𝐴 ∈ 𝜎, let 𝐶𝐴 be the convex hull of 𝑓(𝐴). Then 𝑋 is homeomorphic to the simplicial complex 𝐂S = ⋃𝐴∈S 𝐶𝐴 . • The definition of triangulation can also be made more flexible by allowing curvilinear polyhedra. This is given by a 𝑘-disc 𝐷𝑘 ⊂ ℝ𝑘 where the boundary 𝜕𝐷𝑘 = ⋃ 𝑃𝑖 , for finitely many closed subsets 𝑃𝑖 , each of which is homeomorphic to a curvilinear (𝑘 − 1)-polyhedron (defined by induction), and the polyhedra in the boundaries of the different 𝑃𝑖 agree. We call this a pseudotriangulation. A curvilinear 2-polyhedron is a disc in which the boundary 𝑆 1 has been divided into edges by several vertices. Example 1.48. Let us give examples of triangulations of surfaces. (1) Take the disc 𝐷2 . Choose 𝑄 = (0, 1) and 𝑃 = (0, −1). These vertices determine two edges 𝑎, 𝑎′ , which are two semicircles on 𝜕𝐷2 . Now we identify each point (𝑥, 𝑦) ∈ 𝑎 with (−𝑥, 𝑦) ∈ 𝑎′ . The quotient 𝐷2 /∼ is homeomorphic to 𝑆 2 . Therefore 𝑆 2 admits a pseudo-triangulation with one face 𝐷2 , one edge 𝑎 = 𝑎′ , and two vertices 𝑃 and 𝑄. In this triangulation, there is only one edge which is embedded into the 2-cell (in this case, the face 𝐷2 ) in two different ways. To denote this, we label all the images of the embedding with the same letter. As the embeddings are determined only by the orientation (cf. Lemma 1.34), we just depict it with an arrow (see Exercise 1.29).

1.2. PL structures

23

Q

a

c

c0

b

b0

a0 a P Figure 1.4. Pseudo-triangulation and triangulation of 𝑆2 .

Q

a

c

b0

b

c0

a0

P Figure 1.5. Pseudo-triangulation and triangulation of ℝ𝑃 2 .

To get a triangulation, it is enough to divide each edge into two consecutive edges, as 𝑎 = 𝑏𝑐. We get a square, as in Figure 1.4. (2) Let us describe a pseudo-triangulation for ℝ𝑃 2 (see Figure 1.5). Recall that ℝ𝑃2 = 𝑆 2 /∼ , where (𝑥, 𝑦, 𝑧) ∼ (−𝑥, −𝑦, −𝑧). We consider the hemisphere 𝑆 2+ = {(𝑥, 𝑦, 𝑧) ∈ 𝑆 2 |𝑧 ≥ 0} and the induced equivalence relation, given by (𝑥, 𝑦, 0) ∼ (−𝑥, −𝑦, 0), for (𝑥, 𝑦, 0) ∈ 𝜕𝑆 2+ . The inclusion 𝚤 ∶ 𝑆 2+ → 𝑆 2 induces a map 𝚤 ̄ ∶ 𝑆 2+ / ∼ → 𝑆 2 /∼ , which is clearly continuous and bijective. As the first space is compact and the second is Hausdorff, it is a homeomorphism. Hence ℝ𝑃 2 ≅ 𝑆 2+ / ∼ . Now 𝐷2 ≅ 𝑆 2+ via (𝑥, 𝑦) ↦ (𝑥, 𝑦, √1 − 𝑥2 − 𝑦2 ). So the induced equivalence relation in 𝐷2 is (𝑥, 𝑦) ∼ (−𝑥, −𝑦) for (𝑥, 𝑦) ∈ 𝜕𝐷2 . Therefore 𝑞 ∶ 𝐷2 ⟶ ℝ𝑃 2 ,

(𝑥, 𝑦) ↦ [𝑥, 𝑦, √1 − 𝑥2 − 𝑦2 ]

is a quotient map, and ℝ𝑃2 ≅ 𝐷2 / ∼ . This produces a pseudo-triangulation: take the disc 𝐷2 with vertices 𝑃 and 𝑄 and edges 𝑎 and 𝑎′ as in (1). We identify each point (𝑥, 𝑦) ∈ 𝑎 with the point (−𝑥, −𝑦) ∈ 𝑎′ . For a triangulation, subdivide 𝑎 = 𝑏𝑐 in the two edges. (3) Take the square 𝑃 = [0, 1] × [0, 1] and the map 𝑓 ∶ 𝑃 → 𝑇 2 = 𝑆 1 × 𝑆 1 , 𝑓(𝑥, 𝑦) = (𝑒2𝜋i𝑥 , 𝑒2𝜋i𝑦 ). This gives a triangulation of the torus (see Figure 1.6).

24

1. Topological surfaces

b

b

a

a

a

a

a b

b

b

Figure 1.6. Triangulation and regular triangulation of 𝑇 2 .

To obtain a regular triangulation, we just make a subdivision on the square. See the right picture of Figure 1.6. The Klein bottle can also be triangulated with the square 𝑃 = [0, 1] × [0, 1], with two edges 𝑎, 𝑏 and one vertex. Recall the definition of the Klein bottle as the mapping torus of the map 𝑓 ∶ 𝑆 1 → 𝑆 1 , 𝑓(𝑥, 𝑦) = (−𝑥, 𝑦), given in Example 1.33(8). This means that Kl = (𝑆 1 ×[0, 1])/∼ , (𝑒2𝜋i𝑡 , 0) ∼ (𝑒2𝜋i(1−𝑡) , 1), 𝑡 ∈ [0, 1]. This is equivalent to Kl ≅ 𝑃/∼ , with (𝑡, 0) ∼ (1 − 𝑡, 1) and (0, 𝑦) ∼ (1, 𝑦), 𝑡 ∈ [0, 1], 𝑦 ∈ [0, 1], giving the triangulation in Figure 1.7.

b a

a

a

a

b b Figure 1.7. Triangulation of the Klein bottle.

Remark 1.49. If we have a triangulation (or pseudo-triangulation) 𝜏 on a space 𝑋, we can obtain another triangulation 𝜏′ by subdividing the polyhedra. The typical way of doing this is by taking the barycenters of all the polyhedra of the triangulation as new vertices of 𝜏′ , and form the other 𝑙-polyhedra as convex hulls of successive barycenters. This procedure is called the barycentric subdivision of 𝜏. After one barycentric subdivision, all polyhedra in the triangulation are 𝑘-simplices. If we repeat the barycentric subdivision twice, then we get a regular triangulation even if 𝜏 was a pseudotriangulation. Note that the sizes of the 𝑙-polyhedra are divided at least by two after a barycentric subdivision. So if we repeat this procedure a high number of times, we can make all the polyhedra of the triangulation as small as we want. In particular, by doing barycentric subdivision, we can subordinate the triangulation 𝜏 to any open covering of 𝑋, in the sense that any cell is contained in at least one open set of the cover.

1.2. PL structures

25

Definition 1.50. A piecewise linear (PL) 𝑛-manifold is a topological 𝑛-manifold 𝑀 endowed with a triangulation 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 )}. The triangulation 𝜏 is called a PL structure on 𝑀. For a triangulation 𝜏 on an 𝑛-manifold, all 𝑘-cells with 𝑘 < 𝑛 are contained as a face of an 𝑛-cell. Therefore 𝑞 ∶ ⨆ 𝑃𝛼𝑛 → 𝑀 is also a quotient map. This is proved as follows: suppose that there is a 𝑘-cell 𝑓𝛽 (𝑃𝛽𝑘 ) not contained in the closure of an 𝑙-cell with 𝑙 > 𝑘. Then 𝑓𝛽 (Int 𝑃𝛽𝑘 ) is an open subset (since its complement is the closure of the rest of the cells), and so any point of it has a neighbourhood homeomorphic to an open subset of ℝ𝑘 . This contradicts Theorem 1.2. PL manifolds can be put into a category, but in this case it is somewhat involved to formalize the adequate definitions. Let (𝑀1 , 𝜏1 = {(𝑃𝛼𝑘 , 𝑓𝛼 )}), (𝑀2 , 𝜏2 = {(𝑃𝛽𝑘 , 𝑔𝛽 )}) be triangulated spaces (with regular triangulations). We say that a continuous map ℎ ∶ 𝑀1 → 𝑀2 is simplicial if for every 𝑃𝛼𝑙 ∈ 𝜏1 there exists a 𝑃𝛽𝑚 ∈ 𝜏2 so that ℎ(𝑓𝛼 (𝑃𝛼𝑙 )) ⊂ 𝑔𝛽 (𝑃𝛽𝑚 ) and an affine map 𝐿𝛼𝛽 ∶ 𝑃𝛼𝑙 → 𝑃𝛽𝑚 sending vertices to vertices, such that 𝑔𝛽 ∘ 𝐿𝛼𝛽 = ℎ ∘ 𝑓𝛼 . The weak point about this notion is the scarcity of simplicial maps. So we allow as a PL map a continuous map that is simplicial after a polyhedral subdivision. It is thus natural to define a PL structure on a manifold 𝑀 as an equivalence class of triangulations 𝜏 up to polyhedral subdivisions, in the same vein as a smooth structure is defined as a smooth atlas up to compatibility of atlas. Note that there are tricky situations to have in mind: take 𝑀 = [0, 1] × [−1, 1], and let 𝜏 be a triangulation with the edges of 𝜕𝑀 and adding the edge [0, 1] × {0}, and let 𝜏′ be the triangulation adding the edge {(𝑥, 𝑥 sin(1/𝑥))|𝑥 ∈ [0, 1]}. Then (𝑀, 𝜏), (𝑀, 𝜏′ ) are different PL structures (Id ∶ (𝑀, 𝜏) → (𝑀, 𝜏′ ) is not a PL map) but there is a homeomorphism 𝑓 ∶ (𝑀, 𝜏) → (𝑀, 𝜏′ ) which is a PL isomorphism. Definition 1.51. We define the category of PL 𝑛-manifolds 𝐏𝐋𝐌𝐚𝐧𝑛 , by taking as objects pairs (𝑀, 𝜏), where 𝑀 is a topological 𝑛-manifold and 𝜏 is a triangulation of 𝑀, and as morphisms the PL maps. The category 𝐏𝐋𝐌𝐚𝐧𝑛𝑐 has objects compact PL 𝑛-manifolds. Remark 1.52. We can define the connected sum in 𝐏𝐋𝐌𝐚𝐧𝑛 . Let (𝑀1 , 𝜏1 ), (𝑀2 , 𝜏2 ) be two connected PL manifolds, and subdivide the triangulations until they are regular. Take two 𝑛-polyhedra 𝑃𝛼𝑛 ∈ 𝜏1 , 𝑃𝛽𝑛 ∈ 𝜏2 , which are 𝑛-simplices. Then there is an isomorphism 𝑓 ∶ 𝜕𝑃𝛼𝑛 → 𝜕𝑃𝛽𝑛 . We take 𝑋1𝑜 = 𝑋1 − 𝑓𝛼 (Int 𝑃𝛼𝑛 ) and 𝑋2𝑜 = 𝑋2 − 𝑓𝛽 (Int 𝑃𝛽𝑛 ). Then we define 𝑋1 #𝑓 𝑋2 = 𝑋1𝑜 ∪𝑓 𝑋2𝑜 . Remark 1.53. Let us briefly discuss the classification problem in the PL category. For 𝑐𝑜 𝑐𝑜 low dimensions, we have 𝕃𝐏𝐋𝐌𝐚𝐧0 = {⋆}, 𝕃𝐏𝐋𝐌𝐚𝐧1 = {𝑆 1 , ℝ}. In general, any differentiable manifold admits a triangulation unique up to PL isomorphism (proved by Whitehead). The existence appears in Exercise 3.16 for the case 𝑛 = 2. Therefore we have maps 𝕃𝐃𝐌𝐚𝐧𝑛 ⟶ 𝕃𝐏𝐋𝐌𝐚𝐧𝑛 ⟶ 𝕃𝐓𝐌𝐚𝐧𝑛 . For low dimensions 𝑛 = 1, 2, 3, both maps are bijections. For 𝑛 ≥ 4 these maps are in general neither injective nor surjective. That means that not every PL manifold admits

26

1. Topological surfaces

a smooth structure, or that there are topological manifolds which do not admit a triangulation. The differences between these three classes is controlled by some classifying spaces denoted 𝐁𝐓𝐨𝐩, 𝐁𝐏𝐋, and 𝐁𝐃𝐢𝐟𝐟, defined in homotopy theoretic terms. Remark 1.54. Suppose that 𝑀 is a PL 𝑛-manifold (with or without boundary). Let 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 )} be the triangulation of 𝑀, so 𝑞 ∶ ⨆ 𝑃𝛼𝑛 → 𝑀 is a quotient map. Let us see how the 𝑛-cells glue. Pick a (𝑛 − 1)-cell 𝐶𝛽𝑛−1 ⊂ 𝑀. As we have remarked above, this appears in the boundary of some (or several) 𝑛-polyhedra. Let 𝑃𝛼𝑛𝑖 , 𝑖 = 1, . . . , 𝑠, be such polyhedra. We say that 𝑃𝛼𝑛𝑖 are incident on 𝑃𝛽𝑛−1 , and we say that 𝑃𝛼𝑛𝑖 , 𝑃𝛼𝑛𝑗 are adjacent. Let 𝐿𝑖 ∶ 𝑃𝛽𝑛−1 → 𝜕𝑃𝛼𝑛𝑖 with 𝑓𝛼𝑖 ∘ 𝐿𝑖 = 𝑓𝛽 . Pick an interior point 𝑥 ∈ 𝑃𝛽𝑛−1 , and let 𝑥𝑖 = 𝐿𝑖 (𝑥). We can assume, without loss of generality, that all maps 𝐿𝑖 are isometric. Take 𝜖 > 0 small, and let 𝑈 = ⨆(𝑃𝛼𝑛𝑖 ∩ 𝐵𝜖 (𝑥𝑖 )). This is a saturated open set, so that 𝑈 ⋆ = 𝑞(𝑈) is an open neighbourhood of 𝑝 = 𝑞(𝑥𝑖 ) ∈ 𝑀. The set 𝑈 ⋆ is homeomorphic to 𝑅𝜖 = (⨆1≤𝑖≤𝑠 𝐵𝜖+ (0)𝑖 )/ ∼ , of a collection of 𝑠 semiballs with the relation (𝑥, 0)𝑖 ∼ (𝑥, 0)𝑗 , 𝑖 ≠ 𝑗, where (𝑥, 𝑦)𝑖 denotes the point in the 𝑖th copy 𝐵𝜖+ (0)𝑖 ⊂ ℝ𝑛+ . As 𝑀 is a manifold around 𝑝, there is neighbourhood 𝑉 ⋆ ⊂ 𝑈 ⋆ homeomorphic to either 𝐵𝜂 (0) if 𝑝 is interior point or to 𝐵𝜂+ (0) if 𝑝 is a boundary point. Suppose first that 𝑝 is an interior point. Take the preimage of 𝐵𝜂 (0) on 𝑅𝜖 . This contains some 𝑅𝛿 , with 𝛿 < 𝜖. So the preimage of 𝐵𝜂 (0)−{0}, which is homotopy equivalent to 𝑆 𝑛−1 , retracts to 𝜕𝑅̄ 𝛿 , which is the union of 𝑠 discs glued along their boundary. Using homology (section 2.3), there should be an epimorphism 𝐻𝑛−1 (𝑆 𝑛−1 ) = ℤ → 𝐻𝑛−1 (𝜕𝑅̄ 𝛿 ) = ℤ𝑠−1 (for 𝑛 = 2, we can use the fundamental group, 𝜋1 (𝑆 1 ) = ℤ → 𝜋1 (𝜕𝑅̄ 𝛿 ) = 𝐹𝑠−1 , the free group on 𝑠 − 1 generators). We conclude that 𝑠 = 2, that is, only two 𝑛-cells are glued along 𝑃𝛽𝑛−1 . Now suppose that 𝑝 is a boundary point. We do a similar argument with 𝐵𝜂+ (0) instead of 𝐵𝜂 (0). Then we have 𝐵𝜂+ (0) − {0} is contractible, so there is an epimorphism {0} → ℤ𝑠−1 and 𝑠 = 1. Hence only one 𝑛-cell appears at a boundary point. 1.2.4. Existence of triangulations for surfaces. Now we shall prove that any compact topological surface without boundary admits a triangulation. This is a result proven by Radó [Rad]. We leave the case of surfaces with boundary as an exercise (Exercise 1.17). The case of open surfaces can be treated by an argument of exhaustion via compact embedded subsurfaces with boundary. We shall use Theorem 1.36. This is a fundamental result in 2-dimensional topology, that has no analogue in higher dimensions 𝑛 ≥ 3 (as shown by the horned Alexander sphere [Ale]). Recall that a Jordan curve is a set 𝐶 ⊂ ℝ2 homeomorphic to 𝑆 1 , and we call a Jordan arc to a set 𝐶 ⊂ ℝ2 homeomorphic to [0, 1]. Theorem 1.36 implies that if 𝐶 ⊂ ℝ2 is a Jordan curve, then ℝ2 − 𝐶 has two connected components, one bounded and the other unbounded. The bounded component has closure homeomorphic to a disc 𝐷2 . Theorem 1.55. Let 𝑆 be a compact topological surface. Then 𝑆 admits a triangulation.

1.2. PL structures

27

Proof. Step 1. We want to find a finite collection of closed subsets 𝐷𝑖 ⊂ 𝑆, 1 ≤ 𝑖 ≤ 𝑚, which are homeomorphic to closed discs such that ⋃ 𝐷𝑖 = 𝑆 and (1.4)

Γ𝑖 = 𝜕𝐷𝑖 is a collection of Jordan curves such that Γ𝑖 ∩ Γ𝑗 , 𝑖 < 𝑗, is a finite collection of points or Jordan arcs.

Let us construct such collection. We take for each point 𝑝 ∈ 𝑆 an open neighbourhood 𝑈 𝑝 with a homeomorphism 𝜑 ∶ 𝑈 → 𝐵1 (0) ⊂ ℝ2 , 𝜑(𝑝) = 0. We take the sets 𝑈 ′ = 𝜑−1 (𝐵1/2 (0)), which cover 𝑆. By compactness, we have a finite collection ′ of such sets 𝑈1′ , . . . , 𝑈𝑚 covering 𝑆, and denote 𝜑𝑖 ∶ 𝑈 𝑖 → 𝐵1 (0), 𝑖 = 1, . . . , 𝑚. The ′ ′ sets 𝐷𝑖 will be constructed as 𝜑−1 𝑖 (𝐷𝑖 ), where 𝐷𝑖 are closed discs (i.e., subsets homeo′ 2 morphic to 𝐷 ) such that 𝐵1/2 (0) ⊂ 𝐷𝑖 ⊂ 𝐵1 (0), and Γ𝑖 = 𝜕𝐷𝑖 . We do this by induc′ ̄ (0), 𝐷1 = 𝜑−1 tion on 𝑖. For 𝑖 = 1, we take 𝐷1′ = 𝐵1/2 1 (𝐷1 ), Γ1 = 𝜕𝐷1 . Now suppose that 𝐷1 , . . . , 𝐷𝑖−1 have been constructed satisfying the property (1.4). Take the images 𝐴𝑗 = 𝜑𝑖 (Γ𝑗 ∩ 𝑈 𝑖 ) ⊂ 𝐵1 (0), 𝑗 < 𝑖. An obvious consequence of Theorem 1.36 is that any Jordan curve has empty interior. So 𝐴 = ⋃𝑗 0}. Definition 1.66 means that two bases (𝑣 1 , . . . , 𝑣 𝑛 ), (𝑤 1 , . . . , 𝑤 𝑛 ) define the same orientation + if (𝑣 1 , . . . , 𝑣 𝑛 ) = (𝑤 1 , . . . , 𝑤 𝑛 )𝐴, with 𝐴 ∈ GL (𝑛, ℝ). We also have the special linear group SL(𝑛, ℝ) = {𝐴 ∈ GL(𝑛, ℝ)| det(𝐴) = 1}. Remark 1.67. (1) If 𝑉 = {0}, we define formally Or(𝑉) = {+, −}, so choosing an orientation is merely choosing a sign. (2) If dim(𝑉) = 1, choosing an orientation is equivalent to choosing a positive direction. So Or(𝑉) = {[+𝑣], [−𝑣]}, with 𝑣 ∈ 𝑉 any non-zero vector. (3) If dim(𝑉) = 2, choosing an orientation is equivalent to choosing a positive direction of rotation for 𝑆 1 . So Or(𝑉) = {[𝑒 1 , 𝑒 2 ], [𝑒 2 , 𝑒 1 ]}, with (𝑒 1 , 𝑒 2 ) any basis of 𝑉. The first orientation declares that 𝛾1 (𝑡) = cos(𝑡)𝑒 1 + sin(𝑡)𝑒 2 is the positively oriented circle, whereas the second orientation declares that 𝛾2 (𝑡) = cos(𝑡)𝑒 2 + sin(𝑡)𝑒 1 is the positively oriented circle. In general, if Or(𝑉) = {𝑜1 , 𝑜2 }, we denote −𝑜1 = 𝑜2 , that is, the “minus” consists of changing the orientation to its opposite. Note that if 𝑜 = [(𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 )], then −𝑜 = [(−𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 )]. Let 𝑀 be now a smooth manifold. Recall that we can associate to each point 𝑝 ∈ 𝑀 a vector space 𝑇𝑝 𝑀 called the tangent space (see section 3.1.1). In a chart (𝑈, 𝜑) around 𝑝, given as 𝜑 = (𝑥1 , . . . , 𝑥𝑛 ), the tangent space is generated by the coordinate vectors 𝜕 𝜕 , . . . , 𝜕𝑥 . If we have another chart (𝑉, 𝜓), with 𝜓 = (𝑦1 , . . . , 𝑦𝑛 ), then the tangent 𝜕𝑥 1

𝑛

space is also generated by

𝜕 𝜕 , . . . , 𝜕𝑦 , 𝜕𝑦1 𝑛

and the change of basis is given by

𝜕𝑦 𝜕 𝜕 =∑ 𝑖 . 𝜕𝑥𝑗 𝜕𝑥 𝑗 𝜕𝑦 𝑖 𝑗 That is, the matrix of the change of basis is given by the differential of the change of 𝜕𝑦

charts, 𝑑(𝜓 ∘ 𝜑−1 ) = ( 𝜕𝑥𝑖 ). 𝑗

Definition 1.68. An orientation on a differentiable manifold 𝑀 consists of a choice of a orientation 𝑜(𝑥) of 𝑇𝑥 𝑀 for every 𝑥 ∈ 𝑀, such that 𝑥 ↦ 𝑜(𝑥) ∈ Or(𝑇𝑥 𝑀) is smooth. This means that for every 𝑥 ∈ 𝑀 there exists a neighbourhood 𝑈 𝑥 ⊂ 𝑀 and linearly independent smooth vector fields 𝑉1 , . . . , 𝑉𝑛 on 𝑈 𝑥 such that 𝑜(𝑦) = [(𝑉1 (𝑦), . . . , 𝑉𝑛 (𝑦))], for all 𝑦 ∈ 𝑈 𝑥 . A manifold which can be given an orientation is called orientable. We denote by Or(𝑀) the set of orientations of 𝑀. In particular Or(𝑀) = ∅ if 𝑀 is not orientable. If

36

1. Topological surfaces

we make explicit the choice of the orientation 𝑜 ∈ Or(𝑀), then we say that (𝑀, 𝑜) is an oriented manifold. There is a useful characterization of orientable manifold. Suppose that 𝑀 comes with an orientation 𝑜. Let 𝜑 ∶ 𝑈 → ℝ𝑛 be a chart with 𝑈 connected, with vector fields 𝑉1 , . . . , 𝑉𝑛 on 𝑈 such that 𝑜(𝑥) = [(𝑉1 (𝑥), . . . , 𝑉𝑛 (𝑥)] for 𝑥 ∈ 𝑈. Then we write 𝜕 𝑉 𝑗 (𝑥) = ∑ 𝑓𝑖𝑗 (𝑥) 𝜕𝑥 , 𝑗 = 1, . . . , 𝑛. The function det(𝑓𝑖𝑗 ) is smooth and non-zero on 𝑈. 𝑖 Therefore by connectedness, either it is everywhere positive or everywhere negative. 𝜕 𝜕 If det(𝑓𝑖𝑗 ) > 0, then 𝑜(𝑥) = [( 𝜕𝑥 , . . . 𝜕𝑥 )], and we say that (𝑈, 𝜑) is an oriented (or 1

𝑛

𝜕

𝜕

positive) chart. If det(𝑓𝑖𝑗 ) < 0, then 𝑜(𝑥) = − [( 𝜕𝑥 , . . . 𝜕𝑥 )], and we say that (𝑈, 𝜑) 1 𝑛 is a negative chart. Taking the reflection 𝑟(𝑡1 , 𝑡2 , . . . , 𝑡𝑛 ) = (−𝑡1 , 𝑡2 , . . . , 𝑡𝑛 ), the chart 𝜙 = 𝑟 ∘ 𝜑 is positive. This means that 𝒜+ = {(𝑈, 𝜑) positive chart} ⊂ 𝒜 is an atlas of 𝑋. It is called a positive atlas. Proposition 1.69. Let 𝑀 be a smooth manifold. Then 𝑀 is orientable if and only if we can find a smooth atlas 𝒜+ = {(𝑈𝛼 , 𝜑𝛼 )} such that all the changes of coordinates 𝜑𝛼 ∘ 𝜑−1 𝛽 have positive Jacobian, i.e., det(𝑑(𝜑𝛼 ∘ 𝜑−1 𝛽 )) > 0 wherever it is defined. Proof. Suppose 𝑀 is orientable, and take its positive atlas 𝒜+ as constructed above. Then for each non-empty intersection 𝑈𝛼 ∩ 𝑈 𝛽 take the corresponding smooth charts 𝜑𝛼 ∶ 𝑈𝛼 → ℝ𝑛 and 𝜑𝛽 ∶ 𝑈 𝛽 → ℝ𝑛 such that 𝑜(𝑥) = [(

𝜕 𝜕 𝜕 𝜕 , ... , , ... , )] = [( )] 𝜕𝑥1 𝜕𝑥𝑛 𝜕𝑦1 𝜕𝑦𝑛

for all 𝑥 ∈ 𝑈𝛼 ∩ 𝑈 𝛽 . Therefore the determinant of the change of basis, the Jacobian, satisfies 𝜕𝑦 𝑖 𝐽(𝜑𝛽 ∘ 𝜑−1 ) > 0. 𝛼 ) = det( 𝜕𝑥𝑗 Conversely, suppose that there exists an atlas 𝒜+ whose change of coordinates all have 𝜕 𝜕 positive Jacobian. Then we define the orientation on 𝑥 ∈ 𝑀 by 𝑜(𝑥) = [( 𝜕𝑥 , . . . , 𝜕𝑥 )] 1 𝑛 for a chart (𝑈, 𝜑) with 𝑥 ∈ 𝑈. This definition is independent of the chart around 𝑥 chosen, precisely because the change of charts in 𝒜+ have positive Jacobian. We conclude that this is a well defined orientation on 𝑀. □ For a smooth manifold with boundary 𝑀, we say that 𝑀 is orientable if Int 𝑀 is orientable as a manifold without boundary. If 𝑀 is orientable, then the (𝑛−1)-manifold 𝜕𝑀 inherits an orientation form that of 𝑀. This is constructed as follows. Take a chart 𝜕 𝜑 ∶ 𝑈 → 𝐵𝜀+ (0) ⊂ ℝ𝑛+ = ℝ𝑛 ∩ {𝑥1 ≥ 0} around 𝑝 ∈ 𝜕𝑀. Any vector 𝜈 = ∑ 𝜈 𝑖 𝜕𝑥 𝑖 at 𝑝 whose first coordinate satisfies 𝜈1 < 0 is called outward pointing. This notion is well defined, i.e., it is independent of charts. If 𝜑 = (𝑥1 , . . . , 𝑥𝑛 ), 𝜓 = (𝑦1 , . . . , 𝑦𝑛 ) are two charts as above, we note that 𝑦1 = 𝑦1 (𝑥1 , . . . , 𝑥𝑛 ) satisfies that 𝑦1 (0, 𝑥2 , . . . , 𝑥𝑛 ) = 0 𝜕𝑦 𝜕𝑦 and 𝑦1 (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) > 0 for 𝑥1 > 0. So 𝜕𝑥1 > 0 and 𝜕𝑥1 = 0 for 𝑖 > 1, at 𝑝. Then if 1

𝑖

1.4. Orientability

37

𝜕

𝜈 = ∑ 𝜈𝑗′ 𝜕𝑦 in the chart 𝜓, we have 𝑗

𝜈 = 𝜈1

𝜕𝑦 𝜕 𝜕 𝜕 𝜕 +∑𝜈 = 𝜈1 1 + ∑ 𝜈′ , 𝜕𝑥1 𝑗>1 𝑗 𝜕𝑥𝑗 𝜕𝑥1 𝜕𝑦1 𝑗>1 𝑗 𝜕𝑦𝑗

𝜕𝑦

so 𝜈′1 = 𝜈1 𝜕𝑥1 > 0, as well. 1

Proposition 1.70. Let (𝑀, 𝑜) be a oriented manifold with boundary. Then 𝜕𝑀 inherits a canonical orientation, denoted 𝑜|𝜕𝑀 . Proof. The case of dimension 𝑛 = 1 has to be treated separately. In this case 𝜕𝑀 is 0-dimensional. We orient 𝑝 by assigning the sign + if 𝜈 gives the orientation of 𝑀, and the sign − if −𝜈 gives the orientation of 𝑀, where 𝜈 is an outward pointing vector at 𝑝 (see Figure 1.10). Suppose now that 𝑛 ≥ 2. Take a positive chart 𝜑 ∶ 𝑈 → 𝐵𝜀+ (0) near a point 𝑝 ∈ 𝜕𝑀, and let 𝜈 be an outward pointing vector. We can arrange that the chart is positive by changing the sign of the last coordinate if necessary (possible since 𝑛 ≥ 2). Note that 𝜕𝑀 ∩ 𝑈 is mapped by 𝜑 to 𝐵𝜀+ (0) ∩ {𝑥1 = 0}. We establish a positive basis for the orientation of 𝜕𝑀 at 𝑝 as any basis (𝑣 2 , . . . , 𝑣 𝑛 ) of 𝑇𝑝 (𝜕𝑀) such that (𝜈, 𝑣 2 , . . . , 𝑣 𝑛 ) is 𝜕

a positive basis for 𝑇𝑝 𝑀. The outward pointing vector is 𝜈 = − 𝜕𝑥 , so this means that 1 the chart (𝑈 ∩𝜕𝑀, 𝜑|𝜕𝑀 ) is negative for the orientation of 𝜕𝑀. This gives a well defined orientation on 𝜕𝑀 that varies smoothly (see Figure 1.10 for the case 𝑛 = 2). □

@M −

M

+

ν

Figure 1.10. Induced orientations.

1.4.2. Orientation of PL manifolds. The concept of orientation can also be defined in the PL category by means of triangulations. First, an orientation 𝑜 of an 𝑛-polyhedron 𝑃 𝑛 is simply an orientation of its interior Int 𝑃 𝑛 , i.e., an orientation of ℝ𝑛 . This induces an orientation 𝑜|𝑃 𝑛−1 for any (𝑛 − 1)-face 𝑃𝑛−1 ⊂ 𝜕𝑃 𝑛 . More explicitly, note that 𝑃 𝑛 − ⋃ 𝑃𝛾𝑛−2 , where 𝑃𝛾𝑛−2 are all the (𝑛 − 2)-faces of 𝑃 𝑛 , is a smooth 𝑛-manifold with boundary, and the boundary consists of the interiors of the (𝑛 − 1)-faces, which are given the induced orientation as described above. More concretely, 𝑜 |𝑃 𝑛−1 = [(𝑤 2 , . . . , 𝑤 𝑛 )] ⟺ 𝑜 = [(𝜈, 𝑤 2 , . . . , 𝑤 𝑛 )], with 𝜈 a tangent vector of 𝑃 𝑛 which is perpendicular to 𝑃𝑛−1 ⊂ 𝜕𝑃 𝑛 and points outwards 𝑃𝑛 .

38

1. Topological surfaces

Figure 1.11. Compatible orientations.

Recall that for a PL manifold, a (𝑛 − 1)-polyhedron 𝑃𝛾𝑛−1 is the boundary of at most two 𝑛-polyhedra (Remark 1.54). When 𝑃𝛾𝑛−1 consists of interior points, there are two (adjacent) 𝑛-polyhedra 𝑃𝛼𝑛 , 𝑃𝛽𝑛 , so that 𝑃𝛾𝑛−1 is a face of them. If 𝑜𝛼 , 𝑜 𝛽 are the orientations of 𝑃𝛼𝑛 , 𝑃𝛽𝑛 , respectively, then we say that they are compatible along 𝑃𝛾𝑛−1 if they coincide at the points of the (𝑛−1)-face. To be precise, let 𝜈𝑎 , 𝜈 𝑏 be the outward normal to 𝑃𝛾𝑛−1 for 𝑃𝛼𝑛 and 𝑃𝛽𝑛 , respectively, so 𝜈𝛼 = −𝜈 𝛽 . Let (𝑤 2 , . . . , 𝑤 𝑛 ) be a basis of 𝑃𝛾𝑛−1 such that 𝑜𝛼 = [(𝜈𝛼 , 𝑤 2 , . . . , 𝑤 𝑛 )]. Then 𝑜 𝛽 = [(−𝜈 𝑏 , 𝑤 2 , . . . , 𝑤 𝑛 )] = −[(𝜈 𝑏 , 𝑤 2 , . . . , 𝑤 𝑛 )], and thus 𝑜𝛼 |𝑃𝛾𝑛−1 = [𝑤 2 , . . . , 𝑤 𝑛 ], So the orientations are compatible along on 𝑃𝛾𝑛−1 are opposite (see Figure 1.11).

𝑜 𝛽 |𝑃𝛾𝑛−1 = −[𝑤 2 , . . . , 𝑤 𝑛 ]. 𝑃𝛾𝑛−1

if the induced orientations of 𝑃𝛼𝑛 and 𝑃𝛽𝑛

Definition 1.71. Let (𝑀, 𝜏) be a PL manifold of dimension 𝑛. An orientation consists of giving an orientation 𝑜𝛼 to each 𝑛-polyhedron 𝑃𝛼𝑛 such that for each (𝑛−1)-polyhedron 𝑃𝛾𝑛−1 with two incident 𝑛-polyhedra 𝑃𝛼𝑛 , 𝑃𝛽𝑛 , the orientations induced by 𝑜𝛼 , 𝑜 𝛽 on 𝑃𝛾𝑛−1 are opposite. Remark 1.72. If 𝑀 is an orientable smooth manifold, then 𝑀 is also orientable with the induced PL structure (Exercise 1.23). So the smooth and PL notions of orientability agree. If we have a planar representation P𝑆 of a surface 𝑆, we can easily determine whether 𝑆 is orientable or not. Proposition 1.73. A compact connected surface 𝑆 is orientable if and only if for every letter in the word 𝐩𝑆 of 𝑆 that appears twice, it does so with different exponents +1 and −1. Proof. Set an orientation on the interior of the 2-polyhedron P𝑆 . Then the induced orientation is given by going around the boundary in the chosen direction (clockwise or anticlockwise). If two edges with the same letter 𝑎 appear with different exponents, then this means that they appear in the boundary with opposite directions under the identification. Hence the two induced orientations on 𝑎 are opposite to one another, which means that they satisfy the compatibility criterium. □

1.4. Orientability

39

Example 1.74. The previous discussion is illustrated in the following example, where the planar representation of the Möbius band makes it clear that it is not orientable. Note that its planar representation is 𝑎𝑏𝑎𝑐, so there are two letters 𝑎 with the same exponent +1, and therefore it is non-orientable (see Figure 1.12).

a a

a

Figure 1.12. The Möbius band is not orientable.

Example 1.75. We can determine easily the orientability of the surfaces in examples 1.59 and 1.62 using Proposition 1.73, as we have words for them: 𝑆 2 and 𝑇 2 are orientable, ℝ𝑃2 and Kl are non-orientable. Also Σ𝑔 , 𝑔 ≥ 1, are orientable and 𝑋𝑘 , 𝑘 ≥ 1, are non-orientable (this justifies the words orientable and non-orientable in Definition 1.61). Corollary 1.76. A surface is non-orientable if and only if it contains an embedded Möbius band. Proof. If there exists an embedding of a Möbius band in 𝑆, then 𝑆 cannot be orientable, because the Möbius band would inherit the orientation, which is impossible. Note that if 𝑆 is orientable and 𝑆 ′ ⊂ 𝑆 is a subsurface, then 𝑆 ′ is also orientable. Conversely, if the surface is non-orientable, then some of its connected components are non-orientable. So we can suppose that 𝑆 is connected. Take a planar representation P𝑆 of 𝑆. It must have two edges labelled with the letter to the same exponent. Consider a small closed interval in the interior of both edges, which match under the identification of the boundary, and take a band (i.e., topological rectangle) inside P𝑆 that joins these two closed intervals. In the quotient, this rectangle defines a Möbius band inside 𝑆. □ Remark 1.77. For the projective plane, a Möbius band is easily drawn. It turns out to be a small neighbourhood along a projective line inside ℝ𝑃 2 (in the left of Figure 1.13 it is a neighbourhood of a horizontal line). But clearly, it does not matter which line is used, so we can use the line at infinity, which is given as 𝜕𝐷2 in the planar model (see the right of Figure 1.13). A neighbourhood of the line at infinity is a Möbius band and its complement is a ball. 1.4.3. Orientation of topological manifolds. The concept of orientability can be defined for a topological 𝑛-manifold. For 𝑛 ≥ 3, it requires the use of homology groups, that will be introduced in section 2.3. So we will focus here in the case of surfaces (𝑛 = 2) using the fundamental group, and we leave the case of 𝑛 ≥ 3 to Exercise 2.22.

40

1. Topological surfaces

d b a

a

a

c

b

a d

d c

d Figure 1.13. ℝ𝑃2 is the union of a disc and a Möbius band.

Let 𝑆 be a topological surface. Let 𝑝 ∈ 𝑆, and consider a chart 𝜑 ∶ 𝑈 = 𝑈 𝑝 → 𝐵1 (0), 𝜑(𝑝) = 0. Then 𝑈 − {𝑝} has the homotopy type of the circle 𝑆𝜖1 (𝑝) = 𝜑−1 (𝑆𝜀1 (0)), 𝑥 via a deformation retract 𝑟(𝑥) = 𝜀 ||𝑥|| , for 0 < 𝜀 < 1. Thus there is an isomorphism 𝜋1 (𝑈 − {𝑝}) ≅ 𝜋1 (𝑆𝜀1 (𝑝)) ≅ ℤ. The first isomorphism is canonical (it does not depend on the chart 𝜑 chosen), but the second one depends on a choice of a generator of 𝜋1 (𝑆𝜀1 (𝑝)). Note that this is given by a direction (clockwise or anticlockwise) for going around 𝑆𝜀1 (𝑝) (which agrees with an orientation of the image of the chart 𝜑(𝑈) ⊂ ℝ2 as explained in Remark 1.67(3)). First note that for the isomorphism 𝜋1 (𝑈 − {𝑝}) ≅ 𝜋1 (𝑆𝜀1 (𝑝)), we need to choose a basepoint in 𝑈 − {𝑝}. However, for any 𝑞1 , 𝑞2 ∈ 𝑈 − {𝑝}, the isomorphism 𝜋1 (𝑈 − {𝑝}, 𝑞1 ) ≅ 𝜋1 (𝑈 − {𝑝}, 𝑞2 ) is canonical. Secondly, we have to choose the radius of the circle, but clearly for 0 < 𝛿 < 𝜀, the isomorphism 𝜋1 (𝑆𝜀1 (𝑝)) ≅ 𝜋1 (𝑆1𝛿 (𝑝)) is canonical. Finally, it is independent of charts: if we have two charts (𝑈, 𝜑), (𝑈 ′ , 𝜓), take a smaller open subset 𝑉 ⊂ 𝑈 ∩ 𝑈 ′ , and 𝜖 > 0 small enough so that 𝜑−1 (𝑆𝜀1 (0)) ⊂ 𝑉 and 𝜓−1 (𝑆𝜀1 (0)) ⊂ 𝑉. Then we have a canonical isomorphism 𝜋1 (𝜑−1 (𝑆𝜀1 (0))) ≅ 𝜋1 (𝑉 − {𝑝}) ≅ 𝜋1 (𝜓−1 (𝑆𝜀1 (0))). We call orientation at 𝑝 a choice of generator 𝑜(𝑝) = [𝛾] ∈ 𝜋1 (𝑆𝜀1 (𝑝)). To define the continuity, take a small circle 𝑆𝜀1 (𝑝) as above and let 𝑉 = 𝜑−1 (𝐵𝜀 (0)) ⊂ 𝑈. Then for any 𝑞 ∈ 𝑉 we have a canonical isomorphism 𝜋1 (𝑆𝜀1 (𝑞)) ≅ 𝜋1 (𝑈 − {𝑞}) ≅ 𝜋1 (𝑈 − 𝑉) ≅ 𝜋1 (𝑆𝜀1 (𝑝)) ≅ 𝜋1 (𝑈 − {𝑝}). An orientation 𝑜(𝑝) at 𝑝 and 𝑜(𝑞) at 𝑞 are compatible if they coincide under this isomorphism. So 𝑜(𝑝) determines an orientation at all points 𝑞 ∈ 𝑉. Definition 1.78. Let 𝑀 be a topological manifold. An orientation of 𝑀 consists of a choice of orientations {𝑜(𝑝)| 𝑝 ∈ 𝑀} such that for each point 𝑝 ∈ 𝑀 there exists a small neighbourhood 𝑈 𝑝 such that all the orientations {𝑜(𝑞)| 𝑞 ∈ 𝑈 𝑝 } are compatible with 𝑜(𝑝). As expected, the notions of orientation for PL manifolds and for topological manifolds agree (Exercise 2.22). Remark 1.79. Let 𝑀 be a connected manifold (smooth, PL, or topological). If 𝑀 is orientable, then it admits exactly two orientations. We see this as follows: let 𝑜 be one orientation, so −𝑜 is another orientation. If 𝑜′ is an orientation, then the sets 𝑉1 = {𝑥 ∈ 𝑀| 𝑜′ (𝑥) = 𝑜(𝑥)} and 𝑉2 = {𝑥 ∈ 𝑀| 𝑜′ (𝑥) = −𝑜(𝑥)} are open (by the continuity property of orientations). As they are disjoint and cover 𝑀, either 𝑀 = 𝑉1 or 𝑀 = 𝑉2 by

1.5. Classification of compact surfaces

41

z

y

x Figure 1.14. The surface Σ𝑔 .

connectedness, so 𝑜′ = 𝑜 or 𝑜′ = −𝑜. If 𝑀 has 𝑟 connected components, then a choice of orientation for 𝑀 is to give an orientation for each of the connected components. Hence 𝑀 has 2𝑟 possible choices of orientations. Remark 1.80. If (𝑀, 𝑜) is a topological 𝑛-manifold with an orientation, a homeomorphism 𝑓 ∶ 𝑀 → 𝑀 preserves orientation if 𝑓(𝑜(𝑥)) = 𝑜(𝑓(𝑥)), for all 𝑥 ∈ 𝑀 (cf. Exercise 1.24). Otherwise we say that 𝑓 reverses orientation. When 𝑀 is connected, these notions make sense for an orientable manifold, even if an orientation has not been fixed. Also for 𝑀 connected, if 𝑓 preserves the orientation at a point, then it preserves the orientation everywhere. Remark 1.81. The orientable surfaces Σ𝑔 admit an orientation reversing homeomorphism. Just locate Σ𝑔 in ℝ3 lying over the horizontal plane and define 𝑓(𝑥, 𝑦, 𝑧) = (𝑥, 𝑦, −𝑧) (see Figure 1.14). As we will shortly see, Σ𝑔 are all the compact connected orientable surfaces (Theorem 1.82), and Remark 1.37 gives the uniqueness of the connected sum of surfaces.

1.5. Classification of compact surfaces 1.5.1. Classification theorem for compact surfaces. The aim of this section is to prove the classification theorem of compact surfaces. Recall that it is enough to restrict 𝑐𝑜 to connected surfaces. The following theorem gives the list 𝕃𝐏𝐋𝐌𝐚𝐧2 . 𝑐

Theorem 1.82. Let 𝑆 be a compact connected triangulated surface without boundary. Then 𝑆 is one and only one of the following. • If 𝑆 is orientable, then 𝑆 is homeomorphic to Σ𝑔 , for some 𝑔 ≥ 0. • If 𝑆 is non-orientable, then 𝑆 is homeomorphic to 𝑋𝑘 , for some 𝑘 ≥ 1. 𝑐𝑜

In other words, 𝕃𝐏𝐋𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 | 𝑔 ≥ 0, 𝑘 ≥ 1}. 𝑐

Proof. Let 𝑆 be a compact connected surface without boundary. By Proposition 1.58, 𝑆 admits a planar representation P𝑆 given by a word 𝐩𝑆 . We will show that the word

42

1. Topological surfaces

𝐩𝑆 is equivalent to one of the canonical words given in Example 1.62, that is 𝑆 2 ∶ 𝑎𝑎−1 ≈ ∅, −1 −1 −1 −1 −1 Σ𝑔 ∶ 𝑎1 𝑏1 𝑎−1 1 𝑏1 𝑎2 𝑏2 𝑎2 𝑏2 . . . 𝑎𝑔 𝑏𝑔 𝑎𝑔 𝑏𝑔 ,

𝑋𝑘 ∶ 𝑎1 𝑎1 𝑎2 𝑎2 . . . 𝑎𝑘 𝑎𝑘 ,

𝑔 ≥ 1,

𝑘 ≥ 1.

If 𝐩𝑆 ≈ 𝐩𝑐 , where 𝐩𝑐 is one of the canonical words above, then 𝑆 ≅ 𝑆𝑐 , where 𝑆𝑐 is one of the surfaces 𝑆 2 , Σ𝑔 , 𝑋𝑘 , respectively. This will complete the proof of the theorem. Case I. 𝑆 is orientable. Then all letters in 𝐩𝑆 appear with different exponent. Define the distance between two edges with the same letter by counting the number of edges in between at either side and choosing the smaller one. Take an edge 𝑎 such that this distance is minimal. If the distance is zero, this means that the letters 𝑎, 𝑎−1 come together, so 𝐩𝑆 = 𝐩1 𝑎𝑎−1 𝐩2 ≈ 𝑎𝑎−1 𝐩2 𝐩1 (maybe changing the direction of the arrows 𝑎). If there are no more letters than 𝑎, then 𝐩𝑆 = 𝑎𝑎−1 , and we are finished. Otherwise, the following pasting procedure shows that

a

a

a a

a

𝐩𝑆 ≈ 𝑎𝑎−1 𝐩2 𝐩1 ≈ 𝐩2 𝐩1 ≈ 𝐩1 𝐩2 ,

(1.7)

reducing the number of letters. Now suppose that the (minimal) distance between the two edges labelled 𝑎 is not zero. Then there is an edge 𝑏 in between. The other edge 𝑏 cannot be at the same side, since the distance between the edges 𝑏 would then be strictly smaller than the distance between the edges 𝑎, contrary to assumption. So, after rotation and possibly changing the direction of the arrows 𝑎, 𝑏, we can write 𝐩𝑆 ≈ 𝑎𝐩1 𝑏 𝐩2 𝑎−1 𝐩3 𝑏−1 𝐩4 . Now we perform the following cutting and pasting procedure.

p1

b

p2 a

c

a

p4

a

p3 b

c

a p3

p4 p1

d b c

p3 p2

c

a d p2

p4 d p1 c

1.5. Classification of compact surfaces

43

We cut the planar representation along 𝑐, and glue along 𝑏. In the resulting new planar representation, we cut along 𝑑 and glue along 𝑎. All these new planar representations correspond to the same surface 𝑆 (i.e., there is a quotient map to 𝑆). This means that (1.8)

𝐩𝑆 ≈ 𝑎𝐩1 𝑏 𝐩2 𝑎−1 𝐩3 𝑏−1 𝐩4 ≈ 𝑑𝑐𝑑 −1 𝑐−1 𝐩1 𝐩4 𝐩3 𝐩2 .

So 𝑆 = 𝑇 2 #𝑆 ′ , where 𝑆 ′ has word 𝐩1 𝐩4 𝐩3 𝐩2 , which has fewer letters. Proceeding inductively, we get that 𝑆 ≅ 𝑇 2 # ⋯ #𝑇 2 = Σ𝑔 , for some 𝑔 ≥ 0. Case II. 𝑆 is not orientable. In this case, there are letters with the same exponent. Let 𝑎 be one such letter. After rotating and possibly changing the direction of the arrows, we have 𝐩𝑆 ≈ 𝑎𝐩1 𝑎𝐩2 . We perform the following cutting and pasting procedure to get

p1 b

a

a a b

b p1

p2

p2 𝐩𝑆 ≈ 𝑎𝐩1 𝑎𝐩2 ≈ 𝑏𝑏(𝐩1 )−1 𝐩2 .

(1.9)

Therefore 𝑆 = ℝ𝑃 2 #𝑆 2 , for some surface 𝑆 2 . Arguing by induction, we get 𝐩𝑆 ≈ 𝑏1 𝑏1 ⋯ 𝑏𝑘 𝑏𝑘 𝐩′ , where 𝐩′ only has letters with different exponents. This means that 𝑆 = ℝ𝑃 2 # (𝑘) ... #ℝ𝑃 2 #𝑆 ′ , where 𝑆 ′ is an orientable connected surface. If there is no such 𝑆 ′ , then 𝑆 = ℝ𝑃 2 # (𝑘) . . . #ℝ𝑃 2 = 𝑋𝑘 , 𝑘 ≥ 1, and we are finished. Finally, we treat the case where 𝑆′ = Σ𝑔 , 𝑔 ≥ 1. The key point is that ℝ𝑃2 #𝑇 2 ≅ 𝑋3 , which is equivalent to 𝑎𝑎𝑏𝑐𝑏−1 𝑐−1 ≈ 𝑎𝑎𝑏𝑏𝑐𝑐. This is proved by using (1.9), that is 𝑎𝐩1 𝑎𝐩2 ≈ 𝑎𝑎(𝐩1 )−1 𝐩2 , repeatedly. We get the following. 𝑎𝑎𝑏𝑐𝑏−1 𝑐−1

≈ ≈ ≈ ≈

𝑎𝑐−1 𝑏−1 𝑎𝑏−1 𝑐−1 𝑏𝑏𝑎−1 𝑐𝑎𝑐 𝑐𝑐𝑎𝑏−1 𝑏−1 𝑎 𝑎𝑎𝑏𝑏𝑐𝑐

≈ 𝑏−1 𝑎𝑏−1 𝑐−1 𝑎𝑐−1 ≈ 𝑏𝑎𝑏𝑐𝑎𝑐 ≈ 𝑐𝑏𝑏𝑎−1 𝑐𝑎 ≈ 𝑎𝑏−1 𝑏−1 𝑎𝑐𝑐

In the first line we put 𝐩1 = 𝑐−1 𝑏−1 , 𝐩2 = 𝑏−1 𝑐−1 . It is followed by a rotation and a change of direction of the arrow. In the second line we use 𝐩1 = 𝑎, 𝐩2 = 𝑐𝑎𝑐, followed by a rotation. In the third line take 𝐩1 = 𝑏𝑏𝑎−1 , 𝐩2 = 𝑎, and in the fourth line 𝐩1 = 𝑏−1 𝑏−1 , 𝐩2 = 𝑐𝑐. Now we have (for 𝑘 ≥ 1 and 𝑔 ≥ 1) 𝑆 = 𝑋𝑘 #Σ𝑔 = 𝑋𝑘−1 #𝑋1 #𝑇 2 #Σ𝑔−1 = 𝑋𝑘−1 #𝑋3 #Σ𝑔−1 = 𝑋𝑘+2 #Σ𝑔−1 . Repeating this, we get finally 𝑆 = 𝑋𝑘+2𝑔 , as required.

44

1. Topological surfaces

Uniqueness. First, 𝑋𝑘 cannot be homeomorphic to any Σ𝑔 since the later is orientable while the former is not. Second, the Euler-Poincaré characteristic gives a well defined map 𝜒 ∶ 𝕃𝐏𝐋𝐌𝐚𝐧2𝑐 → ℤ. For 𝑔 ≠ 𝑔′ , we have 𝜒(Σ𝑔 ) = 2 − 2𝑔 ≠ 2 − 2𝑔′ = 𝜒(Σ𝑔′ ), hence Σ𝑔 ≇ Σ𝑔′ . Also for 𝑘 ≠ 𝑘′ , we have 𝜒(𝑋𝑘 ) = 2 − 𝑘 ≠ 2 − 𝑘′ = 𝜒(𝑋𝑘′ ), hence 𝑋𝑘 ≇ 𝑋𝑘′ . □ 𝑐𝑜

Remark 1.83. The list of topological surfaces 𝕃𝐓𝐌𝐚𝐧2 is also given by 𝑐

𝑐𝑜 𝕃𝐓𝐌𝐚𝐧2 𝑐

= {Σ𝑔 , 𝑋𝑘 | 𝑔 ≥ 0, 𝑘 ≥ 1}.

As we have proven that any topological surface is triangulable (Theorem 1.55), we only need to see that this list has no repetitions, that is that the surfaces Σ𝑔 , 𝑋𝑘 , 𝑔 ≥ 0, 𝑘 ≥ 1, are topologically distinct. This will follow once we prove that the Euler-Poincaré characteristic 𝜒(𝑆) is a topological invariant (Remark 2.114). So a topological surface 𝑆 is determined by a couple of topological invariants: 𝜒(𝑆) and the orientability. 𝑐𝑜

The list of smooth surfaces is also 𝕃𝐃𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 | 𝑔 ≥ 0, 𝑘 ≥ 1}. This will be 𝑐 completed in Theorem 6.56. Remark 1.84. The rules (1.7), (1.8), and (1.9) hold for any planar representation, so they hold also for surfaces with boundary. However, the rule 𝐩𝑆 = 𝐩𝑆1 𝐩𝑆2 for 𝑆 = 𝑆 1 #𝑆 2 (Proposition 1.60) only holds for surfaces without boundary (see Exercise 1.30). Remark 1.85. (1) The Klein bottle is homeomorphic to 𝑋2 , since 𝑎𝑏𝑎𝑏−1 ≈ 𝑐𝑐𝑏−1 𝑏−1 , by using (1.9). (2) ℝ𝑃2 minus a ball is a Möbius band (cf. Remark 1.77). A Möbius band is given by 𝑎𝑐𝑎𝑑 ≈ 𝑏𝑏𝑐−1 𝑑. We can consider 𝑐−1 𝑑 as a single edge, so that the Möbius band is given by the word 𝑏𝑏𝑒. Now we glue a disc with boundary 𝑒, and we get the surface with word 𝑏𝑏, that is, a projective plane. We can also write ℝ𝑃2 = Mob ∪𝑆1 𝐷2 . (3) Also Kl = Mob ∪𝑆1 Mob, since 𝑋2 = ℝ𝑃 2 #ℝ𝑃2 is obtained by removing a ball from the first ℝ𝑃2 , another ball from the second ℝ𝑃2 , and gluing along the two boundaries. (4) Note also that 𝑋1 #𝑇 2 ≅ 𝑋3 can be rewritten as 𝐾𝑙#ℝ𝑃2 ≅ 𝑇 2 #ℝ𝑃2 . A topological interpretation of this isomorphism appears in Exercise 1.37. Corollary 1.86. The classification of connected compact triangulated surfaces with boundary, 𝐏𝐋𝐌𝐚𝐧2𝜕,𝑐 , is given by the following list. 𝑏

(1) Orientable: Σ𝑔 − ( ⨆ 𝐵𝑖 ), 𝑔, 𝑏 ≥ 0, where 𝐵𝑖 are disjoint balls. 𝑖=1 𝑏

(2) Non-orientable: 𝑋𝑘 − ( ⨆ 𝐵𝑖 ), 𝑘 ≥ 1, 𝑏 ≥ 0, where 𝐵𝑖 are disjoint balls. 𝑖=1

Here 𝑏 = 0 means that the boundary is empty. Proof. Let 𝑆 be a connected compact surface with boundary. The boundary 𝜕𝑆 is a compact 1-manifold, maybe disconnected, and without boundary. Let 𝑏 ≥ 0 be the

1.5. Classification of compact surfaces

45

𝑏

𝑐𝑜

number of connected components of 𝜕𝑆. As 𝕃𝐓𝐌𝐚𝐧1 = {𝑆 1 }, we have 𝜕𝑆 ≅ ⨆𝑖=1 𝑆 1𝑖 , 𝑐

where 𝑆 1𝑖 is a copy of the circle. Consider discs 𝐷𝑖2 and homeomorphisms 𝑓𝑖 ∶ 𝑆 1𝑖 → 𝜕𝐷𝑖2 . Take 𝑆 ̄ = 𝑆 ∪ ( ∪𝑓1 𝐷12 ∪𝑓2 𝐷22 ⋯ ∪𝑓𝑏 𝐷𝑏2 ). This is a compact surface without boundary. Hence 𝑆 ̄ = Σ𝑔 or 𝑋𝑘 , depending on ∘

𝑏 whether it is orientable or not, and 𝑆 = 𝑆 ̄ − ( ⨆𝑖=1 𝐷 2𝑖 ). Observe that the choice of the discs is irrelevant. By Exercise 1.14 it does not matter from where the ball is removed. By the argument of Theorem 1.35, it does not depend on the homeomorphisms 𝜑𝑖 . □

Remark 1.87. There is also a theorem of classification for open surfaces (an open manifold is a non-compact connected manifold without boundary). This appears in [Ric]. 𝑐𝑜

1.5.2. The semigroup (𝕃𝐓𝐌𝐚𝐧𝑛 , #). It is interesting to note that the list of compact 𝑐

𝑐𝑜

connected 𝑛-manifolds has the structure of a commutative semigroup (𝕃𝐓𝐌𝐚𝐧𝑛 , #). The 𝑐 operation connected sum # is commutative and associative, i.e., 𝑋#𝑌 ≅ 𝑌 #𝑋 and (𝑋#𝑌 )#𝑍 ≅ #(𝑌 #𝑍). The neutral element is the sphere 𝑆 𝑛 , since 𝑆 𝑛 #𝑋 ≅ 𝑋. However, in general we do not have inverses. Indeed, given a connected surface 𝑆 different from the sphere, there does not exist another surface 𝑆 ′ with 𝑆#𝑆 ′ ≅ 𝑆 2 , since 𝜒(𝑆#𝑆 ′ ) = 𝜒(𝑆) + 𝜒(𝑆 ′ ) − 2 = 2 implies that 𝜒(𝑆) = 𝜒(𝑆 ′ ) = 2 and hence 𝑆 ≅ 𝑆 ′ ≅ 𝑆 2 . Also the cancellation law is not satisfied, that is 𝑋#𝑌 ≅ 𝑋#𝑍 6⟹ 𝑌 ≅ 𝑍 (see Remark 1.85(4)). 𝑐𝑜

For the case of surfaces, let us determine the semigroup 𝐺 = (𝕃𝐓𝐌𝐚𝐧2 , #). It is 𝑐

generated by the surfaces 𝑢 = 𝑋1 = ℝ𝑃 2 and 𝑣 = Σ1 = 𝑇 2 with the relation 𝑢 + 𝑣 = 𝑋1 #Σ1 ≅ 𝑋3 = 3𝑢. Therefore we have an isomorphism of semigroups 𝑐𝑜

𝐺 = (𝕃𝐓𝐌𝐚𝐧2 , #) ≅ 𝑐

ℕ⟨𝑢, 𝑣⟩ . ⟨𝑢 + 𝑣 = 3𝑢⟩

In general, semigroups are difficult to manipulate. So it is convenient to convert an Abelian semigroup into an Abelian group via the Grothendieck construction 𝐺 ↦ 𝐾(𝐺). To describe it, we return to some notions of categorical nature. Definition 1.88. (1) An initial object in a category 𝒞 is 𝑋𝐼 ∈ Obj(𝒞) such that for every object 𝑋 ∈ Obj(𝒞) there exists a unique arrow 𝑋𝐼 → 𝑋. (2) A final object in 𝒞 is 𝑋𝐹 ∈ Obj(𝒞) such that for every object 𝑋 ∈ Obj(𝒞) there exists a unique arrow 𝑋 → 𝑋𝐹 . Initial and final objects may or may not exist, depending on the category. But if an initial (or final) object exists, then it is unique up to unique isomorphism. Imagine that 𝑋𝐼 , 𝑋𝐼′ are two initial objects. Then as 𝑋𝐼 is initial, there is a unique arrow 𝑓 ∶ 𝑋𝐼 → 𝑋𝐼′ . As 𝑋𝐼′ is initial, then there is a unique arrow 𝑓 ∶ 𝑋𝐼′ → 𝑋𝐼 . So 𝑔 ∘ 𝑓 ∶ 𝑋𝐼 → 𝑋𝐼 is an arrow from an initial object. But 1𝑋𝐼 ∶ 𝑋𝐼 → 𝑋𝐼 is another one, so by uniqueness 𝑔 ∘ 𝑓 = 1𝑋𝐼 . Analogously 𝑓 ∘ 𝑔 = 1𝑋𝐼′ , so 𝑓, 𝑔 are isomorphisms and 𝑋𝐼 ≅𝒞 𝑋𝐼′ .

46

1. Topological surfaces

Example 1.89. We give some examples. • In 𝐒𝐞𝐭 and 𝐓𝐨𝐩, 𝑋𝐼 = ∅ and 𝑋𝐹 = ⋆, the singleton. • In 𝐓𝐨𝐩∗ , 𝑋𝐼 = ⋆ and 𝑋𝐹 = ⋆. • In 𝐕𝐞𝐜𝐭𝐤 , 𝑋𝐹 = 𝑋𝐼 = 0. • In the category of commutative rings 𝐑𝐢𝐧𝐠, 𝑋𝐼 = ℤ but there is no 𝑋𝐹 . • Consider the groupoid category Π1 (𝑋) for a topological space 𝑋. Then if 𝑋 is simply connected, all points are isomorphic, and they are initial and final objects. If 𝑋 is path connected but not simply connected, then all points are isomorphic, but there are neither initial nor final objects. If 𝑋 is not path connected, then not all points are isomorphic and there are neither initial nor final objects. Definition 1.90. Let (𝐺, +) be an Abelian semigroup. We call the 𝐾-theory group of 𝐺 or the Grothendieck group to an Abelian group 𝐾(𝐺) with a homomorphism of semigroups 𝚤 ∶ 𝐺 → 𝐾(𝐺) satisfying that, for any Abelian group 𝐴 and any semigroup homomorphism 𝑓 ∶ 𝐺 → 𝐴, there exists a unique group homomorphism 𝑓 ̄ ∶ 𝐾(𝐺) → 𝐴 such that 𝑓 = 𝑓 ̄ ∘ 𝚤. 𝑓

𝐺 𝑖

 𝐾(𝐺)

/𝐴 =

𝑓̄

The definition of 𝐾(𝐺) is given in terms of what is usually called a universal property. Let us see that this is an initial/final object property in a suitable category. Let 𝒮 be the category defined as: • its objects are semigroup homomorphisms 𝑓 ∶ 𝐺 → 𝐴, where 𝐴 is an Abelian group; • the homomorphisms between objects 𝑓 ∶ 𝐺 → 𝐴 and 𝑔 ∶ 𝐺 → 𝐵 are given by group homomorphisms 𝜙 ∶ 𝐴 → 𝐵 such that the following diagram commutes. 𝐺

𝐺

𝑓

𝑔

/𝐴  /𝐵

𝜙

Such a category is usually called an arrow category, since its objects are arrows of other categories, and its morphisms form diagrams in two levels (the top and bottom levels are the arrows defining the objects “from” and “to” of the morphism). With the above description of 𝒮, an initial object is 𝐺 → 𝐾(𝐺), so it must be unique. The existence has to be proved by a constructive procedure. In this case, we explicitly construct 𝐾(𝐺) as the set of equivalence classes of pairs (𝑥, 𝑦) ∈ 𝐺 × 𝐺 with the equivalence relation ∼ given by (𝑥1 , 𝑦1 ) ∼ (𝑥2 , 𝑦2 ) if and only if there exists 𝑠 ∈ 𝐺 such that 𝑥1 + 𝑦2 + 𝑠 = 𝑥2 + 𝑦1 + 𝑠. This is easily proved to be an equivalence relation (the

Problems

47

insertion of the term 𝑠 ∈ 𝐺 is necessary for the transitivity). The quotient 𝐾(𝐺) = (𝐺 × 𝐺)/∼ is an Abelian group with operation [(𝑥1 , 𝑦1 )] + [(𝑥2 , 𝑦2 )] = [(𝑥1 + 𝑥2 , 𝑦1 + 𝑦2 )], neutral element 0 = [(𝑥, 𝑥)], and symmetric element −[(𝑥, 𝑦)] = [(𝑦, 𝑥)]. The homomorphism 𝚤 ∶ 𝐺 → 𝐾(𝐺) is defined by 𝚤(𝑥) = [(𝑥 + 𝑦, 𝑦)], for any 𝑦 ∈ 𝐺. It is natural to think of the equivalence classes [(𝑥, 𝑦)] formally as the difference 𝑥 − 𝑦, since [(𝑥, 𝑦)] = 𝚤(𝑦) − 𝚤(𝑥). This construction is quite typical, and it appears in the axiomatic constructions of the integers ℤ from the natural numbers ℕ, i.e., ℤ = 𝐾(ℕ, +), and the non-zero rationals ℚ∗ from ℤ, ℚ∗ = 𝐾(ℤ − {0}, ⋅). 𝑐𝑜

For the semigroup 𝐺 = (𝕃𝐓𝐌𝐚𝐧2 , #), the 𝐾-theory is given by 𝑐

ℕ⟨𝑢, 𝑣⟩ ℤ⟨𝑢, 𝑣⟩ 𝐾(𝐺) ≅ 𝐾 ( ≅ ℤ⟨𝑢⟩ ≅ ℤ . )≅ ⟨𝑢 + 𝑣 = 3𝑢⟩ ⟨𝑢 + 𝑣 = 3𝑢⟩ Actually, in 𝐾(𝐺), Σ𝑔 = Σ𝑔 + ℝℙ2 − ℝℙ2 = 𝑋2𝑔+1 − ℝℙ2 = 𝑋2𝑔+1 − 𝑋1 = 𝑋2𝑔 . The isomorphism 𝐾(𝑆) ≅ ℤ can be given using the Euler-Poincaré characteristic. For this, consider the map 𝐹 ∶ 𝐺 → ℤ, 𝐹(𝑆) = 2 − 𝜒(𝑆). This is easily seen to be a homomorphism since 𝐹(𝑆#𝑆 ′ ) = 2−(𝜒(𝑆#𝑆 ′ )) = 2−(𝜒(𝑆)+𝜒(𝑆 ′ )−2) = (2−𝜒(𝑆))+(2−𝜒(𝑆′ )) = 𝐹(𝑆)+𝐹(𝑆 ′ ). It sends 𝐹(𝑢) = 1, 𝐹(𝑣) = 2, hence it is an isomorphism. Definition 1.91. We define the genus of a compact connected surface without bound1 ary 𝑋 as 2 𝐹(𝑋). Therefore Σ𝑔 has genus 𝑔 and 𝑋𝑘 has genus 𝑘/2. It is clearly additive.

Problems Exercise 1.1. Prove that, in a category 𝒞, if a morphism has inverse, then this is unique. Prove that “being isomorphic” is an equivalence relation on Obj(𝒞), and that the set of automorphisms of an object 𝑋 (isomorphisms from 𝑋 to 𝑋) is a group. Exercise 1.2. Prove the theorem of invariance of dimension for 𝑛 = 1, that is, if 𝑈 ⊂ ℝ and 𝑉 ⊂ ℝ𝑚 are open subsets which are homeomorphic, then 𝑚 = 1. Exercise 1.3. Classify connected topological 1-manifolds (directly from the definition). Exercise 1.4. Classify connected smooth 1-manifolds (directly from the definition). Exercise 1.5. Classify connected PL 1-manifolds (directly from the definition). Exercise 1.6. Prove that any topological 1-manifold admits a triangulation (this gives a solution to Exercise 1.3 from Exercise 1.5). Exercise 1.7. Classify (connected) 1-manifolds with (non-empty) boundary (in the topological, PL, and smooth categories). Exercise 1.8. Let Ω be the set of all countable ordinals. In [0, 1] × Ω, glue (1, 𝜔) ∼ (0, 𝜔 + 1), for each 𝜔 ∈ Ω. Prove that the resulting space 𝑋 is Hausdorff and locally Euclidean, but not second countable. This is called the long line. Does it admit a smooth atlas? Exercise 1.9. Let 𝑋 be a topological space. The suspension of 𝑋 is the space S(𝑋) = 𝑋 × [0, 1]/∼ , where (𝑥, 0) ∼ (𝑥′ , 0), (𝑥, 1) ∼ (𝑥′ , 1), for all 𝑥, 𝑥′ ∈ 𝑋. Show that S(𝑆𝑛 ) is homeomorphic to 𝑆𝑛+1 for all 𝑛 ≥ 0.

48

1. Topological surfaces

Exercise 1.10. Prove that the product of two topological manifolds with boundaries is a topological manifold with boundary. Determine its boundary. Show that ((0, 1) × (0, 1)) ∪ ({1} × [0, 1]) is not a manifold with boundary. Exercise 1.11. Let 𝑓 ∶ 𝜕𝐷𝑛 → 𝜕𝐷𝑛 be a homeomorphism. Prove that 𝑀 = 𝐷𝑛 ∪𝑓 𝐷𝑛 is homeomorphic to 𝑆𝑛 . Show that if 𝑓 is a diffeomorphism, then 𝑀 is a smooth manifold. Moreover, if 𝑓 is smoothly isotopic to the identity (that is, there is 𝐻 ∶ 𝜕𝐷𝑛 × [0, 1] → 𝜕𝐷𝑛 smooth, with 𝑓𝑡 = 𝐻( ⋅ , 𝑡) diffeomorphisms, 𝑓0 = Id and 𝑓1 = 𝑓), then 𝑀 is diffeomorphic to 𝑆𝑛 . Exercise 1.12. Let 𝑀 be a manifold, and let 𝑓 ∶ 𝑀 → 𝑀 be a homeomorphism. Prove that the mapping torus 𝑇𝑓 is a manifold. If 𝑀 is a manifold with boundary, then 𝑇𝑓 is a manifold with boundary, and the boundary is the mapping torus of 𝜕𝑀. Exercise 1.13. Prove that if 𝑀 is a smooth manifold and 𝑓 ∶ 𝑀 → 𝑀 is a diffeomorphism, then 𝑇𝑓 is a smooth manifold. Exercise 1.14. Let 𝑀 be a connected topological manifold, and let 𝑝, 𝑞 ∈ 𝑀. Prove that there is a homeomorphism of 𝑀 that sends 𝑝 to 𝑞. If 𝑀 is a smooth manifold, show that there is one such diffeomorphism. Exercise 1.15. Prove that the (differentiable) connected sum of two smooth connected surfaces 𝑆 1 , 𝑆 2 does not depend on the choice of points or charts. Exercise 1.16. Let 𝑋 be a triangulated space. Prove that it is compact if and only if the number of cells is finite. Exercise 1.17. Prove that any compact surface with boundary admits a triangulation. Exercise 1.18. Let 𝑋 be a compact triangulated space. Prove that the Euler-Poincaré characteristic is invariant by any subdivision of a triangulation. Exercise 1.19. Give a formula for 𝜒(𝑀1 #𝑀2 ), where 𝑀1 , 𝑀2 are compact, triangulated, 𝑛-dimensional connected manifolds. Exercise 1.20. Prove that GL(𝑛, ℝ) has two path connected components, which are the sets + − GL (𝑛, ℝ) = {𝐴 ∈ GL(𝑛, ℝ)| det(𝐴) > 0} and GL (𝑛, ℝ) = {𝐴 ∈ GL(𝑛, ℝ)| det(𝐴) < 0} (cf. section 4.1.1). Conclude that for a finite dimensional real vector space 𝑉 of dimension 𝑛 ≥ 1, the set ℬ of all the bases of 𝑉 has two path connected components, and that Or(𝑉) is in bijection with the set of path connected components. Exercise 1.21. Prove that ℝ𝑃𝑛 is orientable if and only if 𝑛 is odd. Exercise 1.22. Let 𝐻 ⊂ ℝ𝑛 be a smooth hypersurface. Prove that an orientation of 𝐻 is equivalent to the choice of a (continuously varying) unitary normal vector 𝑁𝑝 ∈ (𝑇𝑝 𝐻)⟂ , 𝑝 ∈ 𝐻. Exercise 1.23. If 𝑀 is an orientable smooth manifold, then 𝑀 is also orientable with the induced PL structure. Exercise 1.24. Let (𝑀, 𝑜) be an oriented (topological, PL, or smooth) manifold, and let 𝑓 ∶ 𝑀 → 𝑀 be an isomorphism (homeomorphism, PL isomorphism, or diffeomorphism). Define properly the image of the orientation 𝑓(𝑜(𝑥)), for 𝑥 ∈ 𝑀. Exercise 1.25. We define the 𝑛-dimensional Möbius band as Mob𝑛 = Mob × [0, 1]𝑛−2 , for 𝑛 ≥ 2. Prove that, in the PL category, a manifold is non-orientable if and only if it contains an embedded Mob𝑛 . Exercise 1.26. Let 𝑀1 , 𝑀2 be two PL connected 𝑛-manifolds. Prove that if 𝑀1 is non-orientable, then the two connected sums 𝑀1 #Id 𝑀2 and 𝑀1 #𝑟 𝑀2 , are homeomorphic (use Exercise 1.25).

References and extra material

49

Exercise 1.27. Let 𝑀1 , 𝑀2 be two orientable connected 𝑛-manifolds. Show that both connected sums 𝑀1 #𝑓 𝑀2 are orientable. Show that if one of 𝑀1 , 𝑀2 is not orientable, then 𝑀1 #𝑓 𝑀2 is not orientable.3 Exercise 1.28. Give a planar representation (that is, a triangulation with only one curvilinear 3-polyhedron) of ℝ𝑃3 , and prove that it is orientable. Prove that any (compact) triangulable connected 𝑛-manifold has a planar representation. Exercise 1.29. Let 𝑃 ⊂ ℝ2 be a polygon with edges 𝑙1 , . . . , 𝑙𝑛 parametrized by functions 𝛼1 (𝑡), . . . , 𝛼𝑛 (𝑡), Let 𝛽1 (𝑡), . . . , 𝛽𝑛 (𝑡) be another parametrization of the same edges with 𝛽𝑖 (0) = 𝛼𝑖 (0), 𝛽𝑖 (1) = 𝛼𝑖 (1). Identify the edges according to a partition 𝐴1 , . . . , 𝐴𝑟 of the set {1, . . . , 𝑛}, that is 𝛼𝑖 (𝑡) ∼ 𝛼𝑗 (𝑡) if 𝑖, 𝑗 ∈ 𝐴𝑠 , 𝑡 ∈ [0, 1]. Analogously, we define ≈ via 𝛽𝑖 (𝑡) ≈ 𝛽𝑗 (𝑡) if 𝑖, 𝑗 ∈ 𝐴𝑠 , 𝑡 ∈ [0, 1]. Show that 𝑃/∼ and 𝑃/ ≈ are homeomorphic. Exercise 1.30. Give a formula for the word associated to the connected sum of two connected surfaces with boundary given by words 𝐩𝑋 , 𝐩𝑌 , respectively. Exercise 1.31. Prove that always 𝑎𝑎𝐩1 𝐩2 ≈ 𝐩1 𝑎𝑎𝐩2 , where 𝐩1 , 𝐩2 are subwords. However show that 𝐩1 ≈ 𝐩′1 does not imply 𝐩1 𝐩2 ≈ 𝐩′1 𝐩2 , where 𝐩2 does not share letters with 𝐩1 , 𝐩′1 . Explain this in relation to Proposition 1.60. Exercise 1.32. Prove the theorem of classification of connected surfaces with boundary by using only manipulation of words. Exercise 1.33. Determine the surfaces with boundary given by the words, 𝐩𝑋 = 𝑎 𝑏 𝑎 𝑓 𝑐 𝑑 𝑏−1 𝑑 −1 𝑔 𝑒 𝑐 𝑒,

and

𝐩𝑌 = 𝑎 𝑏 𝑎 𝑓 𝑐 𝑑 𝑏−1 𝑑 −1 𝑔 𝑒 𝑐−1 𝑒.

Exercise 1.34. Let 𝑋 be the space parametrizing (unordered) subsets of two points in 𝑆1 . Describe the surface 𝑋. Exercise 1.35. Let 𝒞 be a category. Prove that there is a category, called the opposite category, 𝒞 𝑜𝑝 , with Obj(𝒞 𝑜𝑝 ) = Obj(𝒞) and Mor𝒞𝑜𝑝 (𝑋, 𝑌 ) = Mor𝒞 (𝑌 , 𝑋), for objects 𝑋, 𝑌 . Prove that an initial (resp., final) object for 𝒞 is a final (resp., initial) object for 𝒞 𝑜𝑝 . Finally prove that a covariant (resp., contravariant) functor 𝐹 ∶ 𝒞1 → 𝒞2 is a contravariant (resp., covariant) functor 𝑜𝑝 𝐹 ∶ 𝒞1 → 𝒞2 . Exercise 1.36. Determine the semigroup of the compact connected surfaces with boundary 𝑐𝑜

(𝕃𝐓𝐌𝐚𝐧2 , #) and the corresponding Grothendieck ring. 𝜕,𝑐

Exercise 1.37. Let 𝑀 be a connected 𝑛-manifold. Gluing a handle 𝐻 = 𝑆 𝑛−1 × [0, 1] consists of removing two small disjoint balls 𝐵1 , 𝐵2 ⊂ 𝑀 and gluing 𝜕𝐵1 , 𝜕𝐵2 with the two connected components of 𝜕𝐻. Prove that there are two ways to glue a handle. Moreover, consider the manifolds 𝑆𝑛−1 × 𝑆 1 and the 𝑛-dimensional Klein bottle Kl 𝑛 , which is the mapping torus of the reflection on 𝑆𝑛−1 . Then prove that the two manifolds that are the result of gluing a handle to 𝑀 are homeomorphic to 𝑀#(𝑆 𝑛−1 × 𝑆 1 ) and 𝑀#Kl 𝑛 .

References and extra material Basic reading. The following references have material closely related to what is covered in this chapter. For a general introduction to manifolds, we recommend [Boo] and [Sp1]. The notions related to categories can be found in [McL] and sheaves in chapter II of [Wel]. Topological notions can be found in [Kos]. The important theorem of classification of triangulated surfaces appears in [Ma1], and the existence of triangulations can be read from [G-X]. 3 We suggest that the reader try exercises 1.25, 1.26, and 1.27 in the PL category, or in the topological category for 𝑛 = 2. The results hold in the topological category for all 𝑛 as well, and the exercises can be completed using the theory in section 2.2.2.

50

1. Topological surfaces

[Boo] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Pure and Applied Mathematics, Vol. 120, 2nd Edition, Academic Press, 2002. [G-X] J. Gallier, D. Xu, A Guide to the Classification Theorem for Compact Surfaces, Geometry and Computing, Vol. 9, Springer, 2013. [McL] S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Springer, 1998. [Ma1] W.S. Massey, Algebraic Topology: An Introduction, Graduate Texts in Mathematics, Vol. 56, Springer, 1977. [Mun] J.R. Munkres, Elements of Algebraic Topology, Taylor Francis Inc, 1996. [Sp1] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition, Publish-or-Perish, 1999. [Wel] R.O. Wells, Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, 3rd Edition, 2008. Further reading. When teaching a course based on this book, the professor can propose some topics for a short dissertation or specific study to be done by the students. We recommend the following topics related to the content of this chapter. • Classification of non-compact surfaces. This gives an example of a very rich classification. It can be found in: [Ric] I. Richards, On the classification of noncompact surfaces, Trans. Amer. Math. Soc. 106, 259-269, 1963. • Jordan-Schönflies theorem. This completes the triangulation of surfaces. It can be found at: [Cai] S.S. Cairns, An elementary proof of the Jordan-Schönflies theorem, Proc. Amer. Math. Soc. 2, 860-867, 1951. • Structures 𝐶 𝑟 on manifolds. We recommend the treatment in: [Hir] M.W. Hirsch, Differential Topology, Graduate Texts in Math., Springer, 1997. • Whitney embedding. A smooth manifold can be embedded as a submanifold of ℝ𝑁 . [Ada] M. Adachi, Embeddings and Immersions, Translations of Mathematical Monographs, Vol. 124, American Mathematical Society, 1993. • Exotic spheres. A proof of the first exotic sphere found by Milnor appears in the original article: [Mi1] J.W. Milnor, On manifolds homeomorphic to the 7-sphere, Annals Math. 64, 399-405, 1956. • Alexander horned sphere. This provides an example of a surface in ℝ3 homeomorphic to 𝑆2 but not bounding a ball. This shows that the Jordan-Schönflies theorem fails in high dimensions. [Ale] J.W. Alexander, An example of a simply connected surface bounding a region which is not simply connected, Proc. Nat. Acad. Sci. USA 10, 8-10, 1924. • Topological quantum field theory. A TQFT is a functor from a category of manifolds and cobordisms to an algebraic category. In the case of surfaces, it uses the classification of surfaces with boundary. [Koc] J. Kock, Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society student texts, Cambridge University Press, 2003. References. We add some references that have been mentioned within the text. However, most of them are clearly a level beyond the scope of the book. We also include expository references to historical developments, which are more accessible.

References and extra material

51

[Do1] S. Donaldson, An application of gauge theory to four-dimensional topology, J. Diff. Geom. 18, 279-315, 1983. [Fra] G. Franzoni, The Klein bottle: variations on a theme, Notices Amer. Math. Soc. 59, 1076-1082, 2012. [Fre] M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17, 357-453, 1982. [Har] J. Harris, Algebraic Geometry: A First Course, Graduate Texts in Mathematics, Vol. 133, Springer, 1995. [Mi2] J.W. Milnor, Differential topology forty-six years later, Notices Amer. Math. Soc. 58, 804-809, 2011. [Moi] E. Moise, Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Math. (2), 56, 96-114, 1952. [Nas] J. Nash, Real algebraic manifolds, Annals of Math. (2), 56, 405-421, 1952. [Rad] T. Radó, Über den Begriff der Riemannschen Fläche, Acta Sci. Math. Szeged. 2, 101-121, 1925. [Sco] A. Scorpan, The Wild World of 4-Manifolds, American Mathematical Society, 2005. [Wh1] H. Whitney, The self-intersections of a smooth 𝑛-manifold in 2𝑛-space, Annals of Math. (2), 45, 220-246, 1944. [Wh2] H. Whitney, Differentiable manifolds. Annals of Math. (2), 37, 645-680, 1936.

Chapter 2

Algebraic topology

The aim of algebraic topology is to study topological spaces through algebraic invariants and their properties. For this purpose, we will construct functors from the category of topological spaces (or a suitable subcategory) into a category of algebraic objects, such as groups, vector spaces or rings. These functors will provide new tools for distinguishing (sometimes even classifying) topological objects by means of their algebraic invariants. As a general philosophy, the richer the introduced algebraic structure, the more useful for extracting topological properties. A key notion in algebraic topology is homotopy equivalence, which gives an equivalence of spaces softer than that of homeomorphisms, sitting midway between topology and algebra. We shall start by introducing homotopy groups, which roughly speaking detect holes by surrounding them with spheres. Then we move to covers and ramified covers, which will be very useful in dealing with the first homotopy group, the so-called fundamental group. Afterwards we discuss homology theory, which detect holes by surrounding them with polyhedra. We introduce versions of homology specially suited for the topological, PL, and smooth categories: singular homology for topological spaces, simplicial homology for triangulated spaces, and de Rham cohomology for smooth manifolds.

2.1. Homotopy theory We recall the following categories (Example 1.5): • 𝐓𝐨𝐩. Its objects are topological spaces and the morphisms are continuous mappings between them. We denote Map(𝑋, 𝑌 ) = Mor𝐓𝐨𝐩 (𝑋, 𝑌 ) = {𝑓 ∶ 𝑋 → 𝑌 |𝑓 is continuous}. • 𝐓𝐨𝐩∗ . This is the category of pointed spaces. Its objects are (𝑋, 𝑥0 ), where 𝑋 is a topological space and 𝑥0 ∈ 𝑋. A morphism 𝑓 ∶ (𝑋, 𝑥0 ) → (𝑌 , 𝑦0 ) is a continuous map 𝑓 ∶ 𝑋 → 𝑌 with 𝑓(𝑥0 ) = 𝑦0 . Sometimes we denote 𝑋 ∈ Obj(𝐓𝐨𝐩∗ ) omitting reference to the point 𝑥0 when it is clear from the context. We denote Map∗ (𝑋, 𝑌 ) = Mor𝐓𝐨𝐩∗ (𝑋, 𝑌 ). 53

54

2. Algebraic topology

Both categories are subcategories of 𝐓𝐨𝐩𝐏, the category of pairs of spaces. Its objects are pairs (𝑋, 𝐴) with 𝑋 a topological space and 𝐴 ⊂ 𝑋. Given (𝑋, 𝐴), (𝑌 , 𝐵) ∈ Obj(𝐓𝐨𝐩𝐏), a morphism 𝑓 ∶ (𝑋, 𝐴) → (𝑌 , 𝐵) is a continuous map 𝑓 ∶ 𝑋 → 𝑌 with 𝑓(𝐴) ⊂ 𝐵. Note that 𝑋 ∈ Obj(𝐓𝐨𝐩) corresponds to the pair (𝑋, ∅). We also need the product of pairs, given as (𝑋, 𝐴) × (𝑌 , 𝐵) = (𝑋 × 𝑌 , (𝐴 × 𝑌 ) ∪ (𝑋 ∪ 𝐵)). 2.1.1. The homotopy category. Definition 2.1. Let 𝑓0 , 𝑓1 ∶ (𝑋, 𝐴) → (𝑌 , 𝐵) be morphisms of 𝐓𝐨𝐩𝐏. It is said that 𝑓0 and 𝑓1 are homotopic relative to (𝐴, 𝐵) if there exists a morphism of 𝐓𝐨𝐩𝐏, 𝐻 ∶ (𝑋, 𝐴) × [0, 1] = (𝑋 × [0, 1], 𝐴 × [0, 1]) → (𝑌 , 𝐵), such that 𝑓0 = 𝐻(−, 0) and 𝑓1 = 𝐻(−, 1). The homotopy relation is denoted by 𝑓0 ∼ 𝑓1 . Observe that with 𝐻 ∶ (𝑋 × [0, 1], 𝐴 × [0, 1]) → (𝑌 , 𝐵), a morphism of 𝐓𝐨𝐩𝐏 means that, for all (𝑎, 𝑠) ∈ 𝐴 × [0, 1], we have 𝐻(𝑎, 𝑠) ∈ 𝐵. This is equivalent to 𝐻𝑠 = 𝐻(−, 𝑠) ∶ 𝑋 → 𝑌 being a morphism between (𝑋, 𝐴) and (𝑌 , 𝐵) for all 𝑠 ∈ [0, 1]. Remark 2.2. A morphism 𝑓 ∶ (𝑋, ∅) → (𝑌 , ∅) is just a continuous map 𝑓 ∶ 𝑋 → 𝑌 . Given two continuous maps 𝑓0 , 𝑓1 ∶ (𝑋, ∅) → (𝑌 , ∅), a homotopy relative to (∅, ∅) is just a continuous map 𝐻 ∶ 𝑋 × [0, 1] → 𝑌 with 𝐻0 = 𝑓0 and 𝐻1 = 𝑓1 . In this case, it is said that 𝑓0 and 𝑓1 are homotopic. The relation ∼ is an equivalence relation: • 𝑓 ∼ 𝑓, for any 𝑓 ∶ (𝑋, 𝐴) → (𝑌 , 𝐵), just taking 𝐻(𝑥, 𝑠) = 𝑓(𝑥), for all 𝑠 ∈ [0, 1]. • If 𝑓 ∼ 𝑔 with homotopy 𝐻(𝑥, 𝑠), then 𝑔 ∼ 𝑓 with the homotopy 𝐾(𝑥, 𝑠) = 𝐻(𝑥, 1 − 𝑠). • If 𝑓 ∼ 𝑔 with homotopy 𝐻(𝑥, 𝑠), and 𝑔 ∼ 𝑘 with homotopy 𝐾(𝑥, 𝑠), then 𝑓 ∼ 𝑘 using the homotopy 𝐿(𝑥, 𝑠) = {

𝐻(𝑥, 2𝑠), 𝐾(𝑥, 2𝑠 − 1),

𝑠 ∈ [0, 1/2], 𝑠 ∈ [1/2, 1].

Moreover, it works well with compositions: • If 𝑓0 , 𝑓1 ∶ (𝑋, 𝐴) → (𝑌 , 𝐵) are homotopic with homotopy 𝐻(𝑥, 𝑠), and 𝑔 ∶ (𝑌 , 𝐵) → (𝑍, 𝐶), then 𝑔 ∘ 𝑓0 ∼ 𝑔 ∘ 𝑓1 via the homotopy 𝑔(𝐻(𝑥, 𝑠)). • If 𝑔0 , 𝑔1 ∶ (𝑌 , 𝐵) → (𝑍, 𝐶) are homotopic with homotopy 𝐾(𝑦, 𝑠), and 𝑓 ∶ (𝑋, 𝐴) → (𝑌 , 𝐵), then 𝑔0 ∘ 𝑓 ∼ 𝑔1 ∘ 𝑓 via the homotopy 𝐾(𝑓(𝑥), 𝑠). Definition 2.3. Let 𝒞 be a category. Suppose that for each 𝑋, 𝑌 ∈ Obj(𝒞) there is an equivalence relation ∼ in Mor𝒞 (𝑋, 𝑌 ) such that for 𝑓0 , 𝑓1 ∈ Mor𝒞 (𝑋, 𝑌 ) and 𝑔0 , 𝑔1 ∈ Mor𝒞 (𝑌 , 𝑍), if 𝑓0 ∼ 𝑓1 , 𝑔0 ∼ 𝑔1 , then 𝑔0 ∘ 𝑓0 ∼ 𝑔1 ∘ 𝑓1 . We define the quotient category 𝒞 by • Obj(𝒞) = Obj(𝒞), • Mor𝒞 (𝑋, 𝑌 ) = Mor𝒞 (𝑋, 𝑌 )/∼. Observe that there is a functor, called the quotient functor, 𝒞 ⟶ 𝒞.

2.1. Homotopy theory

55

Definition 2.4. We define the homotopy category 𝐇𝐨𝐓𝐨𝐩 and the pointed homotopy category 𝐇𝐨𝐓𝐨𝐩∗ as the quotient categories of 𝐓𝐨𝐩 and 𝐓𝐨𝐩∗ , respectively, under the equivalence relation of homotopy of maps. So the category 𝐇𝐨𝐓𝐨𝐩 has objects Obj(𝐇𝐨𝐓𝐨𝐩) = Obj(𝐓𝐨𝐩) and morphisms Hom𝐇𝐨𝐓𝐨𝐩 (𝑋, 𝑌 ) = Map(𝑋, 𝑌 )/ ∼ , and similarly for the pointed homotopy category. We will usually denote [𝑋, 𝑌 ] = Map(𝑋, 𝑌 )/∼

and

[𝑋, 𝑌 ]∗ = Map∗ (𝑋, 𝑌 )/∼ ,

for the spaces of classes of homotopy of maps between spaces (or pointed spaces) 𝑋 and 𝑌 . Note that when we say 𝑓 = 𝑔 ∈ [𝑋, 𝑌 ], we mean 𝑓 ∼ 𝑔 as continuous maps 𝑓, 𝑔 ∶ 𝑋 → 𝑌 . Definition 2.5. We say that two topological spaces 𝑋 and 𝑌 are of the same homotopy type if they are isomorphic in 𝐇𝐨𝐓𝐨𝐩, and we write 𝑋 ∼ 𝑌 . We say that 𝑋 is contractible if 𝑋 ∼ ⋆, the singleton. Let us spell out the meaning of 𝑋, 𝑌 being of the same homotopy type. There must exist maps 𝑓 ∶ 𝑋 → 𝑌 , 𝑔 ∶ 𝑌 → 𝑋 such that 𝑔∘𝑓 = 1𝑋 ∈ [𝑋, 𝑋] and 𝑓∘𝑔 = 1𝑌 ∈ [𝑌 , 𝑌 ], i.e., 𝑔 ∘ 𝑓 ∼ 1𝑋 , 𝑓 ∘ 𝑔 ∼ 1𝑌 . Such 𝑓, 𝑔 are called homotopy equivalences, and we say that 𝑔 is the homotopy inverse of 𝑓. By Exercise 1.1, this is unique up to homotopy (i.e., in 𝐇𝐨𝐓𝐨𝐩). If a space 𝑋 is contractible, then there are 𝑓 ∶ ⋆ = {𝑝0 } → 𝑋 and 𝑔 ∶ 𝑋 → ⋆ = {𝑝0 } which are homotopy inverses. Clearly 𝑔(𝑥) = 𝑝0 for all 𝑥 ∈ 𝑋, and let 𝑥0 = 𝑓(𝑝0 ) ∈ 𝑋, then we have that 𝑓 ∘ 𝑔 = 1⋆ and 𝑔 ∘ 𝑓 = 𝑐𝑥0 ∼ 1𝑋 , where 𝑐𝑥0 is the constant map, 𝑐𝑥0 (𝑥) = 𝑥0 , for all 𝑥 ∈ 𝑋. Therefore 𝑋 is contractible if and only if 𝑐𝑥0 ∼ 1𝑋 . Another related notion is that of retract and deformation retract. Definition 2.6. Let 𝑋 be a topological space. A retract is a subset 𝐴 ⊂ 𝑋 and a continuous map 𝑟 ∶ 𝑋 → 𝐴 such that 𝑟|𝐴 = 1𝐴 . A deformation retract is a retract 𝑟 ∶ 𝑋 → 𝐴 such that there is a homotopy 𝐻 ∶ 𝑋 × [0, 1] → 𝑋 with 𝐻(𝑥, 0) = 𝑥, 𝐻(𝑥, 1) = 𝑟(𝑥), for all 𝑥 ∈ 𝑋, and 𝐻(𝑎, 𝑠) = 𝑎, for all 𝑎 ∈ 𝐴 and 𝑠 ∈ [0, 1]. If 𝑟 ∶ 𝑋 → 𝐴 is a retract, then 𝑟 ∘ 𝑖 = 1𝐴 , where 𝑖 ∶ 𝐴 ↪ 𝑋 is the inclusion. If moreover it is a deformation retract, then 𝑖 ∘ 𝑟 ∼ 1𝑋 . In this case 𝑋 ∼ 𝐴. Observe that 𝑋 is contractible if and only if 𝑋 deformation retracts to some 𝑥0 ∈ 𝑋. Remark 2.7. The homotopy category gives rise to a classification problem: the classification of spaces up to homotopy. Note that there are natural maps 𝕃𝐓𝐨𝐩 → 𝕃𝐇𝐨𝐓𝐨𝐩 ,

𝕃𝐓𝐨𝐩∗ → 𝕃𝐇𝐨𝐓𝐨𝐩∗ .

As recalled throughout, we shall focus on manifolds. There is a homotopy category of compact topological 𝑛-manifolds and a pointed homotopy category of compact topological 𝑛-manifolds, denoted 𝐇𝐨𝐌𝐚𝐧𝑛𝑐 and 𝐇𝐨𝐌𝐚𝐧𝑛𝑐,∗ , respectively. There is a bijection (Exercise 2.4) 𝑐𝑜 𝑐𝑜 𝕃𝐇𝐨𝐌𝐚𝐧𝑛 ≅ 𝕃𝐇𝐨𝐌𝐚𝐧𝑛 , 𝑐

𝑐,∗

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2. Algebraic topology

which implies that for manifolds the choice of a basepoint is not relevant. There is also a natural functor 𝑐𝑜 𝑐𝑜 𝕃𝐓𝐌𝐚𝐧𝑛 ⟶ 𝕃𝐇𝐨𝐌𝐚𝐧𝑛 , 𝑐

𝑐

which is very relevant, since the second list classifies (compact, connected) 𝑛-manifolds up to homotopy equivalence. The map being obviously surjective (it comes from a quotient functor), it is clearly a bijection for 𝑛 = 1, and we shall see in Theorem 2.29 that it is a bijection for 𝑛 = 2. In dimensions 𝑛 ≥ 3, it is not injective (for example, there are lens spaces which are homotopy equivalent but not homeomorphic; see Example 4.34(2)). 2.1.2. Homotopy groups. Let (𝑋, 𝑥0 ) be a pointed topological space. Consider the 𝑛-sphere (𝑆𝑛 , 𝑝), with basepoint 𝑝 = (1, 0, . . . , 0) ∈ 𝑆 𝑛 = {(𝑥0 , 𝑥1 , . . . , 𝑥𝑛 ) ∈ ℝ𝑛+1 |𝑥02 + 𝑥12 + . . . + 𝑥𝑛2 = 1}. In the particular case of the circle, we have 𝑆 1 = {(𝑥, 𝑦)|𝑥2 + 𝑦2 = 1} = {𝑒2𝜋i𝑡 |𝑡 ∈ [0, 1]} ⊂ ℂ, and the basepoint is 𝑝 = 1. We define the 𝑛th homotopy group as 𝜋𝑛 (𝑋, 𝑥0 ) = [𝑆 𝑛 , 𝑋]∗ . Remark 2.8. Let 𝒞 be a category and fix 𝑍 ∈ Obj(𝒞). We have a natural functor ℎ𝑍 ∶ 𝒞 ⟶ 𝐒𝐞𝐭, given by ℎ𝑍 (𝑋) = Mor𝒞 (𝑍, 𝑋), and for 𝑓 ∶ 𝑋 → 𝑌 , 𝑓∗ = ℎ𝑍 (𝑓) ∶ ℎ𝑍 (𝑋) = Mor𝒞 (𝑍, 𝑋) → ℎ𝑍 (𝑌 ) = Mor𝒞 (𝑍, 𝑌 ), 𝑓∗ (𝑔) = 𝑓 ∘ 𝑔 . For a particular category 𝒞, it is natural to ask if there is a collection {𝑍𝑖 } such that ℎ𝑍𝑖 (𝑋) ≅ ℎ𝑍𝑖 (𝑌 ), for all 𝑖, implies that 𝑋 ≅ 𝑌 . Such collection would serve to characterize any object 𝑋 by means of the invariants ℎ𝑍𝑖 (𝑋). In the homotopy category, the homotopy groups are functors 𝜋𝑛 = ℎ𝑆𝑛 ∶ 𝐇𝐨𝐓𝐨𝐩∗ → 𝐒𝐞𝐭 associated to the spheres 𝑆 𝑛 , 𝑛 ≥ 0. The sets 𝜋𝑛 (𝑋) capture the holes of 𝑋, since, intuitively, there is a hole when we can surround it with a sphere, and this sphere cannot be shrunk (homotoped to the basepoint). Definition 2.9. We say that 𝑋, 𝑌 are of the same weak homotopy type if there is 𝑓 ∶ 𝑋 → 𝑌 such that 𝑓∗ ∶ 𝜋𝑛 (𝑋) → 𝜋𝑛 (𝑌 ) induces isomorphisms for all 𝑛. So the question in Remark 2.8 amounts here to whether the invariants 𝜋𝑛 (𝑋), 𝑛 ≥ 0, of a space characterize it (see Remark 2.105, and Exercises 2.7 and 2.8). Let us look first at the functor 𝜋0 . We have 𝑆 0 = {−1, 1}, with basepoint 1. A map 𝑓 ∶ (𝑆 0 , 1) → (𝑋, 𝑥0 ) sends 𝑓(1) = 𝑥0 , hence 𝑓 is determined by its image 𝑝𝑓 = 𝑓(−1) ∈ 𝑋. Therefore Map∗ (𝑆0 , 𝑋) = 𝑋. Two maps 𝑓0 , 𝑓1 ∶ 𝑆 0 → 𝑋 are homotopic if there is a homotopy 𝐻 ∶ 𝑆 0 × [0, 1] → 𝑋, which is defined by the path 𝛾(𝑠) = 𝐻(−1, 𝑠),

2.1. Homotopy theory

57

such that 𝛾(0) = 𝑓0 (−1) and 𝛾(1) = 𝑓1 (−1). Therefore 𝑝0 = 𝑓0 (−1) and 𝑝1 = 𝑓1 (−1) belong to the same path connected component, and we write 𝑝0 ∼ 𝑝1 . Therefore 𝜋0 (𝑋) = [𝑆 0 , 𝑋]∗ ≅ 𝑋/∼ = {set of path connected components of 𝑋}. Note that the functorial behaviour of 𝜋0 implies that homotopy equivalent spaces have the same number of path connected components. In general, 𝜋1 (𝑋, 𝑥0 ) is called the fundamental group of 𝑋 and 𝜋𝑛 (𝑋, 𝑥0 ) are called the higher homotopy groups for 𝑛 ≥ 2. Let us see that these have the structure of groups, which are moreover Abelian for 𝑛 ≥ 2. In the category of pointed spaces, there is a well defined pointed union 𝑋 ∨ 𝑌 of two spaces (Example 1.33). We denote 𝑍 ∨ 𝑍 when we do a pointed union of two (disjoint) copies of 𝑍. Definition 2.10. We say that (𝑍, 𝑧0 ) is a co-H-space when we have maps 𝑚 ∶ 𝑍 → 𝑍∨𝑍 (called comultiplication) and 𝐼 ∶ 𝑍 → 𝑍 (called inverse), satisfying the following. (1) (𝑚 ∨ 1𝑍 ) ∘ 𝑚 ∼ (1𝑍 ∨ 𝑚) ∘ 𝑚, as maps 𝑍 → 𝑍 ∨ 𝑍 ∨ 𝑍 (here for 𝑓 ∶ 𝑋 → 𝑌 and 𝑔 ∶ 𝑋 ′ → 𝑌 ′ , we denote 𝑓 ∨ 𝑔 ∶ 𝑋 ∨ 𝑋 ′ → 𝑌 ∨ 𝑌 ′ in the obvious way). (2) 𝑝𝑍 ∘ (1𝑍 ∨ 𝑐𝑧0 ) ∘ 𝑚 ∼ 1𝑍 and 𝑝𝑍 ∘ (𝑐𝑧0 ∨ 1𝑍 ) ∘ 𝑚 ∼ 1𝑍 , where 𝑝𝑍 ∶ 𝑍 ∨ 𝑍 → 𝑍 is the push map (the identity in both components). (3) 𝑝𝑍 ∘ (𝐼 ∨ 1𝑍 ) ∘ 𝑚 ∼ 𝑝𝑍 ∘ (1𝑍 ∨ 𝐼) ∘ 𝑚 ∼ 𝑐𝑧0 . (4) The co-H-space is Abelian if 𝑆 𝑍 ∘ 𝑚 ∼ 𝑚, where 𝑆 𝑍 ∶ 𝑍 ∨ 𝑍 → 𝑍 ∨ 𝑍 is the swap map (interchanging both components). Remark 2.11. The name of co-H-space is explained as follows. In algebraic topology, an H-space is a space 𝑍 with a group structure up to homotopy.1 This is given by a map 𝜇 ∶ 𝑍 × 𝑍 → 𝑍 satisfying the usual associativity, inverses, and neutral element axioms, where the maps are equal up to homotopy. Duality in a category is expressed by changing 𝒞 to 𝒞 𝑜𝑝 (see Exercise 1.35). In 𝐓𝐨𝐩, the dual notion of 𝑍 × 𝑍 is 𝑍 ∨ 𝑍 (see Examples 2.21 and 2.23), and the dual of the map 𝜇 is a map 𝑚 ∶ 𝑍 → 𝑍 ∨ 𝑍 satisfying dual axioms. For instance the H-associativity means that the maps 𝜇×1𝑍

𝜇

1𝑍 ×𝜇

𝜇

𝑍 × 𝑍 × 𝑍 = (𝑍 × 𝑍) × 𝑍 ⟶ 𝑍 × 𝑍 ⟶ 𝑍, 𝑍 × 𝑍 × 𝑍 = 𝑍 × (𝑍 × 𝑍) ⟶ 𝑍 × 𝑍 ⟶ 𝑍 are homotopic. The co-H-associativity means that the maps 𝑚

𝑚∨1𝑍

𝑚

1𝑍 ∨𝑚

𝑍 ⟶ 𝑍 ∨ 𝑍 ⟶ (𝑍 ∨ 𝑍) ∨ 𝑍 = 𝑍 ∨ 𝑍 ∨ 𝑍, 𝑍 ⟶ 𝑍 ∨ 𝑍 ⟶ 𝑍 ∨ (𝑍 ∨ 𝑍) = 𝑍 ∨ 𝑍 ∨ 𝑍 are homotopic, which is (1) in Definition 2.10. The others are similar. Proposition 2.12. If 𝑍 is a co-H-space, then ℎ𝑍 (𝑋) = [𝑍, 𝑋]∗ is a group. If 𝑍 is an Abelian co-H-space, then ℎ𝑍 (𝑋) is an Abelian group. Proof. We define the multiplication as ∗ ∶ ℎ𝑍 (𝑋) × ℎ𝑍 (𝑋) → ℎ𝑍 (𝑋), 1

(𝑓, 𝑔) ↦ 𝑓 ∗ 𝑔 = 𝑝𝑋 ∘ (𝑓 ∨ 𝑔) ∘ 𝑚.

The letter H in “H-space” is named after Hopf, who defined this concept.

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2. Algebraic topology

We check that this gives a group structure: (1) Associative. (𝑓 ∗ 𝑔) ∗ ℎ = 𝑝𝑋 ∘ ((𝑝𝑋 ∘ (𝑓 ∨ 𝑔) ∘ 𝑚) ∨ ℎ) ∘ 𝑚 = 𝑝𝑋 ∘ (𝑓 ∨ 𝑔 ∨ ℎ) ∘ (𝑚 ∨ 1𝑍 ) ∘ 𝑚 ∼ 𝑝𝑋 ∘ (𝑓 ∨ 𝑔 ∨ ℎ) ∘ (1𝑍 ∨ 𝑚) ∘ 𝑚 ∼ 𝑝𝑋 ∘ (𝑓 ∨ (𝑝𝑋 ∘ (𝑔 ∨ ℎ) ∘ 𝑚)) = 𝑓 ∗ (𝑔 ∗ ℎ). (2) Neutral element. Let 𝑒 = [𝑐𝑥0 ] ∈ ℎ𝑍 (𝑋). Then 𝑓 ∗ 𝑒 = 𝑝𝑋 ∘ (𝑓 ∨ 𝑐𝑥0 ) ∘ 𝑚 = 𝑝𝑋 ∘ (𝑓 ∨ 𝑓) ∘ (1𝑍 ∨ 𝑐𝑧0 ) ∘ 𝑚 = 𝑓 ∘ 𝑝𝑍 ∘ (1𝑍 ∨ 𝑐𝑧0 ) ∘ 𝑚 ∼ 𝑓. Analogously 𝑒 ∗ 𝑓 ∼ 𝑓. (3) Inverses: Let 𝑓 ∈ ℎ𝑍 (𝑋) and 𝑔 = 𝑓 ∘ 𝐼 ∈ ℎ𝑍 (𝑋). Then 𝑓 ∗ 𝑔 = 𝑝𝑋 ∘ (𝑓 ∨ (𝑓 ∘ 𝐼)) ∘ 𝑚 = 𝑓 ∘ 𝑝𝑍 ∘ (1𝑍 ∨ 𝐼) ∘ 𝑚 ∼ 𝑓 ∘ 𝑐𝑧0 = 𝑐𝑥0 . So 𝑓 ∗ 𝑔 = 𝑒, and analogously 𝑔 ∗ 𝑓 = 𝑒. (4) If 𝑍 is an Abelian co-H-space, then 𝑔∗𝑓 = 𝑝∘(𝑔∨𝑓)∘𝑚 = 𝑝𝑋 ∘(𝑓∨𝑔)∘𝑆 𝑍 ∘𝑚 ∼ 𝑝𝑋 ∘ (𝑓 ∨ 𝑔) ∘ 𝑚 = 𝑓 ∗ 𝑔. □ Theorem 2.13. 𝑆 1 is a co-H-space. 𝑆 𝑛 is an Abelian co-H-space for 𝑛 ≥ 2. Proof. We have 𝑆 𝑛 = {(𝑥0 , 𝑥1 , . . . , 𝑥𝑛 )|𝑥02 + 𝑥12 + ⋯ + 𝑥𝑛2 = 1}, 𝑝 = (1, 0, . . . , 0). Take the equator 𝐶 = 𝑆 𝑛 ∩ {𝑥𝑛 = 0} and collapse it. This gives the comultiplication map 𝑚 ∶ 𝑆 𝑛 → 𝑆 𝑛 /𝐶 ≅ 𝑆 𝑛 ∨ 𝑆 𝑛 .

m

𝑛 In the latter identification, we take both hemispheres 𝑆 𝑛+ /𝐶 and 𝑆− /𝐶 to 𝑆 𝑛 , where 𝑛 = 𝑆 ∩ {±𝑥𝑛 ≥ 0}, by a rigid motion (a rotation), which preserves the orientation (note that all orientation preserving motions are homotopic since SO(𝑛 + 1) is connected). The inverse 𝐼 ∶ 𝑆 𝑛 → 𝑆 𝑛 is defined as 𝐼(𝑥0 , 𝑥1 , . . . , 𝑥𝑛 ) = (𝑥0 , 𝑥1 , . . . , −𝑥𝑛 ).

𝑆 𝑛±

It is (geometrically) clear that properties (1), (2) of Definition 2.10 hold (see Figure 2.1). For property (1), homotope so that the white area gets reduced, the grey area moves downwards and the striped area expands. For (2), homotope from the identity until the upper hemisphere covers the whole of 𝑆 𝑛 . For property (3), note that 𝐹 = 𝑝𝑆𝑛 ∘(1𝑆𝑛 ∨𝐼)∘𝑚 sends the upper hemisphere 𝑆 𝑛+ /𝐶 𝑛 one-to-one to 𝑆 𝑛 , in an orientation preserving way, and the lower hemisphere 𝑆− /𝐶 via 𝐹(𝑥0 , . . . , 𝑥𝑛 ) = 𝐹(𝑥0 , . . . , −𝑥𝑛 ). We define a homotopy by sending the left 𝑥0 ≤ 2𝑠 − 1 to the endpoints in the same vertical, for each 𝑠 ∈ [0, 1]. Namely, 𝐹(𝑥 , 𝑥′ , 𝑥𝑛 ), 𝑥0 ∈ [2𝑠 − 1, 1], 𝐻((𝑥0 , 𝑥′ , 𝑥𝑛 ), 𝑠) = { 0 𝐹(2𝑠 − 1, 𝑥′ , ±√4𝑠(1 − 𝑠) − ||𝑥′ ||2 ), 𝑥0 ∈ [−1, 2𝑠 − 1],

2.1. Homotopy theory

59

m m 1S n

1S n m m

m

pS n ◦ (1S n _ cp)

Figure 2.1. Properties (1) and (2) of Definition 2.10.

which is well defined by the invariance property of 𝐹. Finally, for property (4) when 𝑛 ≥ 2, consider the homotopy given by rotating around the equator, given as 𝐻((𝑥0 , . . . , 𝑥𝑛 ), 𝑠) = (𝑥0 , . . . , 𝑥𝑛−2 , cos(𝜋𝑠)𝑥𝑛−1 + sin(𝜋𝑠)𝑥𝑛 , − sin(𝜋𝑠)𝑥𝑛−1 + cos(𝜋𝑠)𝑥𝑛 ), between the identity and a map that swaps both half spheres (the swapping preserves the orientation!). Then 𝑚 ∼ 𝑆 𝑆𝑛 ∘ 𝑚 with homotopy 𝑚 ∘ 𝐻. □ As a consequence of Theorem 2.13, we have the functors, 𝜋0 ∶ 𝐇𝐨𝐓𝐨𝐩∗ → 𝐒𝐞𝐭, 𝜋1 ∶ 𝐇𝐨𝐓𝐨𝐩∗ → 𝐆𝐫𝐨𝐮𝐩, 𝜋𝑛 ∶ 𝐇𝐨𝐓𝐨𝐩∗ → 𝐀𝐛𝐞𝐥,

𝑛 ≥ 2.

By composition, we also have functors from 𝐓𝐨𝐩∗ . In particular, if 𝜋𝑛 (𝑋) ≇ 𝜋𝑛 (𝑌 ) for some 𝑛, then 𝑋 ≁ 𝑌 . Remark 2.14. If 𝐴 ⊂ 𝑋 is a deformation retract, then 𝜋𝑛 (𝐴) ≅ 𝜋𝑛 (𝑋). If we have the weaker condition of 𝐴 ⊂ 𝑋 being a retract with retraction 𝑟 ∶ 𝑋 → 𝐴, that is 𝑟 ∘ 𝑖 = 1𝐴 , then 𝑟∗ ∘ 𝑖∗ = Id. In particular, 𝑖∗ ∶ 𝜋𝑛 (𝐴) → 𝜋𝑛 (𝑋) is injective and 𝑟∗ ∶ 𝜋𝑛 (𝑋) → 𝜋𝑛 (𝐴) is surjective, for all 𝑛 ≥ 0.

60

2. Algebraic topology

2.1.3. The fundamental group. We end up by checking that our definition coincides with the usual definition of the fundamental group by loops. Recall that a continuous map 𝛾 ∶ [0, 1] → 𝑋 is a path on 𝑋. Two paths 𝛾0 , 𝛾1 ∶ [0, 1] → 𝑋 are homotopic (relative to its endpoints) if there is a homotopy 𝐻 ∶ [0, 1] × [0, 1] → 𝑋, where 𝛾𝑠 = 𝐻(−, 𝑠) ∶ [0, 1] → 𝑋 is a family of paths with 𝛾𝑠 (0) = 𝑥0 , 𝛾𝑠 (1) = 𝑥1 , for all 𝑠 ∈ [0, 1]. A path is a loop (based at the point 𝑥0 ∈ 𝑋) if 𝛾(0) = 𝛾(1) = 𝑥0 . The space of loops based at 𝑥0 is denoted Ω(𝑋, 𝑥0 ). Note that Ω(𝑋, 𝑥0 ) = Hom𝐓𝐨𝐩∗ ((𝑆1 , 1), (𝑋, 𝑥0 )), since a map 𝛾 ∶ ([0, 1], {0, 1}) → (𝑋, 𝑥0 ) descends to a map in the quotient 𝛾 ̄ ∶ [0, 1]/{0, 1} ≅ (𝑆 1 , 1) → (𝑋, 𝑥0 ) via 𝛾(𝑒 ̄ 2𝜋i𝑡 ) = 𝛾(𝑡). A homotopy 𝐻 ∶ ([0, 1], {0, 1}) × [0, 1] → (𝑋, 𝑥0 ) descends also to a homotopy 𝐻̄ ∶ 𝑆 1 × [0, 1] → (𝑋, 𝑥0 ). Therefore 𝜋1 (𝑋, 𝑥0 ) ≅ Ω(𝑋, 𝑥0 )/∼ . For the product in 𝜋1 (𝑋, 𝑥0 ), the map 𝑚 ∶ 𝑆 1 → 𝑆 1 ∨ 𝑆 1 is defined by 𝑚(𝑒2𝜋i𝑡 ) = {

𝑒4𝜋i𝑡 𝑒4𝜋i(𝑡−1/2)

in the left 𝑆 1 , in the right 𝑆 1 ,

𝑡 ∈ [0, 1/2], 𝑡 ∈ [1/2, 1].

Therefore, for 𝛾1 , 𝛾2 ∶ [0, 1] → 𝑋, the product is defined by the map 𝛾(𝑡) = {

𝛾1 (2𝑡), 𝛾2 (2𝑡 − 1),

𝑡 ∈ [0, 1/2], 𝑡 ∈ [1/2, 1].

which satisfies that 𝛾 ̄ = 𝑝𝑆1 ∘ (𝛾1̄ ∨ 𝛾2̄ ) ∘ 𝑚. This is the usual juxtaposition of loops, which is denoted as 𝛾 = 𝛾1 ∗ 𝛾2 . γ1 γ1 γ1

x0

γ2 γ2

γ2 ←

Finally, the inverse map is 𝐼(𝑒2𝜋i𝑡 ) = 𝑒2𝜋i(1−𝑡) , so the loop 𝛾(𝑡) = 𝛾(1 − 𝑡) defines the inverse of [𝛾]. Proposition 2.15. Let 𝑋 be a path connected topological space and let us take 𝑝0 , 𝑝1 ∈ 𝑋. Then we have an isomorphism 𝜋𝑛 (𝑋, 𝑝0 ) ≅ 𝜋𝑛 (𝑋, 𝑝1 ), for all 𝑛 ≥ 1. Proof. Let us fix a path 𝛼 ∶ [0, 1] → 𝑋 with 𝛼(0) = 𝑝0 and 𝛼(1) = 𝑝1 . Consider the map Ψ𝛼 ∶ 𝜋𝑛 (𝑋, 𝑝0 ) → 𝜋𝑛 (𝑋, 𝑝1 ) defined for [𝑓] ∈ 𝜋𝑛 (𝑋, 𝑝0 ), 𝑓 ∶ (𝑆 𝑛 , 𝑝) → (𝑋, 𝑝0 ), as Ψ𝛼 (𝑓) = 𝑝𝑋 ∘ ((𝛼 ∘ ℎ) ∨ 𝑓) ∘ 𝑚, where ℎ ∶ 𝑆 𝑛 → [0, 1], ℎ(𝑥0 , . . . , 𝑥𝑛 ) = (𝑥0 + 1)/2 is the height function (see Figure 2.2). It is easy to see that if 𝛼 ∼ 𝛽 (relative to the endpoints), then Ψ𝛼 (𝑓) ∼ Ψ𝛽 (𝑓). Also

2.1. Homotopy theory

61

X

p1 α

p0

Figure 2.2. The map Ψ𝛼 .

Ψ𝛽 ∘ Ψ𝛼 (𝑓) ∼ Ψ𝛼∗𝛽 (𝑓), when 𝛼 and 𝛽 are paths that can be concatenated. So Ψ𝛼 is an isomorphism with inverse Ψ← . □ 𝛼

Remark 2.16. For the case of the fundamental group 𝜋1 (𝑋, 𝑥0 ), we have that Ψ𝛼 (𝛾) = ←

𝑝𝑋 ∘ ((𝛼 ∘ ℎ) ∨ 𝛾) ∘ 𝑚 coincides with 𝛼 ∗ 𝛾 ∗ 𝛼, for [𝛾] ∈ 𝜋1 (𝑋, 𝑥0 ) (see Figure 2.3). η η

α

α η

Figure 2.3. The map Ψ𝛼 for the fundamental group.

Remark 2.17. Therefore, for a path connected topological space 𝑋 it is natural to denote as 𝜋𝑛 (𝑋) the abstract group isomorphic to any of the 𝜋𝑛 (𝑋, 𝑥0 ), since all of them are isomorphic. This gives a map Obj(𝐓𝐨𝐩𝐏𝐂 ) → Obj(𝐆𝐫𝐨𝐮𝐩), where 𝐓𝐨𝐩𝐏𝐂 is the category of path connected spaces. However, this map is not a functor since it cannot be defined on the level of morphisms (Exercise 2.5). However, if 𝑋 ∼ 𝑌 , then 𝜋𝑛 (𝑋) ≅ 𝜋𝑛 (𝑌 ). In particular, there is a well defined map 𝜋𝑛 ∶ 𝕃𝐓𝐨𝐩𝐏𝐂 → 𝕃𝐆𝐫𝐨𝐮𝐩 . This is proved as follows: Let 𝑓 ∶ 𝑋 → 𝑌 , 𝑔 ∶ 𝑌 → 𝑋 such that 𝑔 ∘ 𝑓 ∼ 1𝑋 , 𝑓 ∘ 𝑔 ∼ 1𝑌 . Let 𝐻 ∶ 𝑋 × [0, 1] → 𝑋 be the homotopy from 1𝑋 to 𝑔 ∘ 𝑓, and let 𝛼(𝑡) = 𝐻(𝑥0 , 𝑡), where 𝑥0 is the basepoint of 𝑋. Then for all ℎ ∶ (𝑆 𝑛 , 𝑝) → (𝑋, 𝑥0 ), we have that (𝑔 ∘ 𝑓) ∘ ℎ ∼ Ψ𝛼 (ℎ). So 𝑔∗ ∘ 𝑓∗ = Ψ𝛼 , where 𝑓∗ ∶ 𝜋𝑛 (𝑋, 𝑥0 ) → 𝜋𝑛 (𝑌 , 𝑓(𝑥0 )) and 𝑔∗ ∶ 𝜋𝑛 (𝑌 , 𝑓(𝑥0 )) → 𝜋𝑛 (𝑋, 𝑔(𝑓(𝑥0 ))). In particular, such 𝑔∗ is an epimorphism. With a similar argument, 𝑓∗ ∘ 𝑔∗ = Ψ𝛽 , where 𝑔∗ ∶ 𝜋𝑛 (𝑌 , 𝑓(𝑥0 )) → 𝜋𝑛 (𝑋, 𝑔(𝑓(𝑥0 ))), 𝑓∗ ∶ 𝜋𝑛 (𝑋, 𝑔(𝑓(𝑥0 ))) → 𝜋𝑛 (𝑌 , 𝑓(𝑔(𝑓(𝑥0 )))), and 𝛽 is some path from 𝑓(𝑥0 ) to 𝑓(𝑔(𝑓(𝑥0 ))). This implies that 𝑔∗ is a monomorphism, and hence an isomorphism. Definition 2.18. We say that a space 𝑋 is simply connected if 𝑋 is path connected and 𝜋1 (𝑋) = {𝑒}, the trivial group (the group consisting only of the neutral element). Example 2.19. The basic example of non-trivial fundamental group is that of 𝑆 1 . Using the theory of covers, we will prove that 𝜋1 (𝑆 1 , 1) ≅ ℤ, being a generator of the group the class of the identity mapping 𝛾 ∶ (𝑆 1 , 1) → (𝑆 1 , 1) (see Remark 2.49).

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2. Algebraic topology

2.1.4. Seifert-van Kampen theorem. Let 𝒞 be a category. Let us define products and coproducts on 𝒞. We start by the notion of product. Let us suppose this given three objects 𝑋1 , 𝑋2 , 𝑌 and two morphisms 𝑓1 ∶ 𝑋1 → 𝑌 , 𝑓2 ∶ 𝑋2 → 𝑌 . We consider the category 𝐏 whose objects are commutative squares 𝑔2

𝑈 𝑔1

 𝑋1

𝑓1

/ 𝑋2  /𝑌

𝑓2

where 𝑈 ∈ Obj(𝒞) and 𝑔1 , 𝑔2 are morphisms of 𝒞 with 𝑓1 ∘ 𝑔1 = 𝑓2 ∘ 𝑔2 . Morphisms between a square given by (𝑈, 𝑔1 , 𝑔2 ) and a square given by (𝑈 ′ , 𝑔′1 , 𝑔′2 ) consist of maps from all the objects of the first square to the second, where we require that the maps are the identity for the objects that are fixed, 𝑋1 , 𝑋2 , 𝑌 . Thus, a morphism in 𝐏 is determined by 𝛼 ∶ 𝑈 → 𝑈 ′ such that 𝑔′1 ∘ 𝛼 = 𝑔1 , 𝑔′2 ∘ 𝛼 = 𝑔2 . Definition 2.20. A product of 𝑋1 , 𝑋2 over 𝑌 (also called fibered product or pullback) is a final object of the category 𝐏. Equivalently, (𝑍, ℎ1 , ℎ2 ) such that for any (𝑍 ′ , ℎ1′ , ℎ2′ ) in 𝐏, there is a unique 𝑓 ∶ 𝑍 ′ → 𝑍 such that the following diagram commutes. 𝑍′ A

A

ℎ′2 𝑓

A

A 𝑍

ℎ′1

ℎ2

ℎ1

"  𝑋1

𝑓1

 / 𝑋2  /𝑌

𝑓2

Recall that final objects may exist or not, but if they exist, they are unique. Therefore the same holds for products in 𝒞. If the product exists, it is usually denoted as 𝑋1 ×𝑌 𝑋2 . If 𝒞 has an final object 𝑌 𝐹 , and we take 𝑌 = 𝑌 𝐹 , then as maps to 𝑌 𝐹 are unique, 𝑌 𝐹 plays no role in the definition. In this case, we denote the product as 𝑋1 × 𝑋 2 . Note that in principle, a product is an element of 𝐏, but it is customary to say that it is an element of the category 𝒞, forgetting the morphisms. In this context, we call ℎ1 , ℎ2 projections. Example 2.21. The product of two objects 𝑋, 𝑌 in 𝐒𝐞𝐭 or 𝐓𝐨𝐩 is given as the Cartesian product 𝑋 × 𝑌 . Also, the product in 𝐕𝐞𝐜𝐭𝐤 or 𝐀𝐛𝐞𝐥 is given by the Cartesian product 𝑉 × 𝑊, for two objects 𝑉, 𝑊. And similarly, the product of two groups 𝐺 1 , 𝐺 2 in 𝐆𝐫𝐨𝐮𝐩 is 𝐺 1 × 𝐺 2 with the usual group structure. As an example with a different flavour, take the category 𝐎𝐩𝐞𝐧(𝑋) of Definition 1.12. The product of 𝑈, 𝑉 ∈ 𝐎𝐩𝐞𝐧(𝑋) is 𝑈 ∩ 𝑉. The dual notion of a product is that of coproduct. Duality in a category is obtained by reversing the direction of the arrows in the arguments (see Exercise 1.35). Suppose that we are given three objects 𝑌 , 𝑋1 , 𝑋2 and two morphisms 𝑓1 ∶ 𝑌 → 𝑋1 , 𝑓2 ∶ 𝑌 → 𝑋2 .

2.1. Homotopy theory

63

We consider the category 𝐂𝐨𝐏 whose objects are commutative squares 𝑌

𝑓1

/ 𝑋1 𝑔1

𝑓2

 𝑋2

 /𝑈

𝑔2

where 𝑈 ∈ Obj(𝒞) and 𝑔1 , 𝑔2 are morphisms and 𝑔1 ∘𝑓1 = 𝑔2 ∘𝑓2 . A morphism between a square (𝑈, 𝑔1 , 𝑔2 ) and a square (𝑈 ′ , 𝑔′1 , 𝑔′2 ) is given by a morphism 𝛼 ∶ 𝑈 → 𝑈 ′ such that 𝛼 ∘ 𝑔1 = 𝑔′1 , 𝛼 ∘ 𝑔2 = 𝑔′2 . Definition 2.22. A coproduct of 𝑋1 , 𝑋2 over 𝑌 (also called an amalgamated product or pushout) is an initial object of 𝐂𝐨𝐏. Equivalently, a (𝑍, ℎ1 , ℎ2 ) such that for any (𝑍 ′ , ℎ1′ , ℎ2′ ), there is a unique 𝑓 ∶ 𝑍 → 𝑍 ′ such that the following diagram commutes. 𝑌

𝑓2

𝑓1

 𝑋2

/ 𝑋1 ℎ1

ℎ2

 /𝑍 A ℎ′2

ℎ′1

A

𝑓

A

A  / 𝑍′

The coproduct is unique if it exists. It is usually denoted 𝑍 = 𝑋1 ⋆𝑌 𝑋2 . If 𝒞 has an initial object 𝑌 𝐼 and we take 𝑌 = 𝑌 𝐼 , then we denote 𝑋1 ⋆ 𝑋2 . In this context, ℎ1 , ℎ2 are the inclusions. Example 2.23. The coproduct of two objects 𝑋, 𝑌 in 𝐒𝐞𝐭 or 𝐓𝐨𝐩 is 𝑋 ⊔𝑌 . The coproduct of 𝑋, 𝑌 in 𝐓𝐨𝐩∗ is 𝑋 ∨ 𝑌 . The coproduct in 𝐕𝐞𝐜𝐭𝐤 or 𝐀𝐛𝐞𝐥 is given by the direct product 𝑉 ⊕ 𝑊, for two objects 𝑉, 𝑊 (notably, there is an isomorphism 𝑉 × 𝑊 ≅ 𝑉 ⊕ 𝑊). In the category 𝐎𝐩𝐞𝐧(𝑋), the amalgamated product is 𝑈 ⋆𝑈∩𝑉 𝑉 = 𝑈 ∪ 𝑉. Given groups 𝐺 1 , 𝐺 2 , 𝐻 ∈ Obj(𝐆𝐫𝐨𝐮𝐩), and morphisms 𝜑1 ∶ 𝐻 → 𝐺 1 , 𝜑2 ∶ 𝐻 → 𝐺 2 , the amalgamated product of 𝐺 1 and 𝐺 2 by 𝐻, denoted 𝐺 1 ⋆𝐻 𝐺 2 , is the coproduct in the category 𝐆𝐫𝐨𝐮𝐩. The free product is the coproduct 𝐺 1 ⋆ 𝐺 2 with 𝐻 = {𝑒}. The amalgamated product in 𝐆𝐫𝐨𝐮𝐩 exists. To give its construction, recall the notion of presentation of a group 𝐺. First, we define a free group on generators {𝑎𝑖 | 𝑖 ∈ 𝐼} as the group 𝐹(𝑎𝑖 | 𝑖 ∈ 𝐼) formed by all the words on the letters 𝑎𝑖 raised to integer expo𝑛+𝑚 nents, where 𝑎𝑛𝑖 𝑎𝑚 and 𝑎0𝑖 = 𝑒, where the neutral element 𝑒 = ∅ is the empty 𝑖 = 𝑎𝑖 word. The free group satisfies a universal property: any map {𝑎𝑖 } → 𝐺 to a group 𝐺 (i.e., a morphism in 𝐒𝐞𝐭) defines a unique homomorphism of groups 𝐹(𝑎𝑖 | 𝑖 ∈ 𝐼) → 𝐺. The free groups on 𝑟 generators are denoted 𝐹𝑟 = 𝐹(𝑎1 , . . . , 𝑎𝑟 ). For a group 𝐺, we say that {𝑔𝑖 | 𝑖 ∈ 𝐼} ⊂ 𝐺 are generators if the map 𝜑 ∶ 𝐹(𝑎𝑖 | 𝑖 ∈ 𝐼) → 𝐺, 𝜑(𝑎𝑖 ) = 𝑔𝑖 , is an epimorphism. In this case, take 𝑁 = ker 𝜑 ◁ 𝐹(𝑎𝑖 | 𝑖 ∈ 𝐼), and let {𝑅𝑗 | 𝑗 ∈ 𝐽} ⊂ 𝑁 be generators of 𝑁 as a normal subgroup (i.e., 𝑁 is the smallest normal subgroup that contains {𝑅𝑗 }). This is denoted as 𝑁 = ⟨⟨𝑅𝑗 | 𝑗 ∈ 𝐽⟩⟩. Note that 𝑅𝑗 = 𝑅𝑗 (𝑎𝑖 ) are words in

64

2. Algebraic topology

the letters 𝑎𝑖 , and they are called relations. We say that 𝐺 has a presentation 𝐺 = ⟨𝑎𝑖 , 𝑖 ∈ 𝐼 | 𝑅𝑗 (𝑎𝑖 ), 𝑗 ∈ 𝐽⟩ =

𝐹(𝑎𝑖 | 𝑖 ∈ 𝐼) . ⟨⟨𝑅𝑗 | 𝑗 ∈ 𝐽⟩⟩

The group 𝐺 is finitely generated if it admits a finite set of generators, and finitely presented if it admits a finite set of generators and a finite set of relations. Now take two groups 𝐺 1 , 𝐺 2 and presentations 𝐺 1 = ⟨𝑎𝑖 , 𝑖 ∈ 𝐼 | 𝑅𝑗 (𝑎𝑖 ), 𝑗 ∈ 𝐽⟩,

𝐺 2 = ⟨𝑏𝑘 , 𝑘 ∈ 𝐾 | 𝑆 𝑙 (𝑏𝑘 ), 𝑙 ∈ 𝐿⟩.

Consider another group 𝐻 with generators ℎ𝑟 , 𝑟 ∈ 𝑅, and with morphisms 𝜑1 ∶ 𝐻 → 𝐺 1 , 𝜑2 ∶ 𝐻 → 𝐺 2 . Then the amalgamated product is (see Exercise 2.10) (2.1)

𝐺 1 ⋆𝐻 𝐺 2 = ⟨𝑎𝑖 ,𝑏𝑘 , 𝑖 ∈ 𝐼, 𝑘 ∈ 𝐾 | 𝑅𝑗 (𝑎𝑖 ), 𝑆 𝑙 (𝑏𝑘 ), 𝜑1 (ℎ𝑟 )𝜑2 (ℎ𝑟 )−1 , 𝑗 ∈ 𝐽, 𝑙 ∈ 𝐿, 𝑟 ∈ 𝑅⟩.

The importance of the amalgamated product is that it appears as the way of pasting the fundamental group of two spaces to get the fundamental group of the total space. Theorem 2.24 (Seifert-van Kampen). Let (𝑋, 𝑥0 ) be a pointed topological space, and let 𝑈, 𝑉 ⊂ 𝑋 be open sets with 𝑥0 ∈ 𝑈 ∩ 𝑉, 𝑋 = 𝑈 ∪ 𝑉 and 𝑈, 𝑉, 𝑈 ∩ 𝑉 all path connected. Then, the fundamental group of 𝑋 is the amalgamated product 𝜋1 (𝑋, 𝑥0 ) ≅ 𝜋1 (𝑈, 𝑥0 ) ⋆𝜋1 (𝑈∩𝑉 ,𝑥0 ) 𝜋1 (𝑉, 𝑥0 ) . The proof can be found in [Hat]. We just notice that 𝑈 ∪ 𝑉 = 𝑈 ⋆𝑈∩𝑉 𝑉 is an amalgamated product in 𝐎𝐩𝐞𝐧(𝑋). Therefore the Seifert-van Kampen theorem says 𝐏𝐂 that 𝜋1 ∶ 𝐎𝐩𝐞𝐧∗ (𝑋) → 𝐆𝐫𝐨𝐮𝐩 (where the notation means path connected basepoint open sets of 𝑋) sends amalgamated products to amalgamated products. Example 2.25. We give some examples of computations using Theorem 2.24. (1) If 𝑈, 𝑉 are simply connected and 𝑈 ∩ 𝑉 is path connected, then 𝑋 = 𝑈 ∪ 𝑉 is simply connected. (2) For instance, 𝑆 𝑛 is simply connected for 𝑛 ≥ 2. Take 𝑈 = 𝑆 𝑛 ∩ {𝑥𝑛 > −1/2} and 𝑉 = 𝑆 𝑛 ∩ {𝑥𝑛 < 1/2} and apply the previous item. Observe that this argument does not work for 𝑆 1 since in that case 𝑈 ∩ 𝑉 would not be path connected. (3) Suppose (𝑋, 𝑥0 ), (𝑌 , 𝑦0 ) are two path connected pointed spaces. Suppose that 𝑥0 , 𝑦0 have contractible 𝑈 𝑥0 , 𝑉 𝑦0 such that 𝑈 𝑥0 deformation retracts to 𝑥0 and 𝑉 𝑦0 deformation retracts to 𝑦0 (this happens for instance for 𝑛-manifolds, or for triangulated spaces, see Exercise 2.26). Then the fundamental group of the pointed union 𝑋 ∨ 𝑌 is 𝜋1 (𝑋 ∨ 𝑌 , 𝑝0 ) ≅ 𝜋1 (𝑋, 𝑥0 ) ∗ 𝜋1 (𝑌 , 𝑦0 ). Consider the open sets 𝑋 ∨ 𝑉 and 𝑈 ∨ 𝑌 and apply the Seifert-van Kampen theorem. Note that 𝑋 ∨ 𝑉 deformation retracts to 𝑋, 𝑈 ∨ 𝑌 deformation retracts to 𝑌 , and 𝑈 ∨ 𝑉 is contractible. The result follows. (4) We apply the previous case to the wedge of 𝑟 circles 𝑊𝑟 = 𝑆 1 ∨ 𝑆 1 ∨ ⋯ ∨ 𝑆 1 . This space is sometimes called a bouquet of 𝑟 circles. The fundamental group of the 𝑖th copy is 𝜋1 (𝑆1 , 1) = ⟨𝑎𝑖 ⟩ ≅ ℤ, where 𝑎𝑖 denotes the generator. Then 𝜋1 (𝑊𝑟 ) = ⟨𝑎1 ⟩ ∗ ⋯ ∗ ⟨𝑎𝑟 ⟩ = 𝐹(𝑎1 , . . . , 𝑎𝑟 ) = 𝐹𝑟 , the free groups on 𝑟 generators.

2.1. Homotopy theory

65

Now we shall give the fundamental groups of compact surfaces. Consider a compact connected surface 𝑆 without boundary, and let P𝑆 be a planar representation given by a word 𝐩. Let 𝑎1 , . . . , 𝑎𝑟 be the letters of 𝐩. We can suppose that there is a single vertex 𝑝 in the triangulation (see Example 1.62). This means that all vertices of P𝑆 are equivalent under the identification ∼ of 𝜕P𝑆 . Let 𝑞 ∈ Int P𝑆 be the barycenter of the polygon. Consider 𝑈 = 𝑆 − {𝑞} and 𝑉 = 𝐵𝜖 (𝑞) a small ball around 𝑞. Take a point 𝑝0 ∈ 𝑈 ∩ 𝑉. First, 𝑉 is contractible and 𝑈 ∩ 𝑉 is of the homotopy type of a circle. Then 𝜋1 (𝑉) = {𝑒} and 𝜋1 (𝑈 ∩ 𝑉) = ℤ⟨𝛾⟩, where 𝛾 is a loop going once around 𝑞. Second, P𝑆 −{𝑞} deformation retracts to 𝜕P𝑆 , so 𝑈 = 𝑆 −{𝑞} = (P𝑆 −{𝑞})/∼ deformation retracts to (𝜕P𝑆 )/ ∼ . This space is a collection of circles based at 𝑝, so 𝑈 ∼ 𝑊𝑟 , a bouquet of 𝑟 circles. Thus 𝜋1 (𝑈) = 𝐹(𝑎1 , . . . , 𝑎𝑟 ). The map 𝑖∗ ∶ 𝜋1 (𝑈 ∩ 𝑉) = ⟨𝛾⟩ → 𝜋1 (𝑈) sends 𝛾 to a loop that goes around the boundary 𝜕P𝑆 . So 𝑖∗ (𝛾) is the word 𝐩.

The Seifert-van Kampen theorem gives 𝜋1 (𝑉) = {𝑒} UUUU kk5 k UUUU k k k k UUUU k kk k UUUU k k kk * 𝜋1 (𝑈 ∩ 𝑉) = ⟨𝛾⟩ 𝜋1 (𝑋) = ⟨𝑎1 , . . . , 𝑎𝑟 | 𝐩⟩. SSSS i4 SS𝑖S∗S(𝛾)=𝐩 iiii i i i SSSS iiii SSS) iiii 𝜋1 (𝑈) = 𝐹(𝑎1 , . . . , 𝑎𝑟 ) Now the fundamental groups of the compact connected surfaces can be written down by using the canonical words from Example 1.62. • For 𝑆 = Σ𝑔 , 𝑔 ≥ 1, we have the fundamental group 𝜋1 (Σ𝑔 ) = ⟨𝑎1 , . . . , 𝑎𝑔 , 𝑏1 , . . . , 𝑏𝑔 |[𝑎1 , 𝑏1 ] ⋯ [𝑎𝑔 , 𝑏𝑔 ] = 1⟩, −1 where we write [𝑎𝑖 , 𝑏𝑖 ] = 𝑎𝑖 𝑏𝑖 𝑎−1 𝑖 𝑏𝑖 for the group commutator.

• For 𝑆 = 𝑋𝑘 , 𝑘 ≥ 1, it is 𝜋1 (𝑋𝑘 ) = ⟨𝑎1 , . . . , 𝑎𝑘 | 𝑎1 𝑎1 ⋯ 𝑎𝑘 𝑎𝑘 = 1⟩. • For 𝑆 = 𝑆 2 , we cannot use the word 𝑎𝑎−1 since it does not correspond to a planar representation with only one vertex. But for 𝑆 2 we already know that 𝜋1 (𝑆 2 ) = {𝑒} by Example 2.25.

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2. Algebraic topology

• Of course, the argument above to compute 𝜋1 (𝑆) can be applied to planar representations P𝑆 where there are several (classes of) vertices (including surfaces with boundary). But in this case, one has to draw the planar graph corresponding to 𝜕P𝑆 /∼ to know the generators of the group 𝜋1 (𝑈) (see Exercise 2.1). • The trick of decomposing 𝑆 = 𝑈 ∪ 𝑉 decomposes a 2-dimensional space into spaces which are (homotopically) of smaller dimension. This idea can be used for 𝑛-manifolds. Remark 2.26. The fundamental groups appearing for compact connected surfaces (dimension 𝑛 = 2) are very specific. However, for compact manifolds of dimension 𝑛 ≥ 4, any group which is finitely presented can appear as the fundamental group of a compact connected 𝑛-manifold. Remark 2.27. Theorem 2.24 is a useful tool for computing fundamental groups. However, there is no analogous method for computing higher homotopy groups 𝜋𝑛 (𝑋) of a space 𝑋. Moreover, 𝜋𝑛 (𝑋) of a space can be non-zero for arbitrarily large 𝑛 even for simple spaces of small dimension. An important open problem is that of the computation of homotopy groups of spheres, 𝜋𝑛 (𝑆 𝑘 ). For 𝑛 ≤ 𝑘, we have (by the Hurewicz theorem, Theorem 2.73 and by Corollary 2.92), 𝜋𝑛 (𝑆𝑘 ) = {

0, ℤ,

𝑛 < 𝑘, 𝑛 = 𝑘.

But the computation of 𝜋𝑛 (𝑆 𝑘 ) for 𝑛 > 𝑘 is very hard, and in general there are infinitely many 𝜋𝑛 (𝑆 𝑘 ) ≠ 0. For instance, 𝜋3 (𝑆 2 ) = ℤ, generated by the Hopf map ℎ ∶ 𝑆 3 → 𝑆 2 (see Remark 4.30). 2.1.5. Abelianization of the fundamental group. The fundamental group of a topological space is often not an Abelian group. This fact can hinder the effective use of this invariant. For example, recall that in general, the problem of deciding whether two finitely presented groups 𝐺 1 and 𝐺 2 , given by their presentations, are isomorphic is undecidable. Therefore, the problem of distinguishing two topological spaces up to homotopy by means of their fundamental group can be very hard. On the other hand, this problem does not occur for Abelian groups since finitely generated Abelian groups are classified, and distinguishing two Abelian groups is easy. To make use of this, we canonically associate an Abelian group to any group. Definition 2.28. Let 𝐺 be a group. The Abelianization of 𝐺 is an Abelian group Ab(𝐺) together with a group homomorphism 𝜑 ∶ 𝐺 → Ab(𝐺) such that, for any Abelian group 𝐴 and any homomorphism 𝑓 ∶ 𝐺 → 𝐴, there exists a unique homomorphism 𝑓 ̄ ∶ Ab(𝐺) → 𝐴 such that 𝑓 ̄ ∘ 𝜑 = 𝑓. Using categorical jargon, the Abelianization is given as the initial object in the category of diagrams 𝑓 ∶ 𝐺 → 𝐴, where 𝐴 ∈ Obj(𝐀𝐛𝐞𝐥). Therefore the Abelianization, once we prove that it exists, is unique. In particular, the Abelianization of an Abelian group is itself. The Abelianization of a group 𝐺 is easy to construct. Take 𝐺 Ab(𝐺) = , [𝐺, 𝐺]

2.1. Homotopy theory

67

where [𝐺, 𝐺] denotes de commutator subgroup, that is the subgroup generated by the commutators [𝑔, ℎ] = 𝑔ℎ𝑔−1 ℎ−1 , 𝑔, ℎ ∈ 𝐺 (which is automatically normal). With this definition, Ab(𝐺) is automatically Abelian. Also, if we have a homomorphism 𝑓 ∶ 𝐺 → 𝐴 with 𝐴 an Abelian group, then we have 𝑓([𝑔, ℎ]) = 𝑓(𝑔)𝑓(ℎ)𝑓(𝑔)−1 𝑓(ℎ)−1 = 1. Thus [𝐺, 𝐺] ∈ ker 𝑓 so 𝑓 descends to a unique homomorphism 𝑓 ̄ ∶ 𝐺/[𝐺, 𝐺] → 𝐴 of Abelian groups. We define the Abelianization functor Ab ∶ 𝐆𝐫𝐨𝐮𝐩 → 𝐀𝐛𝐞𝐥 . On objects, it is 𝐺 ↦ Ab(𝐺). For a group homomorphism 𝑓 ∶ 𝐺 1 → 𝐺 2 , compose it with the quotient 𝐺 2 → Ab(𝐺 2 ), so we have a group homomorphism 𝑓′ ∶ 𝐺 1 → Ab(𝐺 2 ) to an Abelian group. Factor it through Ab(𝐺 1 ) to get Ab(𝑓) = 𝑓′̄ ∶ Ab(𝐺 1 ) → Ab(𝐺 2 ). In particular, we can consider the composed functor Ab ∘𝜋1 ∶ 𝐓𝐨𝐩∗ → 𝐆𝐫𝐨𝐮𝐩 → 𝐀𝐛𝐞𝐥 to distinguish spaces. This means that the Abelianized fundamental group is an algebraic invariant. We compute this invariant for the compact connected surfaces. • For 𝑆2 , we have that 𝜋1 (𝑆 2 ) = {0}, which is an Abelian group and thus its own Abelianization. • For the torus Σ1 , we know that 𝜋1 (Σ1 ) = ⟨𝑎, 𝑏 | 𝑎𝑏𝑎−1 𝑏−1 = 1⟩ = ⟨𝑎, 𝑏 | 𝑎𝑏 = 𝑏𝑎⟩. Thus 𝜋1 (Σ1 ) is Abelian, so it is its own Abelianization. In fact, we have 𝜋1 (Σ1 ) = ℤ⟨𝑎, 𝑏⟩ ≅ ℤ2 (we denote as ℤ⟨𝑎1 , . . . , 𝑎𝑟 ⟩ the free Abelian group with generators 𝑎1 , . . . , 𝑎𝑟 , i.e., Ab(𝐹(𝑎1 , . . . , 𝑎𝑟 ))). • For the orientable surface Σ𝑔 with 𝑔 ≥ 2, its fundamental group is not Abelian. For that, observe that we can define the mapping 𝑓 ∶ 𝜋1 (Σ𝑔 ) → 𝐹(𝛼, 𝛽) ≅ ℤ ⋆ ℤ, given by 𝑓(𝑎1 ) = 𝛼, 𝑓(𝑎2 ) = 𝛽, 𝑓(𝑎𝑖 ) = 1, 3 ≤ 𝑖 ≤ 𝑔, 𝑓(𝑏𝑗 ) = 1, 1 ≤ 𝑗 ≤ 𝑔. Since the relation of 𝜋1 (Σ𝑔 ) is in ker(𝑓), 𝑓 defines a map from 𝜋1 (Σ𝑔 ) to 𝐹(𝛼, 𝛽), which is clearly an epimorphism. Since ℤ ⋆ ℤ is not Abelian, 𝜋1 (Σ𝑔 ) is not Abelian for 𝑔 ≥ 2. • To compute the Abelianization of 𝜋1 (Σ𝑔 ), observe that, as [𝑎𝑖 , 𝑏𝑖 ] ∈ [𝐺, 𝐺] for all 𝑖, we have that ⟨⟨[𝑎1 , 𝑏1 ] ⋯ [𝑎𝑔 , 𝑏𝑔 ]⟩⟩ ⊂ [𝐺, 𝐺]. Thus Ab(𝜋1 (Σ𝑔 )) = Ab( ≅

𝐹(𝑎1 , . . . , 𝑎𝑔 , 𝑏1 , . . . , 𝑏𝑔 ) ) ⟨⟨[𝑎1 , 𝑏1 ] ⋯ [𝑎𝑔 , 𝑏𝑔 ]⟩⟩

𝐹(𝑎1 , . . . , 𝑎𝑔 , 𝑏1 , . . . , 𝑏𝑔 ) ⟨⟨[𝑎1 , 𝑏1 ] ⋯ [𝑎𝑔 , 𝑏𝑔 ], [𝑎𝑖 , 𝑎𝑗 ], [𝑎𝑖 , 𝑏𝑗 ], [𝑏𝑖 , 𝑏𝑗 ], 1 ≤ 𝑖, 𝑗 ≤ 𝑔⟩⟩

≅ ℤ⟨𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 ⟩ ≅ ℤ2𝑔 . • For the real projective plane ℝ𝑃 2 = 𝑋1 , we have that the fundamental group is 𝜋1 (𝑋1 ) = ⟨𝑎 | 𝑎2 = 1⟩ ≅ ℤ2 , which is Abelian and it is thus its own Abelianization.

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• For the connected sum of 𝑘 projective planes 𝑋𝑘 with 𝑘 ≥ 2, the fundamental group is not Abelian (Exercise 2.13). Its Abelianization, using additive notation for Abelian groups, is given by Ab(𝜋1 (𝑋𝑘 )) = Ab(

𝐹(𝑎1 , . . . , 𝑎𝑘 ) ℤ⟨𝑎1 , . . . , 𝑎𝑘 ⟩ )≅ ⟨2𝑎1 + ⋯ + 2𝑎𝑘 ⟩ ⟨⟨𝑎21 ⋯ 𝑎2𝑘 ⟩⟩

≅ ℤ⟨𝜔, 𝑎2 , . . . , 𝑎𝑘 | 2𝜔 = 0⟩ ≅ ℤ2 × ℤ𝑘−1 , where, in the third equality, we set 𝜔 = 𝑎1 + ⋯ + 𝑎𝑘 . Using the Abelianized fundamental group, we can distinguish all compact connected surfaces. In fact, given a compact connected surface 𝑆, we can identify it by means of the Abelian group Ab(𝜋1 (𝑆)): • If Ab(𝜋1 (𝑆)) has torsion, then 𝑆 ≅ 𝑋𝑟+1 where 𝑟 is the rank of Ab(𝜋1 (𝑆)). • If Ab(𝜋1 (𝑆)) has no torsion, then 𝑆 ≅ Σ𝑔 where 2𝑔 is the rank of Ab(𝜋1 (𝑆)) (see Remark 2.106(6) for the concepts of torsion and rank of an Abelian group). This completes the classification of topological compact surfaces: Theorem 2.29. We have that 𝑐𝑜

𝕃𝐓𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 |𝑔 ≥ 0, 𝑘 ≥ 1}. 𝑐

Also 𝑐𝑜

𝕃𝐇𝐨𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 |𝑔 ≥ 0, 𝑘 ≥ 1}. 𝑐

𝑐𝑜

Proof. In Theorem 1.82, we saw that 𝕃𝐏𝐋𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 |𝑔 ≥ 0, 𝑘 ≥ 1}. By Remark 𝑐𝑜

𝑐

𝑐𝑜

1.83, the map 𝕃𝐏𝐋𝐌𝐚𝐧2 → 𝕃𝐓𝐌𝐚𝐧2 is surjective. Now there are functors 𝑐

𝑐

𝑐𝑜 𝕃𝐏𝐋𝐌𝐚𝐧2 𝑐

𝑐𝑜

𝑐𝑜

𝑐𝑜

↠ 𝕃𝐓𝐌𝐚𝐧2 ⟶ 𝕃𝐇𝐨𝐌𝐚𝐧2 ≅ 𝕃𝐇𝐨𝐌𝐚𝐧2 𝑐

𝑐

𝜋1

𝑐,∗

Ab

↪ 𝕃𝐇𝐨𝐓𝐨𝐩∗ ⟶ 𝐆𝐫𝐨𝐮𝐩 ⟶ 𝐀𝐛𝐞𝐥 . By the above, the composition is injective. In particular, the first map is injective, hence 𝑐𝑜 𝑐𝑜 𝕃𝐓𝐌𝐚𝐧2 ≅ 𝕃𝐏𝐋𝐌𝐚𝐧2 . 𝑐

𝑐

𝑐𝑜

𝑐𝑜

Note also that 𝕃𝐓𝐌𝐚𝐧2 → 𝕃𝐇𝐨𝐌𝐚𝐧2 is obviously surjective (as the categories have the 𝑐

𝑐

𝑐𝑜

𝑐𝑜

same objects). Therefore the same argument gives 𝕃𝐇𝐨𝐌𝐚𝐧2 ≅ 𝕃𝐏𝐋𝐌𝐚𝐧2 . 𝑐

𝑐

□

Remark 2.30. The homotopy classification of compact surfaces with boundary does not coincide with the classification up to homeomorphism. Let 𝐓𝐌𝐚𝐧2𝜕,𝑐 be the category of compact surfaces with boundary and consider 𝐇𝐨𝐌𝐚𝐧2𝜕,𝑐 the corresponding 𝑐𝑜 homotopy category. The proof of Corollary 1.86 works for 𝕃𝐓𝐌𝐚𝐧2 using Theorem 2.29, 𝜕,𝑐 to give 𝑐𝑜 𝕃𝐓𝐌𝐚𝐧2 = {Σ𝑔,𝑏 , 𝑋𝑘,𝑏 |𝑔 ≥ 0, 𝑘 ≥ 1, 𝑏 ≥ 0}, 𝜕,𝑐

𝑏 (⨆𝑖=1

𝑏

where Σ𝑔,𝑏 = Σ𝑏 − 𝐵𝑖 ) and 𝑋𝑘,𝑏 = 𝑋𝑘 − (⨆𝑖=1 𝐵𝑖 ). Now, it is easy to see that if 𝑏 ≥ 1, then the planar representation of Σ𝑔,𝑏 retracts to a wedge of 2𝑔 + 𝑏 − 1 circles

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2.2. Covers

𝑊2𝑔+𝑏−1 . Analogously, if 𝑏 ≥ 1, then 𝑋𝑘,𝑏 retracts to 𝑊 𝑘+𝑏−1 . Also, for 𝑆 2 − 𝐵1 we have a contractible space. All together we have 𝑐𝑜

𝕃𝐇𝐨𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 , 𝑊𝑟 , ⋆|𝑔 ≥ 0, 𝑘 ≥ 1, 𝑟 ≥ 1}. 𝜕,𝑐

In particular, we have spaces, like the sphere minus three balls 𝑆 2 − (𝐵1 ⊔ 𝐵2 ⊔ 𝐵3 ) and the torus minus one ball 𝑇 2 − 𝐵1 , which are homotopy equivalent (both are homotopy equivalent to 𝑊2 = 𝑆 1 ∨ 𝑆 1 ) but which are not homeomorphic. 𝑐𝑜

𝑐𝑜

Remark 2.31. An important instance of the map 𝕃𝐓𝐌𝐚𝐧𝑛 → 𝕃𝐇𝐨𝐌𝐚𝐧𝑛 is the Poincaré 𝑐

𝑐

conjecture (cf. Example 4.34(3) for the original version). This claims that if 𝑀 is a compact 𝑛-dimensional manifold with the same homotopy groups of the 𝑛-sphere (that is 𝑀 ∼ 𝑆 𝑛 , cf. Exercise 2.24), then 𝑀 is homeomorphic to a sphere. For 𝑛 ≥ 5, the problem was solved by Smale by means of the famous ℎ-cobordism theorem (see [Mi4]). For 𝑛 = 4 it was solved by Freedman. Finally, Perelman solved the problem for 𝑛 = 3 by means of a geometric flow known as the Ricci flow (Remark 6.76(7)). See [Sm1], [Fre], [Per]. There are versions of the Poincaré conjecture in the 𝐓𝐌𝐚𝐧𝑛 , 𝐏𝐋𝐌𝐚𝐧𝑛 and 𝐃𝐌𝐚𝐧𝑛 categories. The smooth version claims that a compact 𝑛-dimensional smooth manifold with the homotopy type of 𝑆𝑛 is diffeomorphic to 𝑆 𝑛 . By Remark 1.22(e), we know that this fails for 𝑛 ≥ 7 due to the existence of exotic smooth structures on spheres. The smooth Poincaré conjecture is not known for 𝑛 = 4, and it holds for 𝑛 = 5, 6. It is clearly true for 𝑛 = 3, since in this dimension the topological and smooth classifications coincide.

2.2. Covers In this section we recall the definition and basic properties of covers, and then we work out some interesting examples of covers between surfaces. We also explore a generalization of covers, the so-called ramified covers. 2.2.1. General results on covers. Definition 2.32. Let 𝑋 and 𝑋 ′ be topological spaces. A continuous map 𝜋 ∶ 𝑋 ′ → 𝑋 is called a cover if 𝜋 is surjective and for each 𝑝 ∈ 𝑋 there exists an open neighbourhood 𝑈 𝑝 so that 𝜋−1 (𝑈) = ⨆𝑖∈𝐼 𝑉 𝑖 and 𝜋|𝑉𝑖 ∶ 𝑉 𝑖 → 𝑈 is a homeomorphism for each 𝑖 ∈ 𝐼. Remark 2.33. (1) The fiber of 𝑝 ∈ 𝑋 is 𝐹𝑝 = 𝜋−1 (𝑝). Its cardinal is the cardinal of 𝐼, which depends on the point 𝑝 in Definition 2.32. However, if 𝑋 is connected, then the cardinal of the fiber 𝐹𝑝 is constant on 𝑋, and it is called the number of sheets or the degree of the cover (Exercise 2.14). (2) It is useful to work in the category of pointed topological spaces and consider covers of the form 𝜋 ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝). This is the right setting because lifts of maps become unique when we fix basepoints. (3) Covers are interesting because they give a lot of information about the fundamental group of a space 𝑋. However, for this to be true, it is necessary

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to suppose some mild conditions on 𝑋. From now on, we will suppose that all spaces are Hausdorff, path connected, and locally simply connected, unless otherwise stated. This is obviously true if 𝑋 is a connected manifold, our main object of interest. Nonetheless, some of the results are true with milder conditions. Now we review the main results of covers, whose proofs can be found in [Mun]. The link between covers and homotopy starts with the fact that paths on 𝑋 can be lifted to paths on a cover. Lemma 2.34. Let 𝜋 ∶ 𝑋 ′ → 𝑋 be a cover, let 𝛾 ∶ [0, 1] → 𝑋 be a path with 𝛾(0) = 𝑝, and let 𝑝′ ∈ 𝐹𝑝 . Then there exists a unique path 𝛾′ ∶ [0, 1] → 𝑋 ′ with 𝛾′ (0) = 𝑝′ and 𝜋 ∘ 𝛾′ = 𝛾. The path 𝛾′ above is called the lift of 𝛾. Note that we usually use the notation with primes for objects in the cover. Note that 𝛾′ (1) ∈ 𝐹𝑝 . Therefore 𝛾′ will be a loop in the case that 𝛾′ (1) = 𝑝′ . In that case 𝜋 ∘ 𝛾′ = 𝛾 implies that 𝜋∗ ([𝛾′ ]) = [𝛾]. This allows us to define the group of the cover, 𝐻(𝑋 ′ ,𝜋) = 𝜋∗ (𝜋1 (𝑋 ′ , 𝑝′ )) < 𝜋1 (𝑋, 𝑝). Observe that by Lemma 2.34, 𝐻(𝑋 ′ ,𝜋) is precisely the set of loops of 𝜋1 (𝑋, 𝑝) that lift to loops (and not just paths) in 𝑋 ′ . In general, the problem of lifting a map 𝑓 ∶ 𝑌 → 𝑋 to a cover is controlled by this group. Proposition 2.35. Let 𝑓 ∶ (𝑌 , 𝑞) → (𝑋, 𝑝) be a continuous map with 𝑌 a path connected space, and let 𝜋 ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝) be a cover. Then there exists a unique 𝑓′ ∶ (𝑌 , 𝑞) → (𝑋 ′ , 𝑝′ ) such that 𝜋 ∘ 𝑓′ = 𝑓 if and only if 𝑓∗ (𝜋1 (𝑌 , 𝑞)) ⊂ 𝐻(𝑋 ′ ,𝜋) . In particular, if 𝑌 is a simply connected space, we can always lift 𝑓 in a unique way if we fix the basepoints. Remark 2.36. Suppose that 𝑓0 , 𝑓1 ∶ (𝑌 , 𝑞) → (𝑋, 𝑝) are homotopic maps, with homotopy 𝐻 ∶ 𝑌 × [0, 1] → 𝑋, and 𝑓0′ ∶ 𝑌 → 𝑋 ′ is a lifting of 𝑓0 . Then there exists a map 𝐻 ′ ∶ 𝑌 × [0, 1] → 𝑋 ′ such that 𝜋 ∘ 𝐻 ′ = 𝐻, using Proposition 2.35, since 𝐻∗ (𝜋1 (𝑌 × [0, 1], (𝑞, 0))) = (𝐻0 )∗ (𝜋1 (𝑌 , 𝑞)) = (𝑓0 )∗ (𝜋1 (𝑌 , 𝑞)) = 𝜋∗ ((𝑓0′ )∗ (𝜋1 (𝑌 , 𝑞))) < 𝜋∗ (𝜋1 (𝑋, 𝑝)). In particular, for 𝑞 ∈ 𝑌 , the path 𝑠 ↦ 𝐻 ′ (𝑞, 𝑠) is a lifting of the constant path 𝐻(𝑞, 𝑠) = 𝑝. By uniqueness of lifts, 𝐻 ′ (𝑞, 𝑠) = 𝑝′ . So 𝐻1′ is a lifting of 𝑓1 , and the lifts 𝑓0′ and 𝑓1′ = 𝐻1′ are homotopic. In particular, homotopic paths 𝛾0 , 𝛾1 (relative to the endpoints) lift to homotopic paths 𝛾0′ , 𝛾1′ , and the endpoints are the same 𝛾0′ (1) = 𝛾1′ (1). Corollary 2.37. Let 𝜋 ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝) be a cover. If 𝑛 ≥ 2, then 𝜋∗ ∶ 𝜋𝑛 (𝑋 ′ , 𝑝′ ) → 𝜋𝑛 (𝑋, 𝑝) is a group isomorphism. If 𝑛 = 1, 𝜋∗ is a monomorphism. In particular, 𝐻(𝑋 ′ ,𝜋) ≅ 𝜋1 (𝑋 ′ , 𝑝′ ). Proof. Let [ℎ] ∈ 𝜋𝑛 (𝑋, 𝑝), with ℎ ∶ 𝑆 𝑛 → 𝑋 a continuous map. Since 𝑆 𝑛 is simply connected, there exists a lifting ℎ′ ∶ 𝑆 𝑛 → 𝑋 ′ of ℎ, and therefore 𝜋 ∘ ℎ′ = ℎ. This shows that 𝜋∗ is onto. Let us see that 𝜋∗ is injective. This actually applies to 𝑛 ≥ 1. Take [ℎ′ ] ∈ 𝜋𝑛 (𝑋 ′ , 𝑝′ ) such that 𝜋∗ [ℎ′ ] = 𝑒. Therefore ℎ = 𝜋 ∘ ℎ′ is homotopic to

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𝑐𝑝 . By Remark 2.36, the liftings ℎ′ and 𝑐𝑝′ are also homotopic. Therefore [ℎ′ ] = 0 in 𝜋𝑛 (𝑋 ′ , 𝑝′ ). □ Theorem 2.38. There is a bijection between subgroups 𝐻 < 𝜋1 (𝑋, 𝑝) and covers 𝜋 ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝). The bijection is given by assigning the group 𝐻(𝑋 ′ ,𝜋) = 𝜋∗ (𝜋1 (𝑋 ′ , 𝑝′ )) to each such 𝜋. This says that given a subgroup 𝐻 < 𝜋1 (𝑋, 𝑝), we can construct a cover (𝑋 ′ , 𝑝′ ) so that the loops of 𝑋 that lift to loops in 𝑋 ′ are exactly those of 𝐻, and the coset classes of 𝜋1 (𝑋, 𝑝)/𝐻 act on the fibers. Note that 𝐻 = 𝐻(𝑋 ′ ,𝜋) ≅ 𝜋1 (𝑋 ′ , 𝑝′ ). The number of sheets is the index of the subgroup |𝜋1 (𝑋, 𝑝) ∶ 𝐻|. Definition 2.39. We define the universal cover 𝜋̃ ∶ (𝑋,̃ 𝑝)̃ → (𝑋, 𝑝) as the cover associated to the subgroup 𝐻 = {𝑒}. This is characterized by the fact that 𝑋̃ is simply connected. Let 𝜋̃ ∶ (𝑋,̃ 𝑝)̃ → (𝑋, 𝑝) be the universal cover. Since 𝑋̃ is simply connected, we see that for any other cover 𝜋′ ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝) there exists a unique lifting 𝑓 ∶ 𝑋̃ → 𝑋 ′ so that 𝜋′ ∘ 𝑓 = 𝜋 and 𝑓(𝑝)̃ = 𝑝′ . It is easy to see that 𝑓 is also a cover. This is the reason to call 𝑋̃ universal, since it covers any other cover of 𝑋. Remark 2.40. Let us express the last relation between subgroups of 𝜋1 and covers in the language of categories. For a pointed topological space (𝑋, 𝑝) let us define the category 𝐂𝐨𝐯(𝑋, 𝑝) whose objects are the covers 𝜋 ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝). The arrows Mor(𝜋1 , 𝜋2 ) between two covers 𝜋1 ∶ (𝑋1′ , 𝑝1′ ) → (𝑋, 𝑝) and 𝜋2 ∶ (𝑋2′ , 𝑝2′ ) → (𝑋, 𝑝) are maps 𝑓 ∶ (𝑋1′ , 𝑝1′ ) → (𝑋2′ , 𝑝2′ ) such that 𝜋2 ∘ 𝑓 = 𝜋1 . From this definition it is easy to see that the map 𝑓 is also a cover, and moreover the degree of 𝜋1 is the product of the degrees of 𝑓 and 𝜋2 . The universal cover 𝜋̃ ∶ (𝑋,̃ 𝑝)̃ → 𝑋 is the initial object in this category. In particular, it is unique up to isomorphism. We can define analogously the category 𝐂𝐨𝐯(𝑋) for non-pointed spaces. On the other hand, for a group 𝐺 we define the category 𝐒𝐮𝐛𝐠𝐫(𝐺) whose objects are subgroups 𝐻 < 𝐺 and the arrows between subgroups 𝐻1 and 𝐻2 consist of the inclusion 𝑖 ∶ 𝐻1 ↪ 𝐻2 if 𝐻1 < 𝐻2 , and ∅ otherwise. Now, for each subgroup 𝐻 < 𝜋1 (𝑋, 𝑝) we have a (unique up to isomorphism) cover 𝜋 ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝) with 𝜋∗ (𝜋1 (𝑋 ′ , 𝑝′ )) = 𝐻. For each inclusion 𝐻1 < 𝐻2 we have a unique 𝑓 ∶ (𝑋1 , 𝑝1 ) → (𝑋2 , 𝑝2 ). Therefore we obtain an isomorphism of categories (2.2)

𝐂𝐨𝐯(𝑋, 𝑝) → 𝐒𝐮𝐛𝐠𝐫(𝜋1 (𝑋, 𝑝)).

Example 2.41. • Consider 𝜛 ∶ ℝ → 𝑆 1 , 𝑡 ↦ 𝑒2𝜋i𝑡 . This is the universal cover of 𝑆 1 . Any other cover of 𝑆 1 is of the form 𝜛𝑚 ∶ 𝑆 1 → 𝑆 1 , 𝑒2𝜋i𝑡 ↦ 𝑒2𝜋i𝑚𝑡 associated with the subgroup ⟨𝑚⟩ ⊂ ℤ ≅ 𝜋1 (𝑆 1 ), 𝑚 ∈ ℤ, 𝑚 ≠ 0. From this we also conclude that 𝜋𝑛 (𝑆 1 ) ≅ 𝜋𝑛 (ℝ) = 0, for 𝑛 ≥ 2. • Consider the cover 𝜋 ∶ 𝑆 2 → ℝ𝑃 2 , (𝑥, 𝑦, 𝑧) ↦ [𝑥, 𝑦, 𝑧]. It is the universal cover of ℝ𝑃2 . Since 𝜋1 (ℝ𝑃 2 ) ≅ ℤ2 , this is the only non-trivial cover. Moreover 𝜋2 (ℝ𝑃 2 ) ≅ 𝜋2 (𝕊2 ) ≅ ℤ.

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• The product of covers is also a cover. For example, 𝜋 = 𝜛 × 𝜛′ ∶ ℝ2 → 𝑇 2 = 𝑆1 × 𝑆 1 , (𝑡1 , 𝑡2 ) ↦ (𝑒2𝜋i𝑡1 , 𝑒2𝜋i𝑡2 ) is the universal cover for the torus. As a corollary, 𝜋1 (𝑇 2 ) = ℤ2 (cf. (2.3)) and also 𝜋𝑛 (𝑇 2 ) = 0 for 𝑛 ≥ 2. The theory of covers is in close relationship with the theory of group actions. From a (nice enough) group action we will obtain a cover, and vice versa. Definition 2.42. Let 𝑋 be a topological space, and denote Homeo(𝑋) the homeomorphisms of 𝑋. Let Γ be a group which acts on 𝑋 by homeomorphisms, that is we have a group homomorphism Γ → Homeo(𝑋), 𝑔 ↦ 𝜑𝑔 ∶ 𝑋 → 𝑋. We denote 𝑔 𝑥 = 𝜑𝑔 (𝑥). We say that the action is: (1) Free. If when 𝑥 ∈ 𝑋 and 𝑔 ∈ Γ satisfy 𝑔 𝑥 = 𝑥, then 𝑔 = 𝑒 ∈ Γ (i.e., there are no fixed points). (2) Proper. For every 𝑥, 𝑦 ∈ 𝑋 there are open neighbourhoods 𝑈 𝑥 and 𝑉 𝑦 so that the set {𝑔 ∈ Γ | 𝑔 𝑈 𝑥 ∩ 𝑉 𝑦 ≠ ∅} is finite (a locally finite condition for the action).2 Remark 2.43. • For 𝑥 ∈ 𝑋, the orbit of 𝑥 is [𝑥] = {𝑔 𝑥 | 𝑔 ∈ Γ}. There is an equivalence relation 𝑥 ∼ 𝑦 if 𝑦 = 𝑔 𝑥, for some 𝑔 ∈ Γ. The quotient space 𝑋/Γ = 𝑋/∼ is endowed with the quotient topology. We denote 𝜋 ∶ 𝑋 → 𝑋/Γ the quotient map. • If 𝑋 is Hausdorff and 𝜋(𝑥) ≠ 𝜋(𝑦), then we can reduce 𝑈 𝑥 and 𝑉 𝑦 in condition (2) above so that 𝑔 𝑈 𝑥 ∩ 𝑉 𝑦 = ∅ for all 𝑔 ∈ Γ. Furthermore, we can reduce 𝑈 𝑥 so that 𝑔 𝑈 𝑥 ∩ 𝑈 𝑥 = ∅ for all 𝑔 ≠ 𝑒. Therefore the quotient map 𝑞 ∶ 𝑋 → 𝑋/Γ is a cover of degree |Γ|. • If 𝑋 is a manifold and Γ acts freely and properly on 𝑋, then 𝑋/Γ is a manifold (Exercise 2.11). • Any action of a finite group Γ on a space 𝑋 with no fixed points is automatically free and proper. This follows easily from the fact that Γ is finite. Definition 2.44. Let 𝜋 ∶ 𝑋 ′ → 𝑋 be a cover. We define the group of deck transformations Deck(𝜋) as the morphisms Mor(𝜋, 𝜋) in the category 𝐂𝐨𝐯(𝑋), i.e., the maps 𝑓 ∶ 𝑋 ′ → 𝑋 ′ so that 𝜋 ∘ 𝑓 = 𝜋. Note that we do not require such maps 𝑓 to fix a basepoint. Clearly any such 𝑓 is a cover of degree 1 and hence a homeomorphism. For any 𝑝 ∈ 𝑋 we have a map 𝑓 ∶ 𝐹𝑝 → 𝐹𝑝 which permutes the elements of the fiber of 𝑝. Therefore, for fixed 𝑝′ ∈ 𝐹𝑝 , we have a map Deck(𝜋) → 𝐹𝑝 , 𝑓 ↦ 𝑓(𝑝′ ). This map is injective. Indeed, note that 𝑓 is a lifting to 𝑋 ′ of the map 𝜋 ∶ 𝑋 ′ → 𝑋, so 𝑓 is uniquely determined once we fix the image of a basepoint 𝑝′ . We can also view this as an action of Deck(𝜋) in the fiber 𝐹𝑝 . 2 The terminology for actions is very confusing in the literature. It is customary to call an action satisfying (1) and (2) a properly discontinuous action. The condition of properness can be stated as that the map Γ × 𝑋 → 𝑋 × 𝑋, (𝑔, 𝑥) ↦ (𝑥, 𝑔 𝑥), is proper (i.e., the preimage of compact sets are compact), when 𝑋 is a locally compact Hausdorff space and Γ is discrete (i.e., it is equipped with discrete topology).

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Definition 2.45. A cover 𝜋 ∶ 𝑋 ′ → 𝑋 is called regular if Γ = Deck(𝜋) acts transitively on the fiber 𝐹𝑝 . This means that for any 𝑝1′ , 𝑝2′ ∈ 𝐹𝑝 , there exists a deck transformation 𝑓 so that 𝑓(𝑝1′ ) = 𝑝2′ . Remark 2.46. • A cover is regular if for fixed 𝑝′ ∈ 𝐹𝑝 , the map Deck(𝜋) → 𝐹𝑝 , 𝑓 ↦ 𝑓(𝑝′ ) is a bijection. • We always have |Deck(𝜋)| ≤ 𝑑 = |𝐺 ∶ 𝐻|, the number of sheets of the cover, where 𝐺 = 𝜋1 (𝑋, 𝑝) and 𝐻 = 𝜋1 (𝑋 ′ , 𝑝′ ). Here |𝐺 ∶ 𝐻| denotes the index of the subgoup 𝐻 < 𝐺, so that |𝐺| = |𝐻| ⋅ |𝐺 ∶ 𝐻|. Therefore |Deck(𝜋)| = 𝑑 if and only if 𝜋 is regular (for 𝑑 finite). • For a small open set 𝑈 𝑝 ⊂ 𝑋 so that 𝜋−1 (𝑈) = ⨆𝑖 𝑉 𝑖 , a deck transformation permutes the sets 𝑉 𝑖 . It follows from the uniqueness of liftings that Deck(𝜋) acts freely and properly. • The cover 𝜋 is regular if and only if 𝑋 ≅ 𝑋 ′ /Γ with Γ = Deck(𝜋). Certainly, as Γ acts in the fibers, the map 𝜋 ∶ 𝑋 ′ → 𝑋 descends to a map 𝑋 ′ /Γ → 𝑋, which is obviously a cover. This map is bijective if and only if Γ acts transitively in the fibers, i.e., when the cover is regular. In this case 𝑋 ′ /Γ → 𝑋 is a degree 1 cover, and hence a homeomorphism. Proposition 2.47. A cover 𝜋 ∶ (𝑋 ′ , 𝑝′ ) → (𝑋, 𝑝) is regular if and only if its associated subgroup 𝐻 = 𝜋∗ (𝜋1 (𝑋 ′ , 𝑝′ )) is a normal subgroup of 𝜋1 (𝑋, 𝑝) = 𝐺. In this case Γ = Deck(𝜋) ≅ 𝐺/𝐻. Proof. If 𝑓 ∶ (𝑋 ′ , 𝑝1′ ) → (𝑋 ′ , 𝑝2′ ) is a deck transformation, then we have 𝑓∗ (𝜋1 (𝑋 ′ , 𝑝1′ )) = 𝜋1 (𝑋 ′ , 𝑝2′ ). As 𝜋 ∘ 𝑓 = 𝜋, we see that 𝐻 = 𝜋∗ (𝜋1 (𝑋 ′ , 𝑝1′ )) = 𝜋∗ (𝜋1 (𝑋 ′ , 𝑝2′ )) = ←

𝜋∗ ([𝜂] ∗ 𝜋1 (𝑋 ′ , 𝑝1′ ) ∗ [ 𝜂]) = 𝑔𝐻𝑔−1 , where 𝜂 is a path in 𝑋 ′ from 𝑝2′ to 𝑝1′ , and 𝑔 = 𝜋∗ ([𝜂]) = [𝜋 ∘ 𝜂] is the projected loop in 𝑋. The loops 𝑔 obtained in this way give a complete set of representatives of the cosets 𝐺/𝐻, so it follows that 𝐻 is a normal subgroup. Finally, there is a bijection 𝐹𝑝 ↔ 𝐺/𝐻 and a bijection Γ ↔ 𝐹𝑝 (since the cover is regular). This gives a bijection Γ ↔ 𝐺/𝐻. This is a group homomorphism (the proof is analogous to the one in Remark 2.48 below). □ Remark 2.48. As a corollary, we obtain that the universal cover 𝜋̃ ∶ (𝑋,̃ 𝑝)̃ → (𝑋, 𝑝), which is associated with the trivial subgroup 𝐻 = {𝑒}, is a regular cover. Hence ̃ 𝑋 ≅ 𝑋/Γ, with Γ = Deck(𝜋)̃ being in bijection with the coset classes 𝜋1 (𝑋, 𝑝)/{𝑒} = 𝜋1 (𝑋, 𝑝). Indeed, the bijection (2.3)

Γ = Deck(𝜋)̃ → 𝜋1 (𝑋, 𝑝)

is a group isomorphism (and both are in bijection with 𝐹𝑝 ). The proof goes as follows: 𝑓 ∈ Γ is associated to the point 𝑓(𝑝)̃ ∈ 𝐹𝑝 , which is associated to the loop [𝛾] ∈ 𝜋1 (𝑋, 𝑝) whose lift satisfies 𝛾′ (0) = 𝑝,̃ 𝛾′ (1) = 𝑓(𝑝). ̃ Now take two elements 𝑓1 , 𝑓2 ∈ Γ and the loops [𝛾1 ], [𝛾2 ] ∈ 𝜋1 (𝑋, 𝑝) with 𝛾1′ (0) = 𝑝,̃ 𝛾1′ (1) = 𝑓1 (𝑝), ̃ 𝛾2′ (0) = 𝑝,̃ 𝛾2′ (1) = 𝑓2 (𝑝). ̃ Then 𝑓1 (𝛾2 (𝑡)) is a path from 𝑓1 (𝑝)̃ to 𝑓1 (𝑓2 (𝑝)). ̃ By uniqueness, the lift of 𝛾1 ∗ 𝛾2 is (𝛾1 ∗ 𝛾2 )′ = 𝛾1′ ∗ 𝑓1 (𝛾2′ ) and has endpoint 𝑓1 (𝑓2 (𝑝)). ̃ So it corresponds to 𝑓1 ∘ 𝑓2 , proving the claim.

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̃ Remark 2.49. The isomorphism (2.3) can be understood as 𝜋1 (𝑋/Γ) = Γ, for a simply connected space 𝑋̃ and a group Γ acting freely and properly. For example, for the circle 𝑆 1 = ℝ/ℤ it follows that 𝜋1 (𝑆 1 ) = ℤ. Example 2.50. For the torus 𝑇 2 = ℝ2 /ℤ2 , the universal cover is 𝜋 ∶ (ℝ2 , 𝑞) → (𝑆 1 × 𝑆 1 , 𝑝) with 𝑝 = (1, 1), 𝑞 = (0, 0), 𝜋(𝑡1 , 𝑡2 ) = (𝑒2𝜋i𝑡1 , 𝑒2𝜋i𝑡2 ) (cf. Example 2.41(3)). The group Γ of deck transformations consists of the maps 𝜑𝑛,𝑚 (𝑡1 , 𝑡2 ) = (𝑡1 + 𝑛, 𝑡2 + 𝑚), which clearly satisfy 𝜋 ∘ 𝜑𝑛,𝑚 = 𝜋, The fiber is 𝐹𝑝 = ℤ2 = {(𝑛, 𝑚) | 𝑛, 𝑚 ∈ ℤ} = {𝜑(0, 0) | 𝜑 ∈ Γ}. We have bijections 𝐹𝑝 → 𝜋1 (𝑇 2 , 𝑝) → Γ, which are (𝑛, 𝑚) ↦ [𝜋 ∘ 𝛾𝑛,𝑚 ] ↦ 𝜑𝑛,𝑚 , where 𝛾𝑛,𝑚 is any path in ℝ2 joining (0, 0) and (𝑛, 𝑚). Now let us study more in detail some facts on free and proper actions. Note that whenever we talk about actions on a topological space, we will denote ∼ the equivalence relation induced by the action, whose equivalence classes are the orbits. Definition 2.51. A fundamental domain of a free and proper action on a topological space 𝑋 is a closed subset 𝐷 ⊂ 𝑋 so that the following hold. (1) For all 𝑥 ∈ 𝑋 there exists 𝑦 ∈ 𝐷 with 𝑦 ∼ 𝑥. (2) If 𝑥, 𝑦 ∈ 𝐷, 𝑥 ≠ 𝑦, and 𝑥 ∼ 𝑦, then 𝑥, 𝑦 ∈ 𝜕𝐷, the topological boundary of 𝐷 ⊂ 𝑋. (3) The quotient 𝐷/∼ is homeomorphic to 𝑋/∼ . If 𝐷 is compact and 𝑋 is Hausdorff, then condition (3) follows from (1) and (2). Example 2.52. • Consider 𝑋 = 𝑆 2 and the action of Γ = ℤ2 = ⟨𝜑⟩ given by 𝜑(𝑥, 𝑦, 𝑧) = (−𝑥, −𝑦, −𝑧). Then 𝑋/Γ ≅ ℝ𝑃 2 , and a fundamental domain is a hemisphere, say 𝑆 2+ = 𝑆 2 ∩{𝑧 ≥ 0}. This fundamental domain is homeomorphic to the disc 𝐷2 , which gives the planar representation of ℝ𝑃2 with word 𝑎𝑎. The identification ∼ in 𝐷2 is the identification on the boundary given by the word. • Consider 𝑋 = ℝ2 and the action of Γ = ℤ2 given by 𝜑𝑛,𝑚 (𝑥, 𝑦) = (𝑥+𝑛, 𝑦+𝑚). We saw in Example 2.50 that 𝑋/Γ ≅ 𝑆 1 × 𝑆 1 . A fundamental domain is 𝐷 = [0, 1] × [0, 1], and 𝐷/ ∼ ≅ ℝ2 /ℤ2 . Again the equivalence relation induced in 𝐷 by the action is the same as the one given by the planar representation of the torus Σ1 . In addition, we have a tessellation of ℝ2 given by the translates of 𝐷 by all the elements of ℤ2 . • In general, when 𝐷 is a fundamental domain, we have 𝑋 = ⋃𝑔∈Γ 𝑔 𝐷, and the sets 𝑔 𝐷 tessellate 𝑋. Moreover, they only intersect along the boundaries, i.e., the sets 𝑔 Int(𝐷) are disjoint for 𝑔 ∈ Γ, where Int(𝐷) denotes the topological interior of 𝐷 ⊂ 𝑋. In the quotient 𝑋 ⋆ = 𝑋/Γ, there is a big open set 𝜋(Int(𝐷)) over which the cover 𝜋 ∶ 𝑋 → 𝑋 ⋆ trivializes. • Let 𝑓 ∶ 𝑋 → 𝑋 be a homeomorphism on a topological space. Consider the action of Γ = ℤ on 𝑋 × ℝ given by (𝑥, 𝑡) ↦ (𝑓(𝑥), 𝑡 + 1). Then the quotient (𝑋 × ℝ)/ℤ is homeomorphic to the mapping torus 𝑇 𝑓 defined in Example 1.33(3). For this it is enough to note that 𝑋 × [0, 1] is a fundamental domain for the action.

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Example 2.53. Consider 𝑋 = ℝ2 and maps 𝑔(𝑥, 𝑦) = (𝑥, 𝑦 + 1), 𝑓(𝑥, 𝑦) = (𝑥 + 1, 1 − 𝑦). Let Γ = ⟨𝑓, 𝑔⟩ ⊂ GA(ℝ2 ), the affine group of ℝ2 . Let us see that Γ acts freely and properly, find a fundamental domain, and see that the quotient 𝑋/Γ is a Klein bottle. First note that 𝑓∘𝑔(𝑥, 𝑦) = (𝑥+1, −𝑦) = 𝑔−1 ∘𝑓(𝑥, 𝑦). It follows that 𝑓∘𝑔𝑚 = 𝑔−𝑚 ∘𝑓 for all 𝑚 ∈ ℤ. We also have that 𝑓𝑛 ∘ 𝑔𝑚 = 𝑓𝑛−1 ∘ 𝑓 ∘ 𝑔𝑚 = 𝑓𝑛−1 ∘ 𝑔−𝑚 ∘ 𝑓 = ⋯ = 𝑛 𝑔(−1) 𝑚 𝑓𝑛 . So any ℎ ∈ Γ is of the form ℎ = 𝑓𝑎 ∘ 𝑔𝑏 , for some 𝑎, 𝑏 ∈ ℤ. Now take the tessellation of ℝ2 given by the squares 𝑄𝑛,𝑚 = [𝑛.𝑛 + 1] × [𝑚, 𝑚 + 1]. Then we have ℎ(𝑄𝑛,𝑚 ) = 𝑓𝑎 ∘ 𝑔𝑏 (𝑄𝑛,𝑚 ) = 𝑄𝑛′ ,𝑚′ , where (𝑛′ , 𝑚′ ) = (𝑛 + 𝑎, −𝑚 − 𝑏) if 𝑎 is odd, and (𝑛′ , 𝑚′ ) = (𝑛 + 𝑎, 𝑚 + 𝑏) if 𝑎 is even. If (𝑛, 𝑚) = (𝑛′ , 𝑚′ ), then 𝑎 = 𝑏 = 0 and ℎ = Id. This means that 𝐷 = 𝑄0,0 = [0, 1] × [0, 1] is a fundamental domain. It also follows that Γ acts properly, since a sufficiently small ball 𝐵 of ℝ2 intersects with at most four of the 𝑄𝑛,𝑚 , and therefore there are at most four elements 𝑔 of Γ which satisfy 𝑔𝐵 ∩ 𝐵 ≠ ∅. Finally, Γ acts freely. Any ℎ = 𝑓𝑎 ∘ 𝑔𝑏 ∈ Γ is of the form ℎ(𝑥, 𝑦) = (𝑥+𝑎, 𝑦+𝑏) if 𝑎 is even, and ℎ(𝑥, 𝑦) = (𝑥+𝑎, 1−𝑦−𝑏) if 𝑎 is odd. Hence it has no fixed points unless 𝑎 = Id. The equivalence relation induced by the action on 𝜕𝐷 is the same as the one given in the planar representation of the Klein bottle. We conclude that 𝐷 = 𝑄0,0 is a fundamental domain for the action, and that ℝ2 /Γ is homeomorphic to the Klein bottle Kl (see Figure 2.4).

Q1;1

Q0;1 g

a

b

D

Q2;1

b a Q1;0

a Q2;0

f

b

b

Figure 2.4. Tessellation of the universal cover of Kl .

In particular, we see that ℝ2 is the universal cover of Kl . Hence the higher homotopy groups are 𝜋𝑛 (Kl ) = 𝜋𝑛 (ℝ2 ) = 0, 𝑛 ≥ 2. The fundamental group is given by 𝜋1 (Kl ) = ⟨𝑎, 𝑏 | 𝑎𝑏𝑎𝑏−1 = 𝑒⟩. The deck transformations have a presentation Γ = ⟨𝑓, 𝑔 | 𝑓𝑔𝑓−1 𝑔 = Id⟩. To see it, take a surjective map 𝐺 = ⟨𝛼, 𝛽 | 𝛼𝛽𝛼−1 𝛽⟩ → Γ, 𝛼 ↦ 𝑓, 𝛽 ↦ 𝑔, which is well defined since 𝑓 ∘ 𝑔 = 𝑔−1 ∘ 𝑓. Using the relation in 𝐺, we see that any element 𝛾 ∈ 𝐺 is of the form 𝛾 = 𝛼𝑎 𝛽 𝑏 . If it goes to zero, then 𝑓𝑎 ∘ 𝑔𝑏 = Id, which can happen only if 𝑎 = 𝑏 = 0. Hence 𝛾 = 𝑒, so the map 𝐺 → Γ is injective, proving that Γ ≅ 𝐺, as required. The isomorphisms Γ → 𝐹𝑝 → 𝜋1 (Kl , 𝑝)

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are given as follows. The fiber over 𝑝 = 𝜋(0, 0) is 𝐹𝑝 = {(𝑛, 𝑚) | 𝑛, 𝑚 ∈ ℤ}. The element 𝑓 ↦ 𝑓(0, 0) = (1, 1) ↦ 𝑎𝑏, 𝑔 ↦ 𝑔(0, 0) = (0, 1) ↦ 𝑎. In general ℎ = 𝑓𝑛 ∘ 𝑔𝑚 ↦ ℎ(0, 0), which is equal to (𝑛, 𝑚) if 𝑛 is even, and (𝑛, 1 − 𝑚) if 𝑛 is odd. This is the endpoint of the lifting of the loop (𝑎𝑏)𝑛 𝑎𝑚 , which is equal to 𝑏𝑛 𝑎𝑚 if 𝑛 is even, and 𝑏𝑛 𝑎𝑚−1 if 𝑛 is odd. Remark 2.54. The universal cover of Σ𝑔 for 𝑔 ≥ 2 is also homeomorphic to ℝ2 , but this is difficult to see at first sight since there is no obvious way to tessellate ℝ2 with 4𝑔-gons. This will be done in the hyperbolic plane in Corollary 4.83. As a corollary, 𝜋𝑛 (Σ𝑔 ) = 0 for 𝑛 ≥ 2, 𝑔 ≥ 2. Euler-Poincaré characteristic and covers. Let (𝑋, 𝜏) be a compact triangulated space with triangulation 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 ) | 0 ≤ 𝑘 ≤ 𝑁, 𝛼 ∈ Λ𝑘 }. Consider a cover 𝜋 ∶ 𝑋 ′ → 𝑋 of finite degree 𝑑. Then any 𝑓𝛼 ∶ 𝑃𝛼𝑘 → 𝑋 lifts to 𝑑 different maps ′ 𝑓𝛼,𝑖 ∶ 𝑃𝛼𝑘 → 𝑋 ′ , 1 ≤ 𝑖 ≤ 𝑑 (the maps lift since 𝑃𝛼𝑘 is contractible). Then the collec′ tion 𝜏′ = {(𝑃𝛼𝑘 , 𝑓𝛼,𝑖 ) | 0 ≤ 𝑘 ≤ 𝑁, 1 ≤ 𝑖 ≤ 𝑑, 𝛼 ∈ Λ𝑘 } gives a triangulation of 𝑋 ′ . Let 𝑛𝑘 = |Λ𝑘 | be the number of 𝑘-polyhedra of 𝑋. Then the number of 𝑘-polyhedra of 𝑋 ′ is 𝑛′𝑘 = 𝑑|Λ𝑘 | = 𝑑𝑛𝑘 . So 𝜒(𝑋 ′ ) = ∑(−1)𝑘 𝑛′𝑘 = 𝑑 ∑(−1)𝑘 𝑛𝑘 = 𝑑 𝜒(𝑋).

2.2.2. Oriented cover. Now we discuss a special type of cover which controls the orientability of a topological manifold. Definition 2.55. Let 𝑀 be a (topological, PL, or smooth) manifold. The oriented cover of 𝑀 is the cover 𝜋̂ ∶ 𝑀̂ → 𝑀 defined by 𝑀̂ = {(𝑝, 𝑜) | 𝑝 ∈ 𝑀, 𝑜 ∈ Or(𝑝)}, where Or(𝑝) is the set of orientations at the point 𝑝 (see section 1.4). The map 𝜋̂ is 𝜋(𝑝, ̂ 𝑜) = 𝑝, and it is clear that the fiber 𝜋̂−1 (𝑝) = {(𝑝, 𝑜), (𝑝, −𝑜)} has two points. We endow the set 𝑀̂ with a topology as follows. Each 𝑝 ∈ 𝑀 has a neighbourhood 𝑈 = 𝑈 𝑝 homeomorphic to a ball of ℝ𝑛 , so 𝑈 is orientable and connected. Therefore if we choose an orientation 𝑜𝑝 on 𝑝, then 𝑜𝑝 naturally induces a unique orientation 𝑜𝑞 on all points 𝑞 ∈ 𝑈. We declare the set 𝑈1̂ = {(𝑞, 𝑜𝑞 )|𝑞 ∈ 𝑈} ⊂ 𝑀̂ as an open set, and hence an open neighbourhood of (𝑝, 𝑜𝑝 ) ∈ 𝑀.̂ If we take the opposite orientation −𝑜𝑝 and repeat the construction, we get another set 𝑈2̂ = {(𝑞, −𝑜𝑞 )|𝑞 ∈ 𝑈}, and clearly 𝑈1̂ ∩ 𝑈2̂ = ∅. We also have 𝜋−1 (𝑈) = 𝑈1̂ ⊔ 𝑈2̂ . Then we take on 𝑀̂ the topology generated by the sets 𝑈1̂ , 𝑈2̂ , where 𝑈 is as above. With this topology, 𝜋̂ ∶ 𝑀̂ → 𝑀 is a cover of degree 2. This implies that 𝑀̂ is a manifold (Exercise 2.11). Now we claim that 𝑀̂ has a canonical orientation. To define it, for 𝑝 ̂ = (𝑝, 𝑜𝑝 ) ∈ 𝑀,̂ we set 𝑜𝑝̂ ̂ = (𝜋|−1 ) (𝑜 ), for 𝑈1̂ a small neighbourhood of 𝑝 ̂ as above, translating the 𝑈̂ 1 ∗ 𝑝 orientation of 𝑜𝑝 via the homeomorphism 𝜋|𝑈̂ 1 ∶ 𝑈1̂ → 𝑈. Then 𝑜𝑝̂ ̂ is an orientation at 𝑝,̂ which is defined in an intrinsic way. The fact that such orientations vary continuously as we vary the points 𝑝 ̂ ∈ 𝑀̂ follows from the very definition of the topology of 𝑀.̂

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Remark 2.56. • If 𝑀 is orientable, then giving an orientation of 𝑀 is equivalent to giving a continuous section 𝑠 ∶ 𝑀 → 𝑀,̂ i.e., a continuous map so that 𝑠 ∘ 𝜋 = Id𝑀 . Therefore, 𝑀 is orientable if and only if there exists such section. • There is a map 𝐴 ∶ 𝑀̂ → 𝑀,̂ given by 𝐴(𝑝, 𝑜) = (𝑝, −𝑜). Clearly 𝐴 is a deck ̂ transformation and 𝑀 = 𝑀/{Id, 𝐴}. This map 𝐴 reverses the orientation of ̂ 𝑀. • Let 𝜋 ∶ 𝑀̂ → 𝑀 be the oriented cover of a connected manifold 𝑀. Then 𝑀 is orientable if and only if 𝑀̂ is not connected. Certainly, if 𝑀 is orientable, then there is a section 𝑠 ∶ 𝑀 → 𝑀.̂ The image 𝑠(𝑀) is an open subset, 𝐴(𝑠(𝑀)) is also open and 𝑀̂ = 𝑠(𝑀) ⊔ 𝐴(𝑠(𝑀)), so 𝑀̂ is not connected. Conversely, if 𝑀̂ is not connected, then take 𝐶 to be one of its connected components. As 𝑀̂ is locally connected, then 𝜋|𝐶 ∶ 𝐶 → 𝑀 is also a cover. As 𝜋 is a cover of degree 2, 𝐶 → 𝑀 must be a one-sheeted cover, hence a homeomorphism, and so there are exactly two connected components 𝑀̂ 1 and 𝑀̂ 2 of 𝑀,̂ and both give homeomorphisms 𝜋|𝑀̂ 𝑖 ∶ 𝑀̂ 𝑖 → 𝑀. We define an orientation for 𝑀 by using any of the homeomorphisms with 𝑀̂ 1 or 𝑀̂ 2 . Therefore 𝑀 is orientable. • Let 𝛾 ∶ [0, 1] → 𝑀 be a loop in 𝑀. It lifts to a path 𝛾 ̂ ∶ [0, 1] → 𝑀̂ in the oriented cover. Let us note that 𝛾(𝑡) ̂ = (𝛾(𝑡), 𝑜(𝑡)) consists of points of 𝑀 with orientations on them varying continuously. Then the loop 𝛾 preserves the orientation if 𝑜(1) = 𝑜(0), that is, if 𝛾 ̂ is a loop. • If 𝑀 is non-orientable, then the cover 𝜋 ∶ 𝑀̂ → 𝑀 is connected and of degree 2. The subgroup of the cover is 𝐻̂ = {[𝛾] ∈ 𝜋1 (𝑀, 𝑝) | 𝛾 preserves the orientation}. This is a very specific index 2 subgroup. • Any index 2 subgroup is normal, hence the cover 𝑀̂ → 𝑀 is regular. Its group of deck transformations is Deck(𝜋)̂ = {Id, 𝐴}. • Note that if 𝑀 is a manifold such that 𝜋1 (𝑀, 𝑝) does not admit index two subgroups, then it is automatically orientable. In particular, any simply connected manifold is orientable. • If 𝑀 is non-orientable and 𝜋 ∶ 𝑀 ′ → 𝑀 is a 2-sheeted cover with 𝑀 ′ orientable, then 𝑀 ′ ≅ 𝑀.̂ • Let (𝑀, 𝑜) be an oriented manifold. Recall that a homeomorphism 𝑓 ∶ 𝑀 → 𝑀 preserves the orientation at a point 𝑝 if (𝑓|𝑈 )∗ (𝑜𝑝 ) = 𝑜𝑓(𝑝) , for 𝑈 𝑝 a neighbourhood of 𝑝 homeomorphic to a ball, and 𝑓|𝑈 ∶ 𝑈 → 𝑓(𝑈). It reverses the orientation at 𝑝 if (𝑓|𝑈 )∗ (𝑜𝑝 ) = −𝑜𝑓(𝑝) . Note that the set 𝑉± = {𝑝 ∈ 𝑀| (𝑓|𝑈 )∗ (𝑜𝑝 ) = ±𝑜𝑓(𝑝) } are both open, disjoint, and cover 𝑀. Hence, if 𝑀 connected, then either 𝑉+ = 𝑀 and we say that 𝑓 preserves orientation, or 𝑉− = 𝑀 and we say that 𝑓 reverses orientation. Finally, note that this condition on 𝑓 does not depend on the choice of orientation 𝑜 of 𝑀. Now let us discuss how the oriented cover and the universal cover are related. Let 𝑀 be a connected manifold, let 𝜋̃ ∶ 𝑀̃ → 𝑀 be the universal cover, and let Γ be the

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deck transformation group of 𝑀.̃ Since 𝑀̃ is simply connected, it is orientable. Fix an orientation 𝑜 ̃ for 𝑀.̃ ̃ denotes the homeomorphisms of 𝑀.̃ Consider the sets Recall that Homeo(𝑀) + ̃ of homeomorphisms that respect the orientation 𝑜,̃ and Homeo− (𝑀) ̃ of Homeo (𝑀) homeomorphisms that reverse it. There are two possibilities: (a) There are no homeomorphisms of 𝑀̃ which reverse the orientation, so ̃ = Homeo+ (𝑀). ̃ Homeo(𝑀) ̃ ≠ ∅. Then we have a short exact sequence (b) Otherwise, Homeo− (𝑀) ̃ → Homeo(𝑀) ̃ → ℤ2 → 0. 0 → Homeo+ (𝑀) The first map is the inclusion and the second assigns ±1 according to whether the homeomorphism preserves or reverses the orientation. In particular ̃ ∶ Homeo+ (𝑀)| ̃ = 2. In this case, taking 𝑓− ∈ Homeo− (𝑀) ̃ ≠ ∅, |Homeo(𝑀) + − ̃ → Homeo (𝑀), ̃ 𝜑 ↦ 𝑓− ∘ 𝜑 and a disjoint we have a bijection Homeo (𝑀) ̃ = Homeo+ (𝑀) ̃ ⊔ Homeo− (𝑀). ̃ decomposition Homeo(𝑀) Remark 2.57. There are simply connected manifolds 𝑀 with Homeo(𝑀) = Homeo+ (𝑀) (Exercise 5.19) and simply connected manifolds 𝑀 that have Homeo− (𝑀) ≠ ∅ (e.g. 𝑀 = Σ𝑔 ). ̃ which acts freely and properly. Let Γ+ = Now, let Γ = Deck(𝜋)̃ < Homeo(𝑀), + − ̃ and Γ− = Γ ∩ Homeo (𝑀). ̃ We also have two cases: Γ ∩ Homeo (𝑀) (1) If Γ = Γ+ , then all elements of Γ preserve the orientation. This implies that ̃ is orientable. Indeed, for 𝑈 = 𝑈 𝑝 a small neighbourhood of 𝑝 ∈ 𝑀, 𝑀 = 𝑀/Γ define the orientation 𝑜𝑈 = (𝜋|̃ 𝑈̃ 𝑖 )∗ (𝑜|̃ 𝑈̃ 𝑖 ), being 𝜋̃−1 (𝑈) = ⨆𝑗∈𝐽 𝑈̃ 𝑗 , and 𝑖 ∈ 𝐽 arbitrary. The choice of a preimage 𝑈̃ 𝑖 of 𝑈 does not matter. Indeed, for any other 𝑈̃ 𝑘 there exists a 𝜑 ∈ Γ with 𝜑(𝑈̃ 𝑘 ) = 𝑈̃ 𝑖 , and 𝜑∗ (𝑜|̃ 𝑈̃ 𝑘 ) = 𝑜|̃ 𝑈̃ 𝑖 since Γ preserves orientation. Noting that 𝜋|𝑈̃ 𝑘 = 𝜋|𝑈̃ 𝑖 ∘ 𝜑|𝑈̃ 𝑘 , it follows that the orientation 𝑜𝑈 is well defined. (2) If Γ− ≠ ∅, then for 𝑓− ∈ Γ− , we have a bijection Γ+ → Γ− and a short exact sequence 0 → Γ+ → Γ → ℤ2 → 0 as before, and |Γ ∶ Γ+ | = 2. The ̃ + is oriented, because all homeomorphisms of Γ+ preserve quotient 𝑀̂ = 𝑀/Γ ̃ + → 𝑀 = 𝑀/Γ ̃ is the oriented cover, orientation. So the map 𝜋̂ ∶ 𝑀̂ = 𝑀/Γ since it is a double cover because Γ+ < Γ is an index 2 subgroup. Moreover, 𝑓− descends to a map 𝐴 ∶ 𝑀̂ → 𝑀̂ which must be the deck transformation of 𝑀.̂ It is orientation reversing, and Γ/Γ+ ≅ ℤ2 = {Id, 𝐴}. Example 2.58. (1) Take 𝑀 = Mob the Möbius band. Its universal cover is 𝑀̃ = ℝ × [0, 1], and ̃ = ⟨𝜑⟩ ≅ ℤ, 𝜑 ∶ 𝑀̃ → 𝑀̃ given by 𝜑(𝑥, 𝑦) = (𝑥 + 1, 1 − 𝑦). It is Γ = Deck(𝑀) easy to see that the fundamental domain for the action of Γ is 𝐷 = [0, 1]×[0, 1] and 𝐷/ ∼ is the usual planar representation for Mob, with associated word 𝑎𝑏𝑎𝑐. Also note that 𝜑 reverses the orientation of 𝑀.̃

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It follows that Γ+ = ⟨𝜑2 ⟩, being 𝜑2 (𝑥, 𝑦) = (𝑥 + 2, 𝑦). Therefore the ̃ + , with fundamental domain 𝐷̂ = [0, 2] × [0, 1], oriented cover is 𝑀̂ = 𝑀/Γ ̂ is a cylinder. and therefore 𝑀̂ ≅ 𝐷/∼ (2) Take 𝑀 = Kl the Klein bottle, with 𝑀̃ = ℝ2 , and Γ = ⟨𝑓, 𝑔⟩, with 𝑓(𝑥, 𝑦) = (𝑥 + 1, 1 − 𝑦), 𝑔(𝑥, 𝑦) = (𝑥, 𝑦 + 1). Clearly 𝑓 reverses orientation so Γ+ = ⟨𝑓2 , 𝑔⟩ ≅ ℤ × ℤ. By Example 2.53, a fundamental domain for the action of Γ is 𝐷 = [0, 1] × [0, 1]. Since 𝑓2 (𝑥, 𝑦) = (𝑥 + 2, 𝑦), a fundamental domain 𝐷̂ for Γ+ is 𝐷̂ = [0, 2] × [0, 1]. Therefore the oriented cover for Kl is the torus Σ1 . So we have a degree 2 cover 𝜋 ∶ Σ1 → Kl . The map 𝜋 can be seen as the quotient of the fundamental domain 𝐷̂ of Σ1 by the action that identifies (𝑥, 𝑦) ∈ 𝐷 ⊂ 𝐷̂ with 𝑓(𝑥, 𝑦) ∈ [1, 2] × [0, 1] ⊂ 𝐷.̂ Associated to the fundamental domain 𝐷 of Kl we have the word 𝑎𝑏𝑎𝑏−1 . Then 𝐻̂ = 𝜋∗ (𝜋1 (𝑇 2 )) ≅ ℤ2 , which corresponds to the group Γ+ of deck transformations. So 𝐻̂ = ⟨𝑎, 𝑏2 | 𝑎𝑏2 = 𝑏2 𝑎⟩ < 𝜋1 (Kl ) = ⟨𝑎, 𝑏 | 𝑎𝑏𝑎𝑏−1 ⟩. Note that 𝑎𝑏2 = 𝑏𝑎−1 𝑏 = 𝑏2 𝑎. If we see the torus smoothly embedded in ℝ3 , being symmetric with respect to 0 and the coordinate planes, the map 𝜋 is the quotient that identifies (𝑥, 𝑦, 𝑧) with (−𝑥, −𝑦, −𝑧) (see Figure 2.5).

T2

Kl π

Figure 2.5. Oriented cover of the Klein bottle.

Remark 2.59. • For the orientable surfaces Σ𝑔 there always exists a fixed point free homeomorphism reversing the orientation. It is a modification of the one in Remark 1.81. Take a model of Σ𝑔 smoothly embedded in ℝ3 symmetric with respect to the coordinate planes (Figure 1.14). Then the map 𝑓(𝑥, 𝑦, 𝑧) = (−𝑥, −𝑦, −𝑧) is a homeomorphism of ℝ3 which reverses the orientation of both ℝ3 and Σ𝑔 and 𝑓2 = Id. The map 𝑓 ∶ Σ𝑔 → Σ𝑔 has no fixed points, so 𝑆 = Σ𝑔 /⟨𝑓⟩ is a non-orientable surface. Computing the Euler-Poincaré characteristic, 2𝜒(𝑆) = 𝜒(Σ𝑔 ) = 2 − 2𝑔, so 𝜒(𝑆) = 1 − 𝑔 = 2 − (𝑔 + 1), and thus 𝑆 = 𝑋𝑔+1 . • Therefore, the oriented cover of 𝑋𝑘 is Σ𝑘−1 , for 𝑘 ≥ 1. In particular, the oriented cover of ℝ𝑃 2 is 𝑆2 , and the oriented cover of the Klein bottle is the torus. Note that this implies that there is an exact sequence 0 → 𝜋1 (Σ𝑘−1 ) → 𝜋1 (𝑋𝑘 ) → ℤ2 → 0 which is not algebraically obvious. Also 𝜋𝑛 (𝑋𝑘 ) = 𝜋𝑛 (Σ𝑘−1 ) = 0, for 𝑘 ≥ 2, 𝑛 ≥ 2, by Remark 2.54.

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Σk−1

Xk

Figure 2.6. Oriented cover of 𝑋𝑘 .

• There is a geometric way to see the cover Σ𝑘−1 → 𝑋𝑘 . Take 𝑘 copies of ℝ𝑃2 and the 𝑘 oriented covers 𝑆 2 → ℝ𝑃 2 . Perform the connected sum on the projective planes, which corresponds to joining each 2-sphere with the next one via two tubes (Figure 2.6). The resulting surface is Σ𝑘−1 . 2.2.3. Ramified covers. We define here the notion of ramified covers, which is specific for surfaces (at least in this form). Definition 2.60. Let 𝑆 and 𝑆 ′ be two surfaces. Let 𝜋 ∶ 𝑆 ′ → 𝑆 be a continuous and surjective map. The map 𝜋 is a ramified cover if for all 𝑝 ∈ 𝑆 there exists a chart (𝑈 𝑝 , 𝜑), 𝜑(𝑝) = 0, so that 𝜋−1 (𝑈) = ⨆𝑖∈𝐼 𝑉 𝑖 , and for all 𝑖 ∈ 𝐼, there exists a chart (𝑉 𝑖 , 𝜙𝑖 ) such that the following diagram commutes, 𝑉𝑖 𝜋|𝑉𝑖

𝜙𝑖

/ 𝐵1 (0) 𝜛

 𝑈

𝜑

 / 𝐵1 (0)

where the map 𝜛 has the form 𝜛(𝑟𝑒i𝜃 ) = 𝑟𝑒i𝑚𝑖 𝜃 for some non-zero integer 𝑚𝑖 > 0.

q

$

p

Figure 2.7. Local model of a ramified cover. The point 𝑞 is a ramification point of index 𝑚𝑞 = 8.

2.2. Covers

81

Remark 2.61. • If 𝑚𝑖 < 0, we can change the chart 𝜙𝑖 by 𝑟 ∘ 𝜙𝑖 , where 𝑟(𝑥, 𝑦) = (𝑥, −𝑦). This makes 𝑚𝑖 > 0. We shall assume then that all 𝑚𝑖 are positive. • If 𝑚𝑖 = 1 for all 𝑖, then 𝜋 is a cover. Therefore a cover is a ramified cover. • We say that a point 𝑞 ∈ 𝑆 ′ has ramification index 𝑚 ∈ ℤ if there exists an open neighbourhood 𝑉 𝑖 = 𝑉 𝑞 and charts as in the above definition so that 𝜙𝑖 (𝑞) = 0 and 𝜛(𝑟𝑒i𝜃 ) = 𝑟𝑒i𝑚𝜃 . We denote 𝑚 = 𝑚𝑞 the ramification index of 𝑞 (see Figure 2.7). • If 𝑞 has ramification index 𝑚𝑞 > 1, then for all 𝑞′ in some neighbourhood 𝑉 𝑞 we have 𝑚𝑞′ = 1. This follows from the fact the 𝜛 is a local homeomorphism away from 0 ∈ 𝐵1 (0). Therefore if we denote 𝑅 = {𝑞 ∈ 𝑆 ′ | 𝑚𝑞 > 1} the set of ramification points, 𝑅 is a discrete set of 𝑆 ′ . We call the set 𝜋(𝑅) ⊂ 𝑆 the ramification values, and 𝑆 0 = 𝑆 − 𝜋(𝑅) the regular values. • Given 𝑞 ∈ 𝑆 ′ , the number 𝑚𝑞 > 0 is well defined, i.e., it is independent of the charts. Indeed, we have 𝑚𝑞 = |𝜋−1 (𝑝′ ) ∩ 𝑉|, where 𝑉 𝑞 , 𝑈 𝑝 are neighbourhoods as given in Definition 2.60, with 𝑝 = 𝜋(𝑞), 𝜋 ∶ 𝑉 → 𝑈, for any 𝑝′ ∈ 𝑈 − {𝑝}. • If 𝑆 and 𝑆 ′ are compact, then as 𝑅 ⊂ 𝑆 ′ is discrete, it is finite, so 𝜋(𝑅) is also finite. Since the fibers of 𝜋 are discrete sets, we see that 𝜋−1 (𝜋(𝑅)) is also ′ finite, and it follows that 𝜋0 = 𝜋|𝑆′ 0 ∶ 𝑆 0 = 𝑆 ′ − 𝜋−1 (𝜋(𝑅)) → 𝑆 0 is an ordinary cover of degree 𝑙, with 𝑙 the cardinal of the fiber 𝐹𝑥 for any regular ′ value 𝑥 ∈ 𝑆 0 . Note that 𝑆 0 and 𝑆 0 are connected. • If 𝑝 ∈ 𝜋(𝑅) is a ramification value, take 𝜋−1 (𝑝) = {𝑞1 , . . . , 𝑞𝑘 } and 𝜋−1 (𝑈 𝑝 ) = 𝑘 ⨆𝑖=1 𝑉 𝑖 as in the definition. A point 𝑥 ∈ 𝑈−{𝑝} has 𝑚𝑞1 +⋯+𝑚𝑞𝑘 preimages. Therefore the degree 𝑙 of the cover 𝜋0 above is 𝑙 = 𝑚𝑞1 +⋯+𝑚𝑞𝑘 . We conclude that the sum of ramification indices is constant along the fibers of a ramified cover. Note that |𝐹𝑝 | = 𝑙 if 𝑝 ∉ 𝜋(𝑅) and |𝐹𝑝 | < 𝑙 if 𝑝 ∈ 𝜋(𝑅). • If we change the chart 𝜑 in Definition 2.60 by 𝜑̄ = ℎ ∘ 𝜑 ∶ 𝑈 𝑝 → 𝐵1 (0), where ℎ ∶ 𝐵1 (0) → 𝐵1 (0) is the homeomorphism ℎ(𝑟𝑒i𝜃 ) = 𝑟𝑚 𝑒i𝜃 , then the map 𝜋 is given in charts by 𝜛̄ ∶ 𝐵1 (0) → 𝐵1 (0), 𝜛(𝑟𝑒 ̄ i𝜃 ) = 𝑟𝑚 𝑒i𝑚𝜃 , that is, in complex 𝑚 coordinates, 𝜛(𝑧) ̄ = 𝑧 . Note that now 𝜛̄ is differentiable (although with zero differential at the origin). Definition 2.62. Let 𝑀 be an 𝑛-manifold, and let Γ < Homeo(𝑀) be a subgroup of homeomorphisms acting properly on 𝑀. We say that the action of Γ is locally linearizable if for all 𝑝 ∈ 𝑀 there is a chart 𝜑 ∶ 𝑈 𝑝 → 𝜑(𝑈) ⊂ ℝ𝑛 , 𝜑(𝑝) = 0, and an inclusion 𝚤 ∶ Γ𝑝 → GL(𝑛, ℝ), where Γ𝑝 = {𝑔 ∈ 𝐺|𝑔 𝑝 = 𝑝} is the isotropy group at 𝑝, satisfying that 𝜑(𝑔 𝑞) = 𝚤(𝑔)(𝜑(𝑞)), for 𝑞 ∈ 𝑈 and 𝑔 ∈ Γ𝑝 . For example, if 𝑀 is a smooth manifold and Γ is a finite group acting by diffeomorphisms, then the action is locally linearizable (Exercise 3.24). From now on, we assume all actions to be locally linearizable. Example 2.63. Let 𝑆 ′ be a compact oriented surface, and let Γ be a finite group which acts on 𝑆 ′ preserving the orientation. Consider the quotient map 𝜋 ∶ 𝑆 ′ → 𝑆 = 𝑆 ′ /Γ.

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We saw before that if Γ acts freely and properly, then 𝜋 is a cover. Let us see that if Γ has fixed points, then 𝑆 is a topological surface and 𝜋 is a ramified cover. Take 𝑝 ∈ 𝑆 ′ a fixed point of some element 𝑔 ∈ Γ, 𝑔 ≠ 𝑒. This means that the isotropy Γ𝑝 = {ℎ ∈ 𝐺|ℎ 𝑝 = 𝑝} < Γ is non-trivial. The group Γ𝑝 is finite since it is a subgroup of the finite group Γ. As the action is locally linearizable and preserves + the orientation, we can take charts so that Γ𝑝 < GL (2, ℝ). Moreover, we can take a 1 scalar product 𝑄 on ℝ2 and average it, that is, considering 𝑄0 = |Γ | ∑ 𝑔∗ 𝑄. Then 𝑝

𝑔∈Γ𝑝

Γ𝑝 preserves 𝑄0 , and hence Γ𝑝 < SO(2). Therefore Γ𝑝 is cyclic, so it has a generator Γ𝑝 = ⟨𝑔⟩, where 𝑔 is a rotation of order 𝑚 = |Γ𝑝 |. So we have a commutative diagram 𝑉𝑝

𝑔

≅

 𝐵𝜀 (0)

/ 𝑉𝑝 ≅

𝜌

 / 𝐵𝜀 (0)

for some neighbourhood 𝑉 𝑝 of 𝑝, and 𝜌(𝑟𝑒i𝜃 ) = 𝑟𝑒i(𝜃+2𝜋/𝑚) . Let 𝑘 = |Γ ∶ Γ𝑝 |, so 𝑁 = |Γ| = 𝑚𝑘, and [𝑝] = {𝑔 𝑝 | 𝑔 ∈ Γ} = {𝑝 = 𝑝1 , . . . , 𝑝 𝑘 }. Take representatives 𝑔𝑖 of the coset classes of Γ/Γ𝑝 , 𝑖 = 1, . . . , 𝑘, and denote 𝑝 𝑖 = 𝑔𝑖 𝑝. 𝑘

Reducing 𝑉 𝑝 if necessary, we have 𝑉 ̃ = ⨆𝑖=1 𝑔𝑖 (𝑉 𝑝 ) is a disjoint union. Note that ′ 𝑝 ̃ ̃ Γ𝑝𝑖 = ⟨𝑔𝑖 ∘ 𝑔 ∘ 𝑔−1 𝑖 ⟩, so 𝑉 is an open and saturated subset of 𝑆 . Finally, 𝑉/Γ ≅ 𝑉 /Γ𝑝 ≅ i𝜃 i𝑚𝜃 𝐵𝜀 (0)/⟨𝜌⟩. The map 𝜛(𝑟𝑒 ) = 𝑟𝑒 descends to a homeomorphism 𝐵𝜀 (0)/⟨𝜌⟩ ≅ 𝐵𝜀 (0), giving a chart 𝑉 𝑝 /Γ𝑝 ≅ 𝐵𝜀 (0). This implies that 𝑆 ′ is a topological surface around 𝜋(𝑝). Also there is a commutative diagram 𝑉𝑝

≅

/ 𝐵𝜀 (0)

≅

 / 𝐵𝜀 (0)

𝜋

 𝑉 𝑝 /Γ𝑝

𝜛

which shows that the quotient map 𝜋 is a ramified cover from 𝑆 ′ to 𝑆 = 𝑆 ′ /Γ. Remark 2.64. The quotient 𝑆 = 𝑆 ′ /Γ in Example 2.63 is a topological surface. However, it is not true that if 𝑆 ′ is a smooth surface and Γ acts by diffeomorphisms, then 𝑆 = 𝑆 ′ /Γ has the structure of a smooth manifold. What it is true in general and for any dimension is that if a finite group Γ acts in a manifold 𝑀, then the quotient comes equipped with a so-called orbifold structure (see Definition 3.72). Remark 2.65. The argument in Example 2.63 can be carried out for an action of a group Γ, not necessarily finite, which acts properly (Definition 2.42), is locally linearizable, and preserves orientation. Example 2.66. Consider the torus 𝑆 ′ = ℝ2 /ℤ2 . Let 𝑎(𝑥, 𝑦) = (−𝑥, −𝑦), Γ = ⟨𝑎⟩ ≅ ℤ2 , which acts in 𝑆 ′ and gives the quotient 𝑆 = 𝑆 ′ /⟨𝑎⟩ = ℝ2 /(⟨𝑎⟩ + ℤ2 ). Let us find the fixed points of the action of Γ in 𝑆 ′ . If 𝑎(𝑥, 𝑦) = (𝑥 + 𝑛, 𝑦 + 𝑚) for some 𝑛, 𝑚 ∈ ℤ, 1 then −2𝑥 = 𝑛 and −2𝑦 = 𝑚, so (𝑥, 𝑦) ∈ 2 ℤ2 . We conclude that the only fixed points 1

1

in the fundamental domain 𝐷′ = [0, 1]2 of 𝑆 ′ are 𝑝 = (0, 0), 𝑞 = (0, 2 ), 𝑟 = ( 2 , 0) and

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1 1

𝑠 = ( 2 , 2 ). It is easy to see that the fundamental domain for 𝑆 is 𝐷 = 𝐷′∩{(𝑥, 𝑦) | 𝑥+𝑦 ≤ 1}, and 𝐷 has the associated word 𝛼𝛼−1 𝛾𝛾−1 𝛽𝛽 −1 , so 𝑆 = 𝐷/∼ ≅ 𝑆 2 . If we take as model for 𝑇 2 the torus of revolution in ℝ3 , then the action of Γ becomes the map given by 𝑎(𝑥, 𝑦, 𝑧) = (𝑥, −𝑦, −𝑧), shown in Figure 2.8. This preserves the orientation and has four fixed points. The surface 𝑆 = 𝑇 2 /Γ is called the pillowcase.

a p α

r

β α

p

q

r

α

β

s q

β

s

Figure 2.8. The pillowcase obtained as the quotient of the torus by an involution.

We conclude that there exists 𝜋 ∶ 𝑇 2 → 𝑆 2 a ramified cover of degree 2 with four ramification points, which in particular yields an ordinary cover 𝜋 ∶ 𝑇 2 − {4 points} → 𝑆 2 − {4 points}. Euler-Poincaré characteristic for ramified covers. Let 𝜋 ∶ 𝑆 ′ → 𝑆 be a ramified cover between compact surfaces. Let us take a triangulation 𝜏 of 𝑆 well adapted for 𝜋. Precisely, we require that the ramification values 𝜋(𝑅) are vertices of 𝜏, and that the edges that meet points 𝑝 ∈ 𝜋(𝑅) are radii of the balls 𝐵1 (0) appearing in the definition of ramified cover. We form a triangulation 𝜏′ of 𝑆 ′ by lifting via 𝜋 the triangulation 𝜏 of 𝑆. As 𝜋|𝑆′0 ∶ 𝑆 ′0 = 𝑆 − 𝜋−1 (𝜋(𝑅)) → 𝑆 0 = 𝑆 − 𝜋(𝑅) is an ordinary cover of some degree, say 𝑙, we lift the triangulation of 𝜏 in 𝑆 0 = 𝑆 − 𝜋(𝑅) to a triangulation of 𝑆 ′0 , and we add vertices at the points of 𝜋−1 (𝜋(𝑅)). If we denote 𝑣, 𝑒, 𝑓 the number of vertices, edges, and faces of 𝑆, and 𝑣′ , 𝑒′ , 𝑓′ the corresponding numbers of 𝑆 ′ , then 𝑓′ = 𝑙𝑓,

𝑒′ = 𝑙𝑒,

𝑣′ = 𝑙𝑣 − ∑ (𝑚𝑞 − 1). 𝑞∈𝑅

The last equality follows from the fact that each ramification point 𝑞 would appear 𝑚𝑞 times if 𝜋 were an ordinary cover, but it appears just once, giving a deficit of 𝑚𝑞 − 1 vertices. We thus obtain the formula (2.4)

𝜒(𝑆 ′ ) = 𝑙𝜒(𝑆) − ∑ (𝑚𝑞 − 1). 𝑞∈𝑅

In particular, for an ordinary cover of degree 𝑙 we recover the previous formula 𝜒(𝑆 ′ ) = 𝑙 𝜒(𝑆).

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Example 2.67. • For the degree 2 ramified cover 𝜋 ∶ 𝑇 2 → 𝑆 in Example 2.66, we have 0 = 𝜒(𝑇 2 ) = 2𝜒(𝑆) − (1 + 1 + 1 + 1), since all the ramification indices are 2. It follows that 𝜒(𝑆) = 2, and this is another way to prove that 𝑆 is homeomorphic to 𝑆 2 . Note that 𝑆 is orientable since 𝜋 is obtained as the quotient by a group Γ which acts on 𝑇 2 preserving the orientation. • If we have a ramified cover coming from a group action 𝜋 ∶ 𝑆 ′ → 𝑆 = 𝑆 ′ /Γ, and |Γ| = 𝑑, then 𝜒(𝑆 ′ ) = 𝑑𝜒(𝑆) − ∑

∑

(𝑚𝑞 − 1) = 𝑑𝜒(𝑆) − ∑

𝑝∈𝜋(𝑅) 𝑞∈𝜋−1 (𝑝)

= 𝑑 (𝜒(𝑆) − ∑ (1 − 𝑝∈𝑅

with

𝑑 𝑚𝑞

𝑞∈𝑅

𝑑 (𝑚 − 1) 𝑚𝑞 𝑞

1 )) , 𝑚𝑞

= |Γ ∶ Γ𝑞 | = |𝜋−1 (𝑝)|. Note that in this special case we have 𝑚𝑞1 =

𝑚𝑞2 for any 𝑞1 , 𝑞2 ∈ 𝜋−1 (𝑝). This follows from the fact that the isotropy groups Γ𝑞1 and Γ𝑞2 are conjugate, hence of the same order.

2.3. Singular homology Apart from the homotopy groups, the other important family of algebraic topological invariants are the homology groups. Again, the intuitive idea is to capture the holes of a topological space. But this time, instead of surrounding them by 𝑘-spheres, they will be surrounded by triangulated 𝑘-spaces, which are understood as a collection (a formal sum) of (a specific type of) 𝑘-polyhedra (the so-called 𝑘-chains). This will give the structure of an Abelian group to the homology. The homology groups have some advantages over the homotopy groups: as all of them are Abelian groups, they are more manageable; we do not need a basepoint; and they are computable in some wide sense. As a drawback, the non-Abelianness of the fundamental group may contain more information than the (first) homology group. As in Chapter 1, we will see the theory in the three categories: topological, PL, and differentiable. In the topological category the central version of homology is the singular homology, and we devote the present section to it. In the PL category, we have simplicial homology, and in the smooth category, we shall develop the de Rham cohomology. 2.3.1. Construction of singular homology. To start with, for 𝑋 a topological space, let 𝑄 𝑘 (𝑋) be the free Abelian group generated by the 𝑘-cubes, which are continuous maps 𝜎 ∶ [0, 1]𝑘 → 𝑋, and let 𝐷𝑘 (𝑋) be the subgroup of 𝑄 𝑘 (𝑋) generated by the cubes which are constant along some coordinate 𝑥𝑖 of [0, 1]𝑛 called the degenerate cubes. Let us define 𝐶𝑘 (𝑋) = 𝑄 𝑘 (𝑋)/𝐷𝑘 (𝑋). An element 𝑐 ∈ 𝐶𝑘 (𝑋), called a 𝑘-chain, is a finite sum 𝑐 = ∑𝑖 𝑛𝑖 𝜎 𝑖 , where 𝑛𝑖 ∈ ℤ and 𝜎 𝑖 are non-degenerate 𝑘-cubes. The sum of chains is given by component-wise sum. We also define maps 𝜕𝑘 ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘−1 (𝑋), called

2.3. Singular homology

85

boundary maps, given on 𝑘-cubes by 𝑘

𝜕𝑘 (𝜎) = ∑ (−1)𝑖 (𝐹𝑖 𝜎 − 𝐵𝑖 𝜎) , 𝑖=1

where 𝐹𝑖 𝜎(𝑡1 , . . . , 𝑡 𝑘−1 ) = 𝜎(𝑡1 , . . . , 𝑡 𝑖−1 , 1, 𝑡 𝑖 , . . . , 𝑡 𝑘−1 ), (2.5)

𝐵𝑖 𝜎(𝑡1 , . . . , 𝑡 𝑘−1 ) = 𝜎(𝑡1 , . . . , 𝑡 𝑖−1 , 0, 𝑡 𝑖 , . . . , 𝑡 𝑘−1 ),

are the front and back 𝑖-faces, respectively. For convenience, it is useful to declare 𝐶𝑘 (𝑋) = 0 for 𝑘 < 0 and 𝜕𝑘 ≡ 0 for 𝑘 ≤ 0. We will follow this convention along this section. Proposition 2.68. The boundary maps satisfy that 𝜕𝑘−1 ∘ 𝜕𝑘 = 0 for all 𝑘. Proof. It is just a straightforward computation. We check it in the basis of 𝐶𝑘 (𝑋). Let 𝜎 be any 𝑘-cube, we have 𝑘

𝜕𝑘−1 ∘ 𝜕𝑘 (𝜎) = 𝜕𝑘−1 ( ∑ (−1)𝑖 (𝐹𝑖 𝜎 − 𝐵𝑖 𝜎)) 𝑖=1 𝑘−1 𝑘

= ∑ ∑ (−1)𝑖 (−1)𝑗 (𝐹𝑗 𝐹𝑖 𝜎 − 𝐵𝑗 𝐹𝑖 𝜎 − 𝐹𝑗 𝐵𝑖 𝜎 + 𝐵𝑗 𝐵𝑖 𝜎). 𝑗=1 𝑖=1

For 𝑗 ≤ 𝑖, we have 𝐹𝑗 𝐹𝑖 𝜎 = 𝐹𝑖−1 𝐹𝑗 𝜎. So the terms 𝐹𝑗 𝐹𝑖 𝜎, 𝑗 ≤ 𝑖, and the terms 𝐹𝑗 𝐹𝑖 𝜎, 𝑗 > 𝑖, appear repeatedly with different sign, and hence they cancel in the sum. Analogously, it happens with the other terms. Therefore 𝜕𝑘 (𝜕𝑘−1 (𝜎)) = 0. □ It is customary to denote 𝐶• (𝑋) = ⨁𝑘≥0 𝐶𝑘 (𝑋) and to abuse notation and suppress the subscript 𝑘 of 𝜕𝑘 . With this notation, (𝐶• (𝑋), 𝜕) is called a chain complex. Definition 2.69. Let 𝑋 be a topological space. We define 𝑍𝑘 (𝑋) = ker(𝜕𝑘 ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘−1 (𝑋)), 𝐵𝑘 (𝑋) = im(𝜕𝑘+1 ∶ 𝐶𝑘+1 (𝑋) → 𝐶𝑘 (𝑋)). 𝐻𝑘 (𝑋) = 𝑍𝑘 (𝑋)/𝐵𝑘 (𝑋). We call 𝑍𝑘 (𝑋) the group of 𝑘-cycles, 𝐵𝑘 (𝑋) the group of 𝑘-boundaries, and 𝐻𝑘 (𝑋) is the 𝑘-singular homology group of 𝑋. The 𝑘-homology classes [𝑎] ∈ 𝐻𝑘 (𝑋) are defined by cycles (i.e., 𝜕𝑎 = 0) and two cycles belong to the same class if their difference is a boundary (i.e., [𝑎1 ] = [𝑎2 ] when 𝑎1 = 𝑎2 + 𝜕𝑏, for some 𝑏 ∈ 𝐶𝑘+1 (𝑋)). Remark 2.70. Note that the condition 𝜕𝑘 ∘ 𝜕𝑘+1 = 0 (the content of Proposition 2.68) is equivalent to 𝐵𝑘 (𝑋) ⊂ 𝑍𝑘 (𝑋), which is necessary to take the quotient 𝑍𝑘 (𝑋)/𝐵𝑘 (𝑋). Example 2.71. For the singleton 𝑋 = ⋆, all the 𝑘-cubes with 𝑘 > 0 are degenerate, and there is just one 0-cube. Therefore 𝐶𝑘 (⋆) = 0 for 𝑘 > 0 and 𝐶0 (⋆) = ℤ so, automatically we have 𝐻0 (⋆) = ℤ and 𝐻𝑘 (⋆) = 0 for 𝑘 > 0. This is the reason to annihilate degenerate cubes in the definition of chains. If we had not done it, the homology of the singleton would not vanish for 𝑘 > 0.

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X Z2(X)

B2(X)

Example 2.72. (1) Let 𝑋 be a topological space and let 𝑋 = ⨆𝛼∈Λ 𝑋𝛼 be its decomposition in path connected components. Then we have that any 𝑘-cube 𝜎 ∶ [0, 1]𝑘 → 𝑋 is included in some 𝑋𝛼 . This means that 𝐶𝑘 (𝑋) = ⨁𝛼∈Λ 𝐶𝑘 (𝑋𝛼 ). Clearly, the boundary maps respect this decomposition, that is 𝜕𝑘 ∶ 𝐶𝑘 (𝑋𝛼 ) → 𝐶𝑘−1 (𝑋𝛼 ). Therefore 𝑍𝑘 (𝑋) = ⨁𝛼∈Λ 𝑍𝑘 (𝑋𝛼 ) and 𝐵𝑘 (𝑋) = ⨁𝛼∈Λ 𝐵𝑘 (𝑋𝛼 ). So 𝐻𝑘 (𝑋) =

⨁𝛼∈Λ 𝑍𝑘 (𝑋𝛼 ) 𝑍𝑘 (𝑋) = = 𝐻𝑘 (𝑋𝛼 ). 𝐵𝑘 (𝑋) ⨁𝛼∈Λ 𝐵𝑘 (𝑋𝛼 ) ⨁ 𝛼∈Λ

(2) If 𝑋 is path connected, then 𝐻0 (𝑋) = ℤ. For that, observe that a 0-cube 𝜎 ∶ {0} → 𝑋 is determined by choosing a point 𝜎(0) ∈ 𝑋. On the other hand, a 1-cube is a map 𝛾 ∶ [0, 1] → 𝑋, so 𝛾 is determined by a path on 𝑋. Moreover, we have 𝜕1 (𝛾) = 𝐹1 𝛾 − 𝐵1 𝛾 = 𝛾(1) − 𝛾(0) , where in the last equality, we used the identification of 0-cubes with points. As 𝑋 is path connected, given any 𝑥, 𝑦 ∈ 𝑋, we have a path 𝛾 with 𝜕1 (𝛾) = 𝑥 − 𝑦. Hence 𝑍0 (𝑋) = 𝐶0 (𝑋) =

⨁

ℤ⟨𝑥⟩,

𝑥∈𝑋

𝐵0 (𝑋) = ⟨𝑥 − 𝑦 | 𝑥, 𝑦 ∈ 𝑋⟩ = { ∑ 𝑛𝑖 𝑥𝑖 | ∑ 𝑛𝑖 = 0}, 𝐻0 (𝑋) = 𝑍0 (𝑋)/𝐵0 (𝑋) ≅ ℤ⟨[𝑝0 ]⟩ ≅ ℤ, where 𝑝0 is any fixed point of 𝑋. (3) As a consequence of the previous two items, 𝐻0 (𝑋) = ℤ⟨𝜋0 (𝑋)⟩ ≅ ⨁Λ ℤ, where Λ is the cardinal of the set of path connected components. The group 𝐶1 (𝑋) is the free Abelian group generated by the paths on 𝑋. A path 𝛾 ∈ 𝐶1 (𝑋) is a cycle if 𝜕1 (𝛾) = 𝛾(1) − 𝛾(0) = 0, that is, if 𝛾 is a loop. First, if 𝛾 is a path, using 𝜎 ∈ 𝐶2 (𝑋), 𝜎 ∶ [0, 1]2 → 𝑋, given by 𝜎(𝑡1 , 𝑡2 ) = {

𝛾(𝑡1 + 𝑡2 ), 𝛾(1),

𝑡1 + 𝑡2 ≤ 1, 𝑡1 + 𝑡2 ≥ 1,

2.3. Singular homology

87

←

←

we have that 𝜕2 (𝜎) = 𝛾 + 𝛾, so 𝛾 = −𝛾 (mod 𝐵1 (𝑋)). Second, if 𝛾1 , 𝛾2 are two paths with 𝛾1 (1) = 𝛾2 (0), we take 𝜎′ ∈ 𝐶2 (𝑋) given by 𝜎′ (𝑡1 , 𝑡2 ) = {

𝛾1 (2𝑡1 /(2 − 𝑡2 )), 𝛾2 (2𝑡1 + 𝑡2 − 2),

2𝑡0 + 𝑡1 ≤ 2, 2𝑡0 + 𝑡1 ≥ 2,

to get 𝜕2 (𝜎′ ) = 𝛾1 + 𝛾2 − (𝛾1 ∗ 𝛾2 ), so (𝛾1 ∗ 𝛾2 ) = 𝛾1 + 𝛾2 (mod 𝐵1 (𝑋)). Therefore if we take 𝑐 ∈ 𝑍1 (𝑋), then 𝑐 is a sum of paths (with coefficients). With the above rules, we can change the paths (by juxtaposition and reversing) to paths with different endpoints. As 𝜕1 (𝑐) = 0, the components of 𝑐 must be loops. So 𝐻1 (𝑋) = 𝑍1 (𝑋)/𝐵1 (𝑋) is generated by the loops in 𝑋 (without fixed point). Moreover, if two loops 𝛾0 , 𝛾1 are homotopic, then take the homotopy 𝐻 ∶ [0, 1] × [0, 1] → 𝑋 with 𝐻0 = 𝛾0 , 𝐻1 = 𝛾1 . Therefore 𝜕2 (𝐻) = 𝛾1 − 𝛾0 and hence [𝛾0 ] = [𝛾1 ]. Thus 𝐻1 (𝑋) is a kind of Abelian version of 𝜋1 (𝑋). Actually we have the following general result. Theorem 2.73 (Hurewicz). Let 𝑋 be a path connected space, then there are group homomorphisms ℎ𝑘 ∶ 𝜋𝑘 (𝑋) → 𝐻𝑘 (𝑋), called Hurewicz maps, such that the following hold. ≅

• For 𝑘 = 1, ℎ1 gives an isomorphism Ab(𝜋1 (𝑋)) ⟶ 𝐻1 (𝑋). • If 𝑚 ≥ 2 and 𝜋𝑘 (𝑋) = 0 for all 0 < 𝑘 < 𝑚, then 𝐻𝑘 (𝑋) = 0 for 0 < 𝑘 < 𝑚 and ≅

ℎ𝑚 ∶ 𝜋𝑚 (𝑋) ⟶ 𝐻𝑚 (𝑋) is an isomorphism. The map ℎ𝑘 of Theorem 2.73 is given by assigning to 𝑓 ∈ 𝜋𝑘 (𝑋) = [𝑆 𝑘 , 𝑋]∗ , the image ℎ𝑘 (𝑓) = 𝑓∗ ([𝑆 𝑘 ]), of the fundamental class of 𝑆 𝑘 , which is the generator [𝑆 𝑘 ] ∈ 𝐻𝑘 (𝑆 𝑘 ) ≅ ℤ (Remark 2.93) and using the induced map 𝑓∗ ∶ 𝐻𝑘 (𝑆𝑘 ) → 𝐻𝑘 (𝑋) (defined next). Remark 2.74. In homotopy theory jargon, a path connected space 𝑋 such that 𝜋𝑘 (𝑋) = 0 for 0 < 𝑘 ≤ 𝑛 is called 𝑛-connected. The hypothesis of the Hurewicz theorem requires 𝑋 to be (𝑚 − 1)-connected. Functoriality of homology. Let 𝑋, 𝑌 be topological spaces, and take a continuous map ℎ ∶ 𝑋 → 𝑌 . This map induces a group homomorphism at the level of chains ℎ𝑘 ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘 (𝑌 ) given by ℎ𝑘 𝜎 = ℎ ∘ 𝜎 ∶ [0, 1]𝑘 → 𝑌 for any 𝜎 ∶ [0, 1]𝑘 → 𝑋, which extends to 𝑘-chains by linearity. This homomorphism commutes with the boundary operator, since we have for 𝜎 ∈ 𝐶𝑘 (𝑋), 𝑘

𝜕𝑘 (ℎ𝑘 𝜎) = 𝜕𝑘 (ℎ ∘ 𝜎) = ∑ (−1)𝑖 (𝐹𝑖 (ℎ ∘ 𝜎) − 𝐵𝑖 (ℎ ∘ 𝜎)) 𝑖=1 𝑘

(2.6)

= ∑ (−1)𝑖 (ℎ ∘ 𝐹𝑖 𝜎 − ℎ ∘ 𝐵𝑖 𝜎) 𝑖=1 𝑘

= ℎ𝑘−1 ( ∑ (−1)𝑖 (𝐹𝑖 𝜎 − 𝐵𝑖 𝜎)) = ℎ𝑘−1 𝜕𝑘 𝜎. 𝑖=1

88

2. Algebraic topology

Thus we have 𝜕𝑘 ∘ℎ𝑘 = ℎ𝑘−1 ∘𝜕𝑘 , for all 𝑘 ≥ 0. This means that we have a commutative diagram ⋯

𝜕𝑘+2

/ 𝐶𝑘+1 (𝑋) ℎ𝑘+1

⋯

𝜕𝑘+2



/ 𝐶𝑘+1 (𝑌 )

𝜕𝑘+1

/ 𝐶𝑘 (𝑋)

𝜕𝑘

ℎ𝑘

𝜕𝑘+1

 / 𝐶𝑘 (𝑌 )

/ 𝐶𝑘−1 (𝑋)

𝜕𝑘−1

/⋯

𝜕𝑘−1

/ ⋯.

ℎ𝑘−1

𝜕𝑘

 / 𝐶𝑘−1 (𝑌 )

In particular, this implies that ℎ𝑘 (ker 𝜕𝑘 ) ⊂ ker 𝜕𝑘 and ℎ𝑘 (im 𝜕𝑘+1 ) ⊂ im 𝜕𝑘+1 , so it induces a map in homology ℎ∗ ∶ 𝐻𝑘 (𝑋) ⟶ 𝐻𝑘 (𝑌 ),

ℎ∗ [𝑐] = [ℎ𝑘 𝑐].

If we have continuous maps 𝑓 ∶ 𝑋 → 𝑌 and 𝑔 ∶ 𝑌 → 𝑍, then for any 𝑘-cube 𝜎, (𝑔 ∘ 𝑓)𝑘 (𝜎) = 𝑔 ∘ 𝑓 ∘ 𝜎 = 𝑔𝑘 ∘ 𝑓𝑘 (𝜎). So (𝑔 ∘ 𝑓)∗ = 𝑔∗ ∘ 𝑓∗ . Also (1𝑋 )∗ = Id on 𝐻𝑘 (𝑋). Thus the 𝑘-homology is a functor 𝐻𝑘 ∶ 𝐓𝐨𝐩 ⟶ 𝐀𝐛𝐞𝐥 . Now we move to the homotopy category 𝐇𝐨𝐓𝐨𝐩. We want to show that 𝐻𝑘 defines a functor on it. For this, we have to prove that if we have homotopic maps 𝑓, 𝑔 ∶ 𝑋 → 𝑌 , then 𝑓∗ = 𝑔∗ . Consider the homotopy 𝐻 ∶ 𝑋×[0, 1] → 𝑌 with 𝐻(𝑥, 0) = 𝑓(𝑥), 𝐻(𝑥, 1) = 𝑔(𝑥) for 𝑥 ∈ 𝑋. We look at the space 𝑋 × [0, 1] and the inclusions 𝑖𝑠 ∶ 𝑋 → 𝑋 × [0, 1], 𝑖𝑠 (𝑥) = (𝑥, 𝑠), and the projection 𝜋 ∶ 𝑋 × [0, 1] → 𝑋. Then 𝑓 = 𝐻 ∘ 𝑖0 , 𝑔 = 𝐻 ∘ 𝑖1 . From the functoriality in 𝐓𝐨𝐩, we have that 𝑓∗ = 𝐻∗ ∘ (𝑖0 )∗ and 𝑔∗ = 𝐻∗ ∘ (𝑖1 )∗ . Hence if we prove that (𝑖0 )∗ = (𝑖1 )∗ , then 𝑓∗ = 𝐻∗ ∘ (𝑖0 )∗ = 𝐻∗ ∘ (𝑖1 )∗ = 𝑔∗ . Lemma 2.75. For the maps 𝑖0 , 𝑖1 ∶ 𝑋 → 𝑋 ×[0, 1], we have that (𝑖0 )∗ = (𝑖1 )∗ ∶ 𝐻𝑘 (𝑋) → 𝐻𝑘 (𝑋 × [0, 1]). Proof. The idea is to construct a family of group homomorphisms 𝐾𝑘 ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘+1 (𝑋 × [0, 1]) such that (2.7)

(𝑖1 )𝑘 − (𝑖0 )𝑘 = ±𝜕𝑘+1 𝐾𝑘 ± 𝐾𝑘−1 𝜕𝑘 .

This is called a chain homotopy (see Definition 2.83). For 𝜎 ∶ [0, 1]𝑘 → 𝑋, define 𝐾𝑘 𝜎 ∶ [0, 1]𝑘+1 → 𝑋 × [0, 1] by 𝐾𝑘 𝜎(𝑡1 , . . . , 𝑡 𝑘+1 ) = (𝜎(𝑡1 , . . . , 𝑡 𝑘 ), 𝑡 𝑘+1 ). So 𝐹𝑖 𝐾𝑘 𝜎 = 𝐾𝑘−1 𝐹𝑖 𝜎 and 𝐵𝑖 𝐾𝑘 𝜎 = 𝐾𝑘−1 𝐵𝑖 𝜎 for 𝑖 ≤ 𝑘, whereas 𝐹 𝑘+1 𝐾𝑘 𝜎 = 𝑖1 ∘ 𝜎 and 𝐵𝑘+1 𝐾𝑘 𝜎 = 𝑖0 ∘ 𝜎, where 𝐹𝑖 , 𝐵𝑖 are the front and back face operators in (2.5). Then 𝑘

𝜕𝑘+1 (𝐾𝑘 𝜎) = ∑ (−1)𝑖 (𝐹𝑖 𝐾𝑘 𝜎 − 𝐵𝑖 𝐾𝑘 𝜎) + (−1)𝑘+1 (𝐹 𝑘+1 𝐾𝑘 𝜎 − 𝐵𝑘+1 𝐾𝑘 𝜎) 𝑖=1 𝑘

= 𝐾𝑘−1 ∑ (−1)𝑖 (𝐹𝑖 𝜎 − 𝐵𝑖 𝜎) + (−1)𝑘+1 (𝑖1 ∘ 𝜎 − 𝑖0 ∘ 𝜎) 𝑖=1

= 𝐾𝑘−1 𝜕𝑘 (𝜎) + (−1)𝑘+1 ((𝑖1 )𝑘 𝜎 − (𝑖0 )𝑘 𝜎) . Now to prove that (𝑖0 )∗ = (𝑖1 )∗ , take [𝑐] ∈ 𝐻𝑘 (𝑋). Then 𝜕𝑘 (𝑐) = 0. Apply (2.7) to 𝑐, to get (𝑖1 )𝑘 (𝑐) − (𝑖0 )𝑘 (𝑐) = ±𝜕𝑘+1 (𝐾𝑘 (𝑐)). Hence (𝑖0 )∗ [𝑐] = (𝑖1 )∗ [𝑐], as required. □

2.3. Singular homology

89

Remark 2.76. The homotopy invariance of homology would also follow from proving that (2.8)

𝜋∗ ∶ 𝐻𝑘 (𝑋 × [0, 1]) → 𝐻𝑘 (𝑋)

is an isomorphism (such result is called a Poincaré lemma). Certainly, 𝜋 ∘ 𝑖𝑠 = 1𝑋 implies that 𝜋∗ ∘ (𝑖𝑠 )∗ = Id, for any 𝑠 ∈ [0, 1]. If 𝜋∗ is an isomorphism, then we can invert and (𝑖0 )∗ = (𝜋∗ )−1 = (𝑖1 )∗ . Note that 𝜋∗ is clearly surjective, so we only need to see that it is injective. Therefore, we have that the 𝑘-homology group defines a functor 𝐻𝑘 ∶ 𝐇𝐨𝐓𝐨𝐩 → 𝐀𝐛𝐞𝐥 . In particular, if 𝑋 ∼ 𝑌 , then 𝐻𝑘 (𝑋) ≅ 𝐻𝑘 (𝑌 ), for all 𝑘 ≥ 0. Remark 2.77. There is an alternative definition of chains using 𝑘-simplices (Remark 1.47) instead of 𝑘-cubes [Mun], yielding to the same invariant. The advantage is that we do not have to annihilate degenerate simplices to have 𝐻𝑘 (⋆) = 0 for 𝑘 > 0. On the other hand, the proof of Lemma 2.75 gets more involved since we cannot just multiply by [0, 1]. Remark 2.78. Note that if 𝑟 ∶ 𝑋 → 𝐴 is a deformation retract, then by functoriality 𝑟∗ ∶ 𝐻𝑘 (𝑋) → 𝐻𝑘 (𝐴) are isomorphisms for 𝑘 ≥ 0. If 𝑟 ∶ 𝑋 → 𝐴 is only a retract, with inclusion 𝑖 ∶ 𝐴 ↪ 𝑋, then 𝑟∗ ∶ 𝐻𝑘 (𝑋) → 𝐻𝑘 (𝐴) is an epimorphism and 𝑖∗ ∶ 𝐻𝑘 (𝐴) → 𝐻𝑘 (𝑋) is a monomorphism. This follows from 𝑟∗ ∘ 𝑖∗ = Id𝐻𝑘 (𝐴) .

2.3.2. Homological algebra. The definition of homology just seen sits in a much wider context that underpins many different algebraic and topological invariants with similar constructions. The theoretical framework of these constructions is known as homological algebra. Stated in full generality, the notions there are quite general and abstract, but we shall introduce them to the extent of making clear the common structure of our constructions. We shall work on a background Abelian category 𝒜. We shall not define it precisely, but roughly an Abelian category is a category (in general of algebraic objects) such that morphisms 𝑓 ∶ 𝐴 → 𝐵 have images, kernels, and cokernels; there is an initial and final object called the zero object 0, there are zero maps 0 ∶ 𝐴 → 𝐵 (those factoring through the zero object), and there exist products and direct sums (coproducts) for objects of 𝒜; see [McL]. The categories of Abelian groups 𝐀𝐛𝐞𝐥 and of vector spaces 𝐕𝐞𝐜𝐭𝐤 are Abelian categories. Definition 2.79. A chain complex is a family of objects of 𝒜, 𝐴𝑘 for 𝑘 ∈ ℤ, and morphisms 𝜕𝑘 ∶ 𝐴𝑘 → 𝐴𝑘−1 , called boundary maps, such that 𝜕𝑘−1 ∘ 𝜕𝑘 = 0, for all 𝑘. ⋯

𝜕𝑘+2

/ 𝐴𝑘+1

𝜕𝑘+1

/ 𝐴𝑘

𝜕𝑘

/ 𝐴𝑘−1

𝜕𝑘−1

/ ⋯.

90

2. Algebraic topology

Given two chain complexes (𝐴𝑘 , 𝜕𝑘 ) and (𝐵𝑘 , 𝜕𝑘′ ), a chain morphism is a family of morphisms 𝑓𝑘 ∶ 𝐴𝑘 → 𝐵𝑘 such that 𝜕𝑘 ∘ 𝑓𝑘 = 𝑓𝑘−1 ∘ 𝜕𝑘 , ⋯

𝜕𝑘+2

𝑓𝑘+1

⋯

′ 𝜕𝑘+2

𝜕𝑘+1

/ 𝐴𝑘+1

/ 𝐴𝑘

𝜕𝑘

𝑓𝑘



′ 𝜕𝑘+1

/ 𝐵𝑘+1

 / 𝐵𝑘

/ 𝐴𝑘−1

𝜕𝑘−1

/⋯

′ 𝜕𝑘−1

/ ⋯.

𝑓𝑘−1 𝜕𝑘′

 / 𝐵𝑘−1

Remark 2.80. • It is customary to denote 𝐴• = ⨁𝑘 𝐴𝑘 and to remove the subscript of 𝜕𝑘 . In this case, the chain complex is denoted (𝐴• , 𝜕). • With these definitions, we have the category 𝐂𝐡𝒜 of chain complexes with their morphisms. This is again an Abelian category, i.e., the morphisms have kernels, images, and cokernels. Example 2.81. Let us take 𝒜 = 𝐀𝐛𝐞𝐥. Then given a topological space 𝑋, the complex (𝐶• (𝑋), 𝜕) of singular 𝑘-chains is a chain complex. A continuous map ℎ ∶ 𝑋 → 𝑌 induces a chain morphism ℎ ∶ (𝐶• (𝑋), 𝜕) → (𝐶• (𝑌 ), 𝜕), by (2.6). Definition 2.82. Let (𝐴• , 𝜕) be a chain complex on an Abelian category 𝒜. The 𝑘homology of 𝐴• is the object of 𝒜 defined by 𝐻𝑘 (𝐴• , 𝜕) =

ker(𝜕𝑘 ∶ 𝐴𝑘 → 𝐴𝑘−1 ) . im(𝜕𝑘+1 ∶ 𝐴𝑘+1 → 𝐴𝑘 )

Suppose that we have a morphism of chain complexes 𝑓 ∶ (𝐴• , 𝜕) → (𝐵• , 𝜕′ ). Since ′ 𝑓 ∘ 𝜕 = 𝜕′ ∘ 𝑓 we have that 𝑓 (ker 𝜕𝑘 ) ⊂ ker 𝜕𝑘′ and 𝑓 (im 𝜕𝑘+1 ) ⊂ im 𝜕𝑘+1 . Thus 𝑓 descends to the quotient giving a morphism 𝑓∗ ∶ 𝐻𝑘 (𝐴• , 𝜕) → 𝐻𝑘 (𝐵• , 𝜕′ ), called the induced morphism in homology. It is clear that (𝑔 ∘ 𝑓)∗ = 𝑔∗ ∘ 𝑓∗ , for morphisms 𝑓

𝑔

(𝐴• , 𝜕) → (𝐵• , 𝜕′ ) → (𝐶• , 𝜕″ ). Therefore the 𝑘-homology of chain complexes is a functor 𝐻𝑘 ∶ 𝐂𝐡𝒜 → 𝒜. In this language, singular homology is the composition of functors 𝐶 • (-)

𝐻𝑘

𝐓𝐨𝐩 ⟶ 𝐂𝐡𝐀𝐛𝐞𝐥 ⟶ 𝐀𝐛𝐞𝐥 . Moreover, the notion of homotopic maps can be translated to the category of chain complexes. Definition 2.83. Let 𝒜 be an Abelian category and let 𝑓, 𝑔 ∶ (𝐴• , 𝜕) → (𝐵• , 𝜕′ ) be two morphisms of chain complexes. A chain homotopy between 𝑓 and 𝑔 is a family of morphisms 𝐾𝑘 ∶ 𝐴𝑘 → 𝐵𝑘+1 such that 𝑔𝑘 − 𝑓𝑘 = ±𝜕𝑘+1 𝐾𝑘 ± 𝐾𝑘−1 𝜕𝑘 , ⋯

⋯

𝜕𝑘+2

′ 𝜕𝑘+2

𝜕𝑘+1

/ 𝐴𝑘 𝜕𝑘 / 𝐴𝑘−1 z 𝐾 zz 𝐾𝑘 zz z 𝑓,𝑔 𝑘−1zzz 𝑓,𝑔 𝑓,𝑔 z z z  }zz  }zz  / 𝐵𝑘+1 / 𝐵𝑘 / 𝐵𝑘−1 / 𝐴𝑘+1

′ 𝜕𝑘+1

𝜕𝑘′

𝜕𝑘−1

′ 𝜕𝑘−1

/⋯ /⋯

2.3. Singular homology

91

Proposition 2.84. If 𝑓, 𝑔 ∶ (𝐴• , 𝜕) → (𝐵• , 𝜕′ ) are two chain homotopic morphisms of chain complexes, then 𝑓∗ = 𝑔∗ ∶ 𝐻𝑘 (𝐴• , 𝜕) → 𝐻𝑘 (𝐵• , 𝜕). Proof. This is proved as in Lemma 2.75. By definition, there is a chain homotopy 𝐾𝑘 ∶ 𝐴𝑘 → 𝐵𝑘+1 such that 𝑔𝑘 − 𝑓𝑘 = ±𝜕𝑘+1 𝐾𝑘 ± 𝐾𝑘−1 𝜕𝑘 . For [𝑎] ∈ 𝐻𝑘 (𝐴• , 𝜕), we have 𝑔∗ [𝑎] − 𝑓∗ [𝑎] = [𝑔𝑘 𝑎 − 𝑓𝑘 𝑎] = [±𝜕𝑘+1 𝐾𝑘 𝑎 ± 𝐾𝑘−1 𝜕𝑘 𝑎] = [±𝜕𝑘+1 𝐾𝑘 𝑎] = 0, since 𝜕𝑘 𝑎 = 0. Hence 𝑓∗ [𝑎] = 𝑔∗ [𝑎]. □ Now we come to the useful notion of exact sequence. Definition 2.85. Let 𝐴, 𝐵, 𝐶 be three objects in an Abelian category 𝒜, and let 𝑓 ∶ 𝑓

𝑔

𝐴 → 𝐵 and 𝑔 ∶ 𝐵 → 𝐶 be two morphisms. We say that 𝐴 ⟶ 𝐵 ⟶ 𝐶 is exact if im 𝑓 = ker 𝑔. More generally, given a sequence of objects 𝐴𝑘 for 𝑘 ∈ ℤ and morphisms 𝑓 ∶ 𝐴𝑘 → 𝐴𝑘−1 , we say that the sequence ...

𝑓𝑘+2

/ 𝐴𝑘+1

𝑓𝑘+1

/ 𝐴𝑘

𝑓𝑘

/ 𝐴𝑘−1

𝑓𝑘−1

/ ...

is exact if it is exact at every 𝐴𝑘 , that is, if im 𝑓𝑘+1 = ker 𝑓𝑘 for all 𝑘 ∈ ℤ. 𝑓𝑘+1

𝑓𝑘

Remark 2.86. Let ⋯ → 𝐴𝑘+1 ⟶ 𝐴𝑘 ⟶ 𝐴𝑘−1 → ⋯ be a complex, i.e., 𝑓𝑘 ∘𝑓𝑘+1 = 0, for 𝑘 ∈ ℤ. Then im 𝑓𝑘+1 ⊂ ker 𝑓𝑘 , and the discrepancy between them is measured by the homology ker 𝑓𝑘 𝐻𝑘 (𝐴• , 𝑓) = . im 𝑓𝑘+1 In this sense, exact sequences are chain complexes with vanishing homology. Example 2.87. 𝑓

• A sequence 0 → 𝐴 → 𝐵 is exact if and only if 𝑓 is injective. The map 0 → 𝐴 0

is the unique possible map (called the zero morphism 0 ⟶ 𝐴), as 0 is the initial object. This sequence is exact at 𝐴 if ker 𝑓 = im 0 = 0, that is if 𝑓 is injective. 𝑓

• A sequence 𝐴 → 𝐵 → 0 is exact if and only if 𝑓 is surjective. Again 𝐵 → 0 is 0

the unique possible map, the zero morphism 𝐵 ⟶ 0. This sequence is exact at 𝐵 if im 𝑓 = ker 0 = 𝐵, that is if 𝑓 is surjective. 𝑓

𝑔

• A sequence 0 → 𝐴 ⟶ 𝐵 ⟶ 𝐶 → 0 is exact if 𝑓 is injective, 𝑔 is surjective, and im 𝑓 = ker 𝑔. Such an exact sequence is very common in homological algebra and receives the name of short exact sequence. In contraposition, an exact sequence with more objects is called a long exact sequence. 𝑓

• For a morphism 𝑓 ∶ 𝐴 → 𝐵, the sequences 0 → ker 𝑓 → 𝐴 ⟶ 𝐵 → coker 𝑓 → 0 and 0 → ker 𝑓 → 𝐴 → im 𝑓 → 0 are exact. • If 0 → 𝐴 → 𝐵 → 0 is exact, then 𝐴 ≅ 𝐵. • If 𝐴 ⊂ 𝐵 is a subobject, then 0 → 𝐴 → 𝐵 → 𝐵/𝐴 → 0 is an exact sequence.

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2. Algebraic topology

• For objects 𝐴, 𝐶, there is a short exact sequence 0 → 𝐴 → 𝐴 ⊕ 𝐶 → 𝐶 → 0, which is called split. A short exact sequence 0 → 𝐴 → 𝐵 → 𝐶 → 0 is split if it is isomorphic to this one. Not every short exact sequence splits, as is shown 𝑓

by the short exact sequence 0 → ℤ → ℤ → ℤ2 → 0, with 𝑓(𝑥) = 2𝑥. One of the most useful tools in homological algebra is the following result that associates a long exact sequence in 𝒜 to a short exact sequence in 𝐂𝐡A . Proposition 2.88. Let (𝐴• , 𝜕), (𝐵• , 𝜕′ ), and (𝐶• , 𝜕″ ) be chain complexes in 𝒜. Suppose that we have a short exact sequence of chain complexes 𝑓

𝑔

0 ⟶ 𝐴• ⟶ 𝐵• ⟶ 𝐶• ⟶ 0, 𝑓𝑘

𝑔𝑘

i.e., 0 → 𝐴𝑘 → 𝐵𝑘 → 𝐶𝑘 → 0 is exact for each 𝑘. Then there are morphisms 𝜕∗ ∶ 𝐻𝑘 (𝐶• ) → 𝐻𝑘−1 (𝐴• ) (called connecting maps) such that we obtain a long exact sequence in homology 𝜕∗

/ 𝐻𝑘+1 (𝐴• )

𝑓∗

/ 𝐻𝑘+1 (𝐵• )

𝑔∗

/ 𝐻𝑘+1 (𝐶• ) Z[ -,

𝑔∗

/ 𝐻𝑘 (𝐶• ) Z[ -,

𝜕∗

8?9 ^/ 𝐻 (𝐴 ) 𝑘 •

𝑓∗

/ 𝐻𝑘 (𝐵• ) 𝜕∗

(/)^ / 𝐻𝑘−1 (𝐴• )

𝑓∗

/ 𝐻𝑘−1 (𝐵• )

𝑔∗

/ 𝐻𝑘−1 (𝐶• )

𝜕∗

/.

We shall not prove this result, which can be found in [Wei]. We recall the definition of the connecting map, which is 𝜕∗ [𝑐] = [𝑓−1 (𝜕(𝑔−1 (𝑐)))], where 𝑓−1 is a left inverse of 𝑓 and 𝑔−1 is a right inverse of 𝑔. This does not depend on the choices. 2.3.3. Mayer-Vietoris exact sequence. The Mayer-Vietoris exact sequence is a powerful result that allows us to compute the homology of a space 𝑋 in terms of an open covering 𝑋 = 𝑈 ∪ 𝑉. We need first prove an accessory result. Let 𝔘 = {𝑈𝛼 |𝛼 ∈ Λ} be an open covering of a space 𝑋. Define the chain complex of small cubes as the subcomplex 𝐶𝑘 (𝑋; 𝔘) ⊂ 𝐶𝑘 (𝑋) generated by 𝑘-cubes contained in some open set of the covering. The corresponding homology is denoted 𝐻𝑘 (𝑋; 𝔘). We have the following. Proposition 2.89. There is an isomorphism 𝐻𝑘 (𝑋; 𝔘) ≅ 𝐻𝑘 (𝑋), for all 𝑘 ≥ 0. Proof. There is a subdivision operator div ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘 (𝑋) given on 𝑘-cubes as div(𝜎) = ∑𝐼 𝑠𝐼 𝜎, where 𝐼 = (𝜖1 , . . . , 𝜖𝑘 ) ∈ {0, 1}𝑘 , and 𝑠𝐼 𝜎(𝑡1 , . . . , 𝑡 𝑘 ) = 𝜎((𝑡1 + 𝜖1 )/2, . . . , (𝑡 𝑘 + 𝜖𝑘 )/2),

(𝑡1 , . . . , 𝑡 𝑘 ) ∈ [0, 1]𝑘 .

We introduce also the subdivision operator div𝑖 in the 𝑖th coordinate: div𝑖 (𝜎) = 𝑠0𝑖 𝜎 + 𝑠1𝑖 𝜎, where 𝑠𝜖𝑖 𝜎(𝑡1 , . . . , 𝑡𝑛 ) = 𝜎(𝑡1 , . . . , (𝑡 𝑖 + 𝜖)/2, . . . , 𝑡 𝑘 ). Then div = div1 ∘ ⋯ ∘ div𝑘 . As

2.3. Singular homology

93

in the discussion after Example 2.72, the homotopy 𝐻(𝑡1 , 𝑡2 , . . . , 𝑡 𝑘+1 ) = {

(𝑠0𝑖 𝜎)(𝑡1 , . . . , 2𝑡 𝑖 /(2 − 𝑡 𝑘+1 ), . . . , 𝑡 𝑘 ), 2𝑡 𝑖 + 𝑡 𝑘+1 ≤ 2, (𝑠1𝑖 𝜎)(𝑡1 , . . . , 2𝑡 𝑖 + 𝑡 𝑘+1 − 2, . . . , 𝑡 𝑘 ), 2𝑡 𝑖 + 𝑡 𝑘+1 ≥ 2,

produces a chain homotopy operator 𝐾 ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘+1 (𝑋) such that ±𝐾𝜕 ± 𝜕𝐾 = Id −div𝑖 . Therefore div𝑖 , and hence div, yield the identity in homology 𝐻𝑘 (𝑋). Now, given a 𝑘-cube 𝜎, the Lebesgue covering lemma says that there is some 𝑡 > 0 𝑡 such that div (𝜎) has all of its 𝑘-cubes included in some 𝑈𝛼 of the covering. This 𝑡 > 0 depends on the 𝑘-cube but, since they are the same element in homology, we can define ∞ ∞ 𝑡 the map div ∶ 𝐻𝑘 (𝑋) → 𝐻𝑘 (𝑋; 𝔘) by div [𝜎] = [div 𝜎], for 𝑡 large enough. From the ∞ previous discussion, div is an isomorphism. □ Let 𝑋 be a topological space and suppose that we have open sets 𝑈, 𝑉 such that 𝑋 = 𝑈 ∪ 𝑉. Then we can relate the homology of 𝑈 ∩ 𝑉, 𝑈 and 𝑉 to that of 𝑋. For this, consider the covering 𝔘 = {𝑈, 𝑉}, and the inclusions 𝑖1 ∶ 𝑈 → 𝑋, 𝑖2 ∶ 𝑉 → 𝑋, 𝑗1 ∶ 𝑈 ∩ 𝑉 → 𝑈 and 𝑗2 ∶ 𝑈 ∩ 𝑉 → 𝑉. It is a straightforward to check that 0 → 𝐶• (𝑈 ∩ 𝑉)

((𝑗1 )∗ ,−(𝑗2 )∗ )

/ 𝐶• (𝑈) ⊕ 𝐶• (𝑉)

(𝑖1 )∗ +(𝑖2 )∗

/ 𝐶• (𝑋; 𝔘) → 0

is a short exact sequence of chain complexes. Using Proposition 2.88 and Proposition 2.89, we have the following result. Theorem 2.90 (Mayer-Vietoris). Let 𝑋 be a topological space and let 𝑈, 𝑉 ⊂ 𝑋 be open sets. Then, we have a long exact sequence for singular homology / 𝐻𝑘+1 (𝑈) ⊕ 𝐻𝑘+1 (𝑉)

/ 𝐻𝑘+1 (𝑋) JK =
1. Now suppose that 𝑛 ≥ 2. We work by induction on 𝑛. The Mayer-Vietoris exact sequence, with 𝐻𝑘 (𝑈) = 𝐻𝑘 (𝑉) = 0 for 𝑘 > 0 implies that for 𝑘 ≥ 2, 𝐻𝑘 (𝑆 𝑛 ) ≅ 𝐻𝑘−1 (𝑈 ∩ 𝑉) = 𝐻𝑘−1 (𝑆 𝑛−1 ). By induction hypothesis, 𝐻𝑛−1 (𝑆 𝑛−1 ) = ℤ and 𝐻𝑘 (𝑆𝑛−1 ) = 0 for 0 < 𝑘 < 𝑛 − 1. Hence 𝐻𝑛 (𝑆 𝑛 ) = ℤ and 𝐻𝑘 (𝑆𝑛 ) = 0 for 1 < 𝑘 < 𝑛. For 𝑘 = 1 the Mayer-Vietoris exact sequence reads 0 → 𝐻1 (𝑆 𝑛 ) → ℤ → ℤ ⊕ ℤ → ℤ → 0, from where we infer that 𝐻1 (𝑆 𝑛 ) = 0.

□

Remark 2.93. We call a fundamental class of 𝐻𝑛 (𝑆𝑛 ) a generator [𝑆 𝑛 ] of 𝐻𝑛 (𝑆 𝑛 ) ≅ ℤ. Remark 2.94. The homology is computable in the sense that the Mayer-Vietoris sequence allows us to compute the homology 𝐻𝑘 (𝑋) inductively, in terms of the open sets of a covering and the pattern of their intersections. There is no similar result for the homotopy groups 𝜋𝑛 (𝑋), 𝑛 ≥ 2 (see Remark 2.27).

2.4. Simplicial homology For a triangulated space, there is another approach to homology by means of a triangulation. This becomes an easier to compute invariant that turns out to be isomorphic to singular homology. 2.4.1. Construction of simplicial homology. Definition 2.95. Let (𝑋, 𝜏) be a triangulated space, with triangulation 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 ) | 𝛼 ∈ Λ𝑘 , 0 ≤ 𝑘 ≤ 𝑛} (Definition 1.46). Let us choose (arbitrarily) orientations 𝑜𝛼𝑘 for the polyhedron 𝑃𝛼𝑘 (cf. section 1.4.2), and call 𝑒𝑘𝛼 = (𝑃𝛼𝑘 , 𝑜𝛼𝑘 ) the oriented polyhedron. We define 𝐶𝑘𝜏 (𝑋) as the free Abelian group 𝐶𝑘𝜏 (𝑋) = ℤ⟨𝑒𝑘𝛼 | 𝛼 ∈ Λ𝑘 ⟩. 𝜏 We also define the boundary group morphisms 𝜕𝑘 ∶ 𝐶𝑘𝜏 (𝑋) → 𝐶𝑘−1 (𝑋) by 𝑘−1 𝜕𝑘 (𝑒𝑘𝛼 ) = ∑ [𝑒𝑘𝛼 ∶ 𝑒𝑘−1 , 𝛽 ] 𝑒𝛽 𝛽∈Λ𝑘

2.4. Simplicial homology

95

𝑘 where [𝑒𝑘𝛼 ∶ 𝑒𝑘−1 𝛽 ] are called the incidence numbers, and they are defined by [𝑒 𝛼 ∶ 𝑘−1 𝑒𝑘−1 ↪ 𝜕𝑃𝛼𝑘 and 𝜖𝑖 = ±1 𝛽 ] = ∑ 𝜖𝑖 , the sum running over all inclusions 𝐿𝑖 ∶ 𝑃𝛽 𝑘−1 according to whether the orientation 𝑜𝛽 agrees or not with the orientation induced as boundary on 𝐿𝑖 (𝑃𝛽𝑘−1 ) ⊂ 𝑃𝛼𝑘 .

Remark 2.96. • As for singular homology, we declare 𝐶𝑘𝜏 (𝑋) = 0 for 𝑘 < 0 and 𝑘 > 𝑛, and 𝜕𝑘 ≡ 0 for 𝑘 > 𝑛 and 𝑘 ≤ 0. • For a regular triangulation, we have that there is only one inclusion 𝑃𝛽𝑘−1 ⊂ 𝜕𝑃𝛼𝑘 . Therefore [𝑒𝑘𝛼 ∶ 𝑒𝑘−1 𝛽 ] = ±1. • In a regular triangulation, all 𝑃𝛼𝑘 are 𝑘-simplices (the convex hull of 𝑘 + 1 affinely independent points in ℝ𝑘 ). Originally, simplicial homology was developed in this situation, thus its name. • The choice of orientation for each 𝑃𝛼𝑘 is not relevant. Changing its orientation amounts to changing the generator 𝑒𝑘𝛼 by −𝑒𝑘𝛼 . Proposition 2.97. The boundary morphisms satisfy 𝜕𝑘−1 ∘ 𝜕𝑘 = 0 for all 𝑘 ≥ 0. Proof. We shall prove this in the case of a regular triangulation, where the notation is easier. We check it on the elements of the basis of 𝐶𝑘𝜏 (𝑋). Given an oriented polyhedron 𝑒𝑘𝛼 , we have 𝑘−1 𝜕𝑘−1 ∘ 𝜕𝑘 (𝑒𝑘𝛼 ) = 𝜕𝑘−1 (∑[𝑒𝑘𝛼 ∶ 𝑒𝑘−1 𝛽 ] 𝑒𝛽 ) 𝛽

(2.9)

𝑘−1 𝑘−2 = ∑ ( ∑[𝑒𝑘𝛼 ∶ 𝑒𝑘−1 . ∶ 𝑒𝑘−2 𝛾 ]) 𝑒 𝛾 𝛽 ][𝑒 𝛽 𝛾

𝛽

Take any oriented (𝑘−2)-polyhedron 𝑒𝑘−2 included in 𝑒𝑘𝛼 . Then 𝑒𝑘−2 ⊂ 𝜕𝑒𝑘𝛼 in a unique 𝛾 𝛾 𝑘−1 𝑘−1 way. So, there are exactly two oriented (𝑘−1)-polyhedra 𝑒 𝛽 and 𝑒 𝛽′ such that 𝑒𝑘−2 ⊂ 𝛾 𝑘−1 𝑘−1 𝑘−1 𝑘−1 𝑘−1 𝑘−1 𝑘 𝑘 𝑘 𝑘−2 𝑘−2 𝑒 𝛽 , 𝑒 𝛽′ ⊂ 𝑒 𝛼 . If [𝑒 𝛼 ∶ 𝑒 𝛽 ] = [𝑒 𝛼 ∶ 𝑒 𝛽′ ], then [𝑒 𝛽 ∶ 𝑒 𝛾 ] = −[𝑒 𝛽′ ∶ 𝑒 𝛾 ] and 𝑘−1 𝑘 the two summands in (2.9) cancel. On the other hand, if [𝑒𝑘𝛼 ∶ 𝑒𝑘−1 𝛽 ] = −[𝑒 𝛼 ∶ 𝑒 𝛽 ′ ], 𝑘−1 𝑘−1 𝑘−2 𝑘−2 then [𝑒 𝛽 ∶ 𝑒 𝛾 ] = [𝑒 𝛽′ ∶ 𝑒 𝛾 ] and again the two summands cancel. In any case, (2.9) is zero. □ Definition 2.98. Let (𝑋, 𝜏) be a triangulated space. The 𝑘th simplicial homology group is the Abelian group 𝐻𝑘𝜏 (𝑋) defined by 𝜏 𝑍𝑘𝜏 (𝑋) = ker(𝜕𝑘 ∶ 𝐶𝑘𝜏 (𝑋) → 𝐶𝑘−1 (𝑋)), 𝜏 𝐵𝑘𝜏 (𝑋) = im(𝜕𝑘+1 ∶ 𝐶𝑘+1 (𝑋) → 𝐶𝑘𝜏 (𝑋)),

𝐻𝑘𝜏 (𝑋) = 𝑍𝑘𝜏 (𝑋)/𝐵𝑘𝜏 (𝑋). Example 2.99. Simplicial homology has the advantage of being more computable than singular homology. Indeed, if the space is compact, then Λ𝑘 is finite and the computation of simplicial homology is a combinatorial issue. Let us compute the simplicial homology of surfaces. Consider a connected surface 𝑆. For 𝑆 = 𝑆 2 , we already know its

96

2. Algebraic topology

singular homology by Corollary 2.92. So suppose that 𝑆 = Σ𝑔 or 𝑋𝑘 , 𝑔, 𝑘 ≥ 1. Consider a planar representation P𝑆 given by a word 𝐩 in canonical form. Let 𝑎1 , . . . , 𝑎𝑟 be the edges of P𝑆 , and let 𝑣 be the vertex of the triangulation. Then 𝐶0𝜏 (𝑆) = ℤ⟨𝑣⟩,

𝐶1𝜏 (𝑆) = ℤ⟨𝑎1 , . . . , 𝑎𝑟 ⟩,

𝐶2𝜏 (𝑆) = ℤ⟨𝐴⟩,

where 𝐴 is the 2-polyhedron given by the polygon. The boundary maps are 𝜕1 (𝑎𝑗 ) = 𝑣 − 𝑣 = 0,

𝑗 = 1, . . . , 𝑟,

𝜕2 (𝐴) = ab(𝐩),

where ab(𝐩) ∈ 𝐶1𝜏 (𝑆) = ℤ⟨𝑎1 , . . . , 𝑎𝑟 ⟩ is the word 𝐩 written in additive (Abelian) notation. −1 • If 𝑆 = Σ𝑔 , 𝑔 ≥ 1, then 𝐶1𝜏 (Σ𝑔 ) = ℤ⟨𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 ⟩ and 𝐩 = 𝑎1 𝑏1 𝑎−1 1 𝑏1 ⋯ −1 𝑎𝑔 𝑏𝑔 𝑎−1 𝑔 𝑏𝑔 , so ab(𝐩) = 0. This implies that

𝐻0𝜏 (Σ𝑔 ) = ℤ⟨𝑣⟩ ≅ ℤ, 𝐻1𝜏 (Σ𝑔 ) = ℤ⟨𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 ⟩ ≅ ℤ2𝑔 , 𝐻2𝜏 (Σ𝑔 ) = ℤ⟨𝐴⟩ ≅ ℤ. Recall that Ab(𝜋1 (Σ𝑔 )) = ℤ2𝑔 , thus confirming Theorem 2.73. • If 𝑆 = 𝑋𝑘 , 𝑘 ≥ 1, then 𝐶1𝜏 (𝑋𝑘 ) = ℤ⟨𝑎1 , . . . , 𝑎𝑘 ⟩, 𝐩 = 𝑎1 𝑎1 ⋯ 𝑎𝑘 𝑎𝑘 and ab(𝐩) = 2𝑎1 + ⋯ + 2𝑎𝑘 ≠ 0. This implies that 𝑍2𝜏 (𝑋𝑘 ) = 0 and 𝐵1𝜏 (𝑋𝑘 ) = ⟨2𝜔⟩, with 𝜔 = 𝑎1 + ⋯ + 𝑎𝑘 . So 𝐻0𝜏 (𝑋𝑘 ) = ℤ⟨𝑣⟩ ≅ ℤ, 𝐻1𝜏 (𝑋𝑘 ) = ℤ⟨𝑎1 , . . . , 𝑎𝑘 ⟩/⟨2𝜔⟩ ≅ ℤ2 × ℤ𝑘−1 , 𝐻2𝜏 (𝑋𝑘 ) = 0. Again observe that Ab(𝜋1 (𝑋𝑘 )) = ℤ2 × ℤ𝑘−1 . • We can also do the case of 𝑆 = 𝑆 2 using the word 𝐩 = 𝑎𝑎−1 . In this case 𝐶0𝜏 (𝑆 2 ) = ℤ⟨𝑝, 𝑞⟩, 𝐶1𝜏 (𝑆2 ) = ℤ⟨𝑎⟩, and 𝐶2𝜏 (𝑆2 ) = ℤ⟨𝐴⟩, with 𝜕1 (𝑎) = 𝑝 − 𝑞 and 𝜕2 (𝐴) = 𝑎 − 𝑎 = 0. Hence ℤ⟨𝑝, 𝑞⟩ ≅ ℤ, ⟨𝑝 − 𝑞⟩ 𝐻1𝜏 (𝑆 2 ) = 0, 𝐻0𝜏 (𝑆 2 ) =

𝐻2𝜏 (𝑆 2 ) = ℤ⟨𝐴⟩ ≅ ℤ. Remark 2.100. The previous example fits into a collection of general results. (1) If 𝑋 is a triangulated space of dimension 𝑛, then 𝐻𝑘𝜏 (𝑋) = 0 for 𝑘 > 𝑛, since 𝐶𝑘𝜏 (𝑋) = 0. This is in contrast with homotopy groups, where it may be 𝜋𝑘 (𝑋) ≠ 0 for 𝑘 > dim 𝑋 (see Remark 2.27). (2) If 𝑋 is triangulated of dimension 𝑛, then 𝐻𝑛𝜏 (𝑋) is a free Abelian group. Certainly, 𝐵𝑛𝜏 (𝑋) = 0, hence 𝐻𝑛𝜏 (𝑋) coincides with 𝑍𝑛𝜏 (𝑋) ⊂ 𝐶𝑛𝜏 (𝑋), which is a free Abelian group being a subgroup of a free Abelian group. (3) If 𝑋 is a compact triangulated space, then 𝐻𝑘𝜏 (𝑋) is a finitely generated Abelian group. This is a consequence of the fact that 𝐶𝑘𝜏 (𝑋) ≅ ℤ𝑛𝑘 , 𝑛𝑘 = |Λ𝑘 |,

2.4. Simplicial homology

97

is finitely generated, and taking kernels, images, and quotients preserve this condition. (4) If 𝑀 is a compact, connected, oriented (triangulated) 𝑛-manifold without boundary, then 𝐻𝑛𝜏 (𝑀) = ℤ. Since 𝑀 is compact, the triangulation has a finite number of 𝑛-cells, let us call them 𝑒𝑛1 , . . . , 𝑒𝑛𝑟 . As 𝑀 is orientable, we can orient the 𝑛-polyhedra with compatible orientations. Thus 𝐶𝑛𝜏 (𝑀) = ℤ⟨𝑒𝑛1 , . . . , 𝑒𝑛𝑟 ⟩. Consider a general element 𝜔 = 𝑚1 𝑒𝑛1 + . . . + 𝑚𝑟 𝑒𝑛𝑟 . Then (2.10)

𝑛−1 𝜕𝜔 = ∑ ∑ 𝑚𝑖 [𝑒𝑛𝑖 ∶ 𝑒𝑛−1 . 𝛾 ] 𝑒𝛾 𝛾

𝑖

𝑛 𝑛 For any (𝑛−1)-polyhedron 𝑒𝑛−1 𝛾 , there are two 𝑛-polyhedra 𝑒 𝑖1 , 𝑒 𝑖2 incident on it (which may be the same). By the compatibility of the orientations, we have 𝑛 𝑛−1 that [𝑒𝑛𝑖1 ∶ 𝑒𝑛−1 𝛾 ] = −[𝑒 𝑖2 ∶ 𝑒 𝛾 ]. Therefore 𝜕𝜔 = 0 if and only if 𝑚𝑖1 = 𝑚𝑖2 𝑛 𝑛 for all pairs of incident 𝑒 𝑖1 , 𝑒 𝑖2 . The connectedness of 𝑋 implies that 𝑚𝑖 = 𝑚𝑗 for all 𝑖, 𝑗. Hence the cycles must have the form 𝜔 = 𝑚(𝑒𝑛1 + ⋯ + 𝑒𝑛𝑟 ), for some 𝑚 ∈ ℤ. This proves that

𝐻𝑛𝜏 (𝑀) = ℤ⟨[𝑀]⟩ ≅ ℤ, where [𝑀] = 𝑒𝑛1 + ⋯ + 𝑒𝑛𝑟 . This is called the fundamental class of 𝑀. It covers 𝑀 once and in a positively oriented way. This agrees with Remark 2.93 for 𝑀 = 𝑆𝑘 . (5) If 𝑀 is a compact, connected, non-orientable (triangulated) 𝑛-manifold without boundary, then 𝐻𝑛𝜏 (𝑀) = 0. Now 𝐶𝑛𝜏 (𝑀) = ℤ⟨𝑒𝑛1 , . . . , 𝑒𝑛𝑟 ⟩, where 𝑒𝑛𝑖 are the 𝑛-polyhedra oriented in any (arbitrary, non-compatible) way. Take 𝜔 = 𝑚1 𝑒𝑛1 + ⋯ + 𝑚𝑟 𝑒𝑛𝑟 with 𝜕𝜔 = 0. By (2.10), we have that 𝑚𝑖1 [𝑒𝑛𝑖1 ∶ 𝑛 𝑛 𝑛 𝑛−1 𝑒𝑛−1 𝛾 ] + 𝑚𝑖2 [𝑒 𝑖2 ∶ 𝑒 𝛾 ] = 0, for any pair 𝑒 𝑖1 , 𝑒 𝑖2 of adjacent 𝑛-polyhedra. This implies in particular that all absolute values |𝑚𝑖 | are equal, say, to some 𝑚 ∈ ℤ≥0 . Suppose that 𝑚 ≠ 0. If 𝑚𝑖 < 0, we change the orientation of 𝑒𝑛𝑖 to make it positive. Therefore we have that 𝜔 = 𝑚(𝑒𝑛1 + ⋯ + 𝑒𝑛𝑟 ), and also 𝑛 𝑛 𝑛 𝑛−1 [𝑒𝑛𝑖1 ∶ 𝑒𝑛−1 𝛾 ] = −[𝑒 𝑖2 ∶ 𝑒 𝛾 ], for any pair 𝑒 𝑖1 , 𝑒 𝑖2 of adjacent 𝑛-polyhedra, implying that 𝑀 is orientable. This contradiction concludes our assertion. (6) If 𝑀 is a compact, connected 𝑛-manifold with (non-empty) boundary, then 𝐻𝑛𝜏 (𝑀) = 0. In this case, if 𝜔 = 𝑚1 𝑒𝑛1 + ⋯ + 𝑚𝑟 𝑒𝑛𝑟 ∈ 𝐶𝑛𝜏 (𝑀) is a non-zero cycle, then (2.10) implies first that all |𝑚𝑖 | = 𝑚 > 0, for all 𝑖. Now if we take any 𝑒𝑛−1 at the boundary, there is a unique 𝑒𝑛𝑖 incident on it. Therefore the 𝛾 coefficient of 𝑒𝑛−1 in (2.10) is 𝑚𝑖 [𝑒𝑛𝑖 ∶ 𝑒𝑛−1 𝛾 𝛾 ], which is non-zero. Therefore 𝜏 𝐻𝑛 (𝑀) = 0. (7) Item (4) allows us to give an alternative definition of an orientation for a compact connected 𝑛-manifold 𝑀 without boundary. Namely, an orientation is a choice of a generator [𝑀] ∈ 𝐻𝑛 (𝑀).

98

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We have given two concepts of homology. They are in fact the same invariant. Theorem 2.101. Let (𝑋, 𝜏) be a triangulated space. Then 𝐻𝑘𝜏 (𝑋) ≅ 𝐻𝑘 (𝑋), for all 𝑘 ≥ 0. Proof. Let 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 ) | 𝛼 ∈ Λ𝑘 , 0 ≤ 𝑘 ≤ 𝑛} be the triangulation of 𝑋. Define the 𝑘skeleton as 𝑘

𝑋𝑘 =

⋃

𝑓𝛼 (𝑃𝛼𝑟 ).

𝑟=0 ∘

Recall that we call 𝐶𝛼𝑘 = 𝑓𝛼 (𝑃𝛼𝑘 ) the 𝑘-cells of 𝑋. The interior of the 𝑘-cell is 𝐶 𝑘𝛼 = ∘

𝑓𝛼 (Int(𝑃𝛼𝑘 )). Let 𝑝𝛼 ∈ 𝐶 𝑘𝛼 be the barycenter of 𝐶𝛼𝑘 . Then 𝐶𝛼𝑘 − {𝑝𝛼 } retracts to 𝜕𝐶𝛼𝑘 = ∘

𝑓𝛼 (𝜕𝑃𝛼𝑘 ), and 𝐶 𝑘𝛼 − {𝑝𝛼 } retracts to a (𝑘 − 1)-sphere 𝑆𝛼𝑘−1 = 𝑆𝜖𝑘−1 (𝑝𝛼 ). Consider the open subsets, 𝑈 = 𝑋𝑘 − {𝑝𝛼 | 𝛼 ∈ Λ𝑘 } ∼ 𝑋𝑘−1 ,

𝑉=

∘

𝑘 ∼ {𝑝𝛼 | 𝛼 ∈ Λ𝑘 }, ⨆ 𝐶𝛼

𝛼∈Λ𝑘

𝑈 ∩𝑉 =

∘

⨆

𝛼∈Λ𝑘

𝑆 𝑘−1 , (𝐶 𝑘𝛼 − {𝑝𝛼 }) ∼ ⨆ 𝛼

𝑈 ∪ 𝑉 = 𝑋𝑘 .

𝛼∈Λ𝑘

Applying the Mayer-Vietoris exact sequence to {𝑈, 𝑉}, we get isomorphisms 𝐻𝑟 (𝑋𝑘 ) ≅ 𝐻𝑟 (𝑋𝑘−1 ) for 𝑟 ≠ 𝑘, 𝑘 − 1, and an exact sequence 0 → 𝐻𝑘 (𝑋𝑘−1 ) → 𝐻𝑘 (𝑋𝑘 ) →

⨁

𝐻𝑘−1 (𝑆𝛼𝑘−1 )

𝛼∈Λ𝑘 𝑖∗

⟶ 𝐻𝑘−1 (𝑋𝑘−1 ) → 𝐻𝑘−1 (𝑋𝑘 ) → 0. Recall that by Corollary 2.92, 𝐻𝑘−1 (𝑆𝛼𝑘−1 ) ≅ ℤ⟨[𝑆𝛼𝑘−1 ]⟩ = 𝐶𝑘𝜏 (𝑋𝑘 ), and note that 𝜕(𝑒𝑘𝛼 ) = 𝑖∗ [𝑆𝛼𝑘−1 ]. Working by induction on the skeleton, we can assume that 𝐻𝑟𝜏 (𝑋𝑘−1 ) ≅ 𝜏 𝐻𝑟 (𝑋𝑘−1 ) for all 𝑟, and in particular, 𝐻𝑘 (𝑋𝑘−1 ) = 0. Also 𝐻𝑘−1 (𝑋𝑘−1 ) = 𝐻𝑘−1 (𝑋𝑘−1 ) = 𝜏 𝑍𝑘−1 (𝑋𝑘−1 ). Hence we have an exact sequence 𝜕

𝜏 0 → 𝐻𝑘 (𝑋𝑘 ) → 𝐶𝑘𝜏 (𝑋𝑘 ) ⟶ 𝑍𝑘−1 (𝑋𝑘−1 ) → 𝐻𝑘−1 (𝑋𝑘 ) → 0, 𝜏 𝜏 𝜏 from where 𝐻𝑘−1 (𝑋𝑘 ) ≅ 𝑍𝑘−1 (𝑋𝑘−1 )/𝐵𝑘−1 (𝑋𝑘−1 ) ≅ 𝐻𝑘−1 (𝑋𝑘−1 ), and 𝐻𝑘 (𝑋𝑘 ) ≅ 𝑍𝑘𝜏 (𝑋𝑘 ) = 𝜏 𝐻𝑘 (𝑋𝑘 ). □

Remark 2.102. Theorem 2.101 implies that the simplicial homology 𝐻𝑘𝜏 (𝑋) does not depend on the triangulation of 𝑋. In Exercise 2.23 we see that the homology 𝐻𝑘𝜏 (𝑋) is independent of the triangulation for a given PL structure. However, Theorem 2.101 is stronger: if a topological space 𝑋 admits two inequivalent triangulations, then the simplicial homologies are still equal. Remark 2.103. Theorem 2.29 can be proved by using the homology of surfaces. By Theorem 2.101 and the computations of Example 2.99, we see that the first homology 𝑐𝑜 𝑐𝑜 𝐻1 ∶ 𝕃𝐓𝐌𝐚𝐧2 → 𝐀𝐛𝐞𝐥 is injective. Therefore 𝕃𝐓𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 |𝑔 ≥ 0, 𝑘 ≥ 1}. 𝑐

𝑐

Triangulated spaces can be generalized to a very important wider class of spaces as follows.

2.4. Simplicial homology

99

Definition 2.104. A CW-complex is a topological space 𝑋 with a filtration 𝑋0 ⊂ 𝑋1 ⊂ 𝑋2 ⊂ ⋯ by closed subsets where 𝑋 = ⋃𝑘≥0 𝑋𝑘 . The spaces 𝑋𝑘 are called 𝑘-skeleta of 𝑋. The 0-skeleton 𝑋0 is a discrete set of points. The 𝑘-skeleta are defined inductively for 𝑘 > 0 as 𝑋𝑘 = 𝑋𝑘−1 ∪ (⋃𝜑 𝐷𝛼𝑘 ), where 𝑋𝑘 is obtained from 𝑋𝑘−1 by attaching 𝑘-cells 𝛼

(Example 1.33(2)) via continuous maps 𝜑𝛼 ∶ 𝜕𝐷𝛼𝑘 → 𝑋𝑘−1 , 𝛼 ∈ Λ𝑘 . We call 𝑘-cell of 𝑋 to the image of 𝑖𝛼 ∶ 𝐷𝛼𝑘 → 𝑋𝑘 ⊂ 𝑋. We give 𝑋 the weak topology, that is, the topology coherent with all the 𝑘-cells: a subset 𝑈 ⊂ 𝑋 is open if and only if 𝑖𝛼−1 (𝑈) ⊂ 𝐷𝛼𝑘 is open for all 𝛼 and all 𝑘 ≥ 0 (see Definition 1.42). A triangulated space is an example of a CW-complex. If a CW-complex satisfies that 𝑋 = 𝑋𝑛 , for some 𝑛, we say that 𝑋 is of dimension 𝑛 (with 𝑛 the minimum such that 𝑋 = 𝑋𝑛 ). However, the dimension of a CW-complex may be ∞ if there are cells of arbitrarily large dimension. Cellular homology is defined for a CW-complex similarly to simplicial homology. For this, we declare 𝐶𝑘𝐶𝑊 (𝑋) = ⨁𝛼∈Λ ℤ⟨𝐷𝛼𝑘 ⟩ and the boundary map is 𝜕(𝐷𝛼𝑘 ) = 𝑘

∑𝛽 [𝐷𝛼𝑘 ∶ 𝐷𝛽𝑘−1 ] 𝐷𝛽𝑘−1 , defined by taking (𝜑𝛼 )∗ ([𝜕𝐷𝛼𝑘 ]) ∈ 𝐻𝑘−1 (𝑋𝑘−1 ), and pushing 𝑘−1 down to 𝑋𝑘−1 /𝑋𝑘−2 = ⋁𝛽 𝑆 𝑘−1 = 𝐷𝛽𝑘−1 /𝜕𝐷𝛽𝑘−1 , to get finally an element 𝛽 , where 𝑆 𝛽 𝑘−1 in (𝜑𝛼 )∗ ([𝜕𝐷𝛼𝑘 ]) ∈ 𝐻𝑘−1 (⋁𝛽 𝑆 𝑘−1 𝛽 ) ≅ ⨁𝛽 ℤ⟨𝐷𝛽 ⟩. It can be proven that 𝐻𝑘 (𝑋) ≅

𝐻𝑘𝐶𝑊 (𝑋) for any CW-complex. Remark 2.105. The most suitable category to do algebraic topology is that of spaces of the same homotopy type as CW-complexes. This is so because of the following results of Whitehead. • If 𝑓 ∶ 𝑋 → 𝑌 is a weak homotopy equivalence between connected CWcomplexes (cf. Definition 2.9), that is if 𝑓∗ ∶ 𝜋𝑛 (𝑋) → 𝜋𝑛 (𝑌 ) is an isomorphism for all 𝑛 ≥ 1, then 𝑋 ∼ 𝑌 (where 𝑓 is the homotopy equivalence). • If 𝑋, 𝑌 are simply connected CW-complexes and 𝑓 ∶ 𝑋 → 𝑌 induces isomorphisms 𝑓∗ ∶ 𝐻𝑛 (𝑋) → 𝐻𝑛 (𝑌 ) for all 𝑛 ≥ 2, then 𝑓 is a weak homotopy equivalence and hence a homotopy equivalence. 2.4.2. Homology with other coefficients. We want to introduce homology (both for singular homology and for simplicial homology) with coefficients in any commutative ring. Let us first recall the concept of tensor product from commutative algebra. Consider a ground ring 𝐴, and focus on the category Mod𝐴 consisting of 𝐴-modules (this is also an Abelian category in the sense of section 2.3.2). Given two 𝐴-modules 𝑀, 𝑁, the tensor product, denoted 𝑀 ⊗𝐴 𝑁 is the 𝐴-module satisfying the following properties. • There exists an 𝐴-bilinear mapping Ψ ∶ 𝑀 × 𝑁 → 𝑀 ⊗𝐴 𝑁. We denote 𝑚 ⊗ 𝑛 = Ψ(𝑚, 𝑛), for 𝑚 ∈ 𝑀, 𝑛 ∈ 𝑁. • For any 𝐴-module 𝑃 and any bilinear map 𝜑 ∶ 𝑀 × 𝑁 → 𝑃, there exists a unique 𝐴-module homomorphism 𝜑̄ ∶ 𝑀 ⊗𝐴 𝑁 → 𝑃 such that 𝜑 = 𝜑̄ ∘ Ψ. By the very definition, the space of bilinear maps 𝑀 × 𝑁 → 𝑃 is isomorphic to Hom𝐴 (𝑀 ⊗𝐴 𝑁, 𝑃). Since this is a universal property, the tensor product, if it exists,

100

2. Algebraic topology

is unique up to isomorphism. The proof of existence of the tensor product is constructive. We define 𝑀 ⊗𝑅 𝑁 = 𝐴⟨𝑀 × 𝑁⟩/𝐼, where 𝐴⟨𝑀 × 𝑁⟩ is the free 𝐴-module generated by 𝑀 × 𝑁, that is, its elements are formal finite sums ∑ 𝑟 𝑖 (𝑚𝑖 , 𝑛𝑖 ) with 𝑟 𝑖 ∈ 𝐴, 𝑚𝑖 ∈ 𝑀, and 𝑛𝑖 ∈ 𝑁, and 𝐼 is the submodule of 𝐴⟨𝑀 × 𝑁⟩ generated by the elements: (𝑚1 + 𝑚2 , 𝑛) − (𝑚1 , 𝑛) − (𝑚2 , 𝑛), (𝑚, 𝑛1 + 𝑛2 ) − (𝑚, 𝑛1 ) − (𝑚, 𝑛2 ), (𝑟𝑚, 𝑛) − 𝑟(𝑚, 𝑛) and (𝑚, 𝑟𝑛) − 𝑟(𝑚, 𝑛), where 𝑚, 𝑚1 , 𝑚2 ∈ 𝑀, 𝑛, 𝑛1 , 𝑛2 ∈ 𝑁, 𝑟 ∈ 𝐴. The elements of 𝑀 ⊗𝐴 𝑁 are thus finite sums ∑𝑖 𝑚𝑖 ⊗ 𝑛𝑖 with 𝑚𝑖 ∈ 𝑀 and 𝑛𝑖 ∈ 𝑁 with relations of the type (𝑟𝑚) ⊗ 𝑛 = 𝑟(𝑚 ⊗ 𝑛), and (𝑚1 + 𝑚2 ) ⊗ 𝑛 = 𝑚1 ⊗ 𝑛 + 𝑚2 ⊗ 𝑛. Remark 2.106. (1) If the base ground ring is 𝐴 = ℤ, then 𝐴-modules are just Abelian groups. (2) When the ground ring is clear from the context, we will denote 𝑀 ⊗ 𝑁 = 𝑀 ⊗𝐴 𝑁. Always 𝐴 ⊗ 𝑁 = 𝑁. (3) If there is a ring homomorphism 𝑓 ∶ 𝐴 → 𝑅, then 𝑅 has an 𝐴-module structure via 𝑎 𝑟 = 𝑓(𝑎)𝑟, for 𝑎 ∈ 𝐴, 𝑟 ∈ 𝑅. Moreover, if 𝑀 is any 𝐴-module, the 𝑀 ⊗𝐴 𝑅 is endowed with an 𝑅-module structure via 𝑟(𝑚 ⊗ 𝑟′ ) = 𝑚 ⊗ (𝑟𝑟′ ). This process is called extension of scalars. In particular, since ℤ is an initial object of 𝐑𝐢𝐧𝐠, we can extend scalars for any Abelian group to become an 𝑅-module. (4) If we have ring homomorphisms 𝐴 → 𝑅 → 𝑅′ , then (𝑀 ⊗𝐴 𝑅) ⊗𝑅 𝑅′ ≅ 𝑀𝐴 ⊗ 𝑅′ . (5) In the case where 𝑅 = 𝐤 is a field of zero characteristic (e.g., 𝐤 = ℚ, ℝ, ℂ), then ℤ⊗𝐤 = 𝐤. Also, ℤ𝑚 ⊗𝐤 = 0, since any element 𝑥 ⊗𝜆 ∈ ℤ𝑚 ⊗𝐤 satisfies 1 𝑥 ⊗ 𝜆 = 𝑚𝑥 ⊗ 𝑚 𝜆 = 0. (6) Let 𝑀 be a finitely generated Abelian group. Then 𝑀 = ℤ𝑟 ⊕ ℤ𝑚1 ⊕ ⋯ ⊕ ℤ𝑚𝑠 , where 𝑟 is the rank of 𝑀 and 𝑇 = ℤ𝑚1 ⊕ ⋯ ⊕ ℤ𝑚𝑟 is the torsion part. Then 𝑀 ⊗ 𝐤 = (ℤ𝑟 ⊕ 𝑇) ⊗ 𝐤 = 𝐤𝑟 . In particular, rank 𝑀 = dim(𝑀 ⊗ 𝐤). (7) If 𝑀 is an Abelian group and we take 𝑅 = ℚ, then 𝑀 ⊗ℚ is the rationalization 𝑚 of 𝑀, that is 𝑀 ⊗ ℚ ≅ 𝑀ℚ , where 𝑀ℚ = { 𝑞 |𝑚 ∈ 𝑀, 𝑞 ∈ ℤ − {0}}, satisfying that

𝑚 𝑞

=

𝑚′ 𝑞′

if and only if there exists 𝑝 ∈ ℤ−{0} such that 𝑝 (𝑞𝑚′ −𝑞′ 𝑚) = 0.

Now let us fix a ring 𝑅 (our ground ring is 𝐴 = ℤ). Let 𝑋 be a topological space. We define homology with coefficients in 𝑅, denoted 𝐻𝑘 (𝑋, 𝑅), as follows. First we extend the scalars for chains by defining 𝐶𝑘 (𝑋, 𝑅) = 𝐶𝑘 (𝑋) ⊗ℤ 𝑅, which is then the free 𝑅module generated by (non-degenerate) 𝑘-cubes in 𝑋. By extension of scalars, we also have well defined operators 𝜕𝑘 ∶ 𝐶𝑘 (𝑋, 𝑅) → 𝐶𝑘−1 (𝑋, 𝑅). With this extension, 𝐶• (𝑋, 𝑅) has an 𝑅-module structure and 𝜕 is an 𝑅-linear map. We define 𝐻𝑘 (𝑋, 𝑅) =

ker(𝜕𝑘 ∶ 𝐶𝑘 (𝑋, 𝑅) → 𝐶𝑘−1 (𝑋, 𝑅)) . im(𝜕𝑘+1 ∶ 𝐶𝑘+1 (𝑋, 𝑅) → 𝐶𝑘 (𝑋, 𝑅))

This 𝐻𝑘 (𝑋, 𝑅) has a natural 𝑅-module structure, and it is usually called the singular homology of 𝑋 with coefficients in 𝑅. Clearly, 𝐻𝑘 (𝑋, ℤ) = 𝐻𝑘 (𝑋). The homology with coefficients in 𝑅 gives functors 𝐻𝑘 (−, 𝑅) ∶ 𝐇𝐨𝐓𝐨𝐩 ⟶ Mod𝑅 .

2.4. Simplicial homology

101

We also have simplicial homology with coefficients in 𝑅 for a triangulated space (𝑋, 𝜏), and an isomorphism 𝐻𝑘𝜏 (𝑋, 𝑅) ≅ 𝐻𝑘 (𝑋, 𝑅). Example 2.107. Let us compute the homology with coefficients in ℤ2 for compact connected surfaces. • For 𝑆 = 𝑆 2 by using the word 𝐩 = 𝑎𝑎−1 , we have 𝐶0𝜏 (𝑆2 , ℤ2 ) = ℤ2 ⟨𝑝, 𝑞⟩, 𝐶1𝜏 (𝑆 2 , ℤ2 ) = ℤ2 ⟨𝑎⟩, 𝐶2𝜏 (𝑆 2 , ℤ2 ) = ℤ2 ⟨𝐴⟩, with 𝜕1 (𝑎) = 𝑝 − 𝑞 and 𝑏2 (𝐴) = 𝑎 − 𝑎 = 0. Hence ℤ⟨𝑝, 𝑞⟩ 𝐻0𝜏 (𝑆 2 , ℤ2 ) = ≅ ℤ2 , 𝐻1𝜏 (𝑆 2 , ℤ2 ) = 0, 𝐻2𝜏 (𝑆 2 , ℤ2 ) = ℤ2 ⟨𝐴⟩ ≅ ℤ2 . ⟨𝑝 − 𝑞⟩ −1 −1 −1 • For 𝑆 = Σ𝑔 , 𝑔 ≥ 1, using the word 𝐩 = 𝑎1 𝑏1 𝑎−1 1 𝑏1 ⋯ 𝑎𝑔 𝑏𝑔 𝑎𝑔 𝑏𝑔 , we have 𝐶0𝜏 (Σ𝑔 , ℤ2 ) = ℤ2 ⟨𝑣⟩, 𝐶1𝜏 (Σ𝑔 , ℤ2 ) = ℤ2 ⟨𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 ⟩, 𝐶2𝜏 (Σ𝑔 , ℤ2 ) = ℤ2 ⟨𝐴⟩, with 𝜕1 ≡ 0, 𝜕2 (𝐴) = 0. Hence

𝐻0𝜏 (Σ𝑔 , ℤ2 ) = ℤ2 ⟨𝑣⟩ ≅ ℤ2 , 2𝑔

𝐻1𝜏 (Σ𝑔 , ℤ2 ) = ℤ2 ⟨𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 ⟩ ≅ ℤ2 , 𝐻2𝜏 (Σ𝑔 , ℤ2 ) = ℤ2 ⟨𝐴⟩ ≅ ℤ2 . • For 𝑆 = 𝑋𝑘 , 𝑘 ≥ 1, using the word 𝐩 = 𝑎1 𝑎1 . . . 𝑎𝑘 𝑎𝑘 , we have 𝐶0𝜏 (𝑋𝑘 , ℤ2 ) = ℤ2 ⟨𝑣⟩, 𝐶1𝜏 (𝑋𝑘 , ℤ2 ) = ℤ2 ⟨𝑎1 , . . . , 𝑎𝑘 ⟩, 𝐶2𝜏 (𝑋𝑘 , ℤ2 ) = ℤ2 ⟨𝐴⟩ with 𝜕1 ≡ 0, 𝜕2 (𝐴) = 2𝑎1 + . . . + 2𝑎𝑘 = 0. Hence 𝐻0𝜏 (𝑋𝑘 , ℤ2 ) = ℤ2 ⟨𝑣⟩ ≅ ℤ2 , 𝐻1𝜏 (𝑋𝑘 , ℤ2 ) = ℤ2 ⟨𝑎1 , . . . , 𝑎𝑘 ⟩ ≅ ℤ𝑘2 , 𝐻2𝜏 (𝑋𝑘 , ℤ2 ) = ℤ2 ⟨𝐴⟩ ≅ ℤ2 . With coefficients in ℤ2 , orientable and non-orientable manifolds do not behave differently. This is in contrast with Remark 2.100 for the homology 𝐻𝑘 (𝑋). It is a fact (called the universal coefficient theorem) that the homology 𝐻𝑘 (𝑋, 𝑅) can be computed from the homology 𝐻𝑘 (𝑋), and hence it does not contain more information. We shall not give the formula in full generality, but we give it in the case where 𝑅 = 𝐤 is a field of zero characteristic (like 𝐤 = ℚ, ℝ, ℂ), since this homology will be used later. Definition 2.108. Let 𝒜, ℬ be Abelian categories. A functor 𝐹 ∶ 𝒜 → ℬ is exact if 𝑓

𝐹(𝑓)

𝑔

𝐹(𝑔)

for any exact sequence 𝐴 → 𝐴′ → 𝐴″ in 𝒜, the induced sequence 𝐹(𝐴) ⟶ 𝐹(𝐴′ ) ⟶ 𝐹(𝐴″ ) is exact Remark 2.109. • If 𝐹 ∶ 𝒜 → ℬ is an exact functor, then it preserves kernels, images, cokernels, and quotients. Indeed, let 𝑓 ∶ 𝐴 → 𝐵 be a morphism in 𝒜. Then we have short exact sequences 0 → ker 𝑓 → 𝐴 → im 𝑓 → 0 and 0 → im 𝑓 → 𝐵 → coker 𝑓 → 0. Applying 𝐹, we have short exact sequences 0 → 𝐹(ker 𝑓) → 𝐹(𝐴) → 𝐹(im 𝑓) → 0 and 0 → 𝐹(im 𝑓) → 𝐹(𝐵) → 𝐹(coker 𝑓) → 0. This implies that we have an exact sequence 𝐹(𝑓)

0 → 𝐹(ker 𝑓) → 𝐹(𝐴) ⟶ 𝐹(𝐵) → 𝐹(coker 𝑓) → 0.

102

2. Algebraic topology

Therefore, ker 𝐹(𝑓) = 𝐹(ker 𝑓) and coker 𝐹(𝑓) = 𝐹(coker 𝑓). From the exact 𝜋

sequence 0 → im 𝑓 → 𝐵 → coker 𝑓 → 0, we have that im 𝑓 = ker 𝜋, thus im 𝐹(𝑓) = ker 𝐹(𝜋) = 𝐹(ker 𝜋) = 𝐹(im 𝑓). Finally, for quotients of 𝐴 ⊂ 𝐵, we have that 𝐵/𝐴 = coker 𝑖, for 𝑖 ∶ 𝐴 → 𝐵 the inclusion. So 𝐹(𝐵/𝐴) = 𝐹(coker 𝑖) = coker 𝐹(𝑖) = 𝐹(𝐵)/𝐹(𝐴). • If a functor preserves short exact sequences, then it is an exact functor. Because in such a case it preserves kernels and images, and thus in Definition 2.108 we would have im 𝐹(𝑓) = 𝐹(im 𝑓) = 𝐹(ker 𝑔) = ker 𝐹(𝑔). • If 𝐹 ∶ 𝒜 → ℬ is a functor and (𝐴• , 𝜕) is a complex, then (𝐹(𝐴• ), 𝐹(𝜕)) is a complex, since 𝐹(𝜕𝑘−1 ) ∘ 𝐹(𝜕𝑘 ) = 𝐹(𝜕𝑘−1 ∘ 𝜕𝑘 ) = 𝐹(0) = 0. If 𝐹 is an exact functor, then 𝐹 commutes with 𝐻𝑘 , ker 𝐹(𝜕𝑘 ) 𝐹 (ker 𝜕𝑘 ) ≅ im 𝐹(𝜕𝑘+1 ) 𝐹 (im 𝜕𝑘+1 ) ker 𝜕𝑘 ≅𝐹( ) = 𝐹 (𝐻𝑘 (𝐴• , 𝜕)) . im 𝜕𝑘+1

𝐻𝑘 (𝐹(𝐴• ), 𝐹(𝜕)) = (2.11)

Lemma 2.110. Let 𝐤 be a field of zero characteristic. Then, the functor − ⊗ 𝐤 ∶ 𝐀𝐛𝐞𝐥 → 𝐕𝐞𝐜𝐭𝐤 is an exact functor. 𝑓

𝑔

Proof. First we discuss the case 𝐤 = ℚ. Let 𝐴 → 𝐴′ → 𝐴″ be an exact sequence of 𝑓ℚ

𝑔ℚ

Abelian groups. Then we have maps of ℚ-vector spaces, 𝐴ℚ ⟶ 𝐴′ℚ ⟶ 𝐴″ℚ . We have to see that im(𝑓ℚ ) = ker(𝑔ℚ ). As 𝑔∘𝑓 = 0, we have 𝑔ℚ ∘𝑓ℚ = 0, hence im(𝑓ℚ ) ⊂ ker(𝑔ℚ ). 𝑏 𝑔(𝑏) Now take 𝑥 = 𝑞 ∈ ker(𝑔ℚ ) ⊂ 𝐴′ℚ . Then 𝑔ℚ (𝑥) = 𝑞 = 0, so there is some 𝑝 ∈ ℤ − {0} such that 𝑝 𝑔(𝑏) = 0. Hence 𝑔(𝑝𝑏) = 0, 𝑝𝑏 ∈ ker 𝑔 = im 𝑓, so there exists 𝑎 ∈ 𝐴 with 𝑏 𝑓(𝑎) 𝑎 𝑝𝑏 = 𝑓(𝑎) and thus 𝑥 = 𝑞 = 𝑝𝑞 = 𝑓ℚ ( 𝑝𝑞 ). Now for another field of zero characteristic, we start with a short exact sequence 0 → 𝐴→𝐴′ →𝐴″ → 0 of Abelian groups. So 0 → 𝐴ℚ →𝐴′ℚ →𝐴″ℚ → 0 is a short exact sequence of ℚ-vector spaces. But all short exact sequences of vector spaces split. So 𝐴′ℚ ≅ 𝐴ℚ ⊕ 𝐴″ℚ . We tensor with 𝐤 and note that 𝐴 ⊗ 𝐤 = (𝐴 ⊗ ℚ) ⊗ℚ 𝐤 = (𝐴ℚ ) ⊗ℚ 𝐤, so 𝐴′𝐤 ≅ 𝐴𝐤 ⊕ 𝐴″𝐤 , and hence 0 → 𝐴𝐤 →𝐴′𝐤 →𝐴″𝐤 → 0 is exact. □ As a consequence of (2.11), for a chain complex of Abelian groups (𝐴• , 𝜕) and a field 𝐤 of characteristic zero, we have 𝐻𝑘 (𝐴• ⊗ 𝐤, 𝜕 ⊗ 𝐤) ≅ 𝐻𝑘 (𝐴• , 𝜕) ⊗ 𝐤 . In particular, the 𝑘th singular homology of a topological space 𝑋 with coefficients in 𝐤 is (2.12)

𝐻𝑘 (𝑋, 𝐤) = 𝐻𝑘 (𝑋) ⊗ 𝐤 .

This is not true for fields like ℤ2 as shown in Example 2.107. 2.4.3. Betti numbers and the Euler-Poincaré characteristic. Definition 2.111. Let 𝑋 be a topological space. We define the 𝑘th Betti number of 𝑋 as 𝑏𝑘 = 𝑏𝑘 (𝑋) = rank 𝐻𝑘 (𝑋).

2.4. Simplicial homology

103

The Betti number is a topological invariant. Note that when 𝑋 is a compact triangulated space, the homology 𝐻𝑘 (𝑋) is finitely generated, and hence 𝑏𝑘 (𝑋) is finite. If 𝐤 is a field of characteristic zero, then 𝑏𝑘 (𝑋) = dim 𝐻𝑘 (𝑋, 𝐤), by (2.12). We introduce the Euler-Poincaré characteristic for topological spaces. We say that 𝑋 has finite homological dimension if there exists 𝑁 with 𝐻𝑘 (𝑋) = 0 for 𝑘 > 𝑁. We say that 𝑋 has finitely generated homology if it has finite homological dimension and 𝐻𝑘 (𝑋) are finitely generated Abelian groups. Definition 2.112. Let 𝑋 be a topological space with finitely generated homology, with 𝐻𝑘 (𝑋) = 0 for 𝑘 > 𝑁. Then the Euler-Poincaré characteristic of 𝑋 is 𝑁

𝜒(𝑋) = ∑ (−1)𝑘 𝑏𝑘 (𝑋). 𝑘=0

This gives an invariant 𝜒 ∶ 𝕃𝐓𝐨𝐩𝐟𝐠𝐡 → ℤ, where 𝐓𝐨𝐩𝐟𝐠𝐡 is the category of topological spaces with finitely generated homology. We want to check that this agrees with our previous Definition 1.63. Suppose that 𝑋 is a compact triangulated space with triangulation 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 )|𝛼 ∈ Λ𝑘 , 0 ≤ 𝑘 ≤ 𝑁}, and consider the complex of simplicial chains (𝐶•𝜏 (𝑋), 𝜕). We tensor it with a field 𝐤 of characteristic zero. Then the number of 𝑘-polyhedra is 𝑛𝑘 = |Λ𝑘 | = dim 𝐶𝑘𝜏 (𝑋, 𝐤). Denote for a while the PL Euler-Poincaré characteristic of Definition 1.63 and the topological Euler-Poincaré characteristic of Definition 2.112 as 𝑁

𝑁

𝜒PL (𝑋) = ∑ (−1)𝑘 𝑛𝑘 ,

𝜒Top (𝑋) = ∑ (−1)𝑘 𝑏𝑘 .

𝑘=0

𝑘=0

Lemma 2.113. Let (𝑉• , 𝜕) be a chain complex of finite dimensional 𝐤-vector spaces of the form 0 → 𝑉 𝑁 → ⋯ → 𝑉1 → 𝑉0 → 0. Then 𝑁

𝑁

∑ (−1)𝑘 dim 𝑉 𝑘 = ∑ (−1)𝑘 dim 𝐻𝑘 (𝑉• , 𝜕). 𝑘=0

𝑘=0

Proof. Let us denote 𝑛𝑘 = dim 𝑉 𝑘 , 𝑧𝑘 = dim 𝑍𝑘 (𝑉• ), 𝛽 𝑘 = dim 𝐵𝑘 (𝑉• ) and 𝑏𝑘 = dim 𝐻𝑘 (𝑉• ). The exact sequence 0 → 𝑍𝑘 (𝑉• ) → 𝑉 𝑘 → 𝐵𝑘−1 (𝑉• ) → 0 implies that 𝑛𝑘 = 𝑧𝑘 + 𝛽 𝑘−1 . The definition of homology 𝐻𝑘 (𝑉• ) = 𝑍𝑘 (𝑉• )/𝐵𝑘 (𝑉• ) implies that 𝑏𝑘 = 𝑧𝑘 − 𝛽 𝑘 . Now 𝑁

𝑁

𝑁

𝑁

∑ (−1)𝑘 𝑛𝑘 = ∑ (−1)𝑘 (𝑧𝑘 + 𝛽 𝑘−1 ) = ∑ (−1)𝑖 𝑧𝑖 − ∑ (−1)𝑗−1 𝛽𝑗−1 𝑘=0

𝑘=0 𝑁

𝑖=0

𝑗=0

𝑁

= ∑ (−1)𝑖 (𝑧𝑖 − 𝛽 𝑖 ) = ∑ (−1)𝑖 𝑏𝑖 , 𝑖=0

where we use 𝛽−1 = 𝛽𝑁 = 0.

𝑖=0

□

104

2. Algebraic topology

Lemma 2.113 applied to 𝑉 𝑘 = 𝐶𝑘𝜏 (𝑋, 𝐤) gives 𝑛

𝑛

𝜒PL (𝑋) = ∑ (−1)𝑘 𝑛𝑘 = ∑ (−1)𝑘 dim 𝐻𝑘𝜏 (𝑋, 𝐤) 𝑘=0

𝑘=0

𝑛

= ∑ (−1)𝑘 dim 𝐻𝑘 (𝑋, 𝐤) = 𝜒Top (𝑋), 𝑘=0

using Theorem 2.101 as well. We shall denote the Euler-Poincaré characteristic simply as 𝜒(𝑋) in the future. As a conclusion, the Euler-Poincaré characteristic is a topological invariant, that is, it does not depend on the triangulation. Another consequence is that the Euler-Poincaré characteristic is a homotopy invariant. If 𝑋 ∼ 𝑌 , then 𝜒(𝑋) = 𝜒(𝑌 ). Remark 2.114. We can rephrase the above by saying that the map (1.5) refines to 𝜒 ∶ 𝜒

𝕃𝐓𝐌𝐚𝐧𝑛𝑐 → ℤ, that is we have a commutative diagram 𝜒 ∶ 𝕃𝐏𝐋𝐌𝐚𝐧𝑛𝑐 → 𝕃𝐓𝐌𝐚𝐧𝑛𝑐 ⟶ ℤ. In particular, we can use the Euler-Poincaré characteristic to finish the classification theorem of topological compact surfaces (Theorem 2.29) as done for the triangulated surfaces (Remark 1.83). Remark 2.115. Historically, Betti defined initially 𝑏𝑘 (𝑋) as a refinement to 𝜒(𝑋) with triangulations. Later Poincaré realized that the ideas of Betti allowed us to define Abelian groups which were invariants of 𝑋 with rank 𝑏𝑘 (𝑋), thereby introducing simplicial homology. It was later that singular homology was introduced following the philosophy of throwing in all imaginable triangulations when you do not have a given triangulation at your disposal. This explains that 𝑘-cubes are all continuous maps 𝜎 ∶ [0, 1]𝑘 → 𝑋.

2.5. De Rham cohomology Now we move to the smooth category. We will see how we can recover the homology with the differentiable structure of a smooth manifold. This follows the original ideas of de Rham, who realized that the integration of differential forms on polyhedra allowed one to transfer the homology to the spaces of forms. The result is the de Rham cohomology. First we will notice that we are forced to use real coefficients 𝐤 = ℝ. Second, the resulting invariant is contravariant, that is it is a dual notion to that of homology (see Remark 2.11 for another instance of duality). On the positive side, de Rham cohomology will have a richer structure, as it has a product which makes it an algebra. Moreover, when the manifold is oriented and compact, this algebra satisfies stronger properties (see section 2.6). 2.5.1. Differential forms and the de Rham complex. Given a smooth manifold 𝑀, (𝑘)

a differential 𝑘-form 𝛼 is an antisymmetric multilinear map 𝛼𝑝 ∶ 𝑇𝑝 𝑀× ⋯ ×𝑇𝑝 𝑀 → ℝ, for each 𝑝 ∈ 𝑀, which is smooth (see section 3.1.1, and specifically Example 3.3(2)). Smoothness is described locally, in terms of a chart (𝑈, 𝜑), with coordinates 𝜑 = 𝜕 𝜕 (𝑥1 , . . . , 𝑥𝑛 ). Let ( 𝜕𝑥 , . . . , 𝜕𝑥 ) be the basis of coordinate vector fields in 𝑈. Then 𝛼 1

𝑛

𝜕

𝜕

is determined in 𝑈 by the images of the basis, 𝛼𝑖1 ⋯𝑖𝑘 (𝑥) = 𝛼 ( 𝜕𝑥 , . . . , 𝜕𝑥 ), and the 𝑖1

𝑖𝑘

requirement that 𝛼 is smooth means that 𝛼𝑖1 ⋯𝑖𝑘 ∈ 𝐶 ∞ (𝑈). The antisymmetry makes

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105

it only necessary to know those 𝛼𝑖1 ⋯𝑖𝑘 with 𝑖1 < ⋯ < 𝑖𝑘 . We write the local expression of 𝛼 as 𝛼|𝑈 =

∑

𝛼𝑖1 ⋯𝑖𝑘 (𝑥)𝑑𝑥𝑖1 ∧ ⋯ ∧ 𝑑𝑥𝑖𝑘 ,

𝑖1 0 and 𝐻𝑑𝑅 (⋆) = ℝ. 0 (4) Consider the 1-dimensional manifold 𝑀 = ℝ. For 𝑘 = 0 we have 𝐻𝑑𝑅 (ℝ) = 2 ℝ, since it is connected. For 𝑘 = 1 observe that, since Ω (ℝ) = 0, every 1form is closed. Let us take 𝛼 ∈ Ω1 (ℝ), that can be written using the global coordinate as 𝛼 = 𝑔(𝑥) 𝑑𝑥 for some 𝑔 ∈ 𝐶 ∞ (ℝ). Now, if we take 𝑓(𝑥) = 𝑥 ∫0 𝑔(𝑡) 𝑑𝑡, we have that 𝑑𝑓 = 𝛼. Therefore, every 1-form in ℝ is exact, so 1 𝐻𝑑𝑅 (ℝ) = 0. 0 (5) Let 𝑀 = 𝑆 1 . Then 𝐻𝑑𝑅 (𝑆 1 ) = ℝ since it is connected. Take the angular coordinate 𝜃, which is only well defined locally. However, the 1-form 𝑑𝜃 is defined globally on 𝑆 1 . Let 𝛼 = 𝑔(𝜃)𝑑𝜃 be a 1-form, where 𝑔(𝜃) is a periodic function on [0, 2𝜋]. Then 𝛼 is exact if and only if 𝛼 = 𝑑𝑓 = 𝑓′ (𝜃)𝑑𝜃, for a 𝜃 periodic function 𝑓(𝜃). Hence 𝑓′ (𝜃) = 𝑔(𝜃) and so 𝑓(𝜃) = 𝐶 + ∫0 𝑔(𝑢)𝑑𝑢, 2𝜋 for a constant 𝐶. The function 𝑓 is periodic if and only if ∫0 𝑔(𝑢)𝑑𝑢 = 0. 2𝜋 1 1 1 So 𝐵𝑑𝑅 (𝑆 1 ) = {𝑔 𝑑𝜃| ∫0 𝑔 = 0} and 𝑍𝑑𝑅 (𝑆1 ) = Ω1 (𝑆 1 ). Hence 𝐻𝑑𝑅 (𝑆 1 ) ≅ ℝ, 2𝜋 with the isomorphism given by 𝑔 𝑑𝜃 ↦ ∫0 𝑔. Note that 𝑑𝜃 is not exact, i.e., 1 𝑑𝜃 ≠ 𝑑𝑓 for any function 𝑓 on 𝑆 .

(6) The de Rham complex consists of functional spaces, which are infinite dimensional as ℝ-vector spaces. This is a drawback with respect to simplicial homology. However, in contrast to singular homology, if 𝑀 has dimension

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𝑘 𝑛, then Ω𝑘 (𝑀) = 0 for 𝑘 > 𝑛. Therefore it follows directly that 𝐻𝑑𝑅 (𝑀) = 0 for 𝑘 > 𝑛.

Remark 2.122. De Rham cohomology can be equally defined for differentiable manifolds with boundary, or even for manifolds with corners. These are manifolds 𝑀 such that any point 𝑝 ∈ 𝑀 has a chart 𝜑 ∶ 𝑈 𝑝 → 𝐵𝜖 (0) ∩ ([0, ∞)𝑘 × ℝ𝑛−𝑘 ) with 𝜑(𝑝) = 0. Remark 2.123. 𝑘 • The de Rham cohomology group 𝐻𝑑𝑅 (𝑀) gives the obstruction for a closed 𝑘 𝑘-form to be exact. So if 𝐻𝑑𝑅 (𝑀) = 0, then any closed 𝑘-form 𝛼 has a (𝑘 − 1)form 𝛽 with 𝛼 = 𝑑𝛽 (the form 𝛽 is called a primitive of 𝛼).

• De Rham cohomology is related to the well known physical problems of finding a potential for a vector field. Let 𝑋 = (𝐴1 , . . . , 𝐴𝑛 ) be a vector field on a open set 𝑈 ⊂ ℝ𝑛 . We say that 𝑋 is conservative if 𝑋 = grad (𝑓). According to Remark 2.117, this means that the 1-form 𝛼 = ∑ 𝐴𝑖 𝑑𝑥𝑖 is exact. A necessary 𝜕𝐴 𝜕𝐴 condition is that 𝛼 is closed, i.e., 𝜕𝑥 𝑖 = 𝜕𝑥𝑗 for 𝑖 ≠ 𝑗. A well known result 𝑗 𝑖 asserts that if 𝑈 is simply connected, then this is a sufficient condition. The 1 technical point is that if 𝑈 is simply connected, then 𝐻𝑑𝑅 (𝑈) = 0, so 𝛼 closed implies 𝛼 exact (Exercise 2.37). • For 𝑈 ⊂ ℝ3 a convex open set, a well known result proved in calculus courses is that a vector field 𝑋 with zero divergence is the rotational of another vector field, and an irrotational vector field (that is, rot(𝑋) = 0) is the gradient of a 3 2 function. This is due to the fact that 𝐻𝑑𝑅 (𝑈) = 0 and 𝐻𝑑𝑅 (𝑈) = 0, respec3 tively (and it is true for any open simply connected 𝑈 ⊂ ℝ ). 2.5.2. Poincaré lemma and homotopy invariance. De Rham cohomology has similar properties to singular cohomology, so it is natural that it defines an invariant from the homotopy category. To see that, we need to prove a Poincaré lemma as mentioned • in Remark 2.76. As we are in the smooth category, we shall prove that 𝐻𝑑𝑅 (𝑀 × ℝ) ≅ • 𝐻𝑑𝑅 (𝑀) for a differentiable manifold 𝑀. To agree with (2.8), we could equally prove the ∗ ∗ analogous version 𝐻𝑑𝑅 (𝑀 × [0, 1]) ≅ 𝐻𝑑𝑅 (𝑀), although this requires using manifolds with boundary (or with corners if 𝑀 has boundary). Theorem 2.124 (Poincaré lemma). Let 𝜋 ∶ 𝑀 × ℝ → 𝑀 be the projection. Then the 𝑘 𝑘 map 𝜋∗ ∶ 𝐻𝑑𝑅 (𝑀) → 𝐻𝑑𝑅 (𝑀 × ℝ) is an isomorphism. Proof. Consider the maps 𝑖𝑠 ∶ 𝑀 → 𝑀 × ℝ, 𝑖𝑠 (𝑥) = (𝑥, 𝑠), for 𝑠 ∈ ℝ. We have 𝜋 ∘ 𝑖0 = 1𝑀 , so 𝑖0∗ ∘ 𝜋∗ = Id. We consider the map 𝑖0 ∘ 𝜋 ∶ 𝑀 × ℝ → 𝑀 × ℝ, (𝑖0 ∘ 𝜋)(𝑥, 𝑡) = (𝑥, 0). This induces a morphism of cochain complexes 𝜋∗ ∘𝑖0∗ ∶ Ω• (𝑀×ℝ) → Ω• (𝑀×ℝ). Let us check that this is chain homotopic to the identity, that is, there exists 𝐾 ∶ Ω𝑘 (𝑀 ×ℝ) → Ω𝑘−1 (𝑀 × ℝ) such that (2.17)

Id −𝜋∗ 𝑖0∗ = ±𝑑𝐾 ± 𝐾𝑑.

Let 𝑡 be the coordinate of the factor ℝ, and let 𝑑𝑡 be the (pullback of the) global generator of Ω1 (ℝ). Therefore, any 𝜔 ∈ Ω𝑘 (𝑀 × ℝ) can be written as 𝜔 = 𝜂1 + 𝜂2 ∧ 𝑑𝑡, where

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111

𝜂1 (𝑥, 𝑡) is a 𝑘-form on 𝑀 dependent on 𝑡, and 𝜂2 (𝑥, 𝑡) is a (𝑘 − 1)-form on 𝑀 dependent on 𝑡. We define 𝑡

(𝐾𝜔)(𝑥, 𝑡) = ∫ 𝜂2 (𝑥, 𝜉) 𝑑𝜉. 0

The differential 𝑑 is written, in terms of the product 𝑀 × ℝ, as 𝑑𝜔 = 𝑑𝑥 𝜔 + 𝑑𝑡 ∧ where 𝑑𝑥 is the differential in the 𝑥-directions. Then

𝑑 𝜔, 𝑑𝑡

𝑡

𝑑 𝑑(𝐾𝜔) = 𝑑𝑥 (𝐾𝜔) + 𝑑𝑡 ∧ (𝐾𝜔) = 𝑑𝑥 (∫ 𝜂2 (𝑥, 𝜉)𝑑𝜉) + (−1)𝑘−1 𝜂2 ∧ 𝑑𝑡, 𝑑𝑡 0 𝐾(𝑑𝜔) = 𝐾(𝑑𝑥 𝜂1 + 𝑑𝑡 ∧ 𝑡

= ∫ ((−1)𝑘 0

𝑑 𝜂 + 𝑑𝑥 𝜂2 ∧ 𝑑𝑡) 𝑑𝑡 1

𝑑 𝜂 (𝑥, 𝜉) + 𝑑𝑥 𝜂2 (𝑥, 𝜉)) 𝑑𝜉 𝑑𝑡 1 𝑡

= (−1)𝑘 (𝜂1 (𝑥, 𝑡) − 𝜂1 (𝑥, 0)) + 𝑑𝑥 (∫ 𝜂2 (𝑥, 𝜉)𝑑𝜉) . 0

Since 𝜋∗ 𝑖0∗ 𝜔 = 𝜂1 (𝑥, 0), we have 𝑑(𝐾𝜔) − 𝐾(𝑑𝜔) = (−1)𝑘−1 (𝜔 − 𝜋∗ 𝑖0∗ 𝜔), proving (2.17). 𝑘 𝑘 By Proposition 2.84, we have 𝜋∗ ∘ 𝑖0∗ = Id ∶ 𝐻𝑑𝑅 (𝑀 × ℝ) → 𝐻𝑑𝑅 (𝑀 × ℝ), completing ∗ the proof that 𝜋 is an isomorphism. □ Theorem 2.124 readily implies that the de Rham cohomology is invariant under smooth homotopies (Remark 2.76), thereby defining a functor ∗ 𝐻𝑑𝑅 ∶ 𝐇𝐨𝐃𝐌𝐚𝐧 ⟶ GrAlgℝ ,

where 𝐇𝐨𝐃𝐌𝐚𝐧 denotes the homotopy category of differentiable manifolds. Certainly, if 𝐻 ∶ 𝑀 × [0, 1] → 𝑁 is a smooth homotopy between 𝑓0 = 𝐻 ∘ 𝑖0 and 𝑓1 = 𝐻 ∘ 𝑖1 , then take a smooth function 𝜌 ∶ ℝ → [0, 1] such that 𝜌(𝑠) = 0 for 𝑠 ≤ 0 and 𝜌(𝑠) = 1 for 𝑠 ≥ 1. Then 𝐾(𝑥, 𝑠) = 𝐻(𝑥, 𝜌(𝑠)) defines a smooth function 𝐾 ∶ 𝑀 × ℝ → 𝑁 with 𝑓0 = 𝐾 ∘ 𝑖0 , 𝑓1 = 𝐾 ∘ 𝑖1 . Now, by Remark 2.76, since 𝑖0∗ = 𝑖1∗ = (𝜋∗ )−1 , then 𝑓0∗ = 𝑓1∗ . Remark 2.125. De Rham cohomology is in principle restricted to dealing with smooth manifolds and smooth maps. However, it can be extended to a larger class of spaces, including all compact triangulated spaces. This follows from a series of considerations: (1) If 𝑀, 𝑁 are smooth manifolds, and 𝑓 ∶ 𝑀 → 𝑁 is a continuous map, then there exists a smooth map 𝑓′ ∶ 𝑀 → 𝑁 homotopic to 𝑓. This can be proven by embedding 𝑁 into a Euclidean space ℝ𝑑 in such a way that 𝑖 ∶ 𝑁 ↪ ℝ𝑑 is a closed smooth submanifold [Ada]. Then a tubular neighbourhood 𝑈 ⊃ 𝑁 can be defined via the exponential map exp ∶ (𝑇𝑁)⟂ → ℝ𝑑 (see page 153). So there is a projection 𝑟 ∶ 𝑈 → 𝑁 which is a deformation retract. Now approach 𝑖 ∘ 𝑓 by a smooth map 𝐹 ∶ 𝑀 → 𝑈 ⊂ ℝ𝑑 , and consider 𝑓′ = 𝑟 ∘ 𝐹 ∶ 𝑀 → 𝑁. (2) If 𝑓, 𝑔 ∶ 𝑀 → 𝑁 are smooth maps, and 𝐻 ∶ 𝑀 × [0, 1] → 𝑁 is a continuous homotopy between them, then there is a smooth homotopy 𝐻 ′ ∶ 𝑀 ×[0, 1] → 𝑁 with 𝐻 ′ (𝑥, 0) = 𝑓(𝑥), 𝐻 ′ (𝑥, 1) = 𝑔(𝑥), for all 𝑥 ∈ 𝑀. This is constructed as in (1).

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(3) If 𝑓 ∶ 𝑀 → 𝑁 is continuous, consider 𝑓′ ∶ 𝑀 → 𝑁 smooth as in (1). Then ∗ ∗ define 𝑓∗ ∶ 𝐻𝑑𝑅 (𝑁) → 𝐻𝑑𝑅 (𝑀) as 𝑓∗ = (𝑓′ )∗ . If 𝑓1′ , 𝑓2′ are two choices of maps as in (1), then they are homotopic since 𝑓1′ ∼ 𝑓 ∼ 𝑓2′ . By (2), they are smoothly homotopic, and hence (𝑓1′ )∗ = (𝑓2′ )∗ . This means that the definition of 𝑓∗ is consistent. (4) By the previous considerations, if 𝑀, 𝑁 are two smooth manifolds of the same ∗ ∗ homotopy type, then 𝐻𝑑𝑅 (𝑀) ≅ 𝐻𝑑𝑅 (𝑁). Indeed, if 𝑓 ∶ 𝑀 → 𝑁 is a contin∗ ∗ uous homotopy equivalence, then 𝑓∗ ∶ 𝐻𝑑𝑅 (𝑁) → 𝐻𝑑𝑅 (𝑀) is the required isomorphism. (5) Consider a topological space 𝑋 that is of the same homotopy type as a smooth manifold, and let ℎ ∶ 𝑋 → 𝑀 be a homotopy equivalence. Then we define ∗ ∗ 𝐻𝑑𝑅 (𝑋) abstractly as 𝐻𝑑𝑅 (𝑀), together with a canonical isomorphism ℎ∗ ∶ ∗ ∗ 𝐻𝑑𝑅 (𝑀) → 𝐻𝑑𝑅 (𝑋). This is easily seen to be consistent. (6) If 𝑓 ∶ 𝑋 → 𝑌 is a continuous map between two such spaces 𝑋, 𝑌 , then consider homotopy equivalences ℎ1 ∶ 𝑋 → 𝑀 and ℎ2 ∶ 𝑌 → 𝑁 to smooth manifolds and let 𝑔 ∶ 𝑀 → 𝑁, 𝑔 = ℎ2 ∘ 𝑓 ∘ ℎ1′ , where ℎ1′ is a homotopy inverse ∗ ∗ ∗ of ℎ1 . Then we define 𝑓∗ ∶ 𝐻𝑑𝑅 (𝑌 ) → 𝐻𝑑𝑅 (𝑋) to be equal to 𝑔∗ ∶ 𝐻𝑑𝑅 (𝑁) → ∗ 𝐻𝑑𝑅 (𝑀), using the canonical isomorphisms. (7) Compact triangulated spaces 𝑋 can be PL-embedded into a Euclidean space 𝑋 ⊂ ℝ𝑑 (cf. Remark 1.47). A small neighbourhood 𝑈 = 𝐵𝜖 (𝑋) is an open subset of ℝ𝑑 (in particular, it has the structure of a smooth manifold) which de∗ ∗ formation retracts to 𝑋 (Exercise 2.38). So one can define 𝐻𝑑𝑅 (𝑋) = 𝐻𝑑𝑅 (𝑈) as in (5). Mayer-Vietoris exact sequence. Analogously to simplicial homology, we also have a Mayer-Vietoris exact sequence for de Rham cohomology. Let 𝑀 be a differentiable manifold. Suppose that we have two open sets 𝑈, 𝑉 ⊂ 𝑀 such that 𝑋 = 𝑈 ∪ 𝑉. Let 𝑖1 ∶ 𝑈 ↪ 𝑀, 𝑖2 ∶ 𝑉 ↪ 𝑀, 𝑗1 ∶ 𝑈 ∩ 𝑉 → 𝑈, and 𝑗2 ∶ 𝑈 ∩ 𝑉 → 𝑉 be the inclusions. Then there is a short exact sequence of cochain complexes (𝑖1∗ ,𝑖2∗ )

𝑗1∗ −𝑗2∗

0 ⟶ Ω• (𝑀) ⟶ Ω• (𝑈) ⊕ Ω• (𝑉) ⟶ Ω• (𝑈 ∩ 𝑉) ⟶ 0. Only the surjectivity is not obvious. For that, take {𝜌𝑈 , 𝜌𝑉 } to be a smooth partition of unity subordinated to {𝑈, 𝑉}. Then given 𝛼 ∈ Ω• (𝑈 ∩ 𝑉), 𝜌𝑈 𝛼 can be extended by zero to 𝑉 and 𝜌𝑉 𝛼 can be extended by zero to 𝑈. Hence (𝜌𝑉 𝛼, −𝜌𝑈 𝛼) ∈ Ω• (𝑈) ⊕ Ω• (𝑉) maps to 𝛼 under 𝑗1∗ − 𝑗2∗ . Applying Proposition 2.88, we obtain a long exact sequence in de Rham cohomology. Theorem 2.126 (Mayer-Vietoris). Let 𝑀 be a differentiable manifold, and let 𝑈, 𝑉 ⊂ 𝑀 be open sets. Then we have a long exact sequence for de Rham cohomology ⋯ ON @A

/ 𝐻 𝑘−1 (𝑀) 𝑑𝑅

/ 𝐻 𝑘−1 (𝑈) ⊕ 𝐻 𝑘−1 (𝑉) 𝑑𝑅 𝑑𝑅

/ 𝐻 𝑘 (𝑀) 𝑑𝑅

/ 𝐻 𝑘 (𝑈) ⊕ 𝐻 𝑘 (𝑉) 𝑑𝑅 𝑑𝑅

/ 𝐻 𝑘−1 (𝑈 ∩ 𝑉) 𝑑𝑅 JK =< / 𝐻 𝑘 (𝑈 ∩ 𝑉) 𝑑𝑅

/ ⋯.

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113

2.5.3. Integration on manifolds and de Rham theorem. Let 𝑀 be an oriented differentiable 𝑛-manifold. The natural objects to be integrated are 𝑛-forms with compact support (see section 2.6.1). The support of 𝜔 ∈ Ω𝑛 (𝑀) is supp(𝜔) = {𝑝 ∈ 𝑀|𝜔𝑝 ≠ 0}, where the overline denotes topological closure. The forms with compact support are denoted Ω𝑛𝑐 (𝑀) ⊂ Ω𝑛 (𝑀). Let 𝜔 ∈ Ω𝑛𝑐 (𝑀). If the support of 𝜔 lies in a positive coordinate chart (𝑈, (𝑥1 , . . . , 𝑥𝑛 )), write 𝜔 = 𝑓 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 . We define ∫ 𝜔 = ∫ 𝑓𝑑𝑥1 ⋯ 𝑑𝑥𝑛 . ℝ𝑛

𝑈

To see that this is well defined, suppose that we have another positive chart (𝑉, (𝑦1 , . . . , 𝑦𝑛 )) containing the support of 𝜔. By (2.14), we have 𝜔 = 𝑓(𝑦1 , . . . , 𝑦𝑛 ) det(

𝜕𝑥𝑖 ) 𝑑𝑦1 ∧ ⋯ ∧ 𝑑𝑦𝑛 . 𝜕𝑦𝑗

Hence, by the change of coordinates for integration, we have | 𝜕𝑥 | ∫ 𝑓(𝑥1 , . . . , 𝑥𝑛 )𝑑𝑥1 ⋯ 𝑑𝑥𝑛 = ∫ 𝑓(𝑦1 , . . . , 𝑦𝑛 ) |det( 𝑖 )| 𝑑𝑦1 ⋯ 𝑑𝑦𝑛 𝜕𝑦 | 𝑗 | 𝑛 𝑛 ℝ ℝ = ∫ 𝑓(𝑦1 , . . . , 𝑦𝑛 ) det( ℝ𝑛

𝜕𝑥𝑖 ) 𝑑𝑦1 ⋯ 𝑑𝑦𝑛 , 𝜕𝑦𝑗

where the last equality holds because the change of charts has positive Jacobian. Note that for the integral to make sense, it is only necessary that 𝜔 is integrable, which is defined as 𝑓 being integrable (a notion which is independent of the chart). Now if 𝜔 is an 𝑛-form on 𝑀, we decompose it as 𝜔 = ∑ 𝜔𝑖 , where the support of 𝜔𝑖 is compact and lies inside a positive coordinate chart 𝑈 𝑖 . We define ∫ 𝜔 = ∑ ∫ 𝜔𝑖 𝑀

𝑖

𝑈𝑖

in case this sum is well defined. If the support of 𝜔 is compact, then it can be covered by a finite number of charts, and hence the sum is finite. This definition is independent on the chosen partition (if one has two coverings, one can take a common refinement). A set 𝐴 ⊂ 𝑀 is a measurable set (resp., a set of zero Lebesgue measure) if 𝐴 ∩ 𝑈 is measurable (resp., of zero Lebesgue measure) on each coordinate chart 𝑈 ⊂ 𝑀. We define then ∫𝐴 𝜔 = ∫𝑀 (𝜒𝐴 𝜔), for a measurable set 𝐴 ⊂ 𝑀, where 𝜒𝐴 is the characteristic function of 𝐴 (that is, 𝜒𝐴 (𝑥) = 1 if 𝑥 ∈ 𝐴, 𝜒𝐴 (𝑥) = 0 if 𝑥 ∉ 𝐴). Definition 2.127. A volume form is an 𝑛-form 𝜈 ∈ Ω𝑛 (𝑀) such that 𝜈𝑝 ≠ 0 for all 𝑝 ∈ 𝑀. Recall that, given a locally finite covering {𝑈𝛼 } of 𝑀, a partition of unity is a collection of smooth functions {𝜌𝛼 } with 𝜌𝛼 ∶ 𝑀 → ℝ such that supp(𝜌𝛼 ) ⊂ 𝑈𝛼 and ∑ 𝜌𝛼 (𝑝) = 1 for all 𝑝 ∈ 𝑀 (note that this sum is finite in a neighbourhood of any point). Lemma 2.128. A smooth manifold 𝑀 is orientable if and only if there exists a volume form on it. An orientation (Definition 1.68) is equivalent to a choice of volume form 𝜈 up to multiplication by everywhere positive functions.

114

2. Algebraic topology

Proof. Suppose that a volume form 𝜈 exists. Then we define an orientation on 𝑇𝑝 𝑀 by a basis (𝑣 1 , . . . , 𝑣 𝑛 ) such that 𝜈𝑝 (𝑣 1 , . . . , 𝑣 𝑛 ) > 0. Clearly, if 𝜈′ = 𝑓 𝜈, where 𝑓 ∶ 𝑀 → (0, ∞), then 𝜈′ defines the same orientation as 𝜈. Note that a chart (𝑈, 𝜑 = (𝑥1 , . . . , 𝑥𝑛 )) 𝜕 𝜕 is positive if and only if 𝜈 ( 𝜕𝑥 , . . . , 𝜕𝑥 ) > 0, i.e., 𝜈 = 𝑓 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 , with 𝑓 > 0 on 1 𝑛 𝑈. Conversely, if 𝑀 admits an orientation, then it has a positive atlas 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )}. Consider a volume form on 𝑈𝛼 , 𝜈𝛼 = 𝜑∗𝛼 (𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 ), such that 𝜈𝛼 (𝑣 1 , . . . , 𝑣 𝑛 ) > 0 for a positive basis (𝑣 1 , . . . , 𝑣 𝑛 ). Let {𝜌𝛼 } be a partition of unity subordinated to {𝑈𝛼 }. The 𝑛-form 𝜈 = ∑𝛼 𝜌𝛼 𝜈𝛼 satisfies that at each 𝑝 ∈ 𝑇𝑝 𝑀, 𝜈𝑝 (𝑣 1 , . . . , 𝑣 𝑛 ) > 0 for a positive basis (𝑣 1 , . . . , 𝑣 𝑛 ), since at least there is one 𝛼 such that 𝜌𝑎 (𝑝) > 0, and all the others are ≥ 0. This 𝜈 is a volume form defining the given orientation of 𝑀. □ To integrate functions on a manifold, we need to fix a volume form. Given a manifold with a volume form (𝑀, 𝜈), we define ∫ 𝑓 = ∫ 𝑓𝜈.

(2.18)

𝑀

𝑀

𝑛

In this sense, integration on ℝ is integration on (ℝ𝑛 , 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 ). For a compact region 𝑅 ⊂ 𝑀, we define the volume of 𝑅 as Vol(𝑅) = ∫𝑅 1. In particular, if 𝑀 is compact, then Vol(𝑀) = ∫𝑀 1 is the volume of 𝑀. In the case of a surface 𝑆, it is customary to call it the area and to denote it area(𝑆). An important result for integration is the Stokes theorem. Theorem 2.129 (Stokes). Let 𝑀 be an oriented manifold with boundary 𝜕𝑀, which is given the induced orientation (Proposition 1.70). Let 𝜔 ∈ Ω𝑛−1 (𝑀) be an (𝑛 − 1)-form 𝑐 with compact support. Then ∫ 𝑑𝜔 = ∫ 𝜔. 𝑀

𝜕𝑀

Proof. Suppose first that supp(𝜔) lies in a coordinate chart 𝜑 ∶ 𝑈 → ℝ𝑛+ = {(𝑥1 , . . . , 𝑥𝑛 )|𝑥1 ≥ 0}. ˆ 𝑖 ∧ ⋯ ∧ 𝑑𝑥𝑛 , where the symbol ˆ means that the term is Write 𝜔 = ∑ 𝑓𝑖 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥 ∞ 𝜕𝑓 𝜕𝑓 omitted. Then 𝑑𝜔 = ∑(−1)𝑖−1 𝜕𝑥𝑖 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 . For 𝑖 > 1, we have ∫−∞ 𝜕𝑥𝑖 𝑑𝑥𝑖 = 0, 𝑖

since 𝑓𝑖 has compact support. This means that ∫ℝ𝑛+ ∞ 𝜕𝑓1 𝑑𝑥1 𝜕𝑥1

have ∫0

𝜕𝑓𝑖 𝑑𝑥1 𝜕𝑥𝑖

𝑖

⋯ 𝑑𝑥𝑛 = 0. For 𝑖 = 1, we

= −𝑓1 (0, 𝑥2 , . . . , 𝑥𝑛 ), since 𝑓1 = 0 for 𝑥1 large. So

∫ 𝑑𝜔 = ∫ ℝ𝑛 +

ℝ𝑛 +

𝜕𝑓1 𝑑𝑥 ⋯ 𝑑𝑥𝑛 = − ∫ 𝜔(0, 𝑥2 , . . . , 𝑥𝑛 ). 𝜕𝑥1 1 ℝ𝑛−1

Observe that, by our convention on the orientation of 𝜕𝑀 given in Proposition 1.70, we have that (𝑥2 , . . . , 𝑥𝑛 ) gives the opposite orientation for ℝ𝑛−1 = 𝜕ℝ𝑛+ . Hence ∫𝑀 𝑑𝜔 = ∫𝜕𝑀 𝜔. If the support of 𝜔 does not lie in a coordinate chart, we cover it by finitely many coordinate charts 𝑈1 ∪ ⋯ ∪ 𝑈𝑟 , using the compactness of supp(𝜔). We take a partition of unity {𝜌𝑖 } subordinated to {𝑈 𝑖 } and write 𝜔 = ∑ 𝜔𝑖 with 𝜔𝑖 = 𝜌𝑖 𝜔. □

2.5. De Rham cohomology

115

Remark 2.130. The Stokes theorem encompasses well known classical results. 𝑏

(1) Let [𝑎, 𝑏] ⊂ ℝ be a closed interval. Then ∫[𝑎,𝑏] 𝑑𝑓 = ∫𝑎 𝑓′ (𝑡)𝑑𝑡 = ∫{𝑎,𝑏} 𝑓 = 𝑓(𝑏) − 𝑓(𝑎), for a smooth function 𝑓. This is the fundamental theorem of calculus. (2) Let 𝐷 ⊂ ℝ2 be a simply connected domain (i.e., an open and bounded set) whose boundary 𝜕𝐷 is a smooth curve. Then 𝐷 is a surface with boundary. Let 𝑋 = (𝑃, 𝑄) be a vector field on 𝐷. Green’s theorem reads ∫( 𝐷

𝜕𝑄 𝜕𝑃 − ) 𝑑𝑥𝑑𝑦 = ∫ 𝑃𝑑𝑥 + 𝑄𝑑𝑦. 𝜕𝑥 𝜕𝑦 𝜕𝐷

This is just the Stokes theorem applied to the 1-form 𝛼 = 𝑃𝑑𝑥 + 𝑄𝑑𝑦. (3) In (2) the boundary can be piecewise 𝐶 1 , since corners are of measure zero (Exercise 2.39). This will be used later in Theorem 3.62. (4) Let 𝐷 ⊂ ℝ3 be a domain with smooth boundary 𝑆 = 𝜕𝐷. Let 𝑋 = (𝑃, 𝑄, 𝑅) be a vector field on 𝐷. The Gauss divergence theorem reads ∫𝐷 ÷(𝑋) = ∫𝑆 ⟨𝑋, 𝐍⟩, where 𝐍 is the outer normal vector field to 𝑆 = 𝜕𝐷. This is the Stokes theorem applied to the 2-form 𝛽 = 𝑃 𝑑𝑦 ∧ 𝑑𝑧 + 𝑄 𝑑𝑧 ∧ 𝑑𝑥 + 𝑅 𝑑𝑥 ∧ 𝑑𝑦. The integral ∫𝑆 𝛽 coincides with the integral of the flux of the vector field along 𝜕𝑆, given by ⟨𝑋, 𝐍⟩ (Remark 3.47(4)). Now we move on to relate de Rham cohomology with singular homology via the de Rham theorem. Let 𝑀 be an 𝑛-dimensional manifold. Now we do not assume that 𝑀 is orientable. For 𝑘 ≥ 0, consider the chain complex of smooth 𝑘-cubes with real coefficients, denoted 𝐶𝑘𝑠𝑚 (𝑀, ℝ). This is the real vector space generated by 𝑘-cubes 𝜎 ∶ [0, 1]𝑘 → 𝑀 which are smooth. Its homology will be denoted 𝐻𝑘𝑠𝑚 (𝑀, ℝ). We define the integration on 𝑘-cubes by ∫ ∶ Ω𝑘 (𝑀) × 𝐶𝑘𝑠𝑚 (𝑀, ℝ)

⟶

ℝ

(2.19) (𝜔, 𝜎)

↦

∫𝜔 = ∫ 𝜍

𝜎∗ 𝜔.

[0,1]𝑘

This map is well defined because [0, 1]𝑘 is oriented. The Stokes theorem implies that for a (𝑘 − 1)-form 𝛼, 𝜎∗ (𝑑𝛼) = ∫

∫ 𝑑𝛼 = ∫ 𝜍

[0,1]𝑘

[0,1]𝑘

𝑑(𝜎∗ 𝛼) = ∫ 𝜕([0,1]𝑘 )

𝜎∗ 𝛼 = ∫ 𝛼, 𝜕𝜍

since the definition 𝜕𝜎 = ∑(−1)𝑖 (𝐹𝑖 𝜎 − 𝐵𝑖 𝜎) in section 2.3 gives a natural orientation to the boundary components of 𝜕([0, 1]𝑘 ). Note that the effect of the (𝑘 − 2)-faces of [0, 1]𝑘 is negligible since they are of measure zero for both integrals. The map (2.19) is ℝ-bilinear and induces a map at the level of homology, (2.20)

𝑘 ∫ ∶ 𝐻𝑑𝑅 (𝑀) × 𝐻𝑘𝑠𝑚 (𝑀, ℝ) ⟶ ℝ.

116

2. Algebraic topology

𝑘 This is well defined since if 𝜔 ∈ 𝑍𝑑𝑅 (𝑀) and 𝑐 ∈ 𝐵𝑘𝑠𝑚 (𝑀, ℝ), then 𝑐 = 𝜕𝑐′ for 𝑐′ ∈ 𝑘 𝑠𝑚 𝐶𝑘+1 (𝑀, ℝ) and hence ∫𝑐 𝜔 = ∫𝜕𝑐′ 𝜔 = ∫𝑐′ 𝑑𝜔 = 0. Analogously, if 𝜔 ∈ 𝐵𝑑𝑅 (𝑀) and 𝑐 ∈ 𝑍𝑘𝑠𝑚 (𝑀, ℝ), then 𝜔 = 𝑑𝜂 for 𝜂 ∈ Ω𝑘−1 (𝑀) and hence ∫𝑐 𝜔 = ∫𝑐 𝑑𝜂 = ∫𝜕𝑐 𝜂 = 0.

The use of smooth chains is technical, and should not worry us. The 𝑘-chains can be substituted by smooth 𝑘-chains in practice (Remark 2.140). Theorem 2.131. There is a natural isomorphism 𝐻𝑘 (𝑀, ℝ) ≅ 𝐻𝑘𝑠𝑚 (𝑀, ℝ), for 𝑘 ≥ 0. Now we come to our main result, which shows that de Rham cohomology is dual to singular homology with real coefficients. Definition 2.132. Let 𝑉, 𝑊 be ℝ-vector spaces, and let 𝜑 ∶ 𝑉 × 𝑊 → ℝ be a bilinear mapping. We say that 𝜑 is a perfect pairing if the map 𝑉 → 𝑊 ∗ , 𝑣 ↦ 𝜑(𝑣, −), is an isomorphism. Remark 2.133. • The above definition is in general asymmetric on 𝑉 and 𝑊. • If either of 𝑉 or 𝑊 is finite dimensional, then both are and dim 𝑉 = dim 𝑊, since dim 𝑉 = dim 𝑉 ∗ for a finite dimensional vector space. In this case, the definition of perfect pairing is symmetric since 𝑉 ≅ 𝑊 ∗ if and only if 𝑊 ≅ 𝑉 ∗ . • Let 𝑉, 𝑊 be finite dimensional vector spaces of dimension 𝑚. Take basis 𝑉 = ⟨𝑣 1 , . . . , 𝑣 𝑚 ⟩, 𝑊 = ⟨𝑤 1 , . . . , 𝑤 𝑚 ⟩. The bilinear map 𝜑 is given by a matrix 𝐴 = (𝑎𝑖𝑗 ), 𝑎𝑖𝑗 = 𝜑(𝑣 𝑖 , 𝑤𝑗 ). Then 𝜑 is a perfect pairing when det 𝐴 ≠ 0. Theorem 2.134 (de Rham). The bilinear map (2.20) is a perfect pairing. In particular, 𝑘 𝐻𝑑𝑅 (𝑀) ≅ 𝐻𝑘 (𝑀, ℝ)∗ , for all 𝑘 ≥ 0. Remark 2.135. 𝑘 (1) If 𝑀 is a compact manifold, then 𝐻𝑑𝑅 (𝑀) is finite dimensional (Exercise 3.15). In this case 𝐻𝑘 (𝑀, ℝ) is also finite dimensional and the Betti number can be 𝑘 computed via de Rham cohomology, 𝑏𝑘 (𝑀) = dim 𝐻𝑘 (𝑀, ℝ) = dim 𝐻𝑑𝑅 (𝑀).

(2) The de Rham cohomology of compact connected surfaces follows easily from Theorem 2.134 and the computations of Example 2.99. For the surface Σ𝑔 , 0 1 2 we have 𝐻𝑑𝑅 (Σ𝑔 ) = ℝ, 𝐻𝑑𝑅 (Σ𝑔 ) = ℝ2𝑔 , and 𝐻𝑑𝑅 (Σ𝑔 ) = ℝ. For the surface 0 1 2 𝑘−1 𝑋𝑘 , we have that 𝐻𝑑𝑅 (𝑋𝑘 ) = ℝ, 𝐻𝑑𝑅 (𝑋𝑘 ) = ℝ , and 𝐻𝑑𝑅 (𝑋𝑘 ) = 0. (3) If 𝑀 is an 𝑛-dimensional compact connected oriented manifold, then we have 𝐻𝑛 (𝑀) = ℤ⟨[𝑀]⟩, where the generator [𝑀] is the fundamental class of 𝑀 (Remark 2.100(4)). Therefore 𝐻𝑛 (𝑀, ℝ) = ⟨[𝑀]⟩ is a 1-dimensional vector space. 𝑛 Therefore 𝐻𝑑𝑅 (𝑀) is 1-dimensional. Since 𝑀 is oriented, it has a volume form 𝜈 compatible with the orientation. As 𝜈 > 0, 𝜅 = ∫𝑀 𝜈 > 0. We normalize 1 𝑛 𝜔 = 𝜅 𝜈 so that ∫𝑀 𝜈 = 1. Therefore [𝜔] ∈ 𝐻𝑑𝑅 (𝑀) is sent, via (2.20), to the 𝑛 𝑛 ∗ generator of 𝐻 (𝑀, ℝ) . Hence 𝐻𝑑𝑅 (𝑀) = ⟨[𝜔]⟩. 𝑛 Note that any 𝑛-form 𝛼 with ∫𝑀 𝛼 ≠ 0 is a generator of 𝐻𝑑𝑅 (𝑀). If ∫𝑀 𝛼 = ′ 0, then the form 𝛼 is exact, so ∫𝑀 𝛼 = ∫𝑀 𝛼 if and only if [𝛼] = [𝛼′ ] ∈ 𝑛 𝑛 𝐻𝑑𝑅 (𝑀). A volume form is a generator of 𝐻𝑑𝑅 (𝑀) spread along the manifold. On a different direction, we can construct a very concentrated 𝑛-form. Let

2.5. De Rham cohomology

117

𝑝 ∈ 𝑀, and let 𝑈 𝑝 ≅ 𝐵1 (0) be a neighbourhood of 𝑝. Consider a bump function 𝜌(𝑥) with supp(𝜌) ⊂ 𝐵1 (0) and ∫𝐵1 (0) 𝜌(𝑥)𝑑𝑥1 ⋯ 𝑑𝑥𝑛 = 1. For 𝜖 > 0 we let 𝜌𝜖 (𝑥) = 𝜖−𝑛 𝜌(𝑥/𝜖), and 𝜔𝜖 = 𝜌𝜖 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 . Then supp(𝜔𝜀 ) ⊂ 𝐵𝜀 (0) 𝑛 𝑛 and ∫𝑈 𝜔𝜀 = 1. So [𝜔𝜀 ] ∈ 𝐻𝑑𝑅 (𝑀) is a generator of 𝐻𝑑𝑅 (𝑀). (4) In a sense, 𝜔𝜀 is an approximation of the Dirac delta 𝛿𝑝 at 𝑝. That is, 𝜔𝜀 → 𝛿𝑝 , with a convergence as distributions. This means that for a function 𝑓 ∈ 𝐶 ∞ (𝑀), ∫𝑀 𝑓 𝜔𝜀 → 𝑓(𝑝) = ⟨𝛿𝑝 , 𝑓⟩, for 𝜀 → 0. In this sense, 𝜔𝜀 is a kind of smoothing of the Dirac delta at 𝑝, and 𝜔 is a spreading of the mass (integral) of it throughout the manifold. Remark 2.136. Let 𝑈 ⊂ ℝ𝑛 be an open set. A vector field 𝑋 = (𝐴1 , . . . , 𝐴𝑛 ) on 𝑈 is conservative if 𝑋 = grad (𝑓) for a function 𝑓. In this case, for a path 𝛾 ∶ [𝑎, 𝑏] → 𝑈 𝑏 𝑏 from a point 𝑝 to a point 𝑞, the integral ∫𝛾 𝑋 = ∫𝑎 ⟨𝑋(𝑡), 𝛾′ (𝑡)⟩𝑑𝑡 = ∫𝑎 𝑑𝑓(𝛾′ (𝑡))𝑑𝑡 = 𝑏

∫𝑎 (𝑓 ∘ 𝛾)′ (𝑡)𝑑𝑡 = 𝑓(𝛾(𝑏)) − 𝑓(𝛾(𝑎)) = 𝑓(𝑞) − 𝑓(𝑝). So ∫𝛾 𝑋 does not depend on the path but only on the endpoints. In particular ∫𝛾 𝑋 = 0 for a loop. Also note that 𝑓 is given 𝑝

as 𝑓(𝑝) = 𝑓(𝑝0 ) + ∫𝑝0 𝑋, where the last integral means “any path from 𝑝0 to 𝑝”, and the point 𝑝0 ∈ 𝑈 has been fixed. 𝜕𝐴𝑖 𝜕𝑥𝑗

Let 𝛼 = ∑ 𝐴𝑖 𝑑𝑥𝑖 be the associated 1-form as in Remark 2.123. The condition 𝜕𝐴 − 𝜕𝑥𝑗 = 0, for all 𝑖, 𝑗, means that 𝛼 is closed. In this case ∫𝛾 𝑋 = ∫𝛾 𝛼, for any loop 𝑖

𝛾. Theorem 2.134 says that if ∫𝛾 𝑋 = 0 for any loop, then [𝛼] = 0, and so 𝛼 is exact and 𝑋 is conservative (here we use the discussion after Example 2.72 that says that any element of 𝐻1 (𝑈) can be given by a loop), confirming the previous discussion. Note that for any closed 𝛼, it holds that ∫𝛾 𝑋 = 0 for loops homotopic to zero, since in this case [𝛾] = 0. The central idea in proving Theorem 2.134 is that the homology and cohomology functors satisfy Mayer-Vietoris and coincide in the basic piece, the ball 𝐵 ⊂ ℝ𝑛 in ℝ𝑛 . We formalize the argument algebraically to use it repeatedly. We need the notion of natural transformations of functors. Definition 2.137. Let us consider categories 𝒞1 and 𝒞2 and two covariant functors 𝐹, 𝐺 ∶ 𝒞1 → 𝒞2 . A natural transformation 𝜏 ∶ 𝐹 → 𝐺 is a collection of morphisms 𝜏(𝑋) ∶ 𝐹(𝑋) → 𝐺(𝑋) in 𝒞2 for each 𝑋 ∈ Obj(𝒞1 ) such that, for every morphism 𝑓 ∶ 𝑋 → 𝑌 in 𝒞1 , the following diagram commutes. 𝐹(𝑋)

𝜏(𝑋)

𝐹(𝑓)

 𝐹(𝑌 )

/ 𝐺(𝑋) 𝐺(𝑓)

𝜏(𝑌 )

 / 𝐺(𝑌 )

A similar definition is made for two contravariant functors. Now let 𝑀 be a smooth manifold. Consider the category 𝐎𝐩𝐞𝐧(𝑀) of open subsets of 𝑀 (Definition 1.12), and another Abelian category 𝒜. A collection (𝐹 𝑘 ) of covariant functors 𝐹 𝑘 ∶ 𝐎𝐩𝐞𝐧(𝑀) → 𝒜 is said to satisfy the Mayer-Vietoris property if, for any

118

2. Algebraic topology

open sets 𝑈, 𝑉 we have a long exact sequence ⋯

/ 𝐹 𝑘+1 (𝑈 ∩ 𝑉)

/ 𝐹 𝑘+1 (𝑈) ⊕ 𝐹 𝑘+1 (𝑉)

ON @A ``````0 𝐹 𝑘 (𝑈 ∩ 𝑉)

/ 𝐹 𝑘+1 (𝑈 ∪ 𝑉) JK =
1. Then 𝜏• (𝑈), 𝜏• (𝑉), 𝜏• (𝑈 ∩𝑉) are

2.5. De Rham cohomology

119

isomorphisms for 𝑈 = ⨆ 𝑈2𝑁 , 𝑉 = ⨆ 𝑈2𝑁−1 , and 𝑈 ∩ 𝑉 = ⨆(𝑈𝑁 ∩ 𝑈𝑁+1 ), using the property for the disjoint union of open sets. Now the argument with the Mayer-Vietoris property yields that 𝜏(𝑈 ∪ 𝑉) = 𝜏(𝑀) is an isomorphism. □ Remark 2.139. The analogous version of Lemma 2.138 for contravariant functors holds, but in this case we need to require 𝐹 𝑘(

⨆

𝑈𝛼 ) = ∏ 𝐹 𝑘 (𝑈𝛼 ), 𝐺 𝑘 ( 𝛼

⨆

𝑈𝛼 ) = ∏ 𝐺 𝑘 (𝑈𝛼 ). 𝛼

Now the proof of Theorem 2.134 can be completed. Consider the contravariant functors • 𝐹 = 𝐻𝑑𝑅 ∶ 𝐎𝐩𝐞𝐧(𝑀) → 𝐕𝐞𝐜𝐭ℝ ,

𝐺 = 𝐻•𝑠𝑚 (−, ℝ)∗ ∶ 𝐎𝐩𝐞𝐧(𝑀) → 𝐕𝐞𝐜𝐭ℝ ,

𝑘 𝐹 𝑘 (𝑈) = 𝐻𝑑𝑅 (𝑈),

𝐺 𝑘 (𝑈) = 𝐻𝑘𝑠𝑚 (𝑈, ℝ)∗ .

The natural transformation is given by the integral 𝜏(𝑈) ∶

𝑘 𝐻𝑑𝑅 (𝑈)

⟶

𝐻𝑘𝑠𝑚 (𝑈, ℝ)∗

[𝜔]

↦

([𝜎] ↦ ∫ 𝜔) . 𝜍

The functor 𝐹 satisfies the (contravariant) Mayer-Vietoris property (Theorem 2.126). The functor 𝐻𝑘 (−, ℝ) satisfies the covariant Mayer-Vietoris property (Theorem 2.90), hence 𝐺 = 𝐻𝑘 (𝑀, ℝ)∗ satisfies the contravariant Mayer-Vietoris property by dualizing the long exact sequence of Theorem 2.90. The properties on the disjoint union of open sets follow from Example 2.121(2) and Example 2.72(1) (note that the dual of a direct 𝑘 sum is the product of the duals). Finally, for a ball 𝐵 𝑛 ⊂ ℝ𝑛 , we have that 𝐻𝑑𝑅 (𝐵𝑛 ) = 0 𝑛 ∗ 𝑛 𝐻𝑘 (𝐵 , ℝ) = 0 for 𝑘 > 0. For 𝑘 = 0 we have 𝐻𝑑𝑅 (𝐵 ) = ℝ given by the constant functions, and 𝐻0 (𝐵𝑛 , ℝ) = ℝ⟨𝑝⟩, generated by any point 𝑝 ∈ 𝐵 𝑛 . The integration map 0 𝐻𝑑𝑅 (𝐵 𝑛 ) → 𝐻0 (𝐵 𝑛 , ℝ)∗ is an isomorphism since ∫ 𝑓 = 𝑓(𝑝) , 𝑝

for any 𝑓 ∈

0 𝐻𝑑𝑅 (𝐵 𝑛 ).

Remark 2.140. We also can prove Theorem 2.131 using Lemma 2.138. For this, consider the covariant functors 𝐹1 = 𝐻• (−, ℝ) and 𝐹2 = 𝐻•𝑠𝑚 (−, ℝ). All the theory of singular homology works similarly for smooth singular chains, giving the Mayer-Vietoris exact sequence, and the homotopy property. Therefore 𝐻𝑘 (𝑀, ℝ) ≅ 𝐻𝑘𝑠𝑚 (𝑀, ℝ), for all 𝑘 ≥ 0. Remark 2.141. Lemma 2.138 is a paradigm of homology theories. A homology functor is a graded functor satisfying a Mayer-Vietoris property. If it is homotopy invariant, then the value of the functor on the 𝑛-dimensional ball is that of a point. Hence once this value is fixed, the homology functor is determined. This is why all (co)homology theories generally compute the same algebraic invariant. This is the philosophy of the formalism of the so-called sheaf cohomology that comprises all these (co)homologies (section 5.1.4).

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2. Algebraic topology

Remark 2.142. There is a notion of singular cohomology groups 𝐻 𝑘 (𝑋) (see [Ma2]) for a topological space 𝑋, dual of singular homology. When using real coefficients, 𝑘 𝐻 𝑘 (𝑋, ℝ) = 𝐻𝑘 (𝑋, ℝ)∗ . So Theorem 2.134 says that 𝐻𝑑𝑅 (𝑀) ≅ 𝐻 𝑘 (𝑀, ℝ) for a smooth manifold 𝑀. 𝑘 Remark 2.143. We denote the pairing (2.20) as ⟨[𝜔], [𝑐]⟩ = ∫𝑐 𝜔, for [𝜔] ∈ 𝐻𝑑𝑅 (𝑀), [𝑐] ∈ 𝐻𝑘 (𝑀, ℝ). Note that for 𝑓 ∶ 𝑀 → 𝑁, a smooth map between differentiable manifolds, we have for a smooth 𝑘-cube 𝜎 and 𝜔 ∈ Ω𝑘 (𝑁), that ∫𝑓∗ 𝜍 𝜔 = ∫𝑓∘𝜍 𝜔 = ∫[0,1]𝑘 (𝑓 ∘ 𝜎)∗ 𝜔 = ∫[0,1]𝑘 𝜎∗ (𝑓∗ 𝜔) = ∫𝜍 𝑓∗ 𝜔. Therefore ⟨[𝜔], 𝑓∗ [𝑐]⟩ = ⟨𝑓∗ [𝜔], [𝑐]⟩, for 𝑘 [𝜔] ∈ 𝐻𝑑𝑅 (𝑁), [𝑐] ∈ 𝐻𝑘 (𝑀, ℝ). This property is called adjointness.

2.6. Poincaré duality For compact oriented smooth manifolds, integration of forms has some striking consequences on the topology of the manifold through its de Rham cohomology, which we shall explore. 2.6.1. Cohomology with compact support. We start by introducing cohomology with compact support. Despite the similarities with de Rham cohomology, this is not a cohomology in the usual sense, since it is not invariant under homotopies. Far from being a disadvantage, this non-invariance under homotopies is a source of differentiation unavailable from other invariants. For instance, it can be used to distinguish between non-homeomorphic spaces which are homotopic. In particular, cohomology with compact support can be used to prove that ℝ𝑛 is not homeomorphic to ℝ𝑚 for 𝑛 ≠ 𝑚, yielding the theorem of invariance of dimension (Theorem 1.2). Let 𝑀 be a (not necessarily compact) smooth manifold. Given a 𝑘-form 𝜔 ∈ Ω𝑘 (𝑀), we define the support of 𝜔, supp(𝜔), as the subset supp(𝜔) = {𝑝 ∈ 𝑀 | 𝜔𝑝 ≠ 0} ⊂ 𝑀, where the overline denotes the topological closure. Within this section, we will focus on the forms with compact support Ω𝑘𝑐 (𝑀) = {𝜔 ∈ Ω𝑘 (𝑀) | supp(𝜔) is compact} , and we set Ω•𝑐 (𝑀) = ⨁𝑘 Ω𝑘𝑐 (𝑀). Observe that, for any 𝜔 ∈ Ω𝑘 (𝑀), supp(𝑑𝜔) ⊂ supp(𝜔). Thus, (Ω•𝑐 (𝑀), 𝑑) is a cochain complex. Definition 2.144. We define the 𝑘th de Rham cohomology with compact support, 𝐻𝑐𝑘 (𝑀), as the cohomology of the cochain complex (Ω•𝑐 (𝑀), 𝑑), that is 𝐻𝑐𝑘 (𝑀) =

ker(𝑑 ∶ Ω𝑘𝑐 (𝑀) → Ω𝑘+1 (𝑀)) 𝑐 . 𝑘 im(𝑑 ∶ Ω𝑘−1 (𝑀) → Ω (𝑀)) 𝑐 𝑐

Remark 2.145. • For a compact manifold 𝑀, we have that all 𝑘-forms are compact, and hence 𝑘 𝐻𝑐𝑘 (𝑀) = 𝐻𝑑𝑅 (𝑀), for 𝑘 ≥ 0.

2.6. Poincaré duality

121

• Let 𝑀 be a connected manifold. For 𝑘 = 0, we have that Ω0𝑐 (𝑀) = 𝐶𝑐∞ (𝑀), the space of smooth functions with compact support. If 𝑑𝑓 = 0 for 𝑓 ∈ Ω0𝑐 (𝑀), then 𝑓 is locally constant. If 𝑀 is not compact, then 𝐻𝑐0 (𝑀) = 0. Otherwise, 0 if 𝑀 is compact, then 𝐻𝑐0 (𝑀) = 𝐻𝑑𝑅 (𝑀) = ℝ. • If 𝑀 = ⨆𝜆∈Λ 𝑀𝜆 is the decomposition into connected components, then Ω𝑘𝑐 (𝑀) = ⨁𝜆∈Λ Ω𝑘𝑐 (𝑀𝜆 ), since any 𝑘-form 𝜔 in 𝑀 of compact support can only be non-zero on finitely many of the 𝑀𝜆 . As a consequence, 𝐻𝑐𝑘 (𝑀) = ⨁𝜆 𝐻𝑐𝑘 (𝑀𝜆 ) (cf. Example 2.121(2)). • Let us take 𝑀 = ℝ. Since 𝑀 is not compact, we have 𝐻𝑐0 (ℝ) = 0. For 𝑘 = 1, consider a compactly supported 1-form 𝜔 = 𝑔(𝑡)𝑑𝑡. If 𝜔 is exact, say 𝜔 = 𝑡 𝑑𝑓 = 𝑓′ (𝑡)𝑑𝑡, then we have that 𝑓(𝑡) = ∫0 𝑔(𝜉)𝑑𝜉 + 𝐾, for some 𝐾 ∈ ℝ. As 𝑓, 𝑔 are both of compact support, they satisfy that 𝑓(𝑡) = 0, 𝑔(𝑡) = 0 for 𝑡 𝑡 ∈ (−∞, −𝑅] for some 𝑅 large enough. So we can write 𝑓(𝑡) = ∫−∞ 𝑔(𝜉)𝑑𝜉. But also 𝑓(𝑡) = 0, 𝑔(𝑡) = 0 for 𝑡 ∈ [𝑅, ∞), and the condition of 𝑓 being ∞ compactly supported is that ∫−∞ 𝑔(𝜉)𝑑𝜉 = 0. This means that 𝐻𝑐1 (ℝ) ≅ ℝ, under the map 𝜔 ↦ ∫ℝ 𝜔. Remark 2.146. (1) The cohomology with compact support is invariant under diffeomorphisms. However, it is not functorial from the category 𝐃𝐌𝐚𝐧, since for a smooth map 𝑓 ∶ 𝑀 → 𝑁 between two differentiable manifolds and 𝜔 ∈ Ω•𝑐 (𝑁), the pullback 𝑓∗ 𝜔 in general has no compact support. (2) We can restrict ourselves to proper maps. A map 𝑓 ∶ 𝑀 → 𝑁 is proper if the preimage of a compact set is compact. For such 𝑓, we have an induced map 𝑓∗ ∶ 𝐻𝑐𝑘 (𝑁) → 𝐻𝑐𝑘 (𝑀) by pullback. This gives functoriality from the subcategory of 𝐃𝐌𝐚𝐧 whose morphisms are proper maps. (3) Analogously to the case of the de Rham cohomology, there is an invariance of compactly supported cohomologies under proper homotopies. (4) We can work out an extension of the definition of 𝑓∗ to continuous proper maps as in Remark 2.125. As a homeomorphism is proper, this can serve to prove that homeomorphic spaces have isomorphic compactly supported cohomology. (5) We shall restrict in this text to using morphisms in compactly supported cohomology induced by inclusions. Let 𝑈 ⊂ 𝑉 be two open sets, and let 𝑖 ∶ 𝑈 ↪ 𝑉 denote the inclusion. Then given 𝜔 ∈ Ω𝑘𝑐 (𝑈), we can extend 𝜔 by zero outside supp(𝜔). The extension gives us a well defined form on 𝑉 that we will denote 𝑖∗ 𝜔. As supp(𝑖∗ 𝜔) = supp(𝜔), we have a map 𝑖∗ ∶ Ω𝑘𝑐 (𝑈) → Ω𝑘𝑐 (𝑉). This gives a covariant functor 𝐻𝑐𝑘 ∶ 𝐎𝐩𝐞𝐧(𝑀) → 𝐕𝐞𝐜𝐭ℝ . The following is the analogue of Theorem 2.124 for compactly supported cohomology. Lemma 2.147 (Poincaré lemma). There is an isomorphism 𝐻𝑐𝑘+1 (𝑀) ≅ 𝐻𝑐𝑘 (𝑀 × ℝ).

122

2. Algebraic topology

Proof. We consider the map 𝜋∗ ∶ Ω𝑘+1 (𝑀 × ℝ) → Ω𝑘𝑐 (𝑀), defined as 𝜋∗ (𝜔) = 𝑐 ∫ℝ 𝜂1 (𝑥, 𝑡)𝑑𝑡, where 𝜔 = 𝜂1 ∧ 𝑑𝑡 + 𝜂2 , where 𝜂1 is a 𝑘-form and 𝜂2 is a (𝑘 + 1)-form, both compactly supported on 𝑀, and dependent on 𝑡. The map 𝜋∗ is called integration along fibers. Then 𝜋∗ (𝑑𝜔) = 𝜋∗ (𝑑𝑥 𝜂1 ∧ 𝑑𝑡 + 𝑑𝑥 𝜂2 + (−1)𝑘+1 = ∫(𝑑𝑥 𝜂1 )𝑑𝑡 + (−1)𝑘+1 ∫ ℝ

ℝ

𝑑𝜂2 ∧ 𝑑𝑡) 𝑑𝑡

𝑑𝜂2 𝑑𝑡 = ∫(𝑑𝑥 𝜂1 )𝑑𝑡 = 𝑑𝑥 𝜋∗ (𝜔). 𝑑𝑡 ℝ

So 𝜋∗ defines a homomorphism 𝜋∗ ∶ 𝐻𝑐𝑘+1 (𝑀 × ℝ) → 𝐻𝑐𝑘 (𝑀). This is easily seen to be surjective. If [𝛼] ∈ 𝐻𝑐𝑘 (𝑀), then consider 𝛽 = 𝛼 ∧ 𝜌(𝑡)𝑑𝑡, where 𝜌(𝑡) is a compactly supported function with ∫ℝ 𝜌(𝑡)𝑑𝑡 = 1. Then 𝜋∗ (𝛽) = 𝛼 and 𝛽 is closed. To see that 𝜋∗ is injective, take 𝜔 = 𝜂1 ∧ 𝑑𝑡 + 𝜂2 ∈ Ω𝑘+1 (𝑀 × ℝ) closed such that 𝜋∗ (𝜔) = 𝑑𝑥 𝛾, for a 𝑐 (𝑘 − 1)-form 𝛾. So 𝜂1 ∧ 𝑑𝑡 and 𝑑𝑥 𝛾 ∧ 𝜌(𝑡)𝑑𝑡 have the same integral in the 𝑡-direction. 𝜕𝛼 Hence its difference is exact, i.e., 𝜂1 ∧ 𝑑𝑡 − 𝑑𝑥 𝛾 ∧ 𝜌(𝑡)𝑑𝑡 = 𝜕𝑡 ∧ 𝑑𝑡, where 𝛼(𝑥, 𝑡) is a compactly supported 𝑘-form on 𝑀 dependent on 𝑡. As 𝜔 is closed, 𝑑𝑥 𝜂1 = (−1)𝑘 𝑘

𝑑𝑥 𝛼 = (−1) 𝜂2 . Then 𝜔 =

𝜕𝛼 ∧𝑑𝑡+(−1)𝑘 𝑑𝑥 𝛼+𝑑𝑥 𝛾∧𝜌(𝑡)𝑑𝑡 𝜕𝑡

𝑘

𝜕𝜂2 , 𝜕𝑡

= 𝑑((−1) 𝛼+𝛾∧𝜌(𝑡)𝑑𝑡).

so □

As a corollary of Lemma 2.147, we have that 𝐻𝑐𝑘 (ℝ𝑛 ) = {

0, ℝ,

𝑘 ≠ 𝑛, 𝑘 = 𝑛,

where the map ∫ ∶

𝐻𝑐𝑛 (ℝ𝑛 ) →

ℝ

ℝ𝑛

[𝜔]

↦

∫ 𝜔 ℝ𝑛

is the isomorphism. In particular, a generator of 𝐻𝑐𝑛 (ℝ𝑛 ) is given by a form 𝜔 = 𝑓(𝑥)𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 , where 𝑓(𝑥) is a compactly supported function with ∫ℝ𝑛 𝑓(𝑥)𝑑𝑥1 ⋯ 𝑑𝑥𝑛 = 1. Remark 2.148. Note that 𝐻𝑐• (ℝ𝑛 ) ≇ 𝐻𝑐• (ℝ𝑚 ), hence 𝐻𝑐• is not a homotopy invariant. Also, this fact combined with Remark 2.146(4) proves the theorem of invariance of dimension (Theorem 1.2). A (covariant) Mayer-Vietoris theorem also holds for compactly supported cohomology. Let 𝑈, 𝑉 ⊂ 𝑀 be two open subsets such that 𝑀 = 𝑈 ∪ 𝑉. Let 𝑖1 ∶ 𝑈 ↪ 𝑀, 𝑖2 ∶ 𝑉 ↪ 𝑀, 𝑗1 ∶ 𝑈 ∩ 𝑉 → 𝑈, and 𝑗2 ∶ 𝑈 ∩ 𝑉 → 𝑉 be the inclusions. Then we have a short exact sequence of chain complexes, (𝑗1 ∗ ,𝑗2 ∗ )

𝑖1 ∗ −𝑖2 ∗

0 ⟶ Ω•𝑐 (𝑈 ∩ 𝑉) ⟶ Ω•𝑐 (𝑈) ⊕ Ω•𝑐 (𝑉) ⟶ Ω•𝑐 (𝑀) ⟶ 0. For the surjectivity of the last map, take 𝛼 ∈ Ω𝑘𝑐 (𝑀). Let {𝜌𝑈 , 𝜌𝑉 } be a partition of unity subordinated to {𝑈, 𝑉}. Then (𝜌𝑈 𝛼, −𝜌𝑉 𝛼) ∈ Ω𝑘𝑐 (𝑈) ⊕ Ω𝑘𝑐 (𝑉) is mapped to 𝛼. Applying Proposition 2.88, we obtain the following.

2.6. Poincaré duality

123

Theorem 2.149 (Mayer-Vietoris). Let 𝑀 be a differentiable manifold, and let 𝑈, 𝑉 ⊂ 𝑀 be open sets with 𝑀 = 𝑈∪𝑉. Then, we have a long exact sequence for compactly supported cohomology ⋯

/ 𝐻𝑐𝑘−1 (𝑈 ∩ 𝑉)

ON @A ______/ 𝐻𝑐𝑘 (𝑈 ∩ 𝑉)

/ 𝐻𝑐𝑘−1 (𝑈) ⊕ 𝐻𝑐𝑘−1 (𝑉)

/ 𝐻𝑐𝑘−1 (𝑀) JK =
0. Then 𝐻𝑛 (𝑀) = ℤ⟨[𝑀]⟩, where [𝑀] is the fundamental class of 𝑀 (see Remark 2.100(4)). Definition 2.152. Let 𝑓 ∶ 𝑀 → 𝑁 be a continuous map. We define the degree of 𝑓, 𝑑 = deg(𝑓) ∈ ℤ as the integer such that 𝑓∗ [𝑀] = 𝑑[𝑁]. Remark 2.153. 𝑓

𝑔

• If 𝑀 ⟶ 𝑁 ⟶ 𝑃 are two continuous maps of (compact, connected, oriented) 𝑛-manifolds, then 𝑔∗ 𝑓∗ [𝑀] = 𝑔∗ (deg(𝑓)[𝑁]) = deg(𝑓) deg(𝑔)[𝑃]. Hence deg(𝑔 ∘ 𝑓) = deg(𝑓) deg(𝑔). • If 𝑓, 𝑓′ ∶ 𝑀 → 𝑁 are homotopic maps, then deg(𝑓) = deg(𝑓′ ). • If 𝑓 ∶ 𝑀 → 𝑁 is a homotopy equivalence, then deg(𝑓) = ±1. For this, consider a homotopy inverse 𝑔 ∶ 𝑁 → 𝑀 so that 𝑔∘𝑓 ∼ 1𝑁 . Then deg(𝑓) deg(𝑔) = deg(𝑔 ∘ 𝑓) = deg(1𝑁 ) = 1. Therefore, as the degrees are integers, we have that deg(𝑓) = ±1.

2.6. Poincaré duality

125

• The degree of the constant map 𝑐𝑞 ∶ 𝑀 → 𝑁, where 𝑞 ∈ 𝑁 is zero. Therefore if 𝑓 ∼ 𝑐𝑞 , then deg(𝑓) = 0. • Suppose that 𝑀, 𝑁 are (compact, connected, oriented) differentiable 𝑛-mani𝑛 folds and 𝑓 ∶ 𝑀 → 𝑁 is a smooth map. Then 𝐻𝑑𝑅 (𝑀) = ℝ⟨[𝜈𝑀 ]⟩ and 𝑛 𝐻𝑑𝑅 (𝑁) = ℝ⟨[𝜈𝑁 ]⟩, generated by volume forms such that ∫𝑀 𝜈𝑀 = 1, ∫𝑁 𝜈𝑁 = 1 (see Remark 2.135(3)). So ⟨[𝜈𝑀 ], [𝑀]⟩ = 1 and ⟨[𝜈𝑁 ], [𝑁]⟩ = 1. Then using Remark 2.143, ∫ 𝑓∗ 𝜈𝑁 = ⟨[𝑀], 𝑓∗ [𝜈𝑁 ]⟩ = ⟨𝑓∗ [𝑀], [𝜈𝑁 ]⟩ = ⟨deg(𝑓)[𝑁], [𝜈𝑁 ]⟩ = deg(𝑓). 𝑀

In particular, 𝑓∗ [𝜈𝑁 ] = deg(𝑓)[𝜈𝑀 ]. So the degree of 𝑓 can be also computed with de Rham cohomology. Let 𝑓 ∶ 𝑀 → 𝑁 be a smooth map between differentiable 𝑛-manifolds. We say that 𝑝 ∈ 𝑀 is a regular point if 𝑑𝑝 𝑓 ∶ 𝑇𝑝 𝑀 → 𝑇 𝑓(𝑝) 𝑁 is an isomorphism (Remark 3.5 for the definition of the differential). In this case, there are neighbourhoods 𝑈 𝑝 , 𝑉 𝑓(𝑝) such that 𝑓 ∶ 𝑈 𝑝 → 𝑉 𝑓(𝑝) is a diffeomorphism. Let 𝑆 ⊂ 𝑀 be the set of non-regular points. We call 𝑞 ∈ 𝑁 a regular value if 𝑞 ∉ 𝑓(𝑆), that is, if for all 𝑝 ∈ 𝑓−1 (𝑞), 𝑝 is a regular point. The set of regular values are dense on 𝑀 by a renowned result of Sard. Theorem 2.154 (Sard). The set 𝑓(𝑆) is of zero measure. Let 𝑓 ∶ 𝑀 → 𝑁 be a smooth map between compact connected oriented 𝑛-manifolds. Take a regular value 𝑞 ∈ 𝑁 − 𝑓(𝑆). Then 𝑓−1 (𝑞) consists of regular points, and it is a discrete set. By compactness of 𝑀, 𝑓−1 (𝑞) is a finite set. For each 𝑝 ∈ 𝑓−1 (𝑞), we have neighbourhoods 𝑈 𝑝 , 𝑉 𝑞 with 𝑓 ∶ 𝑈 𝑝 → 𝑉 𝑞 a diffeomorphism. We define the sign of 𝑓 at 𝑝 as 𝜖𝑝 = {

1, −1,

if 𝑑𝑝 𝑓 ∶ 𝑇𝑝 𝑀 → 𝑇 𝑓(𝑝) 𝑁 preserves orientation, if 𝑑𝑝 𝑓 ∶ 𝑇𝑝 𝑀 → 𝑇 𝑓(𝑝) 𝑁 reverses orientation.

Proposition 2.155. In the conditions above, deg(𝑓) = ∑𝑝∈𝑓−1 (𝑞) 𝜖𝑝 , for any regular value 𝑞 of 𝑓. Proof. Let 𝑓−1 (𝑞) = {𝑝1 , . . . , 𝑝𝑟 }. We have neighbourhoods 𝑈1 , . . . , 𝑈𝑟 of 𝑝1 , . . . , 𝑝𝑟 , respectively, and 𝑉 = 𝑉 𝑞 such that 𝑓 ∶ 𝑈 𝑖 → 𝑉 is a diffeomorphism. Consider an 𝑛-form 𝜔 ∈ Ω𝑛 (𝑁) with supp(𝜔) ⊂ 𝑉 and ∫𝑉 𝜔 = 1 (cf. Remark 2.135(3)). Then 𝑛 𝐻𝑑𝑅 (𝑁) = ⟨[𝜔]⟩. Then the support of 𝑓∗ 𝜔 is contained in 𝑈1 ∪⋯∪𝑈𝑟 and ∫𝑈𝑖 (𝑓|𝑈𝑖 )∗ 𝜔 = 𝜖𝑝𝑖 ∫𝑉 𝜔 = 𝜖𝑝𝑖 . So 𝑟

(2.22)

𝑀

as required.

𝑟

deg(𝑓) = ∫ 𝑓∗ 𝜔 = ∑ ∫ (𝑓|𝑈𝑖 )∗ 𝜔 = ∑ 𝜖𝑝𝑖 , 𝑖=1 𝑈𝑖

𝑖=1

□

Remark 2.156. (1) If 𝑓 ∶ 𝑀 → 𝑁 is a cover between compact connected oriented manifolds of 𝑑 sheets, then deg 𝑓 = ±𝑑, the sign depending on whether 𝑓 preserves or reverses orientation. Note that the sets 𝑈 + = {𝑝 | 𝑑𝑝 𝑓 preserves orientation},

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2. Algebraic topology

𝑈 − = {𝑝 | 𝑑𝑝 𝑓 reverses orientation} are open and disjoint. As there are no critical points and since 𝑀 is connected, then either 𝑀 = 𝑈 + or 𝑀 = 𝑈 − . (2) If 𝑓 ∶ 𝑀 → 𝑁 is not surjective, then deg(𝑓) = 0. Just take a point 𝑝 ∈ 𝑁 − im(𝑓), which is automatically a regular value, and apply (2.22). (3) Let 𝑞 ∈ 𝑁 be a regular value of 𝑓. Then deg(𝑓) ≡ |𝑓−1 (𝑞)| (mod 2), since by Proposition 2.155, deg(𝑓) = ∑𝑓(𝑝)=𝑞 𝜖𝑝 ≡ ∑𝑓(𝑝)=𝑞 1 = |𝑓−1 (𝑞)| (mod 2). (4) For a regular value 𝑞 of 𝑓, | deg(𝑓)| ≤ ∑𝑓(𝑝)=𝑞 |𝜖𝑝 | = |𝑓−1 (𝑞)|. (5) For a non-regular value 𝑞, we can have a fiber 𝑓−1 (𝑞) with fewer points than | deg(𝑓)|. However, we can define a multiplicity 𝑚𝑞 for a point 𝑝 ∈ 𝑆 as the sum of the 𝜖𝑝𝑖 for the points in a nearby fiber 𝑓−1 (𝑞′ ), with 𝑞′ close to 𝑞, and 𝑝 𝑖 close to 𝑝. Then always deg(𝑓) = ∑𝑝∈𝑓−1 (𝑞) 𝑚𝑝 (see Figure 2.9). +

M − f +

+

q

q0

N Figure 2.9. The points 𝑞 and 𝑞′ are regular values. For both points, ∑𝑓(𝑝)=𝑞 𝜖𝑝 = deg(𝑓) = 1.

2.6.4. Cohomology classes associated to submanifolds. Let 𝑀 be a compact, oriented manifold of dimension 𝑛. The de Rham theorem and Poincaré duality give two isomorphisms: 𝑘 𝐻𝑘 (𝑀, ℝ) ≅ 𝐻𝑑𝑅 (𝑀)∗ ,

[𝑐] ↦ ([𝛼] ↦ ⟨[𝛼], [𝑐]⟩ = ∫𝛼) , 𝑐

𝑛−𝑘 𝑘 𝐻𝑑𝑅 (𝑀) ≅ 𝐻𝑑𝑅 (𝑀)∗ ,

[𝜂] ↦ ([𝛼] ↦ ⟨[𝛼], [𝜂]⟩ = ∫ 𝛼 ∧ 𝜂) . 𝑀

Here we have used the finite dimensionality of the cohomology. This implies that there is a natural isomorphism 𝑛−𝑘 Ψ ∶ 𝐻𝑘 (𝑀, ℝ) ≅ 𝐻𝑑𝑅 (𝑀),

(2.23)

and therefore each homology class [𝑇] ∈ 𝐻𝑘 (𝑀, ℝ) has associated a cohomology class 𝑛−𝑘 𝑘 [𝜈𝑇 ] ∈ 𝐻𝑑𝑅 (𝑀) such that ⟨[𝛼], [𝑇]⟩ = ⟨[𝛼], [𝜈𝑇 ]⟩ for all [𝛼] ∈ 𝐻𝑑𝑅 (𝑀). This reads (2.24)

∫ 𝛼 = ∫ 𝛼 ∧ 𝜈𝑇 , 𝑇

𝑀

for all 𝛼 ∈ Ω𝑘 (𝑀) closed.

2.6. Poincaré duality

127

It is natural to try to construct geometric representatives 𝜈𝑇 of [𝜈𝑇 ]. Two natural cases have already appeared for connected manifolds: 0 (1) The fundamental class [𝑀] ∈ 𝐻𝑛 (𝑀) has Ψ([𝑀]) = 1 ∈ 𝐻𝑑𝑅 (𝑀). Therefore 𝜈𝑀 = 1. 𝑛 (2) The point class [𝑝] ∈ 𝐻0 (𝑀) has Ψ([𝑝]) = [𝜔] ∈ 𝐻𝑑𝑅 (𝑀) such that 𝜔 is an 𝑛 ∫ 𝑛-form with 𝑀 𝜔 = 1, which is the natural generator of 𝐻𝑑𝑅 (𝑀). This is due to the fact that ∫𝑝 1 = ∫𝑀 1 𝜔 for the constant function 𝑓 = 1. There are two preferred choices for an 𝑛-form on 𝑀 with integral one (see Remark 2.135(3)): a normalized volume form 𝜈 (which is spread along the manifold), or a very concentrated 𝑛-form 𝜔𝑝 near the point 𝑝.

We pass to the construction of Ψ([𝑇]) for homology classes [𝑇] ∈ 𝐻𝑘 (𝑀) with 0 < 𝑘 < 𝑛. We shall do it for homology classes associated to submanifolds (Remark 1.19). Let 𝑁 ⊂ 𝑀 be a closed oriented connected submanifold of dimension 𝑘. Then the inclusion 𝑖 ∶ 𝑁 ↪ 𝑀 defines in a natural way the fundamental class of 𝑁 in 𝑀 as [𝑁] = 𝑖∗ ([𝑁]) ∈ 𝐻𝑘 (𝑀), defined as the image of the fundamental class [𝑁] ∈ 𝐻𝑘 (𝑁) under the induced map 𝑖∗ ∶ 𝐻𝑘 (𝑁) → 𝐻𝑘 (𝑀), and denoted again as [𝑁]. The cohomology class [𝜈𝑁 ] = Ψ([𝑁]) is defined by the equality ∫ 𝛼|𝑁 = ∫ 𝛼 ∧ 𝜈𝑁 , 𝑁

𝑀

for all closed 𝑘-forms 𝛼. Remark 2.157. There is an important result of Thom [Tho] that says that all homology classes in 𝐻𝑘 (𝑀, ℚ) can be represented by submanifolds. This is not true for integer coefficients, i.e., for all classes in 𝐻𝑘 (𝑀). Example 2.158. Consider the circle 𝑆 1 , with angular coordinate 𝜃 ∈ [0, 1]. Then 0 1 𝐻𝑑𝑅 (𝑆 1 ) = ⟨1⟩ and 𝐻𝑑𝑅 (𝑆 1 ) = ⟨[𝑑𝜃]⟩ = ⟨[𝜌(𝜃)𝑑𝜃]⟩. Note that 𝑑𝜃 defines the volume 1 form of 𝑆 . We can substitute 𝑑𝜃 by the bump form 𝜌(𝜃)𝑑𝜃, where 𝜌(𝜃) is supported in 1 1 ( 2 − 𝜖, 2 + 𝜖) and has integral 1. Example 2.159. Now we consider the torus 𝑇 2 = 𝑆 1 × 𝑆 1 , with coordinates (𝜃1 , 𝜃2 ) for each of the two factors. We understand the torus as the square [0, 1] × [0, 1] with the usual identifications in the boundaries. Let 𝑑𝜃1 , 𝑑𝜃2 be the two 1-forms defined as the pullback of the form 𝑑𝜃 on 𝑆 1 by the first and second projections, respectively. We have 𝐻0 (𝑇 2 ) = ⟨𝑝⟩,

0 𝐻𝑑𝑅 (𝑇 2 ) = ⟨1⟩,

𝐻1 (𝑇 2 ) = ⟨[𝑎1 ], [𝑎2 ]⟩,

1 𝐻𝑑𝑅 (𝑇 2 ) = ⟨[𝑑𝜃1 ], [𝑑𝜃2 ]⟩,

𝐻2 (𝑇 2 ) = ⟨[𝑇 2 ]⟩,

2 𝐻𝑑𝑅 (𝑇 2 ) = ⟨[𝑑𝜃1 ∧ 𝑑𝜃2 ]⟩,

where 𝑎1 (𝑡) = (𝑡, 1/2), 𝑎2 (𝑡) = (1/2, 𝑡), 𝑡 ∈ [0, 1]. Note that 𝑑𝜃1 ∧ 𝑑𝜃2 is the volume form of 𝑇 2 . Pairing, we obtain ∫ 𝑑𝜃1 = 1, 𝑎1

∫ 𝑑𝜃2 = 0, 𝑎1

∫ 𝑑𝜃1 = 0, 𝑎2

∫ 𝑑𝜃2 = 1, 𝑎2

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2. Algebraic topology

so by the de Rham theorem, we have that [𝑑𝜃1 ], [𝑑𝜃2 ] are actually generators of 1 • 𝐻𝑑𝑅 (𝑇 2 ). With this computation, we identify the ring structure of 𝐻𝑑𝑅 (𝑇 2 ) since, setting 𝐴1 = [𝑑𝜃1 ], 𝐴2 = [𝑑𝜃2 ], and 𝐵 = [𝑑𝜃1 ∧ 𝑑𝜃2 ], we have 𝐴1 ∧ 𝐴2 = −𝐴2 ∧ 𝐴1 = 𝐵. This allows us to describe the map (2.23), 1 Ψ ∶ 𝐻1 (𝑇 2 ) → 𝐻𝑑𝑅 (𝑇 2 ),

𝑎1 ↦ 𝐴 2 , 𝑎2 ↦ −𝐴1 . This holds because ⟨𝐴1 , 𝑎1 ⟩ = 1 = ⟨𝐴1 , 𝐴2 ⟩, ⟨𝐴2 , 𝑎1 ⟩ = 0 = ⟨𝐴2 , 𝐴2 ⟩, ⟨𝐴1 , 𝑎2 ⟩ = 0 = ⟨𝐴1 , −𝐴1 ⟩, and ⟨𝐴2 , 𝑎2 ⟩ = 1 = ⟨𝐴2 , −𝐴1 ⟩.

a2 a1

!2

a2 !1

a1

Figure 2.10. The loops 𝑎1 , 𝑎2 and their corresponding Thom forms 𝜔𝜀2 , 𝜔𝜀1 . 1

1

Now consider a bump function 𝜌𝜖 ∶ [0, 1] → [0, ∞) supported on ( 2 − 𝜖, 2 + 𝜖) and with integral 1. Take 𝜌𝜖1 (𝜃1 , 𝜃2 ) = 𝜌𝜖 (𝜃1 ) and 𝜌𝜖2 (𝜃1 , 𝜃2 ) = 𝜌𝜖 (𝜃2 ), and define 𝜔𝜖1 = 𝜌𝜖1 𝑑𝜃1 , 𝜔𝜖2 = 𝜌𝜖2 𝑑𝜃2 (see Figure 2.10). Then 0 𝐻𝑑𝑅 (𝑇 2 ) = ⟨1⟩,

1 𝐻𝑑𝑅 (𝑇 2 ) = ⟨[𝜔𝜖1 ], [𝜔𝜖2 ]⟩,

2 𝐻𝑑𝑅 (𝑇 2 ) = ⟨[𝜔𝜖1 ∧ 𝜔𝜖2 ]⟩. 1 1

Now the form 𝜔𝜖1 ∧ 𝜔𝜖2 is a bump 2-form concentrated near 𝑝 = ( 2 , 2 ) = 𝑎1 ∩ 𝑎2 , as 1

1

considered above. The form 𝜔𝜖2 is supported in [0, 1] × ( 2 − 𝜖, 2 + 𝜖), a small neighbourhood around 𝑎1 . This agrees with the fact that Ψ(𝑎1 ) = 𝐴2 = [𝜔𝜖2 ] and justifies the geometric description. Actually, if we make 𝜖 → 0, then 𝜔𝜖2 → 𝛿𝑎1 𝑑𝜃2 , where 𝛿𝑎1 is the Dirac delta along 𝑎1 , in the distributional sense. The Dirac delta 𝛿𝑎1 is defined by ⟨𝛿𝑎1 , 𝛽⟩ = ∫𝑎1 𝛽, for a 1-form 𝛽. The form 𝜔𝜖2 is a smoothing of this concentrated 1-form 𝛿𝑎1 . Remark 2.160. Dirac deltas which are 𝑘-forms are usually known as currents. There is a cohomology of currents that comprises the chains of singular homology (where 𝑇 ∈ 𝐶𝑘 (𝑀, ℝ) is interpreted as 𝛿𝑇 ) and the smooth forms of de Rham cohomology 𝑛−𝑘 simultaneously. It explains the isomorphism 𝐻𝑘 (𝑀, ℝ) ≅ 𝐻𝑑𝑅 (𝑀), as this cohomology of currents computes both. Let us describe the general situation, although we shall skip some details. Let 𝑁 ⊂ 𝑀 be an oriented, compact connected 𝑘-dimensional submanifold. Let 𝑈 be a tubular neighbourhood along 𝑁 which is a union of coordinate open sets 𝑈 = ⋃ 𝑈 𝑖 , where 𝑁 has coordinates (𝑥1 , . . . , 𝑥𝑘 , 0, . . . , 0) on 𝑈 𝑖 . Let us take a bump function 𝜌 ∶ ℝ𝑛−𝑘 → [0, ∞) with compact support in the unit ball and integral 1. Set 𝜌𝜖 (𝑥) = 𝜖−𝑘 𝜌(𝑥/𝜖)

2.6. Poincaré duality

129

and 𝜈𝜖𝑖 = 𝜌𝜖 (𝑥𝑘+1 , . . . , 𝑥𝑛 ) 𝑑𝑥𝑘+1 ∧ ⋯ ∧ 𝑑𝑥𝑛 on 𝑈 𝑖 . A careful patching of the 𝜈𝜖𝑖 with a partition of unity yields an (𝑛 − 𝑘)-form 𝜈𝜖𝑁 on 𝑀 which has compact support inside 𝑈 and is closed. This is called a Thom form of 𝑁. ′

Let us see that Ψ([𝑁]) = [𝜈𝜖𝑁 ]. First note that [𝜈𝜖𝑁 ] = [𝜈𝜖𝑁 ], for 𝜖, 𝜖′ > 0 small. Take a closed 𝑘-form 𝛼 = ∑|𝐼|=𝑘 𝛼𝐼 𝑑𝑥𝐼 . Let 𝐼0 = {1, . . . , 𝑘}. We work locally on 𝑈 𝑖 , where we can substitute 𝜈𝜖𝑁 by 𝜈𝜖𝑖 . Then ∫ 𝛼 ∧ 𝜈𝜖𝑖 = lim ∫ 𝛼 ∧ 𝜈𝜖𝑖 𝜖→0

𝑈𝑖

𝑈𝑖

= lim ∫ 𝛼𝐼0 (𝑥1 , . . . , 𝑥𝑛 )𝜌𝜖 (𝑥𝑘+1 , . . . , 𝑥𝑛 ) 𝑑𝑥1 ⋯ 𝑑𝑥𝑛 𝜖→0

ℝ𝑛

= ∫ 𝛼𝐼0 (𝑥1 , . . . , 𝑥𝑘 , 0, . . . , 0) 𝑑𝑥1 ⋯ 𝑑𝑥𝑘 = ∫ ℝ𝑘

𝛼|𝑁 .

𝑁∩𝑈𝑖

Now globally, we obtain ∫𝑀 𝛼 ∧ 𝜈𝜖𝑁 = ∫𝑁 𝛼|𝑁 , as we wanted to prove. 2.6.5. Intersection product. Let 𝑀 be a compact orientable 𝑛-manifold. The product (2.21) 𝑘 𝑛−𝑘 ∫ ∶ 𝐻𝑑𝑅 (𝑀) × 𝐻𝑑𝑅 (𝑀) ⟶ ℝ 𝑀 𝑛−𝑘 induces, via the isomorphism (2.23), Ψ ∶ 𝐻𝑘 (𝑀, ℝ) → 𝐻𝑑𝑅 (𝑀), a bilinear (graded symmetric) product

⟨−, −⟩ ∶ 𝐻𝑛−𝑘 (𝑀, ℝ) × 𝐻𝑘 (𝑀, ℝ) ⟶ ℝ, which is called the intersection product of 𝑀. It is in particular a perfect pairing on the homology. Let us describe it geometrically. Definition 2.161. Let 𝑀 be a differentiable manifold, and let 𝑆 1 , 𝑆 2 be two submanifolds with dim 𝑆 1 + dim 𝑆 2 = dim 𝑀. We say that 𝑆 1 and 𝑆 2 intersect transversally if, for each 𝑝 ∈ 𝑆 1 ∩ 𝑆 2 , we have 𝑇𝑝 𝑀 = 𝑇𝑝 𝑆 1 ⊕ 𝑇𝑝 𝑆 2 . Remark 2.162. • If 𝑆 1 , 𝑆 2 are two closed submanifolds of a smooth manifold, then we can perturb one of them slightly, say 𝑆 1 , so that they intersect transversally. This means that the embedding 𝑖 ∶ 𝑆 1 ↪ 𝑀 is isotoped (homotoped through differentiable embeddings) to another embedding 𝑖′ ∶ 𝑆 1 ↪ 𝑀 so that 𝑆 ′1 = 𝑖′ (𝑆 1 ) and 𝑆 2 intersect transversally. • If 𝑆 1 and 𝑆 2 intersect transversally, then the set 𝑆 1 ∩ 𝑆 2 is discrete. Indeed, by transversality, at every 𝑝 ∈ 𝑆 1 ∩𝑆 2 there is a chart (𝑥1 , . . . , 𝑥𝑛 ) such that 𝑆 1 has coordinates (𝑥1 , . . . , 𝑥𝑘 , 0, . . . , 0) and 𝑆 2 has coordinates (0, . . . , 0, 𝑥𝑘+1 , . . . , 𝑥𝑛 ). So if 𝑀 is compact, then 𝑆 1 ∩ 𝑆 2 is finite. Let 𝑀 be an oriented compact 𝑛-manifold, and let 𝑆 1 , 𝑆 2 be an oriented compact submanifolds of 𝑀 intersecting transversally. For 𝑝 ∈ 𝑆 1 ∩ 𝑆 2 , we define 𝜖𝑝 = {

+1, −1,

if 𝑇𝑝 𝑆 1 ⊕ 𝑇𝑝 𝑆 2 is oriented as 𝑇𝑝 𝑀, if 𝑇𝑝 𝑆 1 ⊕ 𝑇𝑝 𝑆 2 is not oriented as 𝑇𝑝 𝑀.

130

2. Algebraic topology

Then write (2.25)

𝑆1 ⋅ 𝑆2 =

∑

𝜖𝑝 .

𝑝∈𝑆1 ∩𝑆2

Proposition 2.163. Let 𝑀 be a compact oriented differentiable manifold of dimension 𝑛. Let 𝑆 1 , 𝑆 2 be oriented connected submanifolds of dimensions 𝑘, 𝑛 − 𝑘, respectively, and let 𝑛−𝑘 𝑘 [𝜈 𝑆1 ] = Ψ([𝑆 1 ]) ∈ 𝐻𝑑𝑅 (𝑀), [𝜈 𝑆2 ] = Ψ([𝑆 2 ]) ∈ 𝐻𝑑𝑅 (𝑀) be their associated cohomology classes. Then the intersection product of 𝑆 1 , 𝑆 2 is ⟨[𝑆 1 ], [𝑆 2 ]⟩ = ∫ 𝜈 𝑆1 ∧ 𝜈 𝑆2 = 𝑆 1 ⋅ 𝑆 2 . 𝑀

Proof. Consider Thom forms 𝜈 𝑆1 , 𝜈 𝑆2 of 𝑆 1 , 𝑆 2 , respectively, supported in small tubular neighbourhoods. Then 𝜈 𝑆1 ∧ 𝜈 𝑆2 has support on small neighbourhoods of the points in 𝑆 1 ∩ 𝑆 2 . Let 𝑝 ∈ 𝑆 1 ∩ 𝑆 2 and consider an oriented chart (𝑈 𝑝 , (𝑥1 , . . . , 𝑥𝑛 )) at 𝑝 such that 𝑆 1 has coordinates (𝑥1 , . . . , 𝑥𝑘 , 0, . . . , 0) and 𝑆 2 has coordinates (0, . . . , 0, 𝑥𝑘+1 , . . . , 𝑥𝑛 ). Then we have that 𝜈 𝑆1 = 𝜖𝑝,1 𝜌1 (𝑥𝑘+1 , . . . , 𝑥𝑛 )𝑑𝑥𝑘+1 ∧ ⋯ ∧ 𝑑𝑥𝑛 , 𝜈 𝑆2 = 𝜖𝑝,2 (−1)𝑘(𝑛−𝑘) 𝜌2 (𝑥1 , . . . , 𝑥𝑘 )𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑘 , where 𝜖𝑝,1 =±1 depending on whether the chart (𝑥1 , . . . , 𝑥𝑘 , 0, . . . , 0) is oriented for 𝑆 1 , 𝜖𝑝,2 =±1 depending on whether the chart (0, . . . , 0, 𝑥𝑘+1 , . . . , 𝑥𝑛 ) is oriented for 𝑆 2 , and the factor (−1)𝑘(𝑛−𝑘) accounts for the fact that we have to exchange for 𝑆 2 the 𝑘 first coordinates with the last 𝑛 − 𝑘 coordinates. Note that 𝜖𝑝 = 𝜖𝑝,1 𝜖𝑝,2 . Set 𝐼1 = {1, . . . , 𝑘} and 𝐼2 = {𝑘 + 1, . . . , 𝑛}. Then we have ∫ 𝜈 𝑆1 ∧ 𝜈 𝑆2 𝑀

= ∑ ∫ (𝜖𝑝,1 𝜌1 (𝑥𝑘+1 , . . . , 𝑥𝑛 )𝑑𝑥𝐼2 ) ∧ (𝜖𝑝,2 (−1)𝑘(𝑛−𝑘) 𝜌2 (𝑥1 , . . . , 𝑥𝑘 )𝑑𝑥𝐼1 ) 𝑝∈𝑆1 ∩𝑆2 𝑈 𝑝

=

∑ 𝑝∈𝑆1 ∩𝑆2

=

∑

𝜖𝑝 ∫ 𝜌2 (𝑥1 , . . . , 𝑥𝑘 )𝜌1 (𝑥𝑘+1 , . . . , 𝑥𝑛 )𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 𝑈𝑝

𝜖𝑝 = 𝑆 1 ⋅ 𝑆 2 .

□

𝑝∈𝑆1 ∩𝑆2

Remark 2.164. By Proposition 2.163, the intersection product 𝑆 1 ⋅ 𝑆 2 defined in (2.25) only depends on the homology classes of 𝑆 1 , 𝑆 2 (as long as they intersect transversally). This proposition has an important consequence. Suppose that 𝑀 is a compact connected oriented manifold of even dimension 𝑛 = 2𝑚. We define the intersection form of 𝑀 as the bilinear form 𝑄𝑀 ∶ 𝐻𝑚 (𝑀, ℝ) × 𝐻𝑚 (𝑀, ℝ) → ℝ, 𝑄𝑀 (𝐴, 𝐵) = ⟨𝐴, 𝐵⟩. This is symmetric if 𝑚 is even and antisymmetric if 𝑚 is odd (cf. Example 2.151). Let us compute the intersection form of compact connected orientable surfaces. Clearly 1 for 𝑆 2 we have that 𝐻𝑑𝑅 (𝑆 2 ) = 0 so 𝑄𝑆2 ≡ 0. Consider Σ𝑔 with 𝑔 ≥ 1. Then we can take loops 𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 generating 𝐻1 (Σ𝑔 ) such that 𝑎𝑖 , 𝑏𝑖 intersect transversally at one point, and any pair 𝑎𝑖 , 𝑏𝑖 is disjoint with 𝑎𝑗 , 𝑏𝑗 for 𝑖 ≠ 𝑗. We can arrange the orientations

Problems

131

b3

b2

b1

a2

a1

a3

Figure 2.11. Standard generators of 𝐻1 (Σ𝑔 ).

so that 𝑎𝑖 ⋅ 𝑏𝑖 = 1 and thus 𝑏𝑖 ⋅ 𝑎𝑖 = −1 (see Figure 2.11). Hence, the intersection form 𝑄Σ𝑔 has matrix

(2.26)

0 1 ⎛ −1 0 ⎜ ⎜ ⎜ ⎝

⋱

⎞ ⎟ ⎟. 0 1 ⎟ −1 0 ⎠

Remark 2.165. The intersection form can be defined also in singular cohomology (see Remark 2.142). It plays an important role in the classification of compact manifolds, especially in dimension 4. For instance, the homotopy type of a simply connected compact 4-manifold only depends on the intersection form. However, the homeomorphism type depends on the intersection form and a ℤ2 -invariant called the Kirby-Siebenmann invariant. Moreover, if the intersection form is positive definite and not diagonalizable over ℤ, then the 4-manifold does not admit any smooth structure (cf. Remark 1.22(f)) by a result of Donaldson [Do1].

Problems Exercise 2.1. Compute the fundamental group of the surfaces given in Exercise 1.33 directly from the words (using the Seifert-van Kampen theorem). Exercise 2.2. Prove that 𝜋𝑘 (𝑆𝑛 ) = 0, for 0 < 𝑘 < 𝑛. Exercise 2.3. Show that for two pointed spaces (𝑋1 , 𝑥1 ), (𝑋2 , 𝑥2 ), we have 𝜋𝑛 (𝑋1 × 𝑋2 , (𝑥1 , 𝑥2 )) ≅ 𝜋𝑛 (𝑋1 , 𝑥1 ) × 𝜋𝑛 (𝑋2 , 𝑥2 ), for 𝑛 ≥ 0. Exercise 2.4. Show that two connected manifolds 𝑋, 𝑌 are of the same homotopy type if and 𝑐𝑜 𝑐𝑜 only if they are of the same pointed homotopy type. Thus 𝕃𝐇𝐨𝐌𝐚𝐧𝑛 ≅ 𝕃𝐇𝐨𝐌𝐚𝐧𝑛 . 𝑐

𝑐,∗

Exercise 2.5. Show that the map 𝑋 ↦ 𝜋1 (𝑋) from the category of path connected spaces to groups cannot be made a functor. Exercise 2.6. Prove that if 𝑀 is a topological connected manifold, then 𝜋1 (𝑀) is numerable. Exercise 2.7. Find 𝑋, 𝑌 of the same weak homotopy type but not of the same homotopy type (use non-Hausdorff spaces). Exercise 2.8. Find CW-complexes 𝑋, 𝑌 with 𝜋𝑛 (𝑋) ≅ 𝜋𝑛 (𝑌 ) for all 𝑛 ≥ 0, but not of the same weak homotopy type. Exercise 2.9. Write explicit homotopies for the statements in Theorem 2.13.

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Exercise 2.10. Prove that the amalgamated product of two groups 𝐺 1 , 𝐺 2 over a group 𝐻 is given by the presentation (2.1). Exercise 2.11. Show that if 𝜋 ∶ 𝑋 ′ → 𝑋 is a cover of locally connected spaces with a numerable number of sheets, then 𝑋 being second countable implies that 𝑋 ′ is second countable. Prove that if 𝑋 is a topological manifold, so is 𝑋 ′ . Exercise 2.12. Let 𝑀 be a smooth manifold, and let Γ be a group acting freely and properly by diffeomorphism on 𝑀. Show that 𝑀/Γ admits the structure of a smooth manifold. Show that if 𝜋 ∶ 𝑀 ′ → 𝑀 is a cover of connected spaces and 𝑀 is a smooth manifold, then 𝑀 ′ has the structure of a smooth manifold in such a way that the deck transformations are diffeomorphisms. Exercise 2.13. Show that 𝜋1 (𝑋𝑘 ) is not Abelian for 𝑘 ≥ 2. Exercise 2.14. Show that if 𝜋 ∶ 𝑋 ′ → 𝑋 is a cover and 𝑋 is connected, then the cardinal of the fiber 𝐹𝑝 = 𝜋−1 (𝑝), 𝑝 ∈ 𝑋, is constant. Exercise 2.15. Determine the number of (connected) 2-sheeted covers of a compact surface. How many homeomorphism types appear for the covering spaces? Exercise 2.16. Find for which (connected, compact) surfaces 𝑆, 𝑆 ′ there are covers 𝜋 ∶ 𝑆 ′ → 𝑆. Exercise 2.17. Find which (connected, compact) surfaces with boundary 𝑆, 𝑆 ′ admit covers 𝜋 ∶ 𝑆 ′ → 𝑆. Exercise 2.18. Show that, for a triangulated connected space (𝑋, 𝜏), 𝐻1𝜏 (𝑋) ≅ Ab(𝜋1 (𝑋)). Exercise 2.19. Let 𝑋 be a path connected topological space. Prove that [𝑆1 , 𝑋] = 𝜋1 (𝑋, 𝑝)/conjugation. Show that there is a surjective map [𝑆1 , 𝑋] → 𝐻1 (𝑋), and that in general it is not injective. Exercise 2.20. Prove the theorem of invariance of dimension using homology. Exercise 2.21. Prove that the notion of boundary and interior point for an 𝑛-manifold with boundary is well defined. Exercise 2.22. Give a rigorous definition of orientation for a topological 𝑛-manifold 𝑀 using homology. And prove that it agrees with the notion of orientation for the case that 𝑀 is a PLmanifold. Exercise 2.23. Let (𝑋, 𝜏) be a triangulated space, and let (𝑋, 𝜏′ ) be a subdivision of the triangu′ lation. Prove that 𝐻𝑘𝜏 (𝑋) = 𝐻𝑘𝜏 (𝑋), for all 𝑘 ≥ 0. Exercise 2.24. Using Whitehead theorem (Remark 2.105), prove that if 𝑀 is a compact 𝑛dimensional manifold with the same homotopy groups of the 𝑛-sphere, then 𝑀 ∼ 𝑆 𝑛 . Exercise 2.25. Let 𝑀 be a compact oriented 𝑛-manifold. Prove that 𝐻𝑛−1 (𝑀) is a free group. Use this to prove that if 𝑀 is a compact simply connected 3-manifold, then 𝑀 ∼ 𝑆 3 . Exercise 2.26. Show that a triangulated space or a CW-complex is locally contractible. ′

Exercise 2.27. Prove that 𝐻𝑘𝜏 (𝑀) ≅ 𝐻𝑘𝜏 (𝑀 × [0, 1]) for a triangulated space (𝑀, 𝜏), giving a suitable triangulation 𝜏′ for 𝑀 × [0, 1]. Exercise 2.28. Prove that 𝐻• (𝑀 × 𝑁, ℚ) ≅ 𝐻• (𝑀, ℚ) ⊗ 𝐻• (𝑁, ℚ), for triangulated spaces 𝑀 and 𝑁. Exercise 2.29. For which values 𝑛, 𝑚 there is a compact oriented manifold of the homotopy type of 𝑆𝑛 ∨ 𝑆 𝑚 ?

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133

Exercise 2.30. Prove that the long line of Exercise 1.8 is not contractible, but has trivial fundamental group and homology groups. What is its de Rham cohomology? Exercise 2.31. Give two manifolds 𝑀1 , 𝑀2 of the same dimension whose de Rham cohomologies ∗ ∗ 𝐻𝑑𝑅 (𝑀1 ), 𝐻𝑑𝑅 (𝑀2 ) are isomorphic as vector spaces but not as algebras. Exercise 2.32. Let 𝑀 ′ be a smooth manifold, and let Γ be a finite group acting freely and properly ∗ on 𝑀 by diffeomorphisms. Let 𝑀 = 𝑀 ′ /Γ be the quotient manifold. Show that Γ acts on 𝐻𝑑𝑅 (𝑀 ′ ) • • • • ′ Γ ′ and that 𝐻𝑑𝑅 (𝑀) ≅ 𝐻𝑑𝑅 (𝑀 ) = {𝑥 ∈ 𝐻𝑑𝑅 (𝑀 ) | 𝑔 ⋅ 𝑥 = 𝑥, for all 𝑔 ∈ Γ}. Compute 𝐻𝑑𝑅 (𝑋𝑘 ) using that 𝑋𝑘 = Σ𝑘−1 /ℤ2 . Exercise 2.33. Construct maps 𝑆2 → 𝑆 2 of any degree. Use them to construct maps Σ𝑔 → 𝑆 2 of any degree. Conclude that there are maps of degree 1 which are not homotopy equivalences. 2𝑔−2 Show also that3 there are maps 𝑓 ∶ Σ𝑔 → Σℎ of any degree 𝑑 with |𝑑| ≤ 2ℎ−2 for ℎ, 𝑔 > 1. Exercise 2.34. Show that any map 𝑓 ∶ Σ𝑔 → Σℎ , with 𝑔 < ℎ, is of degree 0. Show that any map 𝑓 ∶ 𝑆 2 → Σℎ is homotopic to a constant if ℎ > 0. Find maps 𝑓 ∶ Σ𝑔 → Σℎ not homotopic to a constant with 0 < 𝑔 < ℎ. Exercise 2.35. Show that there are maps 𝑓, 𝑔 ∶ 𝑇 2 → 𝑇 2 of the same degree but not homotopic. Determine [𝑇 2 , 𝑇 2 ] and the map deg ∶ [𝑇 2 , 𝑇 2 ] → ℤ. Exercise 2.36. Let 𝑆 ⊂ ℝ3 be a compact, connected smooth embedded surface. Prove that 𝐻 0 (ℝ3 − 𝑆) is 2-dimensional. Conclude that 𝑆 is orientable using Exercise 1.22. Exercise 2.37. Find a 1-form on ℝ2 −{0} which is closed but not exact. Find a 2-form on ℝ3 −{0} which is closed but not exact. Exercise 2.38. Let 𝑋 ⊂ ℝ𝑑 be a compact simplicial subset. Prove that a small neighbourhood 𝑈 = 𝐵𝜖 (𝑋) deformation retracts to 𝑋. Exercise 2.39. Let 𝑀 be a manifold with corners. Prove that the Stokes theorem holds for 𝑀. Exercise 2.40. Let 𝑉 be a finite dimensional real vector space. Let 𝜑 be a bilinear antisymmetric map on 𝑉. Prove that if 𝜑 is non-degenerate (meaning that it is a perfect pairing as in Definition 2.132), then there is a basis of 𝑉 such that the associated matrix of 𝜑 is given by (2.26).

References and extra material Basic reading. We recommend [Bre], [D-P], [May], and [Mun] for basic notions of algebraic topology, including the fundamental group, homotopy groups, and covers. Singular homology is treated deeply in [Ma2], and de Rham cohomology is covered in [B-T] and [M-T]. [B-T] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer, 1982. [Bre] G.E. Bredon, Topology and Geometry, Graduate Texts in Mathematics, 1997. [D-P] C.T. Dodson, P.E. Parker, A User’s Guide to Algebraic Topology, Mathematics and Its Applications, Springer, 1997. [M-T] I.H. Madsen, J. Tornehave, From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Cambridge University Press, 1997. [Ma2] W.S. Massey, Singular Homology Theory, Graduate Texts in Mathematics, Vol. 70, Springer, 1991. [May] J.P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics Series, University of Chicago Press, 1999. [Mun] J.R. Munkres, Elements of Algebraic Topology, Taylor Francis Inc, 1996. 2𝑔−2

3 Actually the converse holds: one cannot get maps of degree |𝑑| > 2ℎ−2 . However proving this requires harder techniques called simplicial volumes. We thank Antonio Viruel (Universidad de Málaga) for pointing out this to us.

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Further reading. This chapter can be enhanced in many directions. We propose an extensive collection of directions for preparing a short dissertation by the students. • Fibrations. Introduce the concept of a fibration, and the homotopy lifting property. Prove the long exact sequence of homotopy groups for a fibration. [Ste] N. Steenrod, The Topology of Fibre Bundles, Princeton Math. Studies 14, Princeton University Press, 1951. • Seifert-van Kampen theorem. We propose giving a proof of this in the framework of the fundamental groupoid. This can be found at: [Bro] R. Brown, Topology and Groupoids, BookSurge Publishing, 2006. • Actions of discrete groups. To study the different types of actions, including proper and free actions, of discrete groups on topological spaces, see [Lee] J. Lee, Introduction to Topological Manifolds, Graduate Texts in Mathematics, Springer, 2010. • Singular homology. It is worth completing the full details of the theory of singular homology. This can be found in the book [Ma2] mentioned above. • Simplicial homology. A basic approach to homology is given by developing the homology of simplicial complexes, and then extending it to more general spaces. This can be found in [Mun,§1.2]. • De Rham cohomology. The theory of de Rham cohomology is developed fully in [B-T] and [M-T]. It is interesting to introduce de Rham cohomology and then use it to define homology for more general spaces, as outlined in Remark 2.125. • Whitehead theorem. This topic leads to CW-complexes and the associated cellular homology. We recommend looking at the book: [Hat] A. Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. • Čech cohomology. This cohomology theory is a wide generalization of the Mayer-Vietoris principle to arbitrary coverings. It can be found in [B-T,§8-10]. • Abelian categories. These can be found in: [Pop] N. Popescu, Abelian Categories with Applications to Rings and Modules, Academic Press, 1973. [McL] S. MacLane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Springer, 1998. • Homological algebra. The algebraic aspects of homology can be studied from: [Wei] C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, 1995. • Morse theory. Morse theory gives a description of the homotopy type of a manifold by using the critical points of a generic function. It can be found in [B-T,§17] or in: [Mi3] J.W. Milnor, Morse Theory, Annals of Mathematic Studies, Vol. 51, Princeton University Press, 1963. • Handlebody decomposition and classification of surfaces. We propose using Morse theory for giving an alternative proof of the classification of compact surfaces by decomposing them into a handlebody. This can be studied from [Mi3] or from: [Law] T. Lawson, Topology: a Geometric Approach, Oxford Graduate Texts in Mathematics, 9. Oxford University Press, 2003. • h-Cobordism. A beautiful theory that has been very productive in classifying manifolds in dimensions ≥ 5. We recommend: [Mi4] J.W. Milnor, Lectures on the h-Cobordism Theorem, Princeton Legacy Library, Princeton University Press, 2015.

References and extra material

135

• Transversality and the Sard theorem. This can be found in: [Kos] A. Kosinski, Differential Manifolds, Academic Press, Vol. 138, Elsevier, 1992. [G-P] V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1974. • Poincaré-Hopf theorem. This important result relates the sources and sinks of a smooth vector field with the Euler-Poincaré characteristic. It is a large generalization of the hairy ball theorem. [Mi5] J.W. Milnor, Topology from the Differentiable Viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, 1997. • Rational homotopy. This consists of the study of the rational homotopy groups 𝜋𝑘 (𝑋) ⊗ ℚ for a space 𝑋. We have information on this in [B-T,§19] or in: [G-M] P.A. Griffiths, J.W. Morgan, Rational Homotopy Theory and Differential Forms, Birkhäuser, 1981. • Topology of finite spaces. Finite spaces which are not Hausdorff have interesting homotopy properties. [Bar] J. Barmak, Algebraic Topology of Finite Topological Spaces and Applications, Lecture Notes in Mathematics, Vol. 2032, 2011. • Currents. The theory of (Schwartz) distributions (e.g., Dirac deltas) for manifolds and forms give rise to the concept of currents. This allows one to develop a homology theory that includes naturally singular and de Rham theories. [L-Y] F. Lin, X. Yang, Geometric Measure Theory: An Introduction, Advanced Mathematics, Vol. 1, Science Press, 2003. References. [Ada] M. Adachi, Embeddings and immersions, Translations of Mathematical Monographs, Vol. 124, American Mathematical Society, 1993. [Do1] S. Donaldson, An application of gauge theory to four-dimensional topology, J. Diff. Geom. 18, 279-315, 1983. [Fre] M. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17, 357-453, 1982. [Per] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159. [Sco] A. Scorpan, The Wild World of 4-Manifolds, American Mathematical Society, 2005. [Sm1] S. Smale, On the structure of manifolds, Amer. Jour. Math. 84, 387-399, 1962. [Spa] E.H. Spanier, Algebraic Topology, Springer, 1994. [Sch] M. Schwarz, Morse Homology, Birkhäuser, 1993. [Tho] R. Thom, Quelques propriétés globales des variétés différentiables, Commentarii Mathematici Helvetici, 28, 17-86, 1954.

Chapter 3

Riemannian geometry

We move forward in our study of spaces by giving more structure to manifolds and using it to derive topological properties. Following the path initiated in section 2.5, differentiable geometric information can lead to topological properties. As a general philosophy, we consider a differentiable manifold and introduce geometric structure as a way to make measurements, such as lengths, angles, areas, and volumes (to name some of a geometric nature), or as a way to express forces and moments (to name some of a physical nature). Most of these (but of course, not all) can be written as tensors on a manifold. So a geometric structure can be understood as a tensor or collection of tensors that encode a way to make geometrical measurements or pose physical equations. Thus this is extra structure for the manifold. We shall focus here on the historically most prominent geometric structure, a Riemannian metric, and derive notions as the curvature. We prove the Gauss-Bonnet theorem for oriented connected surfaces which gives its genus in terms of the curvature. That is, we get topology from geometry; although it is also an instance of the fact that we can get global properties (the genus is a global invariant) from local properties (the curvature). In the last section, we shall focus on Riemannian manifolds, which are isotropic (that is, they look alike at every place), where the classification problem becomes tractable.

3.1. Riemannian metrics. Curvature. Geodesics We start by reviewing the necessary tools from Riemannian geometry to apply them to the study of the topology of manifolds, and in particular, to surfaces. As in previous chapters, this will be a review of known notions with few proofs. The theory can be found in [Boo], [DC2], [Sp2]. From now on, 𝑀 will be a smooth 𝑛-manifold. 3.1.1. Tensors on manifolds. In this section, we consider vector spaces over the real numbers. We have already defined the tensor product in section 2.4.2. Given real vector spaces 𝑉1 , . . . , 𝑉 𝑘 , their tensor product is denoted 𝑉1 ⊗ ⋯ ⊗ 𝑉 𝑘 , where the ground ring ℝ is understood. There is a multilinear map 𝑉1 × ⋯ × 𝑉 𝑘 → 𝑉1 ⊗ ⋯ ⊗ 𝑉 𝑘 , (𝑣 1 , . . . , 𝑣 𝑘 ) ↦ 𝑣 1 ⊗ ⋯ ⊗ 𝑣 𝑘 . By definition of tensor product, for any vector space 𝑊, there is an isomorphism between the space of multilinear maps 𝑉1 × ⋯ × 𝑉 𝑘 → 𝑊 and 137

138

3. Riemannian geometry

𝑗

𝑗

Hom(𝑉1 ⊗⋯⊗𝑉 𝑘 , 𝑊). Let (𝑒 1 , . . . , 𝑒 𝑛𝑗 ) be a basis of 𝑉 𝑗 , 𝑗 = 1, . . . , 𝑘. Then a multilinear map 𝜑 ∶ 𝑉1 × ⋯ × 𝑉 𝑘 → 𝑊 is determined by the images of the basis vectors 𝑎𝑖1 ⋯𝑖𝑘 = 𝜑(𝑒1𝑖1 , . . . , 𝑒𝑘𝑖𝑘 ) for 1 ≤ 𝑖𝑗 ≤ 𝑛𝑗 and 1 ≤ 𝑗 ≤ 𝑘. Thus the vector space 𝑉1 ⊗ ⋯ ⊗ 𝑉 𝑘 has basis 𝑒1𝑖1 ⊗ ⋯ ⊗ 𝑒𝑘𝑖𝑘 , 1 ≤ 𝑖𝑗 ≤ 𝑛𝑗 , 1 ≤ 𝑗 ≤ 𝑘. It is easy to see that (𝑉1 ⊗ ⋯ ⊗ 𝑉 𝑘 )∗ = 𝑉1∗ ⊗ ⋯ ⊗ 𝑉𝑘∗ , where 𝛼 = 𝛼1 ⊗ ⋯ ⊗ 𝛼𝑘 ∈ 𝑉1∗ ⊗ ⋯ ⊗ 𝑉𝑘∗ acts on a tensor as (𝛼1 ⊗ ⋯ ⊗ 𝛼𝑘 )(𝑣 1 ⊗ ⋯ ⊗ 𝑣 𝑘 ) = 𝛼1 (𝑣 1 ) ⋯ 𝛼𝑘 (𝑣 𝑘 ). Clearly 𝛼 ∶ 𝑉1 × ⋯ × 𝑉 𝑘 → ℝ defines a multilinear map. We apply this to the the tangent and cotangent spaces of a differentiable manifold 𝑀. Recall that the tangent space 𝑇𝑝 𝑀 is the space of derivations of smooth functions around 𝑝, i.e., a vector 𝑋𝑝 ∈ 𝑇𝑝 𝑀 assigns to every ℎ ∈ 𝐶 ∞ (𝑈 𝑝 ), where 𝑈 𝑝 is a neighbourhood of 𝑝, a real number 𝑋𝑝 (ℎ). This is linear and satisfies the Leibniz rule: 𝑋𝑝 (𝑔 ℎ) = 𝑋𝑝 (𝑔)ℎ(𝑝) + 𝑔(𝑝)𝑋𝑝 (ℎ). Given a chart (𝑈 𝑝 , (𝑥1 , . . . , 𝑥𝑛 )), we have a natu𝜕

𝜕

𝜕

ral basis ( 𝜕𝑥 , . . . , 𝜕𝑥 ) of 𝑇𝑝 𝑀 such that 𝜕𝑥 (𝑥𝑗 ) = 𝛿 𝑖𝑗 , the Kronecker delta (that is, 1 𝑛 𝑖 𝛿 𝑖𝑗 = 1 if 𝑖 = 𝑗, and 𝛿 𝑖𝑗 = 0 if 𝑖 ≠ 𝑗). The dual vector space is the cotangent space 𝑇𝑝∗ 𝑀, also denoted Ω𝑝1 (𝑀). In a coordinate chart, there is a basis (𝑑𝑥1 , . . . , 𝑑𝑥𝑛 ) deter𝜕

mined by 𝑑𝑥𝑖 ( 𝜕𝑥 ) = 𝛿 𝑖𝑗 . A vector field is a map 𝑋 ∶ 𝑀 → 𝑇𝑀 = ⨆𝑝∈𝑀 𝑇𝑝 𝑀 such 𝑗 that 𝑋𝑝 ∈ 𝑇𝑝 𝑀 for each 𝑝 ∈ 𝑀, and it varies smoothly: for every coordinate chart, we 𝜕

can write 𝑋 = ∑ 𝑋𝑖 𝜕𝑥 , with 𝑋𝑖 smooth functions. The space of vector fields on 𝑀 is 𝑖 denoted 𝔛(𝑀). Analogously, a 1-form is a map 𝛼 ∶ 𝑀 → 𝑇 ∗ 𝑀 = ⨆𝑝∈𝑀 𝑇𝑝∗ 𝑀 such that 𝛼𝑝 ∈ 𝑇𝑝 𝑀 for each 𝑝 ∈ 𝑀, and locally 𝛼 = ∑ 𝛼𝑖 𝑑𝑥𝑖 with 𝛼𝑖 smooth functions. The space of 1-forms is denoted Ω1 (𝑀). We denote (𝑟)

(𝑠)

𝑇𝑝𝑟,𝑠 𝑀 = 𝑇𝑝 𝑀⊗ ⋯ ⊗𝑇𝑝 𝑀 ⊗ 𝑇𝑝∗ 𝑀⊗ ⋯ ⊗𝑇𝑝∗ 𝑀, (𝑟,𝑠)

and we call its elements (𝑟, 𝑠)-tensors at 𝑝 ∈ 𝑀. By definition an element of 𝑇𝑝 𝑀 is of the form 𝜕 𝜕 𝑖 ⋯𝑖 𝑇 = ∑ 𝑎𝑗11 ⋯𝑗𝑟𝑠 ⊗⋯⊗ ⊗ 𝑑𝑥𝑗1 ⊗ ⋯ ⊗ 𝑑𝑥𝑗𝑠 , 𝜕𝑥𝑖1 𝜕𝑥𝑖𝑟 where the sum runs over 1 ≤ 𝑖1 , . . . , 𝑖𝑟 , 𝑗1 , . . . , 𝑗𝑠 ≤ 𝑛. Observe that there is a natural iso𝑖 ⋯𝑖 morphism 𝑇𝑝𝑟,𝑠 𝑀 ≅ (𝑇𝑝𝑠,𝑟 𝑀)∗ . Under it, the coefficient 𝑎𝑗11 ⋯𝑗𝑟𝑠 ∈ ℝ can be computed by 𝑖 ⋯𝑖

𝑎𝑗11 ⋯𝑗𝑟𝑠 = 𝑇(𝑑𝑥𝑖1 , . . . , 𝑑𝑥𝑖𝑟 , 𝜕𝑥𝑗 , . . . , 𝜕𝑥𝑗𝑠 ). Henceforth, we shall use often the abbreviated notation 𝜕𝑥𝑖 =

𝜕 . 𝜕𝑥𝑖

1

Definition 3.1. A tensor on 𝑀 of type (𝑟, 𝑠) is a map 𝑇 ∶ 𝑀 → ⨆𝑝∈𝑀 𝑇𝑝𝑟,𝑠 𝑀, such (𝑟,𝑠)

that 𝑇(𝑝) ∈ 𝑇𝑝 𝑀, for all 𝑝 ∈ 𝑀, which is smooth in the following sense: for any 1-forms 𝛼1 , . . . , 𝛼𝑟 ∈ Ω1 (𝑀) and vector fields 𝑋1 , . . . , 𝑋𝑠 ∈ 𝔛(𝑀), the function 𝑇(𝛼1 , . . . , 𝛼𝑟 , 𝑋1 , . . . , 𝑋𝑠 ) is smooth. We denote 𝒯 𝑟,𝑠 (𝑀) the space of (𝑟, 𝑠)-tensors on 𝑀. An element 𝑇 ∈ 𝒯 𝑟,𝑠 (𝑀) defines a multilinear map 𝑇 ∶ Ω1 (𝑀)𝑟 × 𝔛(𝑀)𝑠 → 𝐶 (𝑀). It is 𝐶 ∞ (𝑀)-multilinear, that is, 𝑇(𝑓𝛼1 , 𝛼2 , . . . , 𝑋𝑠 ) = 𝑓 𝑇(𝛼1 , 𝛼2 , . . . , 𝑋𝑠 ) for functions 𝑓 ∈ 𝐶 ∞ (𝑀), and similarly on all entries. Actually, the 𝐶 ∞ (𝑀)-multilinearity ∞

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139

is equivalent to the tensorial character (Exercise 3.1). In a chart (𝑈, 𝜑 = (𝑥1 , . . . , 𝑥𝑛 )), 𝑇 is determined by the coefficients 𝑖 ⋯𝑖

𝑇𝑗11⋯𝑗𝑠𝑟 = 𝑇 (𝑑𝑥𝑖1 , . . . , 𝑑𝑥𝑖𝑟 , 𝜕𝑥𝑗 , . . . , 𝜕𝑥𝑗𝑠 ) ∈ 𝐶 ∞ (𝑈), 1

since locally we can write 𝑖 ⋯𝑖

𝑇|𝑈 = ∑ 𝑇𝑗11⋯𝑗𝑠𝑟 𝜕𝑥𝑖 ⊗ ⋯ ⊗ 𝜕𝑥𝑖𝑟 ⊗ 𝑑𝑥𝑗1 ⊗ ⋯ ⊗ 𝑑𝑥𝑗𝑠 .

(3.1)

1

The 𝐶 ∞ (𝑀)-multilinearity says that 𝑖 ⋯𝑖

𝑗

𝑗

𝑇(𝛼1 , . . . , 𝛼𝑟 , 𝑋1 , . . . , 𝑋𝑠 ) = ∑ 𝑇𝑗11⋯𝑗𝑠𝑟 𝛼1,𝑖1 ⋯ 𝛼𝑟,𝑖𝑟 𝑋1 1 ⋯ 𝑋𝑠 𝑠 ,

(3.2)

for 𝛼𝑙 = ∑ 𝛼𝑙,𝑗 𝑑𝑥𝑗 and 𝑋𝑙 = ∑ 𝑋𝑙𝑖 𝜕𝑥𝑖 . This type of formula is characterized by the fact that the coefficients 𝛼𝑙,𝑗 and 𝑋𝑙𝑖 appear with no derivation. Note that 𝒯 𝑟,𝑠 (𝑀) has the structure of a 𝐶 ∞ (𝑀)-module. Remark 3.2. Formula (3.2) is typical in tensor calculus and justifies the position of the indices (in particular, for the coordinates of a vector field). It often occurs that in an expression, if an index 𝑖 appears repeated, then there is a sum where the index ranges 1 ≤ 𝑖 ≤ 𝑛. In that case, the index usually appears once in the top position and the other time in the bottom position. Note that the indices of (𝑟, 𝑠)-tensors should be in top position when they correspond to vectors, and in the bottom position when they correspond to covectors. Moreover, it is also usual to follow Einstein notation which omits the summatory sign when this happens, being implicitly understood. We shall not do this. Example 3.3. (1) A (0, 0)-tensor is a smooth function. A (1, 0)-tensor is a vector field. (2) A (0, 1)-tensor is a 1-form 𝛼 = ∑ 𝛼𝑗 𝑑𝑥𝑗 . A 𝑘-form is an example of a (0, 𝑘)tensor (see section 2.5.1). A 𝑘-form 𝛼 = ∑ 𝛼𝑗1 ⋯𝑗𝑘 𝑑𝑥𝑗1 ∧ ⋯ ∧ 𝑑𝑥𝑗𝑘 is a multilinear map 𝛼 ∶ 𝔛(𝑀)𝑘 → 𝐶 ∞ (𝑀) which is antisymmetric. Hence we have an equality, 𝑑𝑥𝑗1 ∧ ⋯ ∧ 𝑑𝑥𝑗𝑘 = ∑(−1)𝜍 𝑑𝑥𝑗𝜍(1) ⊗ ⋯ ⊗ 𝑑𝑥𝑗𝜍(𝑘) ,

(3.3)

𝜍

where the sum goes over all permutations of 𝑘 elements. The space of 𝑘-forms is denoted Ω𝑘 (𝑀). (3) We shall also use symmetric (0, 𝑘)-tensors, that is 𝑇 ∶ 𝔛(𝑀)𝑘 → 𝐶 ∞ (𝑀) which is a symmetric multilinear map. If we write 𝑇𝑗1 ⋯𝑗𝑘 = 𝑇(𝜕𝑥𝑗 , . . . , 𝜕𝑥𝑗 ) 1 𝑘 and introduce the symbols 1 ∑ 𝑑𝑥𝑗𝜍(1) ⊗ ⋯ ⊗ 𝑑𝑥𝑗𝜍(𝑘) , (3.4) 𝑑𝑥𝑗1 ⋅ . . . ⋅ 𝑑𝑥𝑗𝑘 = 𝑘! 𝜍 then we have ∑ 𝑇𝑗1 ⋯𝑗𝑘 𝑑𝑥𝑗1 1≤𝑗1 ,. . .,𝑗𝑘 ≤𝑛

𝑇=

∑ 𝑇𝑗1 ⋯𝑗𝑘 𝑑𝑥𝑗1 1≤𝑗1 ,. . .,𝑗𝑘 ≤𝑛

⊗ ⋯ ⊗ 𝑑𝑥𝑗𝑘 =

⋅ . . . ⋅ 𝑑𝑥𝑗𝑘 .

For short, we will also denote 𝑑𝑥𝑖2 = 𝑑𝑥𝑖 ⋅ 𝑑𝑥𝑖 , and we will omit the dot when the symmetrization is clear from the context.

140

3. Riemannian geometry

Note the discrepancy between (3.3) and (3.4), since in the first one we do not divide by 𝑘!. However, this will be useful for later formulas. Remark 3.4. Let us explore some natural operations with tensors. ′

′

(1) Product of tensors. Given 𝑇1 ∈ 𝒯 𝑟,𝑠 (𝑀), 𝑇2 ∈ 𝒯 𝑟 ,𝑠 (𝑀), there is a naturally ′ ′ defined tensor 𝑇1 ⊗ 𝑇2 ∈ 𝒯 𝑟+𝑟 ,𝑠+𝑠 (𝑀). This corresponds to the multilinear ′ ′ map 𝑇1 × 𝑇2 ∶ Ω1 (𝑀)𝑟+𝑟 × 𝔛(𝑀)𝑠+𝑠 → 𝐶 ∞ (𝑀) given by the product of ′ ′ 𝑇1 ∶ Ω1 (𝑀)𝑟 × 𝔛(𝑀)𝑠 → 𝐶 ∞ (𝑀) and 𝑇2 ∶ Ω1 (𝑀)𝑟 × 𝔛(𝑀)𝑠 → 𝐶 ∞ (𝑀). (2) Contraction. For vector spaces 𝑉, 𝑉3 , . . . , 𝑉 𝑘 , there is a natural map 𝐶 ∶ 𝑉 ⊗ 𝑉 ∗ ⊗𝑉3 ⊗⋯⊗𝑉 𝑘 → 𝑉3 ⊗⋯⊗𝑉 𝑘 , 𝑣⊗𝛼⊗𝑇 ↦ 𝛼(𝑣)𝑇, called contraction. For tensors of type (𝑟, 𝑠), with 𝑟, 𝑠 ≥ 1, consider the 𝑎th copy of 𝑇𝑝 𝑀 and the 𝑏th (𝑟,𝑠)

copy of 𝑇𝑝∗ 𝑀, 1 ≤ 𝑎 ≤ 𝑟, 1 ≤ 𝑏 ≤ 𝑠. We may contract them 𝐶𝑎𝑏 ∶ 𝑇𝑝

𝑀→

(𝑟−1,𝑠−1) 𝑇𝑝 𝑀,

giving a map on tensors. For instance, we write the contraction 𝐶11 in coordinates. Since 𝐶11 (𝜕𝑥𝑖 ⊗ 𝑑𝑥𝑗 ) = 𝛿 𝑖𝑗 , we get 𝑖 ⋯𝑖

𝐶11 (∑ 𝑇𝑗11⋯𝑗𝑠𝑟 𝜕𝑥𝑖 ⊗ ⋯ ⊗ 𝜕𝑥𝑖𝑟 ⊗ 𝑑𝑥𝑗1 ⊗ ⋯ ⊗ 𝑑𝑥𝑗𝑠 ) 1

𝑛

𝑎 𝑖 ⋯𝑖

= ∑ ∑ 𝑇𝑎 𝑗22⋯𝑗𝑟𝑠 𝜕𝑥𝑖 ⊗ ⋯ ⊗ 𝜕𝑥𝑖𝑟 ⊗ 𝑑𝑥𝑗2 ⊗ ⋯ ⊗ 𝑑𝑥𝑗𝑠 . 2

𝑎=1

All these formulas follow tensor calculus conventions. Remark 3.5. Finally, let us comment on functoriality. (1) Given a differentiable map 𝑓 ∶ 𝑀 → 𝑁 and 𝑝 ∈ 𝑀, there is a linear map 𝑑𝑝 𝑓 ∶ 𝑇𝑝 𝑀 → 𝑇 𝑓(𝑝) 𝑁, called the differential of 𝑓 at 𝑝, given by 𝑑𝑝 𝑓(𝑋𝑝 )(ℎ) = 𝑋𝑝 (ℎ ∘ 𝑓), for 𝑋𝑝 ∈ 𝑇𝑝 𝑀 and ℎ ∈ 𝐶 ∞ (𝑈 𝑓(𝑝) ). We also denote 𝑓∗ (𝑣) = 𝑑𝑝 𝑓(𝑣), for a vector 𝑣 ∈ 𝑇𝑝 𝑀, 𝑝 ∈ 𝑀. (2) If 𝜑 ∶ 𝑀 → 𝑁 is a diffeomorphism and 𝑇 ∈ 𝒯 𝑟,𝑠 (𝑀), then there is a pushforward tensor 𝜑∗ (𝑇) ∈ 𝒯 𝑟,𝑠 (𝑁). For a vector field 𝑋, it is 𝜑∗ (𝑋)𝜑(𝑝) = 𝑑𝑝 𝜑(𝑋𝑝 ), for 𝑝 ∈ 𝑀. For a 1-form 𝛼, 𝜑∗ (𝛼)𝜑(𝑝) = 𝛼𝑝 ∘ (𝑑𝑝 𝜑)−1 . For general tensors, 𝜑∗ (𝑇1 ⊗ 𝑇2 ) = 𝜑∗ (𝑇1 ) ⊗ 𝜑∗ (𝑇2 ). (3) The pullback is defined by 𝜑∗ (𝑇 ′ ) = (𝜑−1 )∗ (𝑇 ′ ), for 𝑇 ′ ∈ 𝒯 𝑟,𝑠 (𝑁). (4) Locally, if (𝑦1 , . . . , 𝑦𝑛 ) = 𝜑(𝑥1 , . . . , 𝑥𝑛 ) in coordinates, then for 𝑇 as in (3.1) we 𝜕𝑥 𝜕𝑦 obtain 𝜑∗ (𝑇) by substituting 𝑑𝑥𝑗 = ∑ 𝜕𝑦𝑗 𝑑𝑦 𝑖 and 𝜕𝑥𝑗 = ∑ 𝜕𝑥𝑖 𝜕𝑦𝑖 . That is 𝑖

𝑖 ⋯𝑖

(3.5)

𝜑∗ (𝑇) = ∑(𝑇𝑗11⋯𝑗𝑠𝑟 ∘ 𝜑−1 )(𝑦1 , . . . , 𝑦𝑛 )

𝑗

𝜕𝑥𝑗𝑠 𝜕𝑦𝑎1 𝜕𝑦𝑎𝑟 𝜕𝑥𝑗1 ⋯ ⋯ 𝜕𝑥𝑖1 𝜕𝑥𝑖𝑟 𝜕𝑦 𝑏1 𝜕𝑦 𝑏𝑠

⋅ 𝜕𝑦𝑎 ⊗ ⋯ ⊗ 𝜕𝑦𝑎𝑟 ⊗ 𝑑𝑦 𝑏1 ⊗ ⋯ ⊗ 𝑑𝑦 𝑏𝑠 . 1

(5) Since changes of coordinates are instances of diffeomorphisms, the same formula holds. Sometimes, mainly in the physics literature, tensors are defined as gadgets that are given locally at every chart and satisfying the above formula for every change of coordinates. (6) Tensors of type (0, 𝑠) admit pullback for smooth maps 𝑓 ∶ 𝑀 → 𝑁. For 𝛼 ∈ 𝒯 0,𝑠 (𝑁), 𝑓∗ 𝛼(𝑋1 , . . . , 𝑋𝑠 )𝑝 = 𝛼𝑓(𝑝) (𝑓∗ (𝑋1,𝑝 ), . . . , 𝑓∗ (𝑋𝑠,𝑝 )). See (2.13) for

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141

the local expression in the case of 𝑠-forms. The local expression is the same (with ⊗ instead of ∧) for general (0, 𝑠)-tensors. 3.1.2. Connections. Tensors serve to encode geometric information on a manifold. For instance, a magnetic force can be given by a vector field. To solve geometrical or physical problems, we usually have to write differential equations. For this we have to take derivatives on the manifold. For functions 𝑓 ∈ 𝐶 ∞ (𝑀), this is the usual differ𝜕𝑓 entiation 𝑋(𝑓) = ∑ 𝑋 𝑖 𝜕𝑥 , for a vector field 𝑋 = ∑ 𝑋 𝑖 𝜕𝑥𝑖 ∈ 𝔛(𝑀). But we also have 𝑖 to do differentiation of tensors. The main obstacle that we encounter is that there is no natural way to define this differentiation. That is, for functions, the derivative is an intrinsic property of the manifold (or better said, it is encoded in the smooth atlas), but the smooth atlas does not encode a way to differentiate other tensors. Let us check this assertion by trying naively to define the differentiation on vector fields and seeing where it leads. Suppose that we are in a chart with coordinates (𝑥1 , . . . , 𝑥𝑛 ), and consider a vector field 𝑌 = ∑ 𝑌 𝑖 (𝑥)𝜕𝑥𝑖 . It seems natural to define the derivative as (3.6)

𝐷𝑣 𝑌 = lim 𝑡→0

𝑌 (𝑝 + 𝑡𝑣) − 𝑌 (𝑝) = ∑ 𝑣(𝑌 𝑖 )𝜕𝑥𝑖 , 𝑡

for a vector 𝑣 ∈ 𝑇𝑝 𝑀 ≅ ℝ𝑛 , where 𝑣(𝑌 𝑖 ) = 𝐷𝑣 𝑌 𝑖 is the directional derivative of the function 𝑌 𝑖 . Now suppose that we have other coordinates (𝑦1 , . . . , 𝑦𝑛 ). Then 𝑌 = 𝜕𝑥 𝜕𝑥 ∑ 𝑌 ̂ 𝑗 𝜕𝑦𝑗 = ∑ 𝑌 ̂ 𝑗 𝜕𝑦 𝑖 𝜕𝑥𝑖 . So 𝑌 𝑖 = ∑ 𝑌 ̂ 𝑗 𝜕𝑦 𝑖 . Then (3.6) implies that 𝑗

𝑗

𝐷𝑣 𝑌 = ∑ 𝑣(𝑌 𝑖 )𝜕𝑥𝑖 = ∑ 𝑣 (𝑌 ̂ 𝑗 = ∑ 𝑣(𝑌 ̂ 𝑗 )

(3.7)

𝜕𝑥𝑖 )𝜕 𝜕𝑦𝑗 𝑥𝑖

𝜕𝑥𝑖 𝜕𝑥 𝜕 + 𝑌 ̂ 𝑗 𝑣 ( 𝑖 ) 𝜕𝑥𝑖 𝜕𝑦𝑗 𝑥𝑖 𝜕𝑦𝑗

= ∑ 𝑣(𝑌 ̂ 𝑗 )𝜕𝑦𝑗 + 𝑌 ̂ 𝑗 𝑣 (

𝜕𝑥𝑖 )𝜕 . 𝜕𝑦𝑗 𝑥𝑖

If the definition above has a chance of being consistent, one would need that 𝐷𝑣 𝑌 = ∑ 𝑣(𝑌 ̂ 𝑗 )𝜕𝑦𝑗 in the coordinates (𝑦1 , . . . , 𝑦𝑛 ). However, there is a spurious term popping out. Fortunately, this is a term not involving derivatives of 𝑌 (it is of order 0, in the terminology of differential equations). Before knowing if there is a way to overcome these difficulties, let us formalize what we are looking for. Definition 3.6. A connection is a map ∇ ∶ 𝔛(𝑀) × 𝒯 𝑟,𝑠 (𝑀) → 𝒯 𝑟,𝑠 (𝑀), for all 𝑟, 𝑠 ≥ 0, such that for 𝑇, 𝑇1 , 𝑇2 ∈ 𝒯 𝑟,𝑠 (𝑀), 𝑓 ∈ 𝐶 ∞ (𝑀) and 𝑋 ∈ 𝔛(𝑀), we have (1) ∇𝑋 (𝑇1 + 𝑇2 ) = ∇𝑋 𝑇1 + ∇𝑋 𝑇2 , (2) ∇𝑓𝑋 𝑇 = 𝑓∇𝑋 𝑇, (3) for functions, ∇𝑋 𝑓 = 𝑋(𝑓), (4) the Leibniz rule, ∇𝑋 (𝑇1 ⊗ 𝑇2 ) = (∇𝑋 𝑇1 ) ⊗ 𝑇2 + 𝑇1 ⊗ (∇𝑋 𝑇2 ), (5) for any contraction 𝐶 ∶ 𝒯 𝑟,𝑠 (𝑀) → 𝒯 𝑟−1,𝑠−1 (𝑀), ∇𝑋 (𝐶(𝑇)) = 𝐶(∇𝑋 𝑇).

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Remark 3.7. (1) Given a vector field 𝑋 over 𝑀, we say that ∇𝑋 ∶ 𝒯 𝑟,𝑠 (𝑀) → 𝒯 𝑟,𝑠 (𝑀) is the derivative in the direction of 𝑋. Note that property (2) means that ∇𝑋 𝑇 is tensorial on 𝑋. Therefore we can define a tensor ∇𝑇 ∈ 𝒯 𝑟,𝑠+1 (𝑀) given by (∇𝑇)(𝑋, −) = ∇𝑋 𝑇, for 𝑇 ∈ 𝒯 𝑟,𝑠 (𝑀). This tensorial character allows us to define ∇𝑣 𝑇 for a vector 𝑣 ∈ 𝑇𝑝 𝑀, 𝑝 ∈ 𝑀, as ∇𝑣 𝑇 = (∇𝑋 𝑇)(𝑝), for a vector field 𝑋 with 𝑋(𝑝) = 𝑣 (cf. Exercise 3.1). (2) Definition 3.6(3) says that ∇𝑓 = 𝑑𝑓. For a function 𝑓, we have that 𝑓 ⊗ 𝑇 = 𝑓 𝑇. So Definition 3.6(4) for 𝑇1 = 𝑓 says that ∇𝑋 (𝑓 𝑇) = 𝑋(𝑓)𝑇 + 𝑓∇𝑋 𝑇. (3) In general, a tensor 𝑇 is a sum of terms which are tensor products of (1, 0)tensors and (0, 1)-tensors. Therefore, thanks to Definition 3.6(4), it is enough to define ∇𝑋 𝑇 for vector fields and 1-forms. Let us see that it actually is enough to know ∇𝑋 𝑌 for vector fields 𝑌 ∈ 𝔛(𝑀). Let 𝛼 ∈ Ω1 (𝑀) and observe that 𝐶(𝛼 ⊗ 𝑌 ) = 𝛼(𝑌 ), where 𝐶 ∶ Ω1 (𝑀) × 𝔛(𝑀) → 𝐶 ∞ (𝑀) is the contraction. Applying ∇𝑋 , we obtain 𝑋(𝛼(𝑌 )) = ∇𝑋 (𝛼(𝑌 )) = ∇𝑋 𝐶(𝛼 ⊗ 𝑌 ) = 𝐶(∇𝑋 (𝛼 ⊗ 𝑌 )) = 𝐶((∇𝑋 𝛼) ⊗ 𝑌 + 𝛼 ⊗ ∇𝑋 𝑌 ) = (∇𝑋 𝛼)(𝑌 ) + 𝛼(∇𝑋 𝑌 ). And thus, we have that the 1-form ∇𝑋 𝛼 ∈ Ω1 (𝑀) is characterized by the formula (3.8)

(∇𝑋 𝛼) (𝑌 ) = 𝑋 (𝛼(𝑌 )) − 𝛼 (∇𝑋 𝑌 ) . This means that the connection ∇ is determined by ∇ ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀). This is the way that a connection is typically defined in the literature. (4) Let us write down a connection in local coordinates (𝑥1 , . . . , 𝑥𝑛 ). The behaviour of ∇ in the basis vector fields is given by ∇𝜕𝑥 𝜕𝑥𝑗 = ∑ Γ𝑖𝑗𝑘 𝜕𝑥𝑘 , 𝑖

𝑘

which is determined by the 𝑛3 functions Γ𝑖𝑗𝑘 , called the Christoffel symbols of the connection. Now, given 𝑋 = ∑ 𝑋 𝑖 𝜕𝑥𝑖 , 𝑌 = ∑ 𝑌 𝑗 𝜕𝑥𝑗 ∈ 𝔛(𝑈), we have ∇𝑋 𝑌 = ∇𝑋 (∑ 𝑌 𝑗 𝜕𝑥𝑗 ) = ∑ 𝑌 𝑗 ∇𝑋 𝜕𝑥𝑗 + 𝑋(𝑌 𝑗 )𝜕𝑥𝑗 𝑗

(3.9)

𝑗

= ∑ 𝑌 𝑗 𝑋 𝑖 ∇𝜕𝑥 𝜕𝑥𝑗 + ∑ 𝑋 𝑖 𝑖

𝑖,𝑗

= ∑ (∑ 𝑋 𝑖 𝑘

𝑖

𝑖,𝑗

𝜕 𝑌𝑗 𝜕 𝜕𝑥𝑖 𝑥𝑗

𝜕 𝑌𝑘 + ∑ 𝑋 𝑖 𝑌 𝑗 Γ𝑖𝑗𝑘 ) 𝜕𝑥𝑘 . 𝜕𝑥𝑖 𝑖,𝑗

In particular, the Christoffel symbols determine the connection. Note the tensorial character on 𝑋, but the non-tensorial character on 𝑌 (that is, the formula contains derivatives of 𝑌 ). So to know (∇𝑋 𝑌 )𝑝 it is enough to know 𝑋(𝑝), but we have to know 𝑌 in a neighbourhood of 𝑝.

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143

(5) We see in the formula (3.9) that the covariant derivative ∇𝑋 𝑌 is the naive derivative (3.6) plus an order 0 term similar to the spurious term in (3.7). In some sense, the connection is defined by allowing a non-trivial order 0 term in every coordinate chart. (6) Let us give the formula for the covariant derivative of a 1-form using (3.8), ∇𝜕𝑥 𝑑𝑥𝑗 = ∑ (∇𝜕𝑥 𝑑𝑥𝑗 ) (𝜕𝑥𝑘 ) 𝑑𝑥𝑘 𝑖

(3.10)

𝑘

𝑖

𝑗

= ∑ (𝜕𝑥𝑖 (𝛿𝑗𝑘 ) − 𝑑𝑥𝑗 (∇𝜕𝑥 𝜕𝑥𝑘 )) 𝑑𝑥𝑘 = − ∑ Γ𝑖𝑘 𝑑𝑥𝑘 . 𝑘

𝑖

𝑘

Before continuing, let us see that there are connections on any manifold. Proposition 3.8. Every differentiable manifold 𝑀 admits a connection, and the space of connections 𝒞(𝑀) is an affine space modelled on 𝒯 1,2 (𝑀). Proof. As we have said before, a connection ∇ is equivalent to an ℝ-bilinear operator ∇ ∶ 𝔛(𝑀)×𝔛(𝑀) → 𝔛(𝑀) such that ∇𝑓𝑋 𝑌 = 𝑓∇𝑋 𝑌 and ∇𝑋 (𝑓𝑌 ) = 𝑋(𝑓)𝑌 +𝑓∇𝑋 𝑌 , for functions 𝑓 and vector fields 𝑋, 𝑌 . Then if ∇, ∇′ are two connections, their difference ′ 𝑆 = ∇ − ∇′ , 𝑆(𝑋, 𝑌 ) = ∇𝑋 𝑌 − ∇𝑋 𝑌 , is an operator 𝑆 ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀) such that 𝑆(𝑓𝑋, 𝑌 ) = 𝑓 𝑆(𝑋, 𝑌 ) and 𝑆(𝑋, 𝑓𝑌 ) = 𝑓 𝑆(𝑋, 𝑌 ). This means that 𝑆 has tensorial character. Actually, if we write 𝑆 as a map 𝑆 ′ ∶ 𝔛(𝑀) × 𝔛(𝑀) × Ω1 (𝑀) → 𝐶 ∞ (𝑀), 𝑆 ′ (𝑋, 𝑌 , 𝛼) = 𝛼(𝑆(𝑋, 𝑌 )), then 𝑆′ ∈ 𝒯 1,2 (𝑀). This means that 𝒞(𝑀) is an affine space modelled in 𝒯 1,2 (𝑀). It remains to see that 𝒞(𝑀) is non-empty. Note that if ∇1 , . . . , ∇𝑟 are connections and 𝜆1 +⋯+𝜆𝑟 = 1, 𝜆𝑖 ∈ 𝐶 ∞ (𝑀), then ∑ 𝜆𝑖 ∇𝑖 is also a connection, since this is an affine combination. To proceed, consider an open covering {𝑈𝛼 } by coordinate charts, and a subordinated partition of unity {𝜌𝛼 }. For each 𝑈𝛼 , we can consider a connection ∇𝛼 on 𝑈𝛼 by taking Γ𝑖𝑗𝛼,𝑘 ≡ 0. This is called the trivial connection on the chart. Then we define ∇ = ∑ 𝜌𝛼 ∇𝛼 , which is a well defined map ∇ ∶ 𝔛(𝑀)×𝔛(𝑀) → 𝔛(𝑀). It is a connection because, locally on every chart, it is an affine combination of connections. □ Torsion of a connection. The notion of torsion of a connection is easily defined but geometrically subtle. Let us first recall that the Lie bracket of two vector fields 𝑋, 𝑌 is the vector field [𝑋, 𝑌 ] defined by [𝑋, 𝑌 ](𝑓) = 𝑋(𝑌 (𝑓)) − 𝑌 (𝑋(𝑓)). Locally, for 𝑋 = ∑ 𝑋 𝑖 𝜕𝑥𝑖 , 𝑌 = ∑ 𝑌 𝑗 𝜕𝑥𝑗 , we have (3.11)

[𝑋, 𝑌 ] = ∑ (𝑋 𝑖

𝜕𝑌 𝑗 𝜕𝑋 𝑗 − 𝑌𝑖 )𝜕 . 𝜕𝑥𝑖 𝜕𝑥𝑖 𝑥𝑗

This is intrinsically defined. It allows us to define a derivation 𝐿𝑋 on tensors, called the Lie derivative, by setting 𝐿𝑋 (𝑌 ) = [𝑋, 𝑌 ], which satisfies (1), (3), (4) and (5) of Definition 3.6 but not (2) (since the formula (3.11) contains also derivatives of 𝑋). This makes 𝐿𝑋 not suitable for differentiating tensors, since there is no way to define 𝐿𝑣 . However, it still has some nice properties (see Exercises 3.4 and 3.5). Given a connection ∇, its torsion 𝜏∇ ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀) is the difference between the antisymmetrization of ∇ and the Lie bracket, 𝜏∇ (𝑋, 𝑌 ) = ∇𝑋 𝑌 − ∇𝑌 𝑋 − [𝑋, 𝑌 ].

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3. Riemannian geometry

From (3.9) and (3.11) we have the local expression 𝜏∇ (𝑋, 𝑌 ) = ∑ 𝑋 𝑖 𝑌 𝑗 (Γ𝑖𝑗𝑘 − Γ𝑗𝑖𝑘 )𝜕𝑥𝑘 . Therefore it is a (1, 2)-tensor 𝜏∇ = ∑(Γ𝑖𝑗𝑘 − Γ𝑗𝑖𝑘 )𝜕𝑥𝑘 ⊗ 𝑑𝑥𝑖 ⊗ 𝑑𝑥𝑗 . A connection is symmetric or torsion free if 𝜏∇ = 0, that is ∇𝑋 𝑌 − ∇𝑌 𝑋 = [𝑋, 𝑌 ] for all 𝑋, 𝑌 . Equivalently, Γ𝑖𝑗𝑘 = Γ𝑗𝑖𝑘 for all 𝑖, 𝑗, 𝑘, or ∇𝜕𝑥 𝜕𝑥𝑗 = ∇𝜕𝑥 𝜕𝑥𝑖 , for all 𝑖, 𝑗. Note that 𝑖 𝑗 this condition mixes the direction of derivation with the object being differentiated. In this sense, its meaning is not geometrically obvious. Covariant derivative and parallel transport. In this section we introduce an alternative point of view to “differentiate” tensors, which is equivalent to that of connections, but it is more geometric. Given a 𝐶 1 curve 𝑐 ∶ 𝐼 ⊂ ℝ → 𝑀, where 𝐼 ⊂ ℝ is an interval, a vector field along 𝑐 is a smooth map 𝑋 ∶ 𝐼 → 𝑇𝑀 such that 𝑋(𝑡) ∈ 𝑇𝑐(𝑡) 𝑀. We denote by 𝔛(𝑐) the space of vector fields along 𝑐. Analogously we define the space of tensors 𝒯 𝑟,𝑠 (𝑐) along 𝑐. Note that when 𝑐 is not regular, then a tensor along 𝑐 is not in general the restriction of a tensor on 𝑀 to 𝑐(𝐼) ⊂ 𝑀. Here 𝑐 ∶ 𝐼 → 𝑀 is a regular curve means that it is an embedding (Exercise 3.8). 𝐷

Definition 3.9. A covariant derivative is a map 𝑑𝑡 ∶ 𝒯 𝑟,𝑠 (𝑐) → 𝒯 𝑟,𝑠 (𝑐), for every curve 𝑐 and 𝑟, 𝑠 ≥ 0, satisfying the following for 𝑇, 𝑇1 , 𝑇2 ∈ 𝒯 𝑟,𝑠 (𝑐), and 𝑓 ∈ 𝐶 ∞ (𝐼): • • •

𝐷 (𝑇 𝑑𝑡 1

𝐷𝑇1 𝐷𝑇 + 𝑑𝑡2 , 𝑑𝑡 𝐷𝑓 for functions, 𝑑𝑡 = 𝑓′ (𝑡), 𝐷 the Leibniz rule, 𝑑𝑡 (𝑇1 ⊗ 𝑇2 )

+ 𝑇2 ) =

• for any contraction 𝐶,

=

𝐷 (𝐶(𝑇)) 𝑑𝑡

𝐷𝑇1 𝑑𝑡

⊗ 𝑇2 + 𝑇1 ⊗

= 𝐶(

𝐷𝑇2 , 𝑑𝑡

𝐷𝑇 ). 𝑑𝑡

Proposition 3.10. A connection ∇ induces a covariant derivative

𝐷 𝑑𝑡

such that:

𝐷𝑇 (𝑡 ) 𝑑𝑡 0

(1) If 𝑐 is a regular curve at 𝑡0 , then = ∇𝑐′ (𝑡0 ) 𝑇,̂ where 𝑇̂ is an extension of ̂ 𝑇 on a neighbourhood of 𝑐(𝑡0 ), that is 𝑇(𝑡) = 𝑇(𝑐(𝑡)) for 𝑡 ∈ (𝑡0 − 𝜖, 𝑡0 + 𝜖). (2) If 𝑐′ (𝑡0 ) = 0, then

𝐷𝑇 (𝑡 ) 𝑑𝑡 0

= 𝑇 ′ (𝑡0 ) (in local coordinates).

Proof. As in Remark 3.7(3), it is enough to define the covariant derivative on vector fields, since this determines the behaviour on all other types of tensors. Let 𝑐 be a 𝐶 1 curve, and let 𝑋 ∈ 𝔛(𝑐). If 𝑐′ (𝑡0 ) ≠ 0, then it is possible to extend 𝑋 to a vector field 𝐷𝑋 𝑋̂ ∈ 𝔛(𝑈) for a neighbourhood 𝑈 𝑐(𝑡0 ) . Let us see which local expression 𝑑𝑡 should have to satisfy (1). Write 𝑐(𝑡) = (𝑥1 (𝑡), . . . , 𝑥𝑛 (𝑡)) and 𝑋̂ = ∑ 𝑋̂ 𝑖 𝜕𝑥𝑖 . Then 𝐷𝑋 𝜕𝑋 ̂ 𝑘 = ∇𝑐′ (𝑡) 𝑋̂ = ∑ (∑ 𝑥𝑖′ (𝑡) + ∑ Γ𝑖𝑗𝑘 (𝑐(𝑡))𝑥𝑖′ (𝑡)𝑋̂ 𝑗 (𝑐(𝑡))) 𝜕𝑥𝑘 𝑑𝑡 𝜕𝑥 𝑖 𝑖 𝑖,𝑗 (3.12)

= ∑(

𝑑𝑥 𝑑𝑋 𝑘 + ∑ Γ𝑖𝑗𝑘 𝑖 𝑋 𝑗 ) 𝜕𝑥𝑘 , 𝑑𝑡 𝑑𝑡 𝑖,𝑗

̂ using the chain rule for 𝑋(𝑡) = 𝑋(𝑐(𝑡)). The expression (3.12) can serve as a definition of the covariant derivative. It sat𝐷𝑋 isfies automatically (2) in the statement: if 𝑐′ (𝑡0 ) = 0, then 𝑑𝑡 (𝑡0 ) = 𝑋 ′ (𝑡0 ). This

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145

definition is consistent since it does not depend on the choices, and it satisfies the required properties. □ Remark 3.11. The covariant derivative

𝐷 𝑑𝑡

determines uniquely the connection ∇.

Definition 3.12. Let 𝑐 ∶ 𝐼 → 𝑀 be a curve in 𝑀, and let 𝑋 ∈ 𝔛(𝑐). We say that 𝑋 is 𝐷𝑋 parallel if 𝐷𝑡 = 0. We denote 𝔛|| (𝑐) the vector space of parallel vector fields along 𝑐. Proposition 3.13. Let 𝑐 ∶ 𝐼 → 𝑀 be a curve, and let 𝑡0 ∈ 𝐼. Then the evaluation map ev𝑡0 ∶ 𝔛∥ (𝑐) → 𝑇𝑐(𝑡0 ) 𝑀,

ev𝑡0 (𝑋) = 𝑋(𝑡0 ),

is an isomorphism. 𝐷𝑋

Proof. A vector field 𝑋 = ∑ 𝑋 𝑘 𝜕𝑥𝑘 ∈ 𝔛(𝑐) is parallel if 𝑑𝑡 = 0. In local coordinates we have the expression (3.12), so 𝑋 satisfies the equations 𝑑𝑥 𝑑𝑋 𝑘 = − ∑ Γ𝑖𝑗𝑘 𝑖 𝑋 𝑗 , 𝑑𝑡 𝑑𝑡 𝑖,𝑗

(3.13)

for 𝑘 = 1, . . . , 𝑛. This is a system of ordinary differential equations (ODE). The Cauchy theorem on the existence and uniqueness of solutions to ODEs says that (3.13) has a unique solution, well defined for all 𝑡 ∈ 𝐼, once we have fixed the initial conditions 𝑋 𝑘 (𝑡0 ) = 𝑣 𝑘 ,

1 ≤ 𝑘 ≤ 𝑛.

Fixing the initial conditions corresponds to fixing ev𝑡0 (𝑋) = 𝑋(𝑡0 ) = (𝑣 1 , . . . , 𝑣 𝑛 ) ∈ 𝑇𝑐(𝑡0 ) 𝑀.

□

Remark 3.14. (1) The first consequence of Proposition 3.13 is that dim 𝔛∥ (𝑐) = 𝑛. (2) Let 𝑡0 , 𝑡1 ∈ 𝐼. Then we have a map, called parallel transport, 𝑡 ,𝑡1

𝑃𝑐 0 𝑡 ,𝑡2

(3) Clearly 𝑃𝑐 0

= ev𝑡1 ∘ (ev𝑡0 )−1 ∶ 𝑇𝑐(𝑡0 ) 𝑀 → 𝑇𝑐(𝑡1 ) 𝑀 . 𝑡 ,𝑡

𝑡 ,𝑡

𝑡 ,𝑡0

= 𝑃𝑐 1 2 ∘𝑃𝑐 0 1 , 𝑃𝑐 0

𝑡 ,𝑡0

= Id and 𝑃𝑐 1

𝑡 ,𝑡

= (𝑃𝑐 0 1 )−1 , for 𝑡0 , 𝑡1 , 𝑡2 ∈ 𝐼.

(4) 𝑐 ∶ [𝑎, 𝑏] → 𝑀 is a piecewise 𝐶 1 curve if there exist 𝑎 = 𝑡0 < 𝑡1 < ⋯ < 𝑡 𝑘 = 𝑏 (called corner points), such that 𝑐|[𝑡𝑖−1 ,𝑡𝑖 ] is a 𝐶 1 curve for 𝑖 = 1, . . . , 𝑘. Then 𝑡 ,𝑡 𝑡 ,𝑡 we define 𝑃𝑐𝑎,𝑏 = 𝑃𝑐 𝑘 𝑘−1 ∘ ⋯ ∘ 𝑃𝑐 0 1 . This extends the notion of parallel vector 1 fields to piecewise 𝐶 curves. 𝑡 ,𝑡

(5) If 𝑣 ∈ 𝑇𝑐(𝑡0 ) 𝑀, then 𝑋(𝑡) = 𝑃𝑐 0 (𝑣) = (ev𝑡0 )−1 (𝑣) defines the parallel vector field with 𝑋(𝑡0 ) = 𝑣. (6) Let (𝑒 1 , . . . , 𝑒 𝑛 ) be a basis of 𝑇𝑐(𝑡0 ) 𝑀. Consider the parallel extensions 𝐸𝑖 (𝑡) = 𝑡,𝑡 𝑃𝑐 0 (𝑒 𝑖 ), 𝑖 = 1, . . . , 𝑛. This is a basis of vector fields along 𝑐, which is called a parallel frame. Let 𝑋 ∈ 𝔛(𝑐). Then 𝑋 = ∑ 𝑓𝑖 (𝑡)𝐸𝑖 , for some 𝑓𝑖 ∈ 𝐶 ∞ (𝐼), and the covariant derivative is 𝑑𝑓 𝐷𝐸 𝐷𝑋 𝐷 (3.14) = ∑ (𝑓𝑖 𝐸𝑖 ) = ∑ ( 𝑖 𝐸𝑖 + 𝑓𝑖 𝑖 ) = ∑ 𝑓𝑖′ (𝑡)𝐸𝑖 . 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡

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Therefore 𝑓 (𝑡 + ℎ) − 𝑓𝑖 (𝑡0 ) 𝐷𝑋 (𝑡 ) = ∑ 𝑓𝑖′ (𝑡0 )𝑒 𝑖 = lim ∑ 𝑖 0 𝑒𝑖 𝑑𝑡 0 ℎ ℎ→0 𝑓 (𝑡 + ℎ)𝑒 𝑖 − 𝑓𝑖 (𝑡0 )𝑒 𝑖 = lim ∑ 𝑖 0 ℎ ℎ→0 𝑡 +ℎ,𝑡0

= lim

𝑃𝑐 0

ℎ→0

(𝑋(𝑡0 + ℎ)) − 𝑋(𝑡0 ) 𝑑 𝑡,𝑡 = || 𝑃𝑐 0 (𝑋(𝑡)) . ℎ 𝑑𝑡 𝑡=𝑡0

(7) Parallel tensors along 𝑐 are defined analogously, and with the same properties. We have the formula (∇𝑋 𝑇)𝑝 = lim 𝑡→0

𝑃𝑐𝑡,0 𝑇𝑐(𝑡) − 𝑇𝑝 𝑑 = || 𝑃𝑐𝑡,0 (𝑇𝑐(𝑡) ) , 𝑡 𝑑𝑡 𝑡=0

for a vector field 𝑋 and a tensor 𝑇, where 𝑐(𝑡) is a curve with 𝑐(0) = 𝑝, 𝑐′ (0) = 𝑋(𝑝). This expression should be compared with the formula in Exercise 3.7 for the Lie derivative 𝐿𝑋 𝑇, where parallel transport is substituted by the flow of 𝑋. Note however that the non-tensorial character of 𝐿𝑋 follows from the fact that pulling back by a diffeomorphism needs local information and not just the value at a point. (8) Parallel transport is independent of the parametrization of the curve. Let 𝑐 ∶ 𝐼 → 𝑀 be a curve, and let ℎ ∶ 𝐽 → 𝐼, 𝑡 = ℎ(𝑢), a change of parameters, so 𝑐(𝑢) ̃ = 𝑐(ℎ(𝑢)) is a reparametrization of 𝑐(𝑡). There is a bijection 𝔛(𝑐) ↔ 𝔛(𝑐),̃ 𝐷𝑋̃ 𝐷𝑋 𝑑𝑡 𝑋 ↦ 𝑋̃ = 𝑋 ∘ ℎ. Indeed equation (3.12) implies that 𝑑ᵆ = 𝑑𝑡 𝑑ᵆ , thus 𝑋 is parallel if and only if 𝑋̃ is parallel. Remark 3.15. Parallel transport serves to compare tensors (physical quantities) at different points of 𝑀. In this way we “connect” the physical or geometrical information at these points. This is the reason for the name connection for ∇. The connection of information between different points is what allows us to say if a tensor varies (i.e., has non-zero derivative), thereby leading to the concept of its derivative. We also define the holonomy of a connection. Let 𝑐 ∶ [0, 1] → 𝑀 be a piecewise 𝐶 1 loop based at 𝑝 ∈ 𝑀. Then we define the holonomy around 𝑐 as the parallel transport ℎ𝑐 = 𝑃𝑐0,1 ∈ GL(𝑇𝑝 𝑀). We define the holonomy group at the point 𝑝 as 𝑝

Hol∇ = {ℎ𝑐 | 𝑐 piecewise 𝐶 1 loop at 𝑝}. 𝑝

If 𝑀 is connected, the groups Hol∇ are isomorphic groups for different points 𝑝 ∈ 𝑀. Hence the holonomy group of 𝑀 is the abstract group isomorphic to any of them. Remark 3.16. The parallel transport moves objects rigidly from one place to another. This has to be done along a curve, and it depends on the chosen curve. The holonomy measures this deviation. If 𝑐 1 , 𝑐 2 ∶ [0, 1] → 𝑀 are two curves with 𝑐 1 (0) = 𝑐 2 (0) = 𝑝 ←

and 𝑐 1 (1) = 𝑐 2 (1) = 𝑞, then ℎ𝑐 = (𝑃𝑐0,1 )−1 ∘ 𝑃𝑐0,1 , with 𝑐 = 𝑐 1 ∗ 𝑐 2 . So ℎ𝑐 (𝑇) = 𝑇 if and 2 1 0,1 0,1 only if 𝑃𝑐2 (𝑇) = 𝑃𝑐1 (𝑇). In some sense, Hol∇ controls the geometric information that can be moved around 𝑀, that is, the geometric invariants on the whole manifold (see Exercise 3.12 and [K-N]).

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147

Example 3.17. Let us consider ℝ𝑛 with the trivial connection ∇ with Christoffel symbols Γ𝑖𝑗𝑘 ≡ 0. Then for 𝑋 = ∑ 𝑋 𝑖 𝜕𝑥𝑖 , 𝑌 = ∑ 𝑌 𝑗 𝜕𝑥𝑗 , we have ∇𝑋 𝑌 = ∑ 𝑋 𝑖 is the naive derivative ∇𝑋 𝑌 = 𝑋(𝑌 ). For a curve 𝑐 ∶ 𝐼 → ℝ𝑛 , we have

𝜕𝑌 𝑘 𝜕 . 𝜕𝑥𝑖 𝑥𝑘

This

𝐷𝑌 𝑑𝑌 𝑘 =∑ 𝜕 = 𝑌 ′ (𝑡). 𝑑𝑡 𝑑𝑡 𝑥𝑘 In particular, a parallel vector field is a constant vector field 𝑌 (𝑡) ≡ ∑ 𝑎𝑗 𝜕𝑥𝑗 . The holonomy group is trivial, Hol∇ = {Id}. 3.1.3. Riemannian metrics. Tensors serve to encode different types of geometric information on a manifold. For instance, an example that we have met before is a volume form (Definition 2.127) which serves to compute volumes. To be able to measure lengths and angles, we have to endow the manifold with a tensor called a Riemannian metric. Definition 3.18. A Riemannian metric in 𝑀 is a symmetric (0, 2)-tensor g ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝐶 ∞ (𝑀) such that for every 𝑝 ∈ 𝑀 the induced map g𝑝 ∶ 𝑇𝑝 𝑀 × 𝑇𝑝 𝑀 → ℝ is a scalar product (i.e., a symmetric and positive definite bilinear map). The pair (𝑀, g) is called a Riemannian manifold. Remark 3.19. • When the Riemannian metric g is clear from the context, we write ⟨𝑋, 𝑌 ⟩ = g(𝑋, 𝑌 ). • We define the norm of a vector 𝑣 ∈ 𝑇𝑝 𝑀 as ||𝑣|| = √⟨𝑣, 𝑣⟩. We also define the angle between two non-zero vectors 𝑣, 𝑤 ∈ 𝑇𝑝 𝑀 as 𝜃 = ∠(𝑣, 𝑤) ∈ [0, 𝜋] given by cos 𝜃 =

⟨𝑣,𝑤⟩ . ||𝑣|| ||𝑤||

• For a piecewise 𝐶 1 curve 𝑐 ∶ [𝑎, 𝑏] → 𝑀, we define its length as ℓ(𝑐) = 𝑏 ∫𝑎 ||𝑐′ (𝑡)||𝑑𝑡. Observe that this makes sense since 𝑐 is differentiable except at the corner points. • Let 𝑐 ∶ [𝑎, 𝑏] → 𝑀 be a curve, and let 𝑐(𝑢) = 𝑐(ℎ(𝑢)) be a reparametrization of 𝑐(𝑡), with ℎ ∶ [𝑎′ , 𝑏′ ] → [𝑎, 𝑏], 𝑡 = ℎ(𝑢), the change of parameters. Note that it is usual to give the same name to the original curve and its reparametriza𝑏 𝑑𝑐 𝑏′ 𝑑𝑐 𝑏′ 𝑑𝑐 𝑑𝑐 𝑑𝑐 𝑑𝑡 𝑑𝑡 tion. Then 𝑑ᵆ = 𝑑𝑡 𝑑ᵆ , so ∫𝑎 || 𝑑𝑡 ||𝑑𝑡 = ∫𝑎′ || 𝑑𝑡 || | 𝑑ᵆ |𝑑𝑢 = ∫𝑎′ || 𝑑ᵆ ||𝑑𝑢, by the formula for the change of variables. This shows that the length is independent of the parametrization. • A regular curve 𝑐 ∶ [𝑎, 𝑏] → 𝑀 is a 𝐶 1 curve with 𝑐′ (𝑡) ≠ 0, for all 𝑡 ∈ [𝑎, 𝑏]. Then we call 𝑠(𝑡) = ℓ(𝑐|[𝑎,𝑡] ) the arc length parameter. We can reparametrize 𝑐 by using the parameter 𝑠, that is we consider 𝑐(𝑠) = 𝑐(𝑡(𝑠)), where 𝑡 = 𝑡(𝑠) is the inverse of the function 𝑠 = 𝑠(𝑡). Then 𝑐(𝑠) is a curve parametrized by arc length. This means that ℓ(𝑐|[𝑠1 ,𝑠2 ] ) = 𝑠2 − 𝑠1 for all 𝑠1 < 𝑠2 , or equivalently, that ||𝑐′ (𝑠)|| = 1 for all 𝑠. • In a connected Riemannian manifold (𝑀, g) we define the Riemannian distance as the function 𝑑 ∶ 𝑀 × 𝑀 → [0, ∞) given by (3.15)

𝑑(𝑝, 𝑞) = inf{ℓ(𝑐) | 𝑐 is a 𝐶 1 piecewise curve from 𝑝 to 𝑞}.

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3. Riemannian geometry

It is easy to see that 𝑑 is symmetric and satisfies the triangle inequality. It is a bit harder to see that if 𝑑(𝑝, 𝑞) = 0, then 𝑝 = 𝑞. This follows from the analysis of geodesic balls (Proposition 3.32). Moreover, the topology associated to (𝑀, 𝑑) (as metric space) is the same as the topology of 𝑀 (as a manifold). Working in a coordinate chart (𝑥1 , . . . , 𝑥𝑛 ), we have g = ∑ 𝑔𝑖𝑗 𝑑𝑥𝑖 ⊗ 𝑑𝑥𝑗 = ∑ 𝑔𝑖𝑗 𝑑𝑥𝑖 ⋅ 𝑑𝑥𝑗 , 𝑖,𝑗

𝑖,𝑗

where 𝑔𝑖𝑗 = g(𝜕𝑥𝑖 , 𝜕𝑥𝑗 ), using (3.4). Remark 3.20. A Riemannian metric also allows us to compute volumes. Let (𝑀, g) be a Riemannian and oriented manifold. Then we define a volume form 𝜈𝑝 ∶ 𝑇𝑝 𝑀 × ⋯ × 𝑇𝑝 𝑀 → ℝ by declaring that 𝜈𝑝 (𝑒 1 , . . . , 𝑒 𝑛 ) = 1 whenever (𝑒 1 , . . . , 𝑒 𝑛 ) is a positive orthonormal basis of 𝑇𝑝 𝑀. Let us give the local expression of the 𝑛-form 𝜈. Take a positive chart (𝑈, (𝑥1 , . . . , 𝑥𝑛 )). Write 𝜈 = 𝑓𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 and take a positive orthonormal basis (𝑒 1 , . . . , 𝑒 𝑛 ) with 𝜕𝑥𝑖 = ∑ 𝑎𝑖𝑗 𝑒𝑗 . The matrix 𝐴 = (𝑎𝑖𝑗 ) gives the change of basis from the orthonormal basis, in which g has matrix Id, to the canonical basis of the chart, in which g has matrix (𝑔𝑖𝑗 ). Therefore 𝐴𝑡 𝐴 = (𝑔𝑖𝑗 ) and hence det(𝐴)2 = det(𝑔𝑖𝑗 ). Thus 𝑓 = 𝜈 (𝜕𝑥1 , . . . , 𝜕𝑥1 ) = (det 𝐴)𝜈(𝑒 1 , . . . , 𝑒 𝑛 ) = det 𝐴 = √det(𝑔𝑖𝑗 ), so (3.16)

𝜈 = √det(𝑔𝑖𝑗 ) 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 .

This is called the Riemannian volume form, and can be used to integrate functions and compute volumes, as mentioned in (2.18). In the case of surfaces, the volume form is called the area form. In the case of curves, i.e., Riemannian 1-manifolds, the volume form is the arc length differential 𝜈 = 𝑑𝑠, where 𝑠 is the arc length parameter (Remark 3.19). A metric g produces isomorphisms (3.17)

(−)♭ ∶ 𝑇𝑝 𝑀 → 𝑇𝑝∗ 𝑀, 𝑣 ↦ 𝑣 ♭ = ⟨𝑣, −⟩, (−)♯ ∶ 𝑇𝑝∗ 𝑀 → 𝑇𝑝 𝑀, 𝛼 ↦ 𝛼♯ , where ⟨𝛼♯ , 𝑤⟩ = 𝛼(𝑤), for 𝑤 ∈ 𝑇𝑝 𝑀,

which are mutually inverse. For vector fields and 1-forms, we have (−)♭ ∶ 𝔛(𝑀) → Ω1 (𝑀) and (−)♯ ∶ Ω1 (𝑀) → 𝔛(𝑀). In local coordinates, if 𝑋 = ∑ 𝑋 𝑖 𝜕𝑥𝑖 and 𝛼 = ∑ 𝛼𝑗 𝑑𝑥𝑗 , then 𝑋♭ = ∑ 𝑋 𝑖 𝑔𝑖𝑗 𝑑𝑥𝑗 ,

𝛼♯ = ∑ 𝛼𝑗 𝑔𝑖𝑗 𝜕𝑥𝑖 ,

where we write (𝑔𝑖𝑗 ) for the inverse of the matrix (𝑔𝑖𝑗 ). If 𝛼 = 𝑋♭ , we have that 𝛼𝑗 = ∑ 𝑔𝑖𝑗 𝑋 𝑖 , which has the indices at the bottom. For this reason, this process is called lowering the indices. Analogously, if 𝑋 = 𝛼♯ , then 𝑋 𝑗 = ∑ 𝑔𝑖𝑗 𝛼𝑖 and this is called raising the indices. In the literature following Einstein notation (Remark 3.2), it is customary not to change the letter used for the tensor, and to write 𝑋𝑗 = ∑ 𝑔𝑖𝑗 𝑋 𝑖 . For general tensors, the operation of lowering one index is as follows. First we have to fix what (top) index we want to lower and where; let us say that it is the first and we move it to

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149

the last one. Then ↓ ∶ 𝒯 𝑟,𝑠 (𝑀) → 𝒯 𝑟−1,𝑠+1 (𝑀) sends 𝑖 ⋯𝑖

𝑇 = ∑𝑇𝑗11⋯𝑗𝑠𝑟 𝜕𝑥𝑖 ⊗ 𝜕𝑥𝑖 ⊗ ⋯ ⊗ 𝑑𝑥𝑗𝑠 ↦ 1

↓𝑇=

2

𝑎𝑖2 ⋯𝑖𝑟 ∑ 𝑔𝑎𝑏 𝑇𝑗1 ⋯𝑗 𝜕𝑥𝑖 𝑠 2

⊗ ⋯ ⊗ 𝑑𝑥𝑗𝑠 ⊗ 𝑑𝑥𝑏 .

𝑖 ⋯𝑖

𝑖 𝑖 ⋯𝑖

2 𝑟 Note that ↓ 𝑇 is the tensor with coefficients 𝑇𝑗12⋯𝑗𝑠𝑟𝑗𝑠+1 = ∑𝑖 𝑔𝑖1 𝑗𝑠+1 𝑇𝑗11⋯𝑗 . Similarly, 𝑠 1 𝑟,𝑠 𝑟+1,𝑠−1 we have operators of raising the indices ↑ ∶ 𝒯 (𝑀) → 𝒯 (𝑀) with formulas of 𝑖 𝑖2 ⋯𝑖𝑟 𝑖𝑟+1 𝑗1 𝑖𝑟+1 𝑖1 𝑖2 ⋯𝑖𝑟 𝑖𝑗 the type 𝑇𝑗21⋯𝑗 = 𝑔 𝑇 . Note that 𝑔 is obtained by raising the two ∑ 𝑗1 ⋯𝑗𝑠 𝑗1 𝑠 indices of 𝑔𝑖𝑗 .

Example 3.21. If 𝑓 ∶ 𝑀 → ℝ is a smooth function, we define the gradient of 𝑓 as grad (𝑓) = (𝑑𝑓)♯ ∈ 𝔛(𝑀). Then for a regular value 𝜆 of 𝑓 (see section 2.6.3), we have grad (𝑓) ⟂ 𝑇𝑝 𝐻𝜆 , for 𝑝 ∈ 𝐻𝜆 , where 𝐻𝜆 = {𝑥 ∈ 𝑀 | 𝑓(𝑥) = 𝜆} is the 𝜆-level hypersurface. An important point is that every differentiable manifold has a Riemannian metric. Proposition 3.22. Every differentiable manifold 𝑀 can be endowed with a Riemannian metric. The space ℳet(𝑀) of metrics of 𝑀 is a convex subset of 𝒯 0,2 (𝑀) (in particular, contractible). Proof. Let {𝑈𝛼 } be a covering of 𝑀 by coordinate open sets. In 𝑈𝛼 with coordinates (𝑥1 , . . . , 𝑥𝑛 ), we can consider the standard metric given by g𝛼 = ∑ 𝑑𝑥𝑖2 . Take a partition of unity {𝜌𝛼 } subordinated to {𝑈𝛼 }. Then we can define g = ∑ 𝜌𝛼 g𝛼 . This is a symmetric (0, 2)-tensor. Clearly for 𝑣 ∈ 𝑇𝑝 𝑀, 𝑝 ∈ 𝑀, we have g(𝑣, 𝑣) = ∑ 𝜌𝑎 (𝑝)g𝛼 (𝑣, 𝑣) ≥ 0. There is some 𝛼0 with 𝜌𝛼0 (𝑝) > 0, so if 𝑣 ≠ 0, then g(𝑣, 𝑣) ≥ 𝜌𝛼0 (𝑝)g𝛼 (𝑣, 𝑣) > 0. Therefore g is a Riemannian metric. The convexity property follows by observing that if g1 , g2 are two Riemannian metrics on 𝑀 and 0 ≤ 𝜆 ≤ 1, then all the tensors 𝜆g1 +(1−𝜆)g2 are Riemannian metrics. □ 𝑛

This allows us to introduce a category 𝐑𝐢𝐞𝐦 , whose objects are Riemannian 𝑛manifolds (𝑀, g). So we have a surjective map 𝑐𝑜

(3.18)

𝑐𝑜

𝕃𝐑𝐢𝐞𝐦𝑛 → 𝕃𝐃𝐌𝐚𝐧𝑛 ,

which can be useful to the smooth classification problem (e.g., the solution of Exercise 3.13 gives an alternative solution to Exercise 1.4). 𝑛

The morphisms of 𝐑𝐢𝐞𝐦 are differentiable maps 𝑓 ∶ (𝑀1 , g1 ) → (𝑀2 , g2 ) which are local isometries, i.e., such that 𝑓∗ g2 = g1 . This means that g1 (𝑢, 𝑣) = 𝑓∗ g2 (𝑢, 𝑣) = g2 (𝑓∗ 𝑢, 𝑓∗ 𝑣), for 𝑢, 𝑣 ∈ 𝑇𝑝 𝑀1 , so that 𝑓∗ = 𝑑𝑝 𝑓 ∶ 𝑇𝑝 𝑀1 → 𝑇 𝑓(𝑝) 𝑀2 is a linear isometry for all 𝑝 ∈ 𝑀1 (that is, it is bijective and preserves the scalar product). In particular 𝑓 is always a local diffeomorphism. As examples of local isometries, we have a Riemannian open immersion, that is (𝑈, g|𝑈 ) ↪ (𝑀, g) where 𝑈 ⊂ 𝑀 is an open subset, and a Riemannian cover, that is a cover 𝜋 ∶ (𝑀 ′ , g′ ) → (𝑀, g) which is a local isometry. The 𝑛 isomorphisms of the category 𝐑𝐢𝐞𝐦 are the isometries, i.e., diffeomorphisms which are local isometries. In particular, for a Riemannian manifold (𝑀, g) we have the group Isom(𝑀, g) consisting of isometries 𝑓 ∶ (𝑀, g) → (𝑀, g).

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The Levi-Civita connection. Definition 3.23. Let (𝑀, g) be a Riemannian manifold. A connection ∇ on 𝑀 is called compatible with the metric g if ∇g = 0. Remark 3.24. (1) We always have 𝑍⟨𝑋, 𝑌 ⟩ = ∇𝑍 (g(𝑋, 𝑌 )) = (∇𝑍 g) (𝑋, 𝑌 ) + g(∇𝑍 𝑋, 𝑌 ) + g(𝑋, ∇𝑍 𝑌 ), for vector fields 𝑋, 𝑌 , 𝑍 ∈ 𝔛(𝑀). Therefore a connection ∇ is compatible with g if and only if ∇𝑍 g = 0 for all 𝑍, that is 𝑍⟨𝑋, 𝑌 ⟩ = ⟨∇𝑍 𝑋, 𝑌 ⟩ + ⟨𝑋, ∇𝑍 𝑌 ⟩, for all 𝑋, 𝑌 , 𝑍 ∈ 𝔛(𝑀). (2) The connection ∇ is compatible with g if and only if, for all curves 𝑐, the covariant derivative satisfies ⟨𝑋, 𝑌 ⟩′ = ⟨

𝐷𝑋 𝐷𝑌 , 𝑌 ⟩ + ⟨𝑋, ⟩, 𝑑𝑡 𝑑𝑡

for all 𝑋, 𝑌 ∈ 𝔛(𝑐). Therefore if 𝑋, 𝑌 ∈ 𝔛∥ (𝑐), then ⟨𝑋, 𝑌 ⟩ is constant. In particular, the property of orthogonality is preserved when transported parallelly (Remark 3.14(6)). (3) Equivalently, if (𝑒 1 , . . . , 𝑒 𝑛 ) is an orthonormal basis of 𝑇𝛾(𝑡0 ) 𝑀 and (𝐸1 , . . . , 𝐸𝑛 ) is the corresponding parallel frame, then ⟨𝐸𝑖 , 𝐸𝑗 ⟩ = ⟨𝑒 𝑖 , 𝑒𝑗 ⟩ = 𝛿 𝑖𝑗 . Hence (𝐸1 (𝑡), . . . , 𝐸𝑛 (𝑡)) is an orthonormal basis of each 𝑇𝑐(𝑡) 𝑀 by item (2). (4) The converse of (2) holds. Take an orthonormal parallel frame (𝐸𝑖 ) and let 𝑋 = ∑ 𝑓𝑖 𝐸𝑖 , 𝑌 = ∑ 𝑔𝑗 𝐸𝑗 ∈ 𝔛(𝑐). Then ⟨𝑋, 𝑌 ⟩′ = (∑ 𝑓𝑖 𝑔𝑖 )′ = ∑ 𝑓𝑖′ 𝑔𝑖 + 𝐷𝑋 𝐷𝑌 ∑ 𝑓𝑖 𝑔′𝑖 = ⟨ 𝑑𝑡 , 𝑌 ⟩ + ⟨𝑋, 𝑑𝑡 ⟩, using (3.14). 𝑡 ,𝑡

(5) Thus ∇ is compatible with g if and only if all parallel transport maps 𝑃𝑐 0 1 ∶ 𝑇𝑐(𝑡0 ) 𝑀 → 𝑇𝑐(𝑡1 ) 𝑀 are linear isometries. This in turn is equivalent to the 𝑝 assertion that Hol∇ < O(𝑇𝑝 𝑀). Proposition and Definition 3.25. Let (𝑀, g) be a Riemannian manifold. There exists a unique connection which is symmetric and compatible with the metric g. This connection is called the Levi-Civita connection of (𝑀, g). Proof. Suppose that ∇ is a connection satisfying ∇g = 0. Using (3.10), we have 0 = ∇𝜕𝑥 g = ∇𝜕𝑥 (∑ 𝑔𝑖𝑗 𝑑𝑥𝑖 ⊗ 𝑑𝑥𝑗 ) 𝑘

𝑘

𝜕𝑔𝑖𝑗 𝑗 𝑖 =∑ 𝑑𝑥 ⊗ 𝑑𝑥𝑗 − 𝑔𝑖𝑗 Γ𝑘𝑎 𝑑𝑥𝑎 ⊗ 𝑑𝑥𝑗 − 𝑔𝑖𝑗 Γ𝑘𝑏 𝑑𝑥𝑖 ⊗ 𝑑𝑥𝑏 𝜕𝑥𝑘 𝑖 𝜕𝑔𝑖𝑗 𝑡 = ∑( − 𝑔𝑡𝑗 Γ𝑘𝑖𝑡 − 𝑔𝑖𝑡 Γ𝑘𝑗 ) 𝑑𝑥𝑖 ⊗ 𝑑𝑥𝑗 . 𝜕𝑥𝑘

3.1. Riemannian metrics. Curvature. Geodesics

151

𝜕𝑔

This implies that 𝜕𝑥𝑖𝑗 = Γ𝑘𝑖𝑗 + Γ𝑘𝑗𝑖 , where Γ𝑘𝑖𝑗 = ∑ 𝑔𝑡𝑗 Γ𝑘𝑖𝑡 is defined by lowering the 𝑘 index. Now if ∇ is symmetric, then Γ𝑘𝑖𝑗 = Γ𝑖𝑘𝑗 , so 𝜕𝑔𝑖𝑘 𝜕𝑔𝑘𝑗 𝜕𝑔𝑖𝑗 + − = (Γ𝑗𝑖𝑘 + Γ𝑗𝑘𝑖 ) + (Γ𝑖𝑘𝑗 + Γ𝑖𝑗𝑘 ) − (Γ𝑘𝑖𝑗 + Γ𝑘𝑗𝑖 ) = 2Γ𝑖𝑗𝑘 . 𝜕𝑥𝑗 𝜕𝑥𝑖 𝜕𝑥𝑘 Hence, raising the indices back, we obtain the closed formula Γ𝑖𝑗𝑘 =

(3.19)

𝜕𝑔𝑡𝑗 𝜕𝑔𝑖𝑗 𝜕𝑔 1 ∑ 𝑔𝑘𝑡 ( 𝑖𝑡 + − ). 2 𝑡 𝜕𝑥𝑗 𝜕𝑥𝑖 𝜕𝑥𝑡

This expression uniquely determines ∇ locally. Furthermore, this uniqueness also implies the global existence. In every coordinate chart we have a unique connection which is symmetric and compatible with g. On the overlap of two open sets, these connections coincide by uniqueness, and hence define a connection globally on 𝑀. □ Example 3.26. Let us consider ℝ𝑛 with the standard metric g𝑠𝑡𝑑 = ∑𝑖 𝑑𝑥𝑖2 , that is, 𝑔𝑖𝑗 = 𝛿 𝑖𝑗 . Then Γ𝑖𝑗𝑘 ≡ 0, so the Levi-Civita connection is the trivial connection of Example 3.17. 3.1.4. Geodesics. Let 𝑀 be a Riemannian manifold, and let ∇ be its Levi-Civita connection. The geodesics are the curves which can be regarded as straight lines in the context of Riemannian geometry, as they locally minimize the Riemannian distance. Definition 3.27. Let 𝛾 ∶ [𝑎, 𝑏] → 𝑀 be a curve, and let

𝐷 𝑑𝑡

2

along 𝛾. We say that 𝛾 is a geodesic if it is piecewise 𝐶 and

be the covariant derivative 𝐷𝛾′ 𝑑𝑡

= 0.

Remark 3.28. (1) We require the curve 𝛾 to be 𝐶 2 in order to be possible to differentiate it twice 𝐷𝛾′ to compute 𝑑𝑡 . However 𝛾 will eventually be 𝐶 ∞ (Proposition 3.29). (2) Note that for a regular curve 𝛼, it is

𝐷𝛼′ 𝑑𝑡

= ∇𝛼′ 𝛼′ . 𝐷𝛼′

(3) For any 𝐶 2 curve 𝛼 ∶ [𝑎, 𝑏] → 𝑀 the vector field 𝐚 = 𝑑𝑡 ∈ 𝔛(𝛼) is called the acceleration of 𝛼. Therefore 𝛼 is a geodesic if it represents a particle which moves along 𝑀 without acceleration, or in physics terminology, if it has inertial movement, or it is a free falling particle. (4) Suppose 𝛼(𝑡) is a regular curve with speed 𝑣 = ||𝛼′ ||. Then it has a tangent vector 𝐭𝛼 = 𝛼′ /||𝛼′ || and there is a decomposition 𝑇𝛼(𝑡) 𝑀 = ⟨𝐭𝛼 (𝑡)⟩ ⊕ 𝑁𝛼(𝑡) , where 𝑁𝛼(𝑡) = ⟨𝛼′ (𝑡)⟩⟂ is the normal space to the curve. The acceleration decomposes into tangential and normal components, 𝐚 = 𝐚𝑡 +𝐚𝑛 , where 𝐚𝑡 = 𝑑𝑣 𝐷𝛼′ ⟨𝐚, 𝐭𝛼 ⟩𝐭𝛼 . We have that 𝑑𝑡 = ||𝐚𝑡 ||, since 𝑣 ||𝐚𝑡 || = ||𝛼′ || ||𝐚𝑡 || = ⟨ 𝑑𝑡 , 𝛼′ ⟩ = 1 𝑑 ⟨𝛼′ , 𝛼′ ⟩ 2 𝑑𝑡

constant.

=

1 𝑑 ||𝛼′ ||2 2 𝑑𝑡

= 𝑣 𝑣′ . Therefore, 𝐚𝑡 = 0 if and only if ||𝛼′ (𝑡)|| is

(5) Suppose 𝛼(𝑠) is a curve parametrized by arc length. Then we define its geo𝐷𝛼′ desic curvature as 𝐤𝑔 = 𝑑𝑠 , which is its normal acceleration (since 𝐚𝑡 = 0 by (4)). A geodesic has 𝐤𝑔 ≡ 0.

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3. Riemannian geometry

(6) A regular curve 𝛼(𝑡) is called a pregeodesic if its normal acceleration vanishes. In this case, the reparametrization by arc length is a geodesic. Certainly, if 𝑑𝛼 𝑑𝛼 𝑑𝑡 𝛼(𝑠) is the arc length reparametrization, then 𝑑𝑠 = 𝑑𝑡 𝑑𝑠 and hence 2

𝐷 𝑑𝛼 𝐷 𝑑𝛼 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝐷 𝑑𝛼 𝑑 2 𝑡 𝑑𝛼 =( ) ∈ ⟨𝐭𝛼 ⟩, ( )= ( ) ( )+ 2 𝑑𝑠 𝑑𝑠 𝑑𝑡 𝑑𝑡 𝑑𝑠 𝑑𝑠 𝑑𝑠 𝑑𝑡 𝑑𝑡 𝑑𝑠 𝑑𝑡 since both vectors are tangential. Thus 𝛼(𝑠) has no normal acceleration. By (4), it has no tangential acceleration either, and hence it is a geodesic. From Definition 3.27 we have that 𝛾 is a geodesic if and only if the vector field 𝛾′ ∈ 𝔛(𝛾) is a parallel vector field along 𝛾. From (3.12) with 𝑋 𝑘 = 𝛾𝑘′ , we have the equations satisfied by a geodesic 𝑑2𝛾𝑘 𝑑𝛾 𝑑𝛾𝑗 (3.20) = − ∑ Γ𝑖𝑗𝑘 (𝛾(𝑡)) 𝑖 , 1 ≤ 𝑘 ≤ 𝑛. 𝑑𝑡 𝑑𝑡 𝑑𝑡2 𝑖,𝑗 Proposition 3.29. Given 𝑝 ∈ 𝑀 and 𝑣 ∈ 𝑇𝑝 𝑀, there exists a unique geodesic 𝛾𝑝,𝑣 ∶ 𝐼 → ′ 𝑀, for some maximal interval 𝐼 ⊂ ℝ, with 𝛾𝑝,𝑣 (0) = 𝑝 and 𝛾𝑝,𝑣 (0) = 𝑣. Moreover, 𝛾𝑝,𝑣 is ∞ 𝐶 . Proof. The equations (3.20) are a system of ODEs of second order and of quasilinear type. Therefore they have a unique solution 𝛾(𝑡) once the initial conditions are fixed, which are defined in a maximal interval 𝐼 ⊂ ℝ with 0 ∈ 𝐼. The initial conditions are given by 𝛾 𝑘 (0) = 𝑝 𝑘 , 𝛾𝑘′ (0) = 𝑣 𝑘 , 1 ≤ 𝑘 ≤ 𝑛. This is the same as fixing 𝛾(0) = 𝑝 = (𝑝1 , . . . , 𝑝𝑛 ) and 𝛾′ (0) = 𝑣 = (𝑣 1 , . . . , 𝑣 𝑛 ). Note also that as Γ𝑖𝑗𝑘 are 𝐶 ∞ , the solution must be 𝐶 ∞ , even if we a priori only require 𝛾 to be only 𝐶 2 piecewise. □ Some basic properties of geodesics are listed below; see also [DC2]. Remark 3.30. (1) The dependence of the solution 𝛾𝑝,𝑣 (𝑡) to (3.20) on the initial parameters 𝑝, 𝑣 is differentiable. This is a well known result of ODEs. (2) If 𝛾 ∶ [𝑎, 𝑏] → 𝑀 is a geodesic, then ||𝛾′ (𝑡)|| is constant (by Remark 3.28(4)). In particular, ℓ(𝛾) = (𝑏 − 𝑎)||𝛾′ ||. (3) Given a geodesic 𝛾, then any affine reparametrization 𝛾(𝑡) ̂ = 𝛾(𝑎𝑡 + 𝑏) of 𝛾 is also a geodesic. To see this, it is enough to check that ∇𝛾̂′ 𝛾′̂ = ∇𝑎𝛾′ 𝑎𝛾′ = 𝑎2 ∇𝛾′ 𝛾′ = 0. Therefore, we can always reparametrize (regular) geodesics by arc length with a linear change of parameters. A unitary geodesic is a geodesic of speed 1. (4) A consequence of (3) is that 𝛾𝑝,𝑣 (𝑐𝑡) = 𝛾𝑝,𝑐𝑣 (𝑡) since both curves are geodesics and they have the same initial data. Remark 3.31. Let us clarify the relation between distance minimizing curves and geodesics. • Geodesics minimize the distance locally, i.e., for each 𝑝 ∈ 𝑀 there exists a neighbourhood 𝑈 𝑝 so that for every 𝑞 ∈ 𝑈 there is a unique unitary geodesic 𝛾 ∶ [0, 𝑙] → 𝑀 with 𝛾(0) = 𝑝, 𝛾(𝑙) = 𝑞, and the Riemannian distance 𝑑(𝑝, 𝑞) =

3.1. Riemannian metrics. Curvature. Geodesics

153

𝑙 = ℓ(𝛾). This property characterizes geodesics as locally minimizing curves (see also Remark 3.33). • Given 𝑝, 𝑞 ∈ 𝑀 in the same connected component, there may not exist any geodesic joining them. For example, in 𝑀 = 𝑆 2 − {(0, 1, 0), (0, −1, 0)}, take 1 the points 𝑝 = (√𝜀(2 − 𝜀), 1 − 𝜀, 0) and 𝑞 = (−√𝜀(2 − 𝜀), 1 − 𝜀, 0), for 𝜀 < 3 . 2 The geodesics of 𝑆 are the maximal circles (Proposition 4.6), so any geodesic in 𝑀 joining 𝑝 and 𝑞 would have to pass through some of the points removed. Hence there are no geodesics joining 𝑝 and 𝑞 in 𝑀. • It may happen that there are geodesics joining 𝑝 and 𝑞 but none of these geodesics are minimizing curves (a curve 𝛼 joining 𝑝 and 𝑞 is called minimizing if 𝑑(𝑝, 𝑞) = ℓ(𝛼)). It may also happen that there are no minimizing curves at all. For example, take 𝑀 = 𝑆 2 − {(0, 1, 0)}, and 𝑝, 𝑞 as in the previous example. We can join 𝑝 and 𝑞 through a maximal circle, but this path does not minimize the distance. Actually, no path joining 𝑝 and 𝑞 attains the infimum in 𝑑(𝑝, 𝑞). • There may be more than one minimizing curve. Take 𝑀 = 𝑆 2 , 𝑝 and 𝑞 the north and south poles. Every meridian has length equal to 𝑑(𝑝, 𝑞), so there are infinitely many minimizing curves. • What actually holds is the following: if there is a minimizing curve 𝛼 joining points 𝑝, 𝑞 ∈ 𝑀, then the reparametrization of 𝛼 by arc length is a geodesic. The exponential map. Let 𝑀 be a Riemannian manifold. First, we consider 𝑇𝑀 = ⨆𝑝∈𝑀 𝑇𝑝 𝑀 the tangent bundle of 𝑀. We give it a natural topology as ≅

follows. Let 𝜑 = (𝑥1 , . . . , 𝑥𝑛 ) ∶ 𝑈 → ℝ𝑛 be a chart of 𝑀, then we define 𝜑̃ ∶ 𝑇𝑈 → ̃ 𝑣) = (𝑥1 , . . . , 𝑥𝑛 , 𝑣 1 , . . . , 𝑣 𝑛 ), for 𝑣 = ∑ 𝑣 𝑖 𝜕𝑥𝑖 . Then we declare that the 𝑈 × ℝ𝑛 , 𝜑(𝑝, open subsets of 𝑇𝑈 are open subsets of 𝑇𝑀 for every chart 𝑈 ⊂ 𝑀. This determines a ̃ we get a smooth atlas, and thus basis for a topology on 𝑇𝑀. Using the charts (𝑇𝑈, 𝜑), 𝑇𝑀 is a differentiable 2𝑛-manifold. For a geodesic 𝛾𝑝,𝑣 with 𝛾(0) = 𝑝 and 𝛾′ (0) = 𝑣, let us call 𝐼𝑝,𝑣 the maximal interval in which 𝛾𝑝,𝑣 is defined. Consider the set 𝐔 = {(𝑝, 𝑣, 𝑡) | (𝑝, 𝑣) ∈ 𝑇𝑀, 𝑡 ∈ 𝐼𝑝,𝑣 } ⊂ 𝑇𝑀 × ℝ. It follows easily from the smooth dependence of the solutions of differential equations on the initial data that 𝐔 is an open subset of 𝑇𝑀 × ℝ and that the map (3.21)

Γ ∶ 𝐔 → 𝑀,

(𝑝, 𝑣, 𝑡) ↦ Γ(𝑝, 𝑣, 𝑡) = 𝛾𝑝,𝑣 (𝑡)

is smooth. We have that Γ(𝑝, 𝑣, 𝑐𝑡) = 𝛾𝑝,𝑣 (𝑐𝑡) = 𝛾𝑝,𝑐𝑣 (𝑡) = Γ(𝑝, 𝑐𝑣, 𝑡), so Γ(𝑝, 𝑣, 𝑡) = Γ(𝑝, 𝑡𝑣, 1) and hence the function Γ is determined by its value on 𝑡 = 1. Let 𝒱 = {(𝑝, 𝑣) ∈ 𝑇𝑀 | (𝑝, 𝑣, 1) ∈ 𝐔} ≅ 𝐔 ∩ (𝑇𝑀 × {1}), which is an open subset of 𝑇𝑀. Moreover 𝒱𝑝 = 𝒱 ∩ 𝑇𝑝 𝑀 is a radial set (i.e., if 𝑣 ∈ 𝒱𝑝 , then 𝑡𝑣 ∈ 𝒱𝑝 for 𝑡 ∈ [0, 1]). If 𝑣 ∈ 𝒱𝑝 , then Γ(𝑝, 𝑣, 1) = 𝛾𝑝,𝑣 (1) is defined, so Γ(𝑝, 𝑡𝑣, 1) = 𝛾𝑝,𝑣 (𝑡) is defined for 𝑡 ∈ [0, 1] and hence 𝑡𝑣 ∈ 𝒱𝑝 . We define the exponential map as exp ∶ 𝒱 → 𝑀,

exp(𝑝, 𝑣) = Γ(𝑝, 𝑣, 1) = 𝛾𝑝,𝑣 (1),

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3. Riemannian geometry

v

p

expp(v) γp;v (t)

Figure 3.1. The exponential map.

which is also 𝐶 ∞ . If we fix 𝑝 ∈ 𝑀, then we have the exponential map at 𝑝 (see Figure 3.1), exp𝑝 ∶ 𝒱𝑝 → 𝑀,

exp𝑝 (𝑣) = exp(𝑝, 𝑣) = 𝛾𝑝,𝑣 (1).

Proposition and Definition 3.32. There is some 𝜀 > 0 such that exp𝑝 ∶ 𝐵𝜀 (0) → 𝐵𝜀𝑑 (𝑝) is a diffeomorphism from a ball 𝐵𝜀 (0) ⊂ 𝑇𝑝 𝑀 to a metric ball 𝐵𝜀𝑑 (𝑝) ⊂ 𝑀, for the Riemannian distance 𝑑 of (3.15). We call 𝐵𝜀𝑑 (𝑝) a geodesic ball and the inverse 𝜑 = (exp𝑝 )−1 ∶ 𝐵𝜀𝑑 (𝑝) → 𝐵𝜀 (0) ⊂ ℝ𝑛 a geodesic chart. Proof. For a unitary vector 𝑢 ∈ 𝑇𝑝 𝑀 we have exp𝑝 (𝑡𝑢) = 𝛾𝑝,𝑡ᵆ (1) = 𝛾𝑝,ᵆ (𝑡). So exp𝑝 maps rays from 0 ∈ 𝑇𝑝 𝑀, 𝑡 ↦ 𝑡𝑢, to geodesics from 𝑝 ∈ 𝑀. From this fact it follows 𝑑

that 𝑑𝑝 exp𝑝 (𝑣) = 𝑑𝑡 exp𝑝 (𝑡𝑣) = 𝑣, i.e., 𝑑𝑝 exp𝑝 = Id as a map 𝑑𝑝 exp𝑝 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝑀. Therefore exp𝑝 is a local diffeomorphism at 0 ∈ 𝑇𝑝 𝑀. Let 𝜀 > 0, small so that exp𝑝 ∶ 𝐵𝜀 (0) → 𝑈 𝑝 ⊂ 𝑀 is a diffeomorphism onto its image. The local minimizing property of the geodesics (Remark 3.31) implies that for 𝑈 𝑝 small and 𝑞 ∈ 𝑈 𝑝 , we have that 𝑑(𝑝, 𝑞) = ℓ(𝛾), where 𝛾 is the (unique) geodesic from 𝑝 to 𝑞. This geodesic is 𝛾 ∶ [0, 1] → 𝑀, 𝛾(𝑡) = exp𝑝 (𝑡𝑣) where 𝛾(1) = 𝑞 = exp𝑝 (𝑣). Then 𝑑(𝑝, 𝑞) = ℓ(𝛾) = ||𝑣|| = ||(exp𝑝 )−1 (𝑞)||. So exp𝑝 (𝐵𝜀 (0)) = {exp𝑝 (𝑣)| ||𝑣|| < 𝜀} = {𝑞 ∈ 𝑀| 𝑑(𝑝, 𝑞) < 𝜀} = 𝐵𝜀𝑑 (𝑝).

□

Remark 3.33. • An open subset 𝑈 ⊂ 𝑀 is geodesically convex if for every 𝑝, 𝑞 ∈ 𝑈 there is a unique minimizing geodesic 𝛾 joining 𝑝 and 𝑞, and moreover it lies in 𝑈. It can be proved that for 𝜀 > 0 small enough, 𝐵𝜀𝑑 (𝑝) is geodesically convex; see [DC2]. • If the manifold 𝑀 is compact, then all geodesics can be defined for 𝐼 = ℝ. This is due to the fact that there is a uniform 𝜀 > 0 such that {(𝑝, 𝑣) ∈ 𝑇𝑀 | ||𝑣|| < 𝜀} ⊂ 𝒱. Hence all unitary geodesics live for at least time 𝜀. Since this is true for any point 𝑝 ∈ 𝑀, we can prolong any geodesic at least by this amount, and so indefinitely.

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155

Then for 𝑀 compact, exp𝑝 is defined on the whole of 𝑇𝑝 𝑀. However exp𝑝 cannot be a global diffeomorphism. The infimum of the radius 𝑟 so that exp𝑝 is a diffeomorphism in 𝐵𝑟 (0) ⊂ 𝑇𝑝 𝑀, 𝑝 ∈ 𝑀, is called the injectivity radius. Definition 3.34. A Riemannian manifold 𝑀 is geodesically complete (or just complete) if for any connected component 𝑀 𝑜 ⊂ 𝑀 there exists a 𝑝 ∈ 𝑀 𝑜 so that the map exp𝑝 ∶ 𝑇𝑝 𝑀 → 𝑀 is defined on all 𝑇𝑝 𝑀. Clearly a manifold is geodesically complete if and only if its connected components are. Geodesic completeness is a sort of analogue to a Euclid postulate that a line can be extended indefinitely (section 4.4). Geodesically complete Riemannian manifolds are characterized in the following theorem (see [DC2] for a proof). Theorem 3.35 (Hopf-Rinow). Let (𝑀, g) be a connected Riemannian manifold, and let 𝑑 be its Riemannian distance. The following are equivalent. • 𝑀 is geodesically complete. • The domain of (3.21) is 𝐔 = 𝑇𝑀 × ℝ, that is exp𝑝 ∶ 𝑇𝑝 𝑀 → 𝑀 is defined on all 𝑇𝑝 𝑀 for all points 𝑝, so all geodesics are defined for all 𝑡 ∈ ℝ. • The metric space (𝑀, 𝑑) is a complete metric space. • A subset 𝐴 ⊂ 𝑀 is compact if and only if it is closed and bounded. If 𝑀 is geodesically complete, then for all 𝑝, 𝑞 ∈ 𝑀 there is a unitary geodesic 𝛾 ∶ [0, 𝑙] → 𝑀 with 𝛾(0) = 𝑝, 𝛾(𝑙) = 𝑞, 𝑙 = ℓ(𝛾) = 𝑑(𝑝, 𝑞). Remark 3.36. The condition of being geodesically complete is related with the existence of metric holes, i.e., holes in 𝑀 that an inertial particle (geodesic) 𝛾 can meet in its trajectory. In particular, if (𝑀, g) is a connected Riemannian manifold and 𝑈 ⊊ 𝑀 is a proper connected open subset, then (𝑈, g|𝑈 ) is a non-complete manifold. For that, take a geodesic in 𝑀 from a point in the topological boundary 𝜕𝑈 ⊂ 𝑀 that enters into 𝑈. Therefore, if 𝑀 is connected and complete, then 𝑀 is inextensible (it cannot be put as an open subset of a larger Riemannian manifold). However, there are inextensible Riemannian manifolds which are not complete (Exercise 3.34). If (𝑀, g) has a topological hole, that is, 𝑀 ⊊ 𝑀 ′ as an open subset of another differentiable manifold, but the metric g is unbounded near 𝜕𝑀 ⊂ 𝑀 ′ , then the hole is invisible from the metric point of view (it is at infinity), and 𝑀 would be geodesically complete. Actually, every smooth manifold admits a geodesically complete metric (Exercise 3.14). Riemannian submanifolds. Let (𝑀, g) be a Riemannian 𝑛-manifold, and let 𝐻 ⊂ 𝑀 be a submanifold of dimension 𝑑 < 𝑛. The Riemannian metric g in 𝑀 induces a Riemannian metric g𝐻 = 𝑖∗ g on 𝐻, where 𝑖 ∶ 𝐻 ↪ 𝑀 is the inclusion. In particular, for 𝑢, 𝑣 ∈ 𝑇𝑝 𝐻, 𝑝 ∈ 𝐻, we have g𝐻 (𝑢, 𝑣) = g(𝑢, 𝑣). The metric g𝐻 is usually called the first fundamental form of 𝐻, and it is sometimes denoted as 𝐈. There is a natural splitting 𝑇𝑝 𝑀 = 𝑇𝑝 𝐻 ⊕ (𝑇𝑝 𝐻)⟂ into the tangent and normal subspaces, for 𝑝 ∈ 𝐻, which induces projections (3.22)

(−)𝑇 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝐻,

such that 𝑣 = 𝑣𝑇 + 𝑣𝑁 , for each 𝑣 ∈ 𝑇𝑝 𝑀.

(−)𝑁 ∶ 𝑇𝑝 𝑀 → (𝑇𝑝 𝐻)⟂ ,

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3. Riemannian geometry

Let ∇̄ be the Levi-Civita connection of 𝑀. We want to define an induced connection ∇ on 𝐻. For 𝑋, 𝑌 ∈ 𝔛(𝐻), we consider (local) extensions 𝑋 ′ , 𝑌 ′ ∈ 𝔛(𝑀) of 𝑋, 𝑌 , respectively. This means that 𝑋 ′ |𝐻 = 𝑋, 𝑌 ′ |𝐻 = 𝑌 (Exercise 3.8). Then we define 𝑇

∇𝑋 𝑌 = (∇̄ 𝑋 ′ 𝑌 ′ ) .

(3.23)

Proposition 3.37. The operator ∇ ∶ 𝔛(𝐻) × 𝔛(𝐻) → 𝔛(𝐻) is the Levi-Civita connection of (𝐻, g𝐻 ). Proof. Clearly, ∇𝑋 𝑌 is tensorial on 𝑋, so (∇𝑋 𝑌 )𝑝 only depends on 𝑋 ′ (𝑝) = 𝑋(𝑝), 𝑝 ∈ 𝐻. This shows the independence on the extension 𝑋 ′ . Second, on 𝐻, 𝑇 ∇𝑋 𝑌 − ∇𝑌 𝑋 = (∇̄ 𝑋 ′ 𝑌 ′ − ∇̄ 𝑌 ′ 𝑋 ′ ) = [𝑋 ′ , 𝑌 ′ ]𝑇 = [𝑋, 𝑌 ],

since the Lie bracket commutes with restrictions, [𝑋 ′ , 𝑌 ′ ]|𝐻 = [𝑋 ′ |𝐻 , 𝑌 ′ |𝐻 ] = [𝑋, 𝑌 ] (this follows from (3.11)). This formula implies that ∇ has zero torsion, but it also proves that ∇𝑋 𝑌 = ∇𝑌 𝑋 + [𝑋, 𝑌 ] is independent of the extension 𝑌 ′ (the right hand side is), and also that it satisfies the Leibniz rule (the right hand side does). It remains to see that ∇ is a metric connection. Let 𝑋, 𝑌 , 𝑍 ∈ 𝔛(𝐻), and consider extensions 𝑋 ′ , 𝑌 ′ , 𝑍 ′ ∈ 𝔛(𝑀). Then 𝑍(g𝐻 (𝑋, 𝑌 )) = 𝑍 ′ (g(𝑋 ′ , 𝑌 ′ ))|𝐻 = g(∇̄ 𝑍 ′ 𝑋 ′ , 𝑌 ′ )|𝐻 + g(𝑋 ′ , ∇̄ 𝑍 ′ 𝑌 ′ )|𝐻 = g((∇̄ 𝑍 ′ 𝑋 ′ )𝑇 , 𝑌 )|𝐻 + g(𝑋, (∇̄ 𝑍 ′ 𝑌 ′ )𝑇 )|𝐻 □

= g𝐻 (∇𝑍 𝑋, 𝑌 ) + g𝐻 (𝑋, ∇𝑍 𝑌 ).

A consequence of Proposition 3.37 is that for a curve 𝑐 ∶ 𝐼 → 𝐻 ⊂ 𝑀, and 𝑋 ∈ 𝔛(𝑐), the covariant derivative in 𝐻 is given by the formula 𝑇

̄ 𝐷𝑋 𝐷𝑋 =( ) , 𝑑𝑡 𝑑𝑡

(3.24) where if (3.25)

𝐷̄ 𝑑𝑡

is the covariant derivative in 𝑀. In particular, 𝛾 is a geodesic of 𝐻 if and only 𝐷𝛾′ ⟂ 𝑇𝛾(𝑡) 𝐻, 𝑑𝑡

for all 𝑡 ∈ 𝐼.

There is another operator, called the second fundamental form that measures extrinsic geometry of 𝐻 inside 𝑀. We denote by 𝔛(𝐻)⟂ the space of vector fields normal to 𝐻, that is 𝑋 ∈ 𝔛(𝐻)⟂ if 𝑋(𝑝) ∈ (𝑇𝑝 𝐻)⟂ , for all 𝑝 ∈ 𝐻. We define the second fundamental form as 𝐈𝐈 ∶ 𝔛(𝐻) × 𝔛(𝐻) → 𝔛(𝐻)⟂ ,

𝐈𝐈(𝑋, 𝑌 ) = (∇̄ 𝑋 ′ 𝑌 ′ )

𝑁

,

where 𝑋, 𝑌 ∈ 𝔛(𝐻) and 𝑋 ′ , 𝑌 ′ ∈ 𝔛(𝑀) are (local) extensions of 𝑋, 𝑌 , respectively. Clearly, 𝐈𝐈(𝑋, 𝑌 ) = (∇̄ 𝑋 ′ 𝑌 ′ )𝑁 is tensorial on 𝑋 and does not depend on the extension 𝑋 ′ of 𝑋. On the other hand 𝐈𝐈(𝑋, 𝑌 ) − 𝐈𝐈(𝑌 , 𝑋) = (∇̄ 𝑋 ′ 𝑌 ′ − ∇̄ 𝑌 ′ 𝑋 ′ )𝑁 = [𝑋 ′ , 𝑌 ′ ]𝑁 = 0, so 𝐈𝐈(𝑌 , 𝑋) = 𝐈𝐈(𝑋, 𝑌 ). This implies that 𝐈𝐈(𝑋, 𝑌 ) is also tensorial on 𝑌 and does not depend on the extension 𝑌 ′ of 𝑌 . Therefore 𝐈𝐈 is a symmetric tensor. Note the important formula that codifies both the first and second fundamental forms: (3.26)

∇̄ 𝑋 𝑌 = ∇𝑋 𝑌 + 𝐈𝐈(𝑋, 𝑌 ).

3.1. Riemannian metrics. Curvature. Geodesics

157

3.1.5. Curvature. Let 𝑀 be a Riemannian manifold, and let ∇ be its Levi-Civita connection. We introduce now one of the most important concepts in differential geometry, the curvature, that controls the shape of the space as a useful tensor. Definition 3.38. The Riemannian curvature (or just curvature) of a Riemannian manifold 𝑀 is the tensor 𝑅 ∶ 𝔛(𝑀) × 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀) defined as 𝑅(𝑋, 𝑌 )𝑍 = − (∇𝑋 ∇𝑌 𝑍 − ∇𝑌 ∇𝑋 𝑍 − ∇[𝑋,𝑌 ] 𝑍) . Remark 3.39. • The minus sign in Definition 3.38 is traditional in Riemannian geometry. The usual definition of curvature of connections in the theory of fiber bundles (gauge theory) uses a plus sign. Whatever convention, the sign of the sectional curvature (Definition 3.44) must be always the same (round spheres will have positive curvature). • 𝑅 is a (1, 3)-tensor. Note that 𝑅(𝑋, 𝑌 ) = −𝑅(𝑌 , 𝑋), so it suffices to prove tensoriality in the first and third entries, i.e., 𝑅(𝑓𝑋, 𝑌 )𝑍 = 𝑓𝑅(𝑋, 𝑌 )𝑍 and 𝑅(𝑋, 𝑌 )𝑓𝑍 = 𝑓𝑅(𝑋, 𝑌 )𝑍, for 𝑓 ∈ 𝐶 ∞ (𝑀) and vector fields 𝑋, 𝑌 , 𝑍, which is an easy exercise. The term ∇[𝑋,𝑌 ] in the definition of 𝑅 is necessary for this. • We have the following geometric interpretation. Fix a point 𝑝 ∈ 𝑀. If we move slightly in the direction of 𝑋(𝑝) (via its flow, Exercise 3.2) and then in the direction of 𝑌 (𝑝), we have a discrepancy to moving first in the direction of 𝑌 (𝑝) and then in that of 𝑋(𝑝). By Exercise 3.4, this discrepancy is given by [𝑋, 𝑌 ](𝑝) (at first order). The curvature 𝑅(𝑋, 𝑌 )𝑍 measures the noncommutativity of the two derivatives ∇𝑋 and ∇𝑌 and corrects with the derivative ∇[𝑋,𝑌 ] in order to close the circuit (see Figure 3.2).

X [X; Y ]

Y

X

Y

Figure 3.2. The Lie bracket [𝑋, 𝑌 ] measures the deviation of following the flows of 𝑋, 𝑌 in different orders.

• Note that one may write 𝑅(𝑋, 𝑌 ) = −[∇𝑋 , ∇𝑌 ] + ∇[𝑋,𝑌 ] as an operator 𝑅(𝑋, 𝑌 ) ∶ 𝔛(𝑀) → 𝔛(𝑀). The operator 𝑅(𝑋, 𝑌 ) can be applied to tensors. Clearly 𝑅(𝑋, 𝑌 )(𝑓) = 0, for functions 𝑓. This means that although partial derivatives on functions commute, this does not happen for tensors, and it is exactly what the curvature measures. • There are other nice geometric interpretations of curvature. For instance, it is the discrepancy between the parallel transport in the direction of 𝑋 and then

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in the direction of 𝑌 , and the parallel transport in the direction of 𝑌 and then in the direction of 𝑋 (supposing that [𝑋, 𝑌 ] = 0, for simplicity, see Exercise 3.17). See Remarks 3.59, 3.65, and 3.66, for interpreting the curvature with lengths of circles, sum of angles of triangles, or holonomy around loops, in surfaces in the directions 𝑋, 𝑌 . Let us give the expression of 𝑅 in coordinates. Write 𝑅 = ∑ 𝑅𝑙𝑖𝑗𝑘 𝑑𝑥𝑖 ⊗ 𝑑𝑥𝑗 ⊗ 𝑑𝑥𝑘 ⊗ 𝜕𝑥𝑙 . Then 𝑅 (𝜕𝑥𝑖 , 𝜕𝑥𝑗 ) 𝜕𝑥𝑘 = − (∇𝜕𝑥 ∇𝜕𝑥 𝜕𝑥𝑘 − ∇𝜕𝑥 ∇𝜕𝑥 𝜕𝑥𝑘 ) 𝑖

= − ∑ ∇𝜕𝑥 = −∑( = −∑(

𝑗

𝑖

𝑗

𝑚 𝜕𝑥𝑚 ) (Γ𝑗𝑘

𝑚 𝜕Γ𝑗𝑘

𝜕𝑥𝑖 𝑙 𝜕Γ𝑗𝑘

𝜕𝑥𝑖

𝑖

𝑚 − ∇𝜕𝑥 (Γ𝑖𝑘 𝜕𝑥𝑚 ) 𝑗

𝑚 𝑏 𝜕𝑥𝑚 − Γ𝑗𝑘 Γ𝑖𝑚 𝜕𝑥𝑏 −

−

𝑚 Γ𝑖𝑘 𝑚 𝑏 𝜕 − Γ𝑖𝑘 Γ𝑗𝑚 𝜕𝑥𝑏 ) 𝜕𝑥𝑗 𝑥𝑚

𝑙 𝜕Γ𝑖𝑘 𝑚 𝑙 𝑚 𝑙 + Γ𝑗𝑘 Γ𝑖𝑚 − Γ𝑖𝑘 Γ𝑗𝑚 ) 𝜕𝑥𝑙 . 𝜕𝑥𝑗

So the components of 𝑅 are 𝑅𝑙𝑖𝑗𝑘 =

(3.27)

𝑙 𝑙 𝜕Γ𝑗𝑘 𝜕Γ𝑖𝑘 𝑚 𝑙 𝑚 𝑙 − + ∑ Γ𝑖𝑘 Γ𝑗𝑚 − Γ𝑗𝑘 Γ𝑖𝑚 . 𝜕𝑥𝑗 𝜕𝑥𝑖 𝑚

Remark 3.40. In gauge theory it is typical to pack formula (3.27) as follows. Let Γ𝑗𝑘 = ∑ Γ𝑖𝑗𝑘 𝑑𝑥𝑖 , and consider the matrix of 1-forms Γ = (Γ𝑗𝑘 ). Take also 𝑅𝑙𝑘 = ∑ 𝑅𝑙𝑖𝑗𝑘 𝑑𝑥𝑖 ∧ 𝑑𝑥𝑗 and the matrix 𝑅 = (𝑅𝑙𝑘 ). Then 𝑅 = 𝑑Γ + Γ ∧ Γ. Remark 3.41. Metrics and connections can be defined on 𝐶 1 manifolds, and the notion of geodesics and curvature on 𝐶 2 manifolds (see Remark 1.20). We can lower the index of 𝑅 to get a (0, 4)-tensor. This is defined as 𝑅(𝑋, 𝑌 , 𝑍, 𝑇) = ⟨𝑅(𝑋, 𝑌 )𝑍, 𝑇⟩. So 𝑅 = 𝑅𝑖𝑗𝑘𝑙 𝑑𝑥𝑖 ⊗ 𝑑𝑥𝑗 ⊗ 𝑑𝑥𝑙 ⊗ 𝑑𝑥𝑘 , with 𝑅𝑖𝑗𝑘𝑙 = ∑ 𝑅𝑚 𝑖𝑗𝑘 𝑔𝑚𝑙 . The curvature is thus determined by 𝑛4 functions, but it satisfies certain symmetries that make many of those functions redundant (Exercise 3.18). Proposition 3.42. For vector fields 𝑋, 𝑌 , 𝑍, 𝑇 ∈ 𝔛(𝑀) we have the following. (1) 𝑅(𝑋, 𝑌 , 𝑍, 𝑇) = −𝑅(𝑌 , 𝑋, 𝑍, 𝑇), i.e., 𝑅(𝑋, 𝑌 ) = −𝑅(𝑌 , 𝑋). (2) 𝑅(𝑋, 𝑌 , 𝑍, 𝑇) = −𝑅(𝑋, 𝑌 , 𝑇, 𝑍), i.e., 𝑅(𝑋, 𝑌 ) is an antisymmetric endomorphism of 𝑇𝑀. (3) 𝑅(𝑋, 𝑌 , 𝑍, 𝑇) = 𝑅(𝑍, 𝑇, 𝑋, 𝑌 ). (4) 𝑅(𝑋, 𝑌 , 𝑍, 𝑇) + 𝑅(𝑌 , 𝑍, 𝑋, 𝑇) + 𝑅(𝑍, 𝑋, 𝑌 , 𝑇) = 0, known as Bianchi identity. Remark 3.43. • Property (1) of Proposition 3.42 holds for any connection and property (2) only requires ∇ to be compatible with the metric. However, properties (3) and (4) are specific to the Levi-Civita connection. Note that these latter formulas mix the direction of derivation and the objects being differentiated. There is also another property, called second Bianchi identity, for ∇𝑅.

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159

2

2

• By properties (1) and (2), we have that 𝑅𝑝 ∈ (⋀ 𝑇𝑝∗ 𝑀)⊗(⋀ 𝑇𝑝∗ 𝑀). Property 2

2

(3) says that there is symmetry in these two factors, so 𝑅𝑝 ∈ Sym (⋀ 𝑇𝑝∗ 𝑀). This means that 𝑅𝑝 is determined by the induced quadratic form. • For 𝑛 = 2, the component 𝑅1212 determines all the others, therefore the curvature is given by a single function. By the second comment in Remark 3.43, 𝑅𝑝 (𝑥, 𝑦, 𝑧, 𝑡), for 𝑥, 𝑦, 𝑧, 𝑡 ∈ 𝑇𝑝 𝑀, is deter2

mined (as a multilinear map) by 𝑅𝑝 (𝑢, 𝑣, 𝑢, 𝑣), for 𝑢 ∧ 𝑣 ∈ ⋀ 𝑇𝑝 𝑀. Note that 𝑢 ∧ 𝑣 is geometrically a parallelogram, which is determined by the plane they span and its area. This leads us to the following definition. Definition 3.44. Given 𝑝 ∈ 𝑀 and 𝑢, 𝑣 ∈ 𝑇𝑝 𝑀 linearly independent vectors, we define the sectional curvature of the plane 𝜎 = ⟨𝑢, 𝑣⟩ ⊂ 𝑇𝑝 𝑀 as 𝐾𝑝 (𝜎) =

𝑅𝑝 (𝑢, 𝑣, 𝑢, 𝑣) , ||𝑢 ∧ 𝑣||2

where ||𝑢 ∧ 𝑣||2 = ||𝑢||2 ||𝑣||2 − ⟨𝑢, 𝑣⟩2 is the square of the area of the parallelogram determined by 𝑢, 𝑣. The sectional curvature is no longer a tensorial object (it is defined pointwise, but it is not a multilinear map), but it contains exactly the same information as the curvature tensor. Note that if (𝑒 1 , 𝑒 2 ) is an orthonormal basis of the plane 𝜎 = ⟨𝑒 1 , 𝑒 2 ⟩, then 𝐾𝑝 (𝜎) = 𝑅𝑝 (𝑒 1 , 𝑒 2 , 𝑒 1 , 𝑒 2 ). Proposition 3.45. Let 𝑝 ∈ 𝑀, and let 𝜎 ⊂ 𝑇𝑝 𝑀 be a plane. Consider a geodesic ball 𝐵𝜀𝑑 (𝑝) and the surface 𝑆 = exp𝑝 (𝜎 ∩ 𝐵𝜀 (0)) inside 𝑀. Then 𝐾𝑝 (𝜎) = 𝐾𝑝𝑆 (𝑇𝑝 𝑆), where 𝐾 𝑆 is the sectional curvature of 𝑆, with the induced metric. Proof. Denote by ∇̄ the Levi-Civita connection of 𝑀 and by ∇ the Levi-Civita connection of 𝑆 given in (3.23). By the definition of 𝑆, we have that the second fundamental form vanishes at 𝑝, 𝐈𝐈𝑝 = 0. This follows since for any 𝑢 ∈ 𝑇𝑝 𝑆, the geodesic 𝑆 𝛾 = 𝛾𝑝,ᵆ is in 𝑆, and 𝐈𝐈𝑝 (𝑢, 𝑢) = (∇̄ 𝛾′ 𝛾′ )𝑝𝑁 = 0. Therefore (∇̄ 𝑋 𝑌 )𝑝 = (∇̄ 𝑋 𝑌 )𝑝 by (3.26). ̄ Moreover, given 𝑋, 𝑌 , 𝑍, 𝑇 ∈ 𝔛(𝑆), we have ⟨∇𝑋 (𝐈𝐈(𝑌 , 𝑍)), 𝑇⟩𝑝 = 𝑋𝑝 (⟨𝐈𝐈(𝑌 , 𝑍), 𝑇⟩) − ⟨𝐈𝐈(𝑌 , 𝑍), ∇̄ 𝑋 𝑇⟩𝑝 = 0, so (∇̄ 𝑋 𝐈𝐈(𝑌 , 𝑍))𝑝𝑇 = 0, and thus (∇𝑋 ∇𝑌 𝑍)𝑝 = (∇̄ 𝑋 ∇̄ 𝑌 𝑍)𝑝𝑇 . Therefore, if 𝑅̄ denotes the curvature tensor of 𝑀 and 𝑅 denotes the curvature tensor of 𝑆, 𝑅𝑝 (𝑋, 𝑌 , 𝑍, 𝑇) = −⟨∇𝑋 ∇𝑌 𝑍 − ∇𝑌 ∇𝑋 𝑍 − ∇[𝑋,𝑌 ] 𝑍, 𝑇⟩𝑝 = −⟨∇̄ 𝑋 ∇̄ 𝑌 𝑍 − ∇̄ 𝑌 ∇̄ 𝑋 𝑍 − ∇̄ [𝑋,𝑌 ] 𝑍, 𝑇⟩𝑝 = 𝑅𝑝̄ (𝑋, 𝑌 , 𝑍, 𝑇), whence the result.

□

Proposition 3.45 means that the sectional curvature is determined by the curvature of surfaces. We shall see in Theorem 3.51 that the sectional curvature of the surface is its Gaussian curvature, providing a geometric interpretation. So the curvature tensor encodes the information of the curvatures of the surfaces inside 𝑀. This means that the curvature is a 2-dimensional feature.

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Remark 3.46. It is customary in Riemannian geometry to cook up other tensors out of 𝑅. They contain less information but are more manageable. 1

(1) The Ricci tensor is the (0, 2)-tensor Ric = 𝑛−1 𝐶24 (𝑅). That is, take an orthonormal basis (𝑒 1 , 𝑒 2 , . . . , 𝑒 𝑛 ) of 𝑇𝑝 𝑀. Then Ric𝑝 (𝑢, 𝑣) =

1 ∑ 𝑅 (𝑢, 𝑒 𝑖 , 𝑣, 𝑒 𝑖 ). 𝑛−1 𝑖 𝑝

This is a symmetric tensor, so it is determined by the Ricci curvature Ric(𝑢) = Ric(𝑢, 𝑢). If 𝑢 is unitary, then we consider an orthonormal basis (𝑒 1 = 𝑢, 𝑒 2 , . . . , 𝑒 𝑛 ) and the planes 𝜎𝑗 = ⟨𝑒 1 , 𝑒𝑗 ⟩, 2 ≤ 𝑗 ≤ 𝑛. It follows that 1 Ric(𝑢) = 𝑛−1 ∑ 𝐾𝑝 (𝜎𝑗 ), the average of the sectional curvatures of planes containing 𝑢. This average can also be expressed as an integral Ric(𝑢) = 1 ∫ 𝐾 (⟨𝑢, 𝑣⟩), where we have put 𝑆 𝑛−2 = {𝑣 ∈ 𝑇𝑝 𝑀 | 𝑣 ⟂ 𝑢, ||𝑣|| = vol(𝑆 𝑛−2 ) 𝑣∈𝑆 𝑛−2 𝑝 1}. (2) The scalar curvature is defined as the function Scal = 1 𝑛

1 𝑛(𝑛−1)

Scal(𝑝) = ∑ Ric𝑝 (𝑒 𝑖 , 𝑒 𝑖 ) = sectional curvatures through 𝑝.

1 𝐶(Ric). 𝑛

That is,

∑𝑖≠𝑗 𝐾𝑝 (⟨𝑒 𝑖 , 𝑒𝑗 ⟩), i.e., the average of all

(3) Arguments of the theory of group representations say that the only intrinsic tensors (that is, coordinate independent) that can be cooked up with 𝑅 are (combinations of) the Ricci tensor, the scalar curvature, and a tensor known as the Weyl tensor (which is conformally invariant). In dimension 𝑛 = 4, the Weyl tensor 𝑊 decomposes into two pieces 𝑊 + , 𝑊 − (see Remark 6.10).

3.2. Riemannian surfaces and the Gauss-Bonnet theorem 3.2.1. Parametrized surfaces in Euclidean space. Now we enter into the study of the Riemannian aspects of surfaces. We shall start by recalling some well known definitions and properties of parametrized surfaces 𝑆 ⊂ ℝ3 (see [DC1]). Let P ∶ 𝑈 ⊂ ℝ2 → 𝑆 ⊂ ℝ3 , P(𝑢, 𝑣) = (𝑥(𝑢, 𝑣), 𝑦(𝑢, 𝑣), 𝑧(𝑢, 𝑣)), be a parametrization of a surface. This means that 𝑑(ᵆ,𝑣) P has maximal rank at every point and that P is a homeomorphism with its image. Then 𝜑 = P−1 ∶ 𝑆 → 𝑈 is a chart for 𝑆, with coordinates (𝑢, 𝑣). We assume in this section that we work only with one chart. We shall denote derivatives 𝜕𝑥 with respect to 𝑢 and 𝑣 by a subindex, e.g., 𝑥ᵆ = 𝜕ᵆ . Therefore Pᵆ = (𝑥ᵆ , 𝑦ᵆ , 𝑧ᵆ ), P𝑣 = (𝑥𝑣 , 𝑦𝑣 , 𝑧𝑣 ) 𝜕

𝜕

are the generators of 𝑇𝑝 𝑆, 𝑝 = P(𝑢, 𝑣), or in the usual notation, 𝜕ᵆ = 𝜕ᵆ = Pᵆ , 𝜕𝑣 = 𝜕𝑣 = P𝑣 . The standard Riemannian metric in ℝ3 is g𝑠𝑡𝑑 = 𝑑𝑥2 + 𝑑𝑦2 + 𝑑𝑧2 , and this induces the first fundamental form of 𝑆, 𝐈 = g𝑆 = P∗ g𝑠𝑡𝑑 . Writing it down, we have 𝐈 = 𝐸 𝑑𝑢2 + 2𝐹 𝑑𝑢 ⋅ 𝑑𝑣 + 𝐺 𝑑𝑣2 ,

𝐸 = ⟨Pᵆ , Pᵆ ⟩, where { 𝐹 = ⟨Pᵆ , P𝑣 ⟩, 𝐺 = ⟨P𝑣 , P𝑣 ⟩,

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161

or in matrix form in the basis (𝜕ᵆ , 𝜕𝑣 ), 𝐸 𝐈=( 𝐹

𝐹 ). 𝐺

For each 𝑝 ∈ 𝑆, there is a unitary normal 𝐍𝑝 =

Pᵆ × P𝑣 , ||Pᵆ × P𝑣 ||

where Pᵆ × P𝑣 is the vector product in ℝ3 . Remark 3.47. (1) Note that 𝑆 is oriented, by considering the orientation induced by the chart. The area form of 𝑆 is given by 𝜈 = √𝐸𝐺 − 𝐹 2 𝑑𝑢 ∧ 𝑑𝑣, by Remark 3.20. Thus, the integral of a function on 𝑆 is given by ∫𝑆 𝑓 = ∫𝑈 𝑓(𝑢, 𝑣)√𝐸𝐺 − 𝐹 2 𝑑𝑢 𝑑𝑣. (2) Note that ||Pᵆ × P𝑣 || = ||Pᵆ || ||P𝑣 || sin(∠(Pᵆ , P𝑣 )) = √𝐸 √𝐺 √1 − cos2 (∠(Pᵆ , P𝑣 )) = √𝐸𝐺 √1 −

𝐹2 = √𝐸𝐺 − 𝐹 2 . 𝐸𝐺

(3) The normal 𝐍𝑝 defines the orientation by Exercise 1.22. This is the same orientation as the one in (1). Note that the area form 𝜈 satisfies that 𝜈(𝑒 1 , 𝑒 2 ) = 1 for an oriented orthonormal basis (𝑒 1 , 𝑒 2 ) of 𝑇𝑝 𝑆. Then (𝐍𝑝 , 𝑒 1 , 𝑒 2 ) is an oriented orthonormal basis of ℝ3 . Let 𝜔 = 𝑑𝑥∧𝑑𝑦∧𝑑𝑧 be the standard volume form of ℝ3 . Then 𝜈(𝑒 1 , 𝑒 2 ) = 𝜔(𝐍, 𝑒 1 , 𝑒 2 ) = 1, so 𝜈 = 𝑖𝐍 𝜔 = 𝜔(𝐍, −, −), where 𝑖𝑋 is the contraction operator (see Remark 6.19(3)). (4) Let 𝑋 = (𝑃, 𝑄, 𝑅) be a vector field on ℝ3 . The flux of 𝑋 across 𝑆 is the integral ∫⟨𝑋, 𝐍⟩ = ∫ ⟨𝑋, 𝑆

𝑈

Pᵆ × P𝑣 ⟩ √𝐸𝐺 − 𝐹 2 𝑑𝑢 𝑑𝑣 ||Pᵆ × P𝑣 ||

𝑃 = ∫ ⟨𝑋, Pᵆ × P𝑣 ⟩𝑑𝑢 𝑑𝑣 = ∫ det( 𝑥ᵆ 𝑈 𝑈 𝑥𝑣

𝑄 𝑦ᵆ 𝑦𝑣

𝑅 𝑧ᵆ ) 𝑑𝑢 𝑑𝑣 = ∫ 𝛽|𝑆 , 𝑆 𝑧𝑣

with 𝛽 = 𝑃 𝑑𝑦 ∧ 𝑑𝑧 + 𝑄 𝑑𝑧 ∧ 𝑑𝑥 + 𝑅 𝑑𝑥 ∧ 𝑑𝑦, proving Remark 2.130(4). The normal defines the Gauss map (Figure 3.3) 𝐍 ∶ 𝑆 ⟶ 𝑆2 ,

𝑝 ↦ 𝐍(𝑝) = 𝐍𝑝 ,

from the surface 𝑆 to the set of unitary vectors 𝑆 2 ⊂ ℝ3 . A region around a point 𝑝 ∈ 𝑆 is more “curved” if the normal vector varies more abruptly around 𝑝. So it is natural to define the curvature as lim 𝜖→0

vol(𝐍(𝐵𝜀𝑑 (𝑝) ∩ 𝑆)) vol(𝐵𝜀𝑑 (𝑝) ∩ 𝑆)

= det(𝑑𝑝 𝐍).

Definition 3.48. We define the Gaussian curvature as 𝜅𝑆 (𝑝) = det(𝑑𝑝 𝐍).

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S S2

N

Figure 3.3. The Gauss map.

As ℝ3 = 𝑇𝑝 ℝ3 = 𝑇𝑝 𝑆 ⊕ ⟨𝐍𝑝 ⟩, we have 𝑇𝑝 𝑆 = ⟨𝐍𝑝 ⟩⟂ . Also 𝑇𝐍𝑝 𝑆 2 = ⟨𝐍𝑝 ⟩⟂ , since the normal vector at 𝑞 ∈ 𝑆 2 to the sphere is equal to 𝑞, the position vector. Then the differential 𝑑𝑝 𝐍 ∶ 𝑇𝑝 𝑆 → 𝑇𝐍𝑝 𝑆 2 = 𝑇𝑝 𝑆 can be interpreted as an endomorphism of 𝑇𝑝 𝑆. The endomorphism 𝒮𝑝 = −𝑑𝑝 𝐍 is called the Weingarten or shape operator. So 𝜅𝑆 (𝑝) = det(𝒮𝑝 ). The Gaussian curvature, with this definition, is an extrinsic concept (it depends on how 𝑆 sits in ℝ3 ). However, we want to see that it is actually intrinsic and it equals the sectional curvature. In matrix form, with respect to the standard basis (𝜕ᵆ , 𝜕𝑣 ), we write 𝒮𝑝 = (

𝑎11 𝑎21

𝑎12 ), 𝑎22

and 𝜅𝑆 = 𝑎11 𝑎22 − 𝑎12 𝑎21 . Note that this matrix gives the derivatives of the normal vector field, since (𝑎11 , 𝑎21 ) = 𝒮𝑝 (𝜕ᵆ ) = −𝑑𝑝 𝐍 (𝜕ᵆ ) = −𝐍ᵆ and (𝑎12 , 𝑎22 ) = −𝐍𝑣 . Let ∇̄ be the Levi-Civita connection of ℝ3 , which is the trivial connection (Example 3.17). Let ∇ denote the Levi-Civita connection of 𝑆, and 𝐈𝐈 the second fundamental form. As the normal space (𝑇𝑝 𝑆)⟂ = ⟨𝐍𝑝 ⟩⟂ ≅ ℝ, we shall identify normal vectors with scalars, so we interpret 𝐈𝐈𝑝 ∶ 𝑇𝑝 𝑆 × 𝑇𝑝 𝑆 → ℝ as a symmetric bilinear map. In matrix form, with respect to the standard basis (𝜕ᵆ , 𝜕𝑣 ), it is customary to denote 𝑒 𝑓 𝐈𝐈 = ( ). 𝑓 𝑔 Recall that ∇̄ 𝜕𝑢 𝑋 = 𝑋ᵆ , so ∇̄ 𝜕𝑢 Pᵆ = Pᵆᵆ . Using the identity (3.26)) for 𝑋 = Pᵆ , 𝑌 = Pᵆ , 1 2 we have Pᵆᵆ = ∇𝜕𝑢 𝜕ᵆ + 𝐈𝐈(𝜕ᵆ , 𝜕ᵆ )𝐍, and we substitute ∇𝜕𝑢 𝜕ᵆ = Γ11 𝜕ᵆ + Γ11 𝜕𝑣 and 𝐈𝐈(𝜕ᵆ , 𝜕ᵆ ) = 𝑒. Analogous formulas hold for other combinations of basis vectors. This

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163

yields ⎧

1 2 Pᵆᵆ = Γ11 𝜕ᵆ + Γ11 𝜕𝑣 + 𝑒 𝐍,

1 2 Pᵆ𝑣 = Γ12 𝜕ᵆ + Γ12 𝜕𝑣 + 𝑓 𝐍, ⎨ 1 2 ⎩ P𝑣𝑣 = Γ22 𝜕ᵆ + Γ22 𝜕𝑣 + 𝑔 𝐍. These are nice formulas in which the second derivatives of the parametrization put together both the Christoffel symbols and the second fundamental form. Note that

(3.28)

𝑒 = ⟨Pᵆᵆ , 𝐍⟩, { 𝑓 = ⟨Pᵆ𝑣 , 𝐍⟩, 𝑔 = ⟨P𝑣𝑣 , 𝐍⟩. The relation between 𝐈𝐈 and 𝒮 is given by the following computation. For a unitary vector 𝑢 ∈ 𝑇𝑝 𝑆, take a curve 𝛼 ∶ (−𝜀, 𝜀) → 𝑆 with 𝛼(0) = 𝑝, 𝛼′ (0) = 𝑢. Then 𝐈𝐈(𝑢, 𝑢) = ⟨∇̄ 𝛼′ 𝛼′ , 𝐍⟩𝑝 = 𝛼′ (⟨𝛼′ , 𝐍⟩)𝑝 − ⟨𝛼′ , ∇̄ 𝛼′ 𝐍⟩𝑝 = ⟨𝑢, −∇̄ ᵆ 𝐍⟩ = ⟨𝑢, 𝒮𝑝 (𝑢)⟩. Therefore we have a matrix equality (

𝑒 𝑓

𝑓 𝐸 )=( 𝑔 𝐹

𝐹 𝑎11 )( 𝐺 𝑎21

𝑎12 ), 𝑎22

and hence 𝑒𝑔 − 𝑓2 . 𝐸𝐺 − 𝐹 2 Remark 3.49. As 𝐈𝐈 is a symmetric matrix, there is an orthonormal basis (𝑒 1 , 𝑒 2 ) of 1 0 𝜆 0 𝑇𝑝 𝑆 in which it diagonalizes. With respect to this basis, 𝐈 = ( ) and 𝐈𝐈 = ( 1 ), 0 1 0 𝜆2 so 𝒮(𝑒 1 ) = 𝜆1 𝑒 1 , 𝒮(𝑒 2 ) = 𝜆2 𝑒 2 . The directions 𝑒 1 , 𝑒 2 (when 𝜆1 ≠ 𝜆2 ) are called principal directions, and 𝜆1 , 𝜆2 are called principal curvatures. Then 𝜅𝑆 (𝑝) = 𝜆1 𝜆2 . (3.29)

𝜅𝑆 (𝑝) = det(𝒮𝑝 ) =

We add a comment on curves on the surface. Let 𝛼 ∶ 𝐼 → 𝑆 be a curve parametrized by arc length. Let us look at the three terms in the equality ∇̄ 𝛼′ 𝛼′ = ∇𝛼′ 𝛼′ +𝐈𝐈(𝛼′ , 𝛼′ )𝐍. The first one is ∇̄ 𝛼′ 𝛼′ = 𝛼″ = 𝐤𝛼 , the curvature of 𝛼 as a curve in ℝ3 . The second one is the geodesic curvature 𝐤𝑔 = ∇𝛼′ 𝛼′ of 𝛼 (the curvature of 𝛼 as a curve in 𝑆, see Remark 3.28(4)). The third one receives the name of normal curvature of 𝛼 and its denoted 𝐤𝑛 = 𝐈𝐈(𝛼′ , 𝛼′ )𝐍. Then 𝐤𝛼 = 𝐤𝑔 + 𝐤𝑛 . Let 𝐭𝛼 = 𝛼′ be the tangent vector to 𝛼. We define the normal vector of 𝛼 in 𝑆 as the unitary vector 𝐧𝛼 ∈ 𝑇𝛼(𝑠) 𝑆 so that (𝐭𝛼 , 𝐧𝛼 ) is a positive orthonormal basis for 𝑇𝛼(𝑠) 𝑆. The orthonormal frame (𝐭𝛼 , 𝐧𝛼 , 𝐍) is called Darboux frame of 𝛼. Then 𝐤𝑔 = 𝑘𝑔 𝐧𝛼 , for some function 𝑘𝑔 ∶ 𝐼 → 𝑆, called the geodesic curvature function. We also write 𝐤𝑛 = 𝑘𝑛 𝐍, where the normal curvature function is 𝑘𝑛 = 𝐈𝐈(𝛼′ , 𝛼′ ). Remark 3.50. There is a beautiful geometric interpretation of 𝐈𝐈 as follows.1 Suppose that we move along a curve 𝛼(𝑠) in 𝑆 parametrized by arc length such that 𝛼′ (𝑠) is a principal direction for all 𝑠 (these are called curvature lines). If at each point we look 1

We thank Jesús Gonzalo (Universidad Autónoma de Madrid) for sharing this idea with us.

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at our zenith (i.e., in the direction of 𝐍𝛼(𝑠) ), then we see the sky balancing in the frontrear direction. This holds since the movement of the sky is given by −𝑑𝛼(𝑠) 𝐍(𝛼′ (𝑠)) = 𝒮𝛼(𝑠) (𝛼′ (𝑠)), which is parallel to 𝛼′ (𝑠), since it is an eigenvector. If the curve 𝛼(𝑠) is an asymptotic line, that is, if 𝐈𝐈𝛼(𝑠) (𝛼′ (𝑠), 𝛼′ (𝑠)) = 0 for all 𝑠, then we see the sky rotating. This holds since the movement of the sky 𝒮𝛼(𝑠) (𝛼′ (𝑠)) ⟂ 𝛼′ (𝑠), for all 𝑠. Despite its definition, the curvature 𝜅𝑆 is actually an intrinsic invariant of 𝑆. This means that it does not depend on the embedding of 𝑆 in ℝ3 . It only depends on the first fundamental form (the Riemannian metric) and not on the second fundamental form (the normal vector 𝐍). In other words, although the formula (3.29) is a formula for 𝜅𝑆 involving 𝐸, 𝐹, 𝐺, 𝑒, 𝑓, 𝑔, there is another formula for 𝜅𝑆 only involving 𝐸, 𝐹, 𝐺. This is the famous Theorem 3.51 (Gauss egregium). Let 𝑆 ⊂ ℝ3 be an embedded surface with its induced Riemannian structure g𝑆 = 𝐈. Then the Gaussian curvature equals the sectional curvature, 𝜅𝑆 (𝑝) = 𝐾𝑝 (𝑇𝑝 𝑆). Proof. By definition of sectional curvature, 𝐾𝑝 (𝑇𝑝 𝑆) =

𝑅 (𝜕ᵆ , 𝜕𝑣 , 𝜕ᵆ , 𝜕𝑣 ) . ||𝜕ᵆ ∧ 𝜕𝑣 ||2

The denominator is ||𝜕ᵆ ∧ 𝜕𝑣 ||2 = ||𝜕ᵆ ||2 ||𝜕𝑣 ||2 − ⟨𝜕ᵆ , 𝜕𝑣 ⟩2 = 𝐸𝐺 − 𝐹 2 . The numerator can be computed using (3.28), 𝑅 (𝜕ᵆ , 𝜕𝑣 , 𝜕ᵆ , 𝜕𝑣 ) = ⟨∇𝜕𝑣 ∇𝜕𝑢 Pᵆ − ∇𝜕𝑢 ∇𝜕𝑣 Pᵆ , P𝑣 ⟩ = ⟨∇𝜕𝑣 (Pᵆᵆ − 𝑒𝐍) − ∇𝜕𝑢 (P𝑣ᵆ − 𝑓𝐍), P𝑣 ⟩ = ⟨∇̄ 𝜕 (Pᵆᵆ − 𝑒𝐍) − ∇̄ 𝜕 (P𝑣ᵆ − 𝑓𝐍), P𝑣 ⟩ 𝑣

𝑢

= ⟨(Pᵆᵆ𝑣 − 𝑒 𝑣 𝐍 − 𝑒𝐍𝑣 ) − (P𝑣ᵆᵆ − 𝑓ᵆ 𝐍 − 𝑓𝐍ᵆ ), P𝑣 ⟩ = ⟨−𝑒𝐍𝑣 + 𝑓𝐍ᵆ , P𝑣 ⟩ = 𝑒𝑔 − 𝑓2 , since Pᵆᵆ𝑣 = P𝑣ᵆᵆ , ⟨𝑁𝑣 , P𝑣 ⟩ = −⟨𝒮 (𝜕𝑣 ) , 𝜕𝑣 ⟩ = −𝐈𝐈 (𝜕𝑣 , 𝜕𝑣 ) = −𝑔 and ⟨𝑁ᵆ , P𝑣 ⟩ = −𝑓. So 𝐾𝑝 (𝑇𝑝 𝑆) =

𝑒𝑔 − 𝑓2 = 𝜅𝑆 (𝑝). 𝐸𝐺 − 𝐹 2

□

3.2.2. Orthogonal coordinates for surfaces. From now on we shall consider any Riemannian surface (𝑆, g), so we do not assume anymore that it is embedded in a Euclidean space. We define its Gaussian curvature as 𝜅𝑆 (𝑝) = 𝐾𝑝 (𝑇𝑝 𝑆), so that it agrees with the notion in the case of a surface in 3-dimensional Euclidean space. We want to compute intrinsic invariants of 𝑆 using the first fundamental form 𝐈 = g. To simplify computations, we shall deal with coordinate charts (𝑈, 𝜑 = (𝑢, 𝑣)) of 𝑆 on which the metric has the form (3.30)

𝐈 = g = (𝑔𝑖𝑗 ) = (

𝐸 0

0 ), 𝐺

for some functions 𝐸, 𝐺 ∶ 𝑈 → ℝ. In this case, (𝜕ᵆ , 𝜕𝑣 ) is an orthogonal frame (but not orthonormal). We say that (𝑈, 𝜑) are orthogonal coordinates. We start proving that such coordinate charts do exist in any surface.

3.2. Riemannian surfaces and the Gauss-Bonnet theorem

165

Let 𝑝 ∈ 𝑆 and let 𝐵𝜀𝑑 (𝑝) be a geodesic neighbourhood of 𝑝. Fix an orthonormal basis (𝑒 1 , 𝑒 2 ) of 𝑇𝑝 𝑆. Then we have polar geodesic coordinates defined by (the inverse of) (3.31)

𝜑 ∶ (0, 𝜀) × (0, 2𝜋) ⟶ 𝑆, 𝜑(𝑟, 𝜃) = exp𝑝 (𝑟((cos 𝜃)𝑒 1 + (sin 𝜃)𝑒 2 )).

This actually covers 𝐵𝜀𝑑 (𝑝) minus a ray, so it does not cover the point 𝑝. However, the images of such coordinate neighbourhoods cover all of 𝑆. Proposition 3.52. The polar geodesic coordinates are orthogonal coordinates. Actually, the first fundamental form is g=(

1 0 ), 0 𝑟2 𝐻(𝑟, 𝜃)2

where 𝐻(𝑟, 𝜃) is a 𝐶 ∞ function on [0, 𝜀)×[0, 2𝜋] with 𝐻(0, 𝜃) = 1, and 𝐻(𝑟, 0) = 𝐻(𝑟, 2𝜋). Proof. For 𝜃0 fixed, the curve 𝛾 𝜃0 (𝑟) = 𝜑(𝑟, 𝜃0 ) is a geodesic emanating from 𝑝, hence 𝐸 = ||𝜑𝑟 ||2 = ||𝛾𝜃′ 0 ||2 = 1. The vector 𝜕𝜃 = 𝑑(𝑟,𝜃) exp𝑝 (−𝑟 sin 𝜃, 𝑟 cos 𝜃) = 𝑟 𝑣(𝑟, 𝜃), where 𝑣(𝑟, 𝜃) = 𝑑(𝑟,𝜃) exp𝑝 (− sin 𝜃, cos 𝜃). If we take 𝐻(𝑟, 𝜃) = ||𝑣(𝑟, 𝜃)||, then we have 𝐺 = ||𝜕𝜃 ||2 = 𝑟2 𝐻(𝑟, 𝜃)2 . At 𝑟 = 0, 𝑑(0,𝜃) exp𝑝 = Id, hence 𝑣(0, 𝜃) = (− sin 𝜃, cos 𝜃) and 𝐻(0, 𝜃) = ||𝑣(0, 𝜃)|| = 1. Finally, the function 𝐹 = ⟨𝜕𝑟 , 𝜕𝜃 ⟩ is constant in the radial direction because 𝑑𝐹 1 𝑑 1 𝑑 = ⟨∇𝜕𝑟 𝜕𝑟 , 𝜕𝜃 ⟩ + ⟨𝜕𝑟 , ∇𝜕𝑟 𝜕𝜃 ⟩ = ⟨𝜕𝑟 , ∇𝜕𝜃 𝜕𝑟 ⟩ = ⟨𝜕𝑟 , 𝜕𝑟 ⟩ = (1) = 0, 𝑑𝑟 2 𝑑𝜃 2 𝑑𝜃 where we have used that ∇𝜕𝑟 𝜕𝑟 = 0 because 𝜑(𝑟, 𝜃0 ) are geodesics, and the torsion free property for ∇. Then 𝐹(𝑟, 𝜃0 ) = 𝑐(𝜃0 ) is a function of 𝜃0 . But |𝐹(𝑟, 𝜃0 )| ≤ ||𝜕𝜃 || ||𝜕𝑟 || = √𝐸𝐺 = 𝑟√𝐻 → 0 as 𝑟 → 0. Then 𝐹(𝑟, 𝜃) ≡ 0. □ Remark 3.53. • Proposition 3.52 holds in dimension 𝑛. This result is called the Gauss lemma. It says that for geodesic coordinates, the radial direction is perpendicular to the spherical directions. Physically, it means that when looking at the space from a point, the only distortion in the view of the sky happens in the transversal direction (galaxies may look closer or further apart than they are), but there is no tilting. • There are other geometric ways to obtain orthogonal coordinates (Exercise 3.20). Proposition 3.54. Let 𝑆 be a surface and let 𝑈 ⊂ 𝑆 be an orthogonal coordinate neighbourhood with coordinates (𝑢, 𝑣). Then the Gaussian curvature equals 𝜅𝑆 = −

1 2√𝐸𝐺

((

𝐸𝑣 √𝐸𝐺

) +( 𝑣

𝐺ᵆ √𝐸𝐺

) ). ᵆ

166

3. Riemannian geometry

Proof. The proof is a computation. First we compute the Christoffel symbols using (3.19). In this case 𝑔11 = 𝐸, 𝑔12 = 𝑔21 = 0 and 𝑔22 = 𝐺 and the coefficients of the 1 1 inverse matrix of (𝑔𝑖𝑗 ) are 𝑔11 = 𝐸 , 𝑔12 = 𝑔21 = 0, and 𝑔22 = 𝐺 . Then 1 𝐸 , 2𝐸 ᵆ 1 2 Γ11 = − 𝐸𝑣 , 2𝐺

1 𝐸 , 2𝐸 𝑣 1 2 2 Γ12 = Γ21 = 𝐺 , 2𝐺 ᵆ

1 Γ11 =

(3.32)

1 𝐺 , 2𝐸 ᵆ 1 2 Γ22 = 𝐺 . 2𝐺 𝑣

1 1 Γ12 = Γ21 =

1 Γ22 =−

Using now the formula for the curvature tensor in (3.27), where the coordinates are (𝑥1 , 𝑥2 ) = (𝑢, 𝑣), we have 2 𝜕Γ2 𝜕Γ11 1 2 2 2 1 2 2 2 − 21 + Γ11 Γ21 + Γ11 Γ22 − Γ21 Γ11 − Γ21 Γ12 𝜕𝑣 𝜕𝑢 𝐺 𝐸 𝐺 𝐺 𝐺 𝐸 𝐸 𝐺 𝐸 𝐸 = − ( 𝑣 ) − ( ᵆ ) + ᵆ ᵆ − 𝑣 2𝑣 + 𝑣 𝑣 − ᵆ 2ᵆ 2𝐺 𝑣 2𝐺 ᵆ 4𝐸𝐺 4𝐸𝐺 4𝐺 4𝐺 𝐺 𝐸 𝐺 𝐺 𝐺 𝐸 𝐸 𝐺 𝐸 𝐸 = − 𝑣𝑣 − ᵆᵆ + ᵆ ᵆ + 𝑣 2𝑣 + 𝑣 𝑣 + ᵆ 2ᵆ . 2𝐺 2𝐺 4𝐸𝐺 4𝐸𝐺 4𝐺 4𝐺

𝑅2121 =

In this way, since 𝑅1212 = ∑𝑙 𝑔2𝑙 𝑅𝑙121 = 𝐺 𝑅2121 we obtain 𝐸 𝐺 𝐺 𝐺 𝐸 𝐺 𝐸 𝐸 1 𝑅1212 = − (𝐸𝑣𝑣 + 𝐺ᵆᵆ ) + ᵆ ᵆ + 𝑣 𝑣 + 𝑣 𝑣 + ᵆ ᵆ . 2 4𝐸 4𝐺 4𝐸 4𝐺 Therefore, we have 𝑅 (𝜕ᵆ , 𝜕𝑣 , 𝜕ᵆ , 𝜕𝑣 ) ||𝜕ᵆ ∧ 𝜕𝑣 ||2 𝐸 𝐺 𝐺 𝐺 𝐸 𝐺 𝐸 𝐸 𝑅 1 = 1212 = − (𝐸 + 𝐺ᵆᵆ − ᵆ ᵆ − 𝑣 𝑣 − 𝑣 𝑣 − ᵆ ᵆ ) 𝐸𝐺 2𝐸𝐺 𝑣𝑣 2𝐸 2𝐺 2𝐸 2𝐺

𝜅𝑆 (𝑝) = 𝐾𝑝 (𝑇𝑝 𝑆) =

=−

=−

1 2√𝐸𝐺 1 2√𝐸𝐺

𝐸𝑣𝑣 √𝐸𝐺 − 𝐸𝑣 (

((

𝐸𝑣 𝐺+𝐸𝐺𝑣 2√𝐸𝐺

𝐸𝐺 𝐸𝑣 √𝐸𝐺

) +( 𝑣

𝐺ᵆ √𝐸𝐺

𝐺ᵆᵆ √𝐸𝐺 − 𝐺ᵆ +

𝐸𝐺

𝐸𝑢 𝐺+𝐸𝐺𝑢 2√𝐸𝐺

)

) ).

□

ᵆ

Remark 3.55. Theorem 3.51 guaranteed the existence of a formula for 𝜅 in terms of 𝐸, 𝐹, 𝐺. The general formula is rather involved (it is called the Brioschi formula [Gra]). Proposition 3.54 provides the formula in the special situation where 𝐹 = 0. Corollary 3.56. Let 𝑆 be a surface, and let 𝑈 be a coordinate neighbourhood in which the Riemannian metric has the form (3.33)

𝑒2𝑓 g=( 0

0 ). 𝑒2𝑓

Then the Gaussian curvature is 𝜅𝑆 = −𝑒−2𝑓 Δ𝑓, where Δ denotes the Laplacian.

3.2. Riemannian surfaces and the Gauss-Bonnet theorem

167

Proof. These are orthogonal coordinates (𝑢, 𝑣) with 𝐸 = 𝐺 = 𝑒2𝑓 or, equivalently, 1 𝑓 = 2 log 𝐸. Using Proposition 3.54, we have 𝐸 𝐸 1 1 (( 𝑣 ) + ( ᵆ ) ) = − ((log 𝐸)𝑣𝑣 + (log 𝐸)ᵆᵆ ) 2𝐸 𝐸 𝑣 𝐸 ᵆ 2𝐸 = −𝑒−2𝑓 (𝑓ᵆᵆ + 𝑓𝑣𝑣 ) = −𝑒−2𝑓 Δ𝑓.

𝜅𝑆 = −

□

Remark 3.57. The coordinates (3.33) are called conformal, harmonic, or isothermal (see Definition 4.14 and Remark 6.54). As for orthogonal coordinates, conformal coordinates always exist at any point on a surface. However this result is considerably harder to prove (see section 6.3.4). We can give the first useful geometric characterization of curvature in terms of the length of geodesic circles. For 𝑝 ∈ 𝑆, fix 𝑟 small enough and draw the circle 𝛼𝑟 (𝜃) = 𝜑(𝑟, 𝜃),

𝜃 ∈ [0, 2𝜋],

where 𝜑 is the parametrization in (3.31). We call geodesic circle of radius 𝑟 and centre 𝑝 to 𝐶𝑟 (𝑝) = 𝛼𝑟 ([0, 2𝜋]). Theorem 3.58. The length of the geodesic circle of radius 𝑟 is ℓ(𝐶𝑟 (𝑝)) = 2𝜋𝑟 −

𝜋𝜅𝑆 (𝑝) 3 𝑟 + 𝑂(𝑟4 ). 3

Proof. We compute the curvature 𝜅𝑆 using the coordinates in Proposition 3.52. These are 𝐸 = 1, 𝐺 = 𝑟2 𝐻 2 , in coordinates (𝑟, 𝜃). Hence the formula for the curvature in Proposition 3.54 reads now 𝜅𝑆 (𝑟, 𝜃) = −

(𝑟2 𝐻 2 )𝑟 2𝐻 + 𝑟𝐻𝑟𝑟 1 1 . ( ) = − (𝑟𝐻)𝑟𝑟 = − 𝑟 2𝑟𝐻 𝑟𝐻 𝑟𝐻 𝑟𝐻 𝑟

As lim𝑟→0 𝜅𝑆 (𝑟, 𝜃) = 𝜅𝑆 (𝑝), we have the 𝐻𝑟 (0, 𝜃) = 0 (otherwise the limit above would not be finite). Moreover, we have 𝜅𝑆 (𝑝) = −2 lim 𝑟→0

𝐻𝑟 𝐻 − lim 𝑟𝑟 = −2𝐻𝑟𝑟 (0, 𝜃) − 𝐻𝑟𝑟 (0, 𝜃) = −3𝐻𝑟𝑟 (0, 𝜃), 𝑟𝐻 𝑟→0 𝐻 1

using that 𝐻(0, 𝜃) = 1. Therefore 𝐻(𝑟, 𝜃) = 1 − 6 𝜅𝑆 (𝑝)𝑟2 + 𝑂(𝑟3 ), where the last term may depend on 𝜃 also. Now 2𝜋

2𝜋

ℓ(𝐶𝑟 (𝑝)) = ∫

||𝛼′𝑟 (𝜃)||𝑑𝜃 = ∫

0

0

𝑟𝐻(𝑟, 𝜃)𝑑𝜃 = 2𝜋𝑟 −

2𝜋𝜅𝑆 (𝑝) 3 𝑟 + 𝑂(𝑟4 ). 6

□

In particular, if 𝜅𝑆 (𝑝) > 0, then ℓ(𝐶𝑟 (𝑝)) < 2𝜋𝑟, and if 𝜅𝑆 (𝑝) < 0, then ℓ(𝐶𝑟 (𝑝)) > 2𝜋𝑟, for 𝑟 small enough. Remark 3.59. A consequence of Theorem 3.58 and Proposition 3.45 is the following formula for the sectional curvature on a Riemannian manifold. Let (𝑀, g) be a Riemannian manifold and fix 𝑝 ∈ 𝑀 and a plane 𝜎 ⊂ 𝑇𝑝 𝑀. Consider an orthonormal basis (𝑒 1 , 𝑒 2 ) of 𝜎 and draw the circle 𝐶𝑟 (𝑝, 𝜎) = 𝛼𝑟 ([0, 2𝜋]), with 𝛼𝑟 (𝜃) = exp𝑝 (𝑟((cos 𝜃)𝑒 1 + (sin 𝜃)𝑒 2 )), 𝜃 ∈ [0, 2𝜋].

168

3. Riemannian geometry

Then 2𝜋𝑟 − ℓ(𝐶𝑟 (𝑝, 𝜎)) . 𝜋𝑟3 /3 𝑟→0 That is, the curvature can be computed by measuring circles. This is another way to see that it is intrinsic. 𝐾𝑝 (𝜎) = lim

Remark 3.60. For an 𝑛-dimensional Riemannian manifold (𝑀, g), we have a formula similar to that of Theorem 3.58 for the scalar curvature (Remark 3.46(2)), by measuring the volumes of small spheres around a point (Exercise 3.19). Note that for a surface 𝑆, the scalar curvature coincides with the Gaussian curvature, Scal = 𝜅𝑆 . Now we move to get a formula for the geodesic curvature of an arc length parametrized curve in orthogonal coordinates. Take orthogonal coordinates (𝑢, 𝑣) in an open set 𝑈 ⊂ 𝑆 so that the first fundamental form is (3.30). Let 𝛼 ∶ 𝐼 → 𝑆, 𝛼(𝑠) = (𝑢(𝑠), 𝑣(𝑠)) be a curve parametrized by arc length. Taking the orthonormal frame in 𝑈 given by 1 1 𝐸1 = 𝜕ᵆ and 𝐸2 = 𝜕𝑣 , we can write √𝐸

√𝐺

𝛼′ (𝑠) = cos 𝜑(𝑠)𝐸1 + sin 𝜑(𝑠)𝐸2 ,

(3.34)

for some continuous function 𝜑 ∶ 𝐼 → ℝ measuring the oriented angle 𝜑(𝑠) = ∠(𝐸1 , 𝛼′ (𝑠)). Proposition 3.61. The geodesic curvature function of 𝛼 is given by 𝑘𝑔 =

1 2√𝐸𝐺

(𝐺ᵆ

𝑑𝜑 𝑑𝑣 𝑑𝑢 − 𝐸𝑣 ) + . 𝑑𝑠 𝑑𝑠 𝑑𝑠

Proof. Write 𝛼′ = 𝑢′ 𝜕ᵆ + 𝑣′ 𝜕𝑣 , and let us compute the covariant derivative of 𝐸1 , 𝐸2 along 𝛼, using the formulas in (3.32). 𝐷𝐸1 1 1 1 = ∇ 𝛼′ ( 𝜕ᵆ ) = 𝛼′ ( ∇𝛼′ 𝜕ᵆ ) 𝜕ᵆ + 𝑑𝑠 √𝐸 √𝐸 √𝐸 𝑢′ 𝐸ᵆ + 𝑣′ 𝐸𝑣 1 =− 𝜕ᵆ + (𝑢′ ∇𝜕𝑢 𝜕ᵆ + 𝑣′ ∇𝜕𝑣 𝜕ᵆ ) √ √ 2𝐸 𝐸 𝐸 𝑢′ 𝐸ᵆ + 𝑣′ 𝐸𝑣 1 1 2 1 2 =− 𝜕ᵆ + 𝜕ᵆ + Γ11 𝜕𝑣 ) + 𝑣′ (Γ21 𝜕ᵆ + Γ21 𝜕𝑣 )) (𝑢′ (Γ11 √𝐸 2𝐸√𝐸 𝑢′ 𝐸ᵆ + 𝑣′ 𝐸𝑣 =− 𝜕ᵆ 2𝐸√𝐸 1 1 1 1 1 + 𝐸 𝜕 + 𝑣′ 𝐸𝑣 𝜕ᵆ + 𝑣′ 𝐺 𝜕 ) (𝑢′ 𝐸ᵆ 𝜕ᵆ − 𝑢′ 2𝐸 2𝐺 𝑣 𝑣 2𝐸 2𝐺 ᵆ 𝑣 √𝐸 = Hence

𝐷𝐸1 𝑑𝑠

1 2𝐺√𝐸

(𝑣′ 𝐺ᵆ − 𝑢′ 𝐸𝑣 ) 𝜕𝑣 =

1 2√𝐸𝐺

(𝑣′ 𝐺ᵆ − 𝑢′ 𝐸𝑣 ) 𝐸2 .

= 𝜆𝐸2 where 𝜆=

1 2√𝐸𝐺

(𝑣′ 𝐺ᵆ − 𝑢′ 𝐸𝑣 ) .

3.2. Riemannian surfaces and the Gauss-Bonnet theorem

By a similar computation, we also have

𝐷𝐸2 𝑑𝑠

= −𝜆𝐸1 . Observe that this also follows

from the fact that (𝐸1 , 𝐸2 ) is orthonormal. Indeed ⟨ −𝜆 and

𝐷𝐸 ⟨ 𝑑𝑠2 , 𝐸2 ⟩

=

1 𝑑 ⟨𝐸 , 𝐸2 ⟩ 2 𝑑𝑠 2

169

𝐷𝐸2 , 𝐸1 ⟩ 𝑑𝑠

=

𝑑 ⟨𝐸 , 𝐸2 ⟩ 𝑑𝑠 1

− ⟨𝐸2 ,

𝐷𝐸1 ⟩ 𝑑𝑠

=

= 0.

By definition and (3.34), the geodesic curvature is 𝐷𝛼′ = 𝑘𝑔 𝐧𝛼 = 𝑘𝑔 (− sin 𝜑 𝐸1 + cos 𝜑𝐸2 ) . 𝑑𝑠 So if we compute the derivative of 𝛼′ = cos 𝜑 𝐸1 + sin 𝜑 𝐸2 , we obtain 𝐤𝑔 =

𝐷𝐸 𝐷𝐸2 𝐷𝛼′ = −(sin 𝜑)𝜑′ 𝐸1 + cos 𝜑 1 + (cos 𝜑)𝜑′ 𝐸2 + sin 𝜑 𝑑𝑠 𝑑𝑠 𝑑𝑠 = −(sin 𝜑)𝜑′ 𝐸1 + 𝜆 cos 𝜑 𝐸2 + (cos 𝜑)𝜑′ 𝐸2 − 𝜆 sin 𝜑 𝐸1 = (𝜆 + 𝜑′ ) (− sin 𝜑 𝐸1 + cos 𝜑𝐸2 ) , □

so 𝑘𝑔 = 𝜆 + 𝜑′ , as required.

3.2.3. Gauss-Bonnet theorem. Gauss-Bonnet theorem is one of the most relevant results in geometry and topology. It serves to relate geometric information (Riemannian metric) with purely topological information (its Euler-Poincaré characteristic) of a manifold. It also sets a bridge between local properties (the curvature of a surface) and global properties (its genus). In both regards, it is a paradigm of important results in modern differential geometry. Before we state it, let us introduce some notions. Let 𝑆 be an oriented Riemannian surface. We consider a compact region 𝑅 ⊂ 𝑆 whose boundary is a collection of piecewise 𝐶 2 loops. This means that every component of 𝜕𝑅 is parametrized by a closed continuous curve 𝛼 ∶ [𝑎, 𝑏] → 𝜕𝑅 with a partition 𝑎 = 𝑠0 < 𝑠1 < ⋯ < 𝑠𝑚 = 𝑏 such that 𝛼|[𝑠𝑖−1 ,𝑠𝑖 ] is a 𝐶 2 curve parametrized by arc length, for 1 ≤ 𝑖 ≤ 𝑚, and 𝛼(𝑠𝑚 ) = 𝛼(𝑠0 ). In particular, this requires that at the corner points 𝛼(𝑠𝑖 ), the lateral limits 𝛼′ (𝑠− 𝑖 ) and ′ + ′ − 𝛼′ (𝑠+ 𝑖 ) exist for 1 ≤ 𝑖 ≤ 𝑚 − 1, and also 𝛼 (𝑠0 ), 𝛼 (𝑠𝑚 ) exist. Let 𝜃 𝑖 be the oriented angle ′ + ′ − ′ + 𝜃𝑖 = ∠(𝛼′ (𝑠− 𝑖 ), 𝛼 (𝑠𝑖 )), for 1 ≤ 𝑖 ≤ 𝑚 − 1, and also denote 𝜃𝑚 = ∠(𝛼 (𝑠𝑚 ), 𝛼 (𝑠0 )). We call 𝜃𝑖 the corner angles. Theorem 3.62 (Gauss-Bonnet, local version). Let 𝑈 ⊂ 𝑆 be an orthogonal chart, and let 𝑅 ⊂ 𝑈 be a compact region homeomorphic to a closed ball and with 𝜕𝑅 parametrized (by arc length) by a piecewise 𝐶 2 curve 𝛼. We assume that 𝛼 is oriented positively (with respect to the orientation induced by 𝑅 on its boundary). Then 𝑚

∫ 𝜅𝑆 + ∫ 𝑘𝑔 + ∑ 𝜃𝑖 = 2𝜋, 𝑅

𝜕𝑅

𝑖=1

where 𝜅𝑆 is the Gaussian curvature of 𝑆, and 𝑘𝑔 is the geodesic curvature of 𝛼. Proof. Let (𝑢, 𝑣) be the coordinates of 𝑈 ⊂ ℝ2 . As this is an orthogonal chart, the metric is g = 𝐸 𝑑𝑢2 + 𝐺 𝑑𝑣2 . As in Proposition 3.61, fix the orthonormal basis given by 𝐸1 = 𝐸 −1/2 𝜕ᵆ and 𝐸2 = 𝐺 −1/2 𝜕𝑣 , and consider the function 𝜑(𝑠) = ∠(𝐸1 , 𝛼′ (𝑠)), 𝑠 ∈ [𝑎, 𝑏], such that 𝛼′ (𝑠) = cos 𝜑(𝑠)𝐸1 + sin 𝜑(𝑠)𝐸2 . The function 𝜑 ∶ [𝑎, 𝑏] → ℝ − ′ + has jump discontinuities at the points 𝑠𝑖 of height 𝜑(𝑠+ 𝑖 ) − 𝜑(𝑠𝑖 ) = ∠(𝐸1 , 𝛼 (𝑠𝑖 )) − + − − ∠(𝐸1 , 𝛼′ (𝑠𝑖 )) = ∠(𝛼′ (𝑠𝑖 ), 𝛼′ (𝑠𝑖 )) = 𝜃𝑖 . For a planar region 𝑅, the boundary 𝜕𝑅 is

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3. Riemannian geometry

homeomorphic to a circle, and the angle function 𝜑(𝑠) has a total increase in value of 2𝜋 − ′ + ′ − (since 𝛼 runs 𝜕𝑅 positively). Then 𝜑(𝑠+ 0 )−𝜑(𝑠𝑚 ) = ∠(𝐸1 , 𝛼 (𝑠0 ))−∠(𝐸1 , 𝛼 (𝑠𝑚 ))−2𝜋 = ′ − ′ + ∠(𝛼 (𝑠𝑚 ), 𝛼 (𝑠0 )) − 2𝜋 = 𝜃𝑚 − 2𝜋. This implies that 𝑏

∫ 𝑎

𝑚

𝑚−1

𝑑𝜑 + + + − − 𝑑𝑠 = ∑ (𝜑(𝑠− 𝑖 ) − 𝜑(𝑠𝑖−1 )) = 𝜑(𝑠𝑚 ) − 𝜑(𝑠0 ) + ∑ (𝜑(𝑠𝑖 ) − 𝜑(𝑠𝑖 )) 𝑑𝑠 𝑖=1 𝑖=1 𝑚−1

𝑚

= 2𝜋 − 𝜃𝑚 − ∑ 𝜃𝑖 = 2𝜋 − ∑ 𝜃𝑖 . 𝑖=1

𝑖=1

To prove the result, we apply Green’s theorem (Remark 2.130(2)), which says ∫𝜕𝑅 (𝑃𝑑𝑢 + 𝑄𝑑𝑣) = ∫𝑅 (𝑄ᵆ − 𝑃𝑣 )𝑑𝑢 𝑑𝑣, to the region 𝑅 ⊂ 𝑈 ⊂ ℝ2 and the 1-form 𝐸𝑣

𝑃𝑑𝑢 + 𝑄𝑑𝑣 = −

2√𝐸𝐺

𝑑𝑢 +

𝐺ᵆ 2√𝐸𝐺

𝑑𝑣 .

On the one hand, Proposition 3.61 says that ∫ (𝑃𝑑𝑢 + 𝑄𝑑𝑣) = ∫ (− (3.35)

𝜕𝑅

𝜕𝑅

𝐺ᵆ 𝑑𝑣 𝑑𝑢 + ) 𝑑𝑠 2√𝐸𝐺 𝑑𝑠 2√𝐸𝐺 𝑑𝑠 𝐸𝑣

𝑚

= ∫ (𝑘𝑔 − 𝜕𝑅

𝑑𝜑 ) 𝑑𝑠 = ∫ 𝑘𝑔 + ∑ 𝜃𝑖 − 2𝜋. 𝑑𝑠 𝜕𝑅 𝑖=1

On the other hand, Proposition 3.54 gives ∫ (𝑄ᵆ − 𝑃𝑣 ) 𝑑𝑢 𝑑𝑣 = ∫ (( (3.36)

𝑅

𝑅

𝐺ᵆ 2√𝐸𝐺

) +( ᵆ

𝐸𝑣 2√𝐸𝐺

) ) 𝑑𝑢 𝑑𝑣 𝑣

= − ∫ 𝜅𝑆 √𝐸𝐺 𝑑𝑢 𝑑𝑣 = − ∫ 𝜅𝑆 , 𝑅

𝑅

using that the area form of (𝑆, g) is 𝜈 = √𝐸𝐺 𝑑𝑢 ∧ 𝑑𝑣 (Remark 3.47). Equating (3.35) and (3.36), we get the result. □ Remark 3.63. In Theorem 3.62, it is natural to introduce a distributional geodesic 𝑚 curvature given as 𝑘𝑔̂ = 𝑘𝑔 + ∑𝑖=1 𝜃𝑖 𝛿𝑠𝑖 , where 𝛿𝑠𝑖 are the Dirac deltas at the corner points. This is actually the distributional derivative ∇𝛾′ 𝛾′ . The integral is thus ∫𝜕𝑅 𝑘𝑔̂ = 𝑚 ∫𝜕𝑅 𝑘𝑔 + ∑𝑖=1 𝜃𝑖 , and the Gauss-Bonnet theorem becomes ∫𝑅 𝜅𝑆 + ∫𝜕𝑅 𝑘𝑔̂ = 2𝜋. Let us give a description of the curvature in terms of the (local) holonomy around closed loops. Theorem 3.64. Let 𝑝 ∈ 𝑆 and consider a (small) piecewise 𝐶 2 loop 𝛼 ∶ [𝑎, 𝑏] → 𝑆, 𝛼(𝑎) = 𝛼(𝑏) = 𝑝, which is the boundary of a region 𝑅 (travelled in the positive direction) inside an orthogonal chart. Take a unitary 𝑣 ∈ 𝑇𝑝 𝑆 and transport it around the loop 𝛼. Then ∠(𝑣, 𝑃𝛼𝑎,𝑏 (𝑣)) = ∫ 𝜅𝑆 , 𝑅

where 𝑃𝛼𝑎,𝑏 (𝑣) is the parallel transport of 𝑣 along 𝛼 (see Remark 3.14(2)).

3.2. Riemannian surfaces and the Gauss-Bonnet theorem

171

Proof. We can assume that 𝛼 is parametrized by arc length (Remark 3.14(8)). Extend 𝑣 to a positive orthonormal basis (𝑣, 𝑤) of 𝑇𝑝 𝑆, and transport it parallelly around the loop obtaining a basis (𝑉, 𝑊) of 𝔛∥ (𝛼). Let 𝛼′ (𝑠) = cos 𝜑(𝑠) 𝑉 + sin 𝜑(𝑠) 𝑊, where 𝜑(𝑠) = 𝑑𝜑 𝐷𝛼′ ∠(𝑉(𝑠), 𝛼′ (𝑠)). The geodesic curvature is 𝑘𝑔 = 𝑑𝑠 since 𝑑𝑠 = 𝜑′ (− sin 𝜑 𝑉 + cos 𝜑 𝑊). Then 𝑏

𝑚−1

∫ 𝑘𝑔 + ∑ 𝜃𝑖 = 𝜑(𝑏) − 𝜑(𝑎) = 2𝜋 + ∠(𝑉(𝑏), 𝛼′ (𝑏)) − ∠(𝑉(𝑎), 𝛼′ (𝑎)) 𝑎

𝑖=1

= 2𝜋 − 𝜃𝑚 − ∠(𝑉(𝑎), 𝑉(𝑏)), since 𝜃𝑚 = ∠(𝛼′ (𝑏), 𝛼′ (𝑎)). As in Theorem 3.62, the increase in 2𝜋 is due to the fact that 𝛼 gives a positive turn around the region 𝑅. Therefore if 𝛼 bounds a region 𝑅, 𝑚

∠(𝑣, 𝑃𝛼𝑎,𝑏 (𝑣)) = ∠(𝑉(𝑎), 𝑉(𝑏)) = − ∫ 𝑘𝑔 − ∑ 𝜃𝑖 + 2𝜋 = ∫ 𝜅𝑆 , 𝜕𝑅

𝑅

𝑖=1

□

using the Gauss-Bonnet Theorem 3.62.

Remark 3.65. This result gives a second geometric description of the sectional curvature. Let 𝑀 be a Riemannian manifold, 𝑝 ∈ 𝑀 and 𝜎 ⊂ 𝑇𝑝 𝑀 a plane. Consider the holonomy around the circle 𝐶𝑟 (𝑝, 𝜎) ⊂ 𝑆 = exp𝑝 (𝐵𝑅𝑑 (𝑝) ∩ 𝜎) ⊂ 𝑀, for 0 < 𝑟 < 𝑅 small, which is a rotation by some (oriented) angle 𝜃(𝑟, 𝑝, 𝜎). Then 𝐾𝑝 (𝜎) = 𝜅𝑆 (𝑝) = lim 𝑟→0

1 area(𝐵𝑟𝑑 (𝑝)

∩ 𝑆)

∫ 𝐵𝑟𝑑 (𝑝)∩𝑆

𝜅𝑆 = lim 𝑟→0

𝜃(𝑟, 𝑝, 𝜎) , 𝜋𝑟2

noting that area(𝐵𝑟𝑑 (𝑝) ∩ 𝑆) ≈ 𝜋𝑟2 . As a particular case of Theorem 3.62, we can consider a geodesic polygon 𝑃 ⊂ 𝑆, which is a compact convex region whose boundary is a collection of 𝑙 geodesic segments (the edges). Let 𝛽1 , . . . , 𝛽 𝑙 be the (interior) angles of the polygon. They are related to 𝑙 the corner angles as 𝛽 𝑖 = 𝜋 − 𝜃𝑖 . Then Theorem 3.62 reads ∫𝑃 𝜅𝑆 + ∑𝑖=1 (𝜋 − 𝛽 𝑖 ) = 2𝜋, since 𝑘𝑔 = 0. Hence 𝑙

(3.37)

∑ 𝛽 𝑖 = (𝑙 − 2)𝜋 + ∫ 𝜅𝑆 , 𝑖=1

𝑃

which says that the discrepancy of the sum of the angles of 𝑃 with that of a Euclidean polygon is given by the integral of the curvature. In particular, for a geodesic triangle 𝑇 of angles 𝛽1 , 𝛽2 , 𝛽3 , we have (3.38)

𝛽1 + 𝛽2 + 𝛽3 = 𝜋 + ∫ 𝜅𝑆 . 𝑇

If 𝜅𝑆 < 0, then 𝛽1 + 𝛽2 + 𝛽3 < 𝜋, and if 𝜅𝑆 > 0, then 𝛽1 + 𝛽2 + 𝛽3 > 𝜋. If 𝜅𝑆 = 0, we have the well known formula for the sum of the angles of a Euclidean triangle. Remark 3.66. This gives our third geometric description of the sectional curvature. If 𝑀 is a Riemannian manifold, 𝑝 ∈ 𝑀, and 𝜎 ⊂ 𝑇𝑝 𝑀 is a plane, then consider small

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3. Riemannian geometry

geodesic triangles 𝑇𝜖 in the surface 𝑆 = exp𝑝 (𝐵𝑅 (0) ∩ 𝜎), and let 𝛽1 (𝜖), 𝛽2 (𝜖), 𝛽3 (𝜖) be their angles. Then 𝛽 (𝜖) + 𝛽2 (𝜖) + 𝛽3 (𝜖) − 𝜋 𝐾𝑝 (𝜎) = lim 1 . area(𝑇𝜖 ) 𝜖→0 It seems that Lobachevsky wanted to prove that the universe has negative curvature by computing the angles of a triangle formed by three specific stars. His measurements gave no definite answer, as the deviation of the sum of the angles from 𝜋 was within the error of measurements. Also Gauss computed the angles of a triangle with vertices at three mountains in Germany, but it is not known if he wanted to compute the curvature of the earth or the curvature of the universe. Remark 3.67. There is a striking consequence of (3.38) if 𝑆 has a metric of constant curvature 𝜅𝑆 ≡ 𝑘0 . In this case 𝛽1 +𝛽2 +𝛽3 = 𝜋+𝑘0 area(𝑇). Then as 0 < 𝛽1 +𝛽2 +𝛽3 < 3𝜋, we have that • If 𝑘0 > 0, then area(𝑇) < 2𝜋/𝑘0 . • If 𝑘0 < 0, then area(𝑇) < 𝜋/|𝑘0 |. So the area of a geodesic triangle is bounded in both cases. This prevents the existence of homotheties (cf. Remarks 4.12 and 4.68). Theorem 3.68 (Gauss-Bonnet, global version). Let 𝑆 be a Riemannian oriented surface, and let 𝑅 ⊂ 𝑆 be a compact region (maybe non-connected) whose boundary 𝜕𝑅 is piecewise 𝐶 2 (maybe non-connected or empty). We give 𝜕𝑅 the induced orientation and let 𝜃𝑖 be the corner angles of 𝜕𝑅. Then (3.39)

∫ 𝜅𝑆 + ∫ 𝑘𝑔 + ∑ 𝜃𝑖 = 2𝜋𝜒(𝑅). 𝑅

𝜕𝑅

Proof. We divide the region 𝑅 into subregions 𝑅𝑎 , 𝑎 ∈ Λ, conforming a triangulation 𝜏 such that (1) each 𝑅𝑎 is contained in a small open set with orthogonal coordinates, (2) each edge is a regular 𝐶 2 curve, (3) the boundaries 𝜕𝑅𝑎 have corner points at the vertices. We may have extra vertices which are not corner points (in this case we set the corner angle 𝜃𝑖 = 0). For each region 𝑅𝑎 , we apply the local Gauss-Bonnet formula 𝑚𝑎

(3.40)

∫ 𝜅𝑆 + ∫ 𝑅𝑎

𝜕𝑅𝑎

𝑘𝑔 + ∑ 𝜃𝑎,𝑖 = 2𝜋, 𝑖=1

where 𝜕𝑅𝑎 is given the induced orientation as the boundary of 𝑅𝑎 , and 𝜃𝑎,𝑖 are the corner angles of 𝜕𝑅𝑎 and 𝑚𝑎 is the number of corners of each 𝜕𝑅𝑎 . We add up (3.40) for 𝑎 ∈ Λ. We have the following contributions: • Faces, ∑𝑎 ∫𝑅𝑎 𝜅𝑆 = ∫𝑅 𝜅𝑆 .

3.2. Riemannian surfaces and the Gauss-Bonnet theorem

173

• Each interior edge of 𝜏 appears twice with different orientations, so the only surviving contributions of the edges are those of 𝜕𝑅, and thus ∑𝑎 ∫𝜕𝑅𝑎 𝑘𝑔 = ∫𝜕𝑅 𝑘𝑔 . • Let 𝑉 be the set of vertices of 𝜏, and we split 𝑉 = 𝑉𝑒 ∪ 𝑉 𝑖 , where 𝑉𝑒 are the exterior vertices (those in 𝜕𝑅) and 𝑉 𝑖 are the interior ones. We rearrange the sum 𝑒𝑝

𝑚𝑎

𝑒𝑝 −1

∑ ∑ 𝜃𝑎,𝑖 = ∑ ∑ 𝜃𝑝,𝑗 + ∑ ∑ 𝜃𝑝,𝑗 , 𝑎 𝑖=1

𝑝∈𝑉𝑖 𝑗=1

𝑝∈𝑉𝑒 𝑗=1

where 𝑒𝑝 is the number of edges incident in 𝑝, and 𝜃𝑝,𝑗 are the corner angles, as shown in Figure 3.4. Note that an exterior vertex has 𝑒𝑝 − 1 corners.

R θp

Ra βp

p

Figure 3.4. Corner angles and interior angles for the regions 𝑅𝑎 .

For any angle 𝜃𝑝,𝑗 , let 𝛽𝑝,𝑗 be the corresponding interior angle so that 𝜃𝑝,𝑗 + 𝛽𝑝,𝑗 = 𝜋. Then 𝑒𝑝 ♢ If 𝑝 ∈ 𝑉 𝑖 , then ∑𝑗=1 𝛽𝑝,𝑗 = 2𝜋, so 𝑒𝑝

𝑒𝑝

𝑒𝑝

𝑒𝑝

𝑒𝑝 𝜋 = ∑ 𝜋 = ∑ 𝜃𝑝,𝑗 + ∑ 𝛽𝑝,𝑗 = ∑ 𝜃𝑝,𝑗 + 2𝜋 𝑗=1

𝑗=1

𝑗=1

𝑗=1

𝑒

𝑝 and thus ∑𝑗=1 𝜃𝑝,𝑗 = (𝑒𝑝 − 2)𝜋.

𝑒 −1

𝑝 ♢ If 𝑝 ∈ 𝑉𝑒 , then ∑𝑗=1 𝛽𝑝,𝑗 = 𝜋 − 𝜃𝑝 , where 𝜃𝑝 is the corner angle of 𝜕𝑅 at 𝑝. Then

𝑒𝑝 −1

𝑒𝑝 −1

𝑒𝑝 −1

𝑒𝑝 −1

(𝑒𝑝 − 1)𝜋 = ∑ 𝜋 = ∑ 𝜃𝑝,𝑗 + ∑ 𝛽𝑝,𝑗 = ∑ 𝜃𝑝,𝑗 + 𝜋 − 𝜃𝑝 . 𝑗=1 𝑒 −1

𝑗=1

𝑗=1

𝑝 Thus ∑𝑗=1 𝜃𝑝,𝑗 = (𝑒𝑝 − 2)𝜋 + 𝜃𝑝 .

𝑗=1

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3. Riemannian geometry

Putting all together, 𝑒𝑝

𝑚𝑎

𝑒𝑝 −1

∑ ∑ 𝜃𝑎,𝑖 = ∑ ∑ 𝜃𝑝,𝑗 + ∑ ∑ 𝜃𝑝,𝑗 𝑎 𝑖=0

𝑝∈𝑉𝑖 𝑗=1

𝑝∈𝑉𝑒 𝑗=1

= ∑ (𝑒𝑝 − 2)𝜋 + ∑ ((𝑒𝑝 − 2)𝜋 + 𝜃𝑝 ) 𝑝∈𝑉𝑖

𝑝∈𝑉𝑒

= −2𝜋𝑣 + 𝜋 ( ∑ 𝑒𝑝 + ∑ 𝑒𝑝 ) + ∑ 𝜃𝑝 = −2𝜋𝑣 + 2𝜋𝑒 + ∑ 𝜃𝑝 , 𝑝∈𝑉𝑖

𝑝∈𝑉𝑒

𝑝∈𝑉𝑒

𝑝∈𝑉𝑒

where 𝑣 = |𝑉| and 𝑒 are the number of vertices and edges of the triangulation. Observe that ∑𝑝∈𝑉 𝑒𝑝 + ∑𝑝∈𝑉 𝑒𝑝 = 2𝑒 since each edge is incident exactly to 𝑖 𝑒 two vertices. Therefore, taking the sum over all 𝑎 ∈ Λ in (3.40), we have ∫ 𝜅𝑆 + ∫ 𝑘𝑔 − 2𝜋𝑣 + 2𝜋𝑒 + ∑ 𝜃𝑝 = 2𝜋𝑓, 𝑅

𝜕𝑅

𝑝∈𝑉𝑒

where 𝑓 = |Λ| is the number of faces. This implies that ∫ 𝜅𝑆 + ∫ 𝑘𝑔 ∑ 𝜃𝑝 = 2𝜋𝑓 − 2𝜋𝑒 + 2𝜋𝑣 = 2𝜋𝜒(𝑅). 𝑅

𝜕𝑅

□

𝑝∈𝑉𝑒

Remark 3.69. Note that in the left hand side of (3.39) there are contributions of the faces, edges, and vertices, through the curvature, geodesic curvature, and angles (which are 2-dimensional, 1-dimensional, and 0-dimensional phenomena, respectively). The right hand side is a topological quantity with 2-dimensional, 1-dimensional, and 0dimensional contributions. Remark 3.70. We can extract some important consequences of Theorem 3.68. (1) For a compact oriented surface without boundary, we can take 𝑅 = 𝑆. Then 𝜕𝑅 = ∅, and there is no contribution from 𝜕𝑅 nor any corner angles. Then 2𝜋𝜒(𝑆) = ∫𝑆 𝜅. (2) Theorem 3.68 says that ∫𝑆 𝜅 is a topological quantity. We may change the metric of 𝑆 so that we increase the curvature in some region, but then it has to be compensated by a decrease in other region.

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175

(3) Let us take the genus 𝑔 orientable connected compact surface 𝑆 = Σ𝑔 , with some Riemannian metric g, and take 𝑅 = 𝑆. In that case, since 𝜕𝑅 = ∅, we have ∫ 𝜅𝑆 = 2𝜋𝜒(Σ𝑔 ) = 2𝜋(2 − 2𝑔). 𝑆

In particular, • If 𝑆 admits a metric with 𝜅𝑆 > 0, then 𝜒(𝑆) > 0 and so 𝑆 is homeomorphic to the sphere 𝑆 2 . • If 𝑆 admits a metric with 𝜅𝑆 ≡ 0, then 𝜒(𝑆) = 0 and so 𝑆 is homeomorphic to the torus 𝑇 2 . • If 𝑆 admits a metric with 𝜅𝑆 < 0, then 𝜒(𝑆) < 0 and so 𝑆 has genus 𝑔 ≥ 2. These are surprisingly exclusive cases. For instance, it follows that a compact connected surface 𝑆 cannot admit one metric with positive curvature and another one with negative curvature! Of course, this does not guarantee the existence of such metrics. Note that the typical situation is that a metric has points with 𝜅𝑆 > 0, points with 𝜅𝑆 = 0, and points with 𝜅𝑆 < 0 (Exercise 3.22). (4) Let 𝑆 be a non-orientable connected surface with a Riemannian metric. If 𝜅𝑆 > 0, then 𝑆 ≅ 𝑋1 , if 𝜅𝑆 ≡ 0, then 𝑆 ≅ 𝑋2 , and if 𝜅𝑆 < 0, then 𝑆 ≅ 𝑋𝑘 with 𝑘 ≥ 3. To prove this, take the oriented cover 𝜋̂ ∶ 𝑆 ̂ → 𝑆 and apply the previous items to 𝑆.̂ (5) If a compact connected oriented surface Σ𝑔 admits a metric 𝑔 of constant curvature 𝜅𝑆 ≡ 𝑘0 , 𝑘0 ∈ ℝ, then (3.41)

2𝜋(2 − 2𝑔) = ∫ 𝑘0 = 𝑘0 area(𝑆). 𝑆

Therefore, we have a topological restriction to the existence of a metric of constant curvature depending on the genus: • If 𝑔 = 0, then 𝑆 2 = Σ0 can only admit metrics of positive constant curvature. • If 𝑔 = 1, then 𝑇 2 = Σ1 can only admit metrics of zero constant curvature. • If 𝑔 ≥ 2, then Σ𝑔 can only admit metrics of negative constant curvature. Again, this does not say that such metrics do exist. This will be proved in Theorem 4.82. Moreover, the total area of 𝑆 is fixed by the constant 𝑘0 for 𝑔 ≠ 1 but not for 𝑔 = 1. Alternatively, the value of the curvature 𝑘0 is fixed by the area of 𝑆 for 𝑔 ≠ 1. In particular, if the area of 𝑆 increases, then the curvature has to decrease (in absolute value). Remark 3.71. The Gauss-Bonnet theorem is just the first example of a series of results relating metric properties of manifolds with purely topological ones. This is studied in the area of global analysis. Let us mention some results for general Riemannian manifolds (see [DC2]): • Bonnet-Myers theorem: let 𝑀 be a complete connected Riemannian manifold whose Ricci tensor (Remark 3.46(1)) satisfies Ric𝑝 (𝑣) ≥ 𝛿, for some 𝛿 > 0, for all 𝑝 ∈ 𝑀 and 𝑣 ∈ 𝑇𝑝 𝑀 unitary. Then 𝑀 is compact.

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As a corollary, 𝜋1 (𝑀) is finite (since the universal cover 𝑀̃ → 𝑀 will be also compact by the same reasons). • Synge theorem: let 𝑀 be a compact connected Riemannian manifold of dimension 𝑛 with sectional curvature 𝐾 > 0 (i.e., 𝐾𝑝 (𝜎) > 0 for all 𝑝 ∈ 𝑀 and all planes 𝜎 ⊂ 𝑇𝑝 𝑀). If 𝑛 is odd, then 𝑀 is orientable, and if 𝑛 is even and 𝑀 is orientable, then 𝑀 is simply connected. • Hadamard theorem: if a complete simply connected Riemannian manifold 𝑀 of dimension 𝑛 has sectional curvature 𝐾 ≤ 0, then it is diffeomorphic to ℝ𝑛 . Actually for any 𝑝 ∈ 𝑀, exp𝑝 ∶ ℝ𝑛 → 𝑀 is a diffeomorphism. As a corollary, if 𝑀 is compact connected and has sectional curvature 𝐾 ≤ 0 then 𝜋1 (𝑀) is infinite and 𝜋𝑘 (𝑀) = 0 for 𝑘 ≥ 2 (by Corollary 2.37). • Preissman theorem: if 𝑀 is a compact connected Riemannian manifold with 𝐾 < 0 and 𝐴 ⊂ 𝜋1 (𝑀) is an Abelian subgroup, then 𝐴 is infinite cyclic. In particular, 𝜋1 (𝑀) cannot be Abelian (Exercise 3.23). As a corollary, (𝑛)

𝑇 𝑛 = 𝑆 1 × ⋯ ×𝑆 1 cannot admit a metric with 𝐾 < 0. 3.2.4. Orbifolds. In this section we introduce a larger class of spaces called orbifolds. These generalize manifolds by allowing a special type of singular points. Orbifolds will permit us to give a unified treatment of some results in this book, and they appear naturally in the study of smooth manifolds (see [Thu]). Definition 3.72. A topological orbifold (or just an orbifold) is a Hausdorff, second countable, topological space 𝑋 such that every point 𝑝 ∈ 𝑋 has a neighbourhood 𝑈𝑝 ⊂ 𝑋 and a surjective continuous map 𝜑𝑝 ∶ 𝑈𝑝̃ → 𝑈𝑝 , where 𝑈𝑝̃ ⊂ ℝ𝑛 is an open subset, 0 ∈ 𝑈𝑝̃ , 𝜑𝑝 (0) = 𝑝, and there is a finite group Γ𝑝 ⊂ GL(𝑛, ℝ) acting on 𝑈𝑝̃ such that 𝜑𝑝 induces a homeomorphism 𝑈𝑝̃ /Γ𝑝 ≅ 𝑈𝑝 . We say that (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ) is an orbifold chart at 𝑝. We say that 𝑛 is the dimension of the orbifold 𝑋, and that 𝑋 is an 𝑛-orbifold. If 𝑋 is an 𝑛-orbifold, we call 𝑝 ∈ 𝑋 a smooth point if Γ𝑝 = {Id}, in which case 𝑈𝑝 ≅ 𝑈𝑝̃ ⊂ ℝ𝑛 is a usual manifold chart. Otherwise 𝑝 is called a singular point. The index of 𝑝 ∈ 𝑋 is 𝑠𝑝 = |Γ𝑝 |. The set of singular points is denoted 𝑆 = 𝑆𝑋 = {𝑝 ∈ 𝑋|𝑠𝑝 > 1}. It is a closed subset, and it is called the singular locus. Observe that 𝑋 − 𝑆 is a topological 𝑛-manifold. Example 3.73. If Γ is a finite group acting on an 𝑛-manifold 𝑀 and the action is locally linearizable (Definition 2.62), then 𝑋 = 𝑀/Γ is an 𝑛-orbifold. For any 𝑝 ∈ 𝑋, pick 𝑝 ̃ ∈ 𝑀 with 𝜋(𝑝)̃ = 𝑝, where 𝜋 ∶ 𝑀 → 𝑋 is the projection, and let Γ𝑝̃ = {𝑔 ∈ Γ| 𝑔 𝑝 ̃ = 𝑝}̃ be the isotropy group of 𝑝.̃ As the action is locally linearizable, there is some neighbourhood 𝑉𝑝̃ ⊂ 𝑀, a chart 𝜓 ∶ 𝑉𝑝̃ → 𝑉 ̃ ⊂ ℝ𝑛 with 𝜓(𝑝)̃ = 0, and an embedding 𝚤 ∶ Γ𝑝̃ ↪ GL(𝑛, ℝ) such that 𝚤(𝑔) ∘ 𝜓 = 𝜓 ∘ 𝑔, for all 𝑔 ∈ Γ𝑝̃ . Then (𝜋(𝑉𝑝̃ ), 𝑉,̃ Γ𝑝̃ , 𝜋 ∘ 𝜓−1 ) is an orbifold chart for 𝑋 at 𝑝. Remark 3.74. Let 𝑝 ∈ 𝑋, and let (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ) be an orbifold chart. Take a scalar 1 product 𝑄 on ℝ𝑛 and make the average 𝑄0 = |Γ | ∑𝑔∈Γ 𝑔∗ 𝑄. Then 𝑄0 is a scalar 𝑝 𝑝 product on ℝ𝑛 and all 𝑔 ∈ Γ𝑝 satisfy 𝑔∗ 𝑄0 = 𝑄0 . This means that we can arrange that Γ𝑝 ⊂ O(𝑛).

3.2. Riemannian surfaces and the Gauss-Bonnet theorem

177

In the situation above, by considering a ball 𝐵𝜀 (0) ⊂ 𝑈𝑝̃ , we may suppose that the chart has 𝑈𝑝̃ = 𝐵𝜖 (0). For any 𝑞 ∈ 𝑈𝑝 , take 𝑞 ̃ ∈ 𝑈𝑝̃ with 𝑞 = 𝜑𝑝 (𝑞). ̃ Consider Γ𝑞 = {𝑔 ∈ Γ𝑝 | 𝑔(𝑞)̃ = 𝑞}̃ < Γ𝑝 , and let 𝑓1 = Id, 𝑓2 , . . . , 𝑓𝑘 ∈ Γ𝑝 be representatives of the classes of Γ𝑝 /Γ𝑞 , where 𝑘 = |Γ𝑝 ∶ Γ𝑞 |. Then there is some small 𝛿 > 0 so that 𝑓𝑖 (𝐵𝛿 (𝑞)) ̃ are disjoint and Γ𝑞 acts on 𝐵𝛿 (𝑞). ̃ Taking 𝑈𝑞 = 𝜑𝑝 (𝐵𝛿 (𝑞)), ̃ 𝑈𝑞̃ = 𝐵𝛿 (0), and ̃ 𝜑𝑞 = 𝜑𝑝 |𝐵𝛿 (𝑞)̃ − 𝜑𝑝 (𝑞), ̃ we have a chart (𝑈𝑞 , 𝑈𝑞 , Γ𝑞 , 𝜑𝑞 ), called the induced chart at 𝑞 by the chart at 𝑝. An orbifold atlas (or just an atlas) for 𝑋 is a collection of charts 𝒜 = {(𝑈𝛼 , 𝑈𝛼̃ , Γ𝛼 , 𝜑𝛼 )} such that {𝑈𝛼 } covers 𝑋, which includes all orbifold charts (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ) as above, and for every 𝑞 ∈ 𝑈𝑝 , it also includes the induced chart. Definition 3.75. A differentiable orbifold is an orbifold 𝑋 with an atlas 𝒜 such that if (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ) and (𝑈𝑝′ , 𝑈𝑝̃ ′ , Γ𝑝′ , 𝜑𝑝′ ) are two charts at 𝑝 ∈ 𝑋, then there is a diffeomorphism 𝐹 ∶ 𝜑𝑝−1 (𝑈𝑝 ∩ 𝑈𝑝′ ) → (𝜑𝑝′ )−1 (𝑈𝑝 ∩ 𝑈𝑝′ ) and a group isomorphism 𝛾 ∶ Γ𝑝 → Γ𝑝′ such that 𝐹(𝑔 𝑥) = 𝛾(𝑔)𝐹(𝑥), for all 𝑔 ∈ Γ𝑝 . This means that 𝐹 commutes with the orbifold charts. In particular 𝜑𝑝′ (𝐹(𝑥)) = 𝜑𝑝 (𝑥). Here 𝐹 is called the change of charts. Remark 3.76. A differentiable orbifold can be defined by a sheaf of orbifold differentiable functions S as in Definition 1.15. For this, we declare S(𝑈𝑝 ) = {𝑓 ∈ 𝐶 ∞ (𝑈𝑝̃ ) | 𝑓(𝑔 𝑥) = 𝑓(𝑥), for all 𝑔 ∈ Γ𝑝 }. All the theory of smooth manifolds can be translated to this more general context. For instance, an orbifold tensor consists on giving a tensor 𝑇𝑝 on 𝑈𝑝̃ which is Γ𝑝 -invariant, i.e., 𝑔∗ 𝑇𝑝 = 𝑇𝑝 , for all 𝑔 ∈ Γ𝑝 , for each chart (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ) ∈ 𝒜, where 𝑔 is seen as a diffeomorphism 𝑔 ∶ 𝑈𝑝̃ → 𝑈𝑝̃ . They have to be compatible in the sense that 𝑇𝑞 = 𝑇𝑝 |𝑈𝑞 for an induced chart 𝑈𝑞 ⊂ 𝑈𝑝 , and 𝐹 ∗ 𝑇𝑝′ = 𝑇𝑝 for any two charts at the same point 𝑝 as in Definition 3.75. We shall focus on the case of orbisurfaces, i.e., 2-orbifolds. An orbisurface 𝑋 is + said to be oriented if the groups Γ𝑝 ⊂ GL (2, ℝ) and also 𝑋 − 𝑆 is an oriented surface. Suppose that 𝑋 is an oriented orbisurface, and let 𝑝 ∈ 𝑋. By Remark 3.74, we can suppose that Γ𝑝 < SO(2) is a finite group acting by positive isometries, so Γ𝑝 is cyclic of some order 𝑠 = 𝑠𝑝 . Hence it consists of the rotations 𝜌2𝜋𝑘/𝑠 , 𝑘 = 0, . . . , 𝑠 − 1, where 𝜃 − sin 𝜃 denotes the rotation of angle 𝜃. Therefore any 𝑝 ∈ 𝑆 is isolated, 𝜌𝜃 = ( cos sin 𝜃 cos 𝜃 ) thus 𝑆 is a discrete collection of points and 𝑋 − 𝑆 is connected. In particular, if 𝑋 is compact, then 𝑆 is finite. Lemma 3.77. An oriented orbisurface 𝑋 is equivalent to a topological oriented surface 𝑋 with a discrete set of points 𝑆 and integer numbers 𝑠𝑝 > 1 for each 𝑝 ∈ 𝑆. Proof. This is similar to the argument in Example 2.63. Let 𝑋 be an oriented orbisurface and take 𝑝 ∈ 𝑋. If 𝑝 is a smooth point, there is nothing to do. For 𝑝 ∈ 𝑆, take a chart (𝑈𝑝 , 𝑈𝑝̃ = 𝐵𝜖 (0), Γ𝑝 = ⟨𝜌2𝜋/𝑠 ⟩, 𝜑𝑝 ), where 𝑠 = 𝑠𝑝 . Then there are homeomorphisms 𝐵𝜖 (0)/Γ𝑝 ≅ 𝑈𝑝 and 𝜛𝑠 ∶ 𝐵𝜖 (0)/Γ𝑝 → 𝐵𝜖 (0), 𝜛𝑠 (𝑟𝑒i𝜃 ) = 𝑟𝑒i𝑠𝜃 , giving a chart for 𝑋 at 𝑝, in the manifold sense. Conversely, if 𝑋 is topological surface with a discrete subset of points 𝑆 ⊂ 𝑋, then we construct the orbifold chart around a point 𝑝 ∈ 𝑆 by taking a neighbourhood 𝑈𝑝 ⊂ 𝑋

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such that 𝑈𝑝 ∩ 𝑆 = {𝑝}, with a homeomorphism 𝜓 ∶ 𝑈𝑝 → 𝐵𝜖 (0) ⊂ ℝ2 . We take a chart (𝑈𝑝 , 𝑈𝑝̃ = 𝐵𝜖 (0), Γ𝑝 = ⟨𝜌2𝜋/𝑠 ⟩, 𝜑𝑝 ), 𝑠 = 𝑠𝑝 , with 𝜑𝑝 ∶ 𝐵𝜀 (0) → 𝑈𝑝 , given by 𝜑𝑝 (𝑟𝑒i𝜃 ) = 𝜓−1 (𝑟𝑒i𝑠𝜃 ). This chart is compatible with the (manifold) charts for 𝑋 − 𝑆. □ Definition 3.78. Let 𝑋 be a compact oriented orbisurface. Take a triangulation of 𝑋 such that all points of 𝑆 are among the vertices. We define the orbifold Euler-Poincaré characteristic of 𝑋 as 1 𝜒orb (𝑋) = ∑ − 𝑒 + 𝑓, 𝑠 𝑝∈𝑉 𝑝 where 𝑉 is the set of vertices, 𝑒 the number of edges and 𝑓 the number of faces. Recalling that 𝑠𝑝 = 1 for a smooth point 𝑝, we have that ∑ 𝑝∈𝑉

1 1 1 = 𝑣 − ∑ (1 − ) = 𝑣 − ∑ (1 − ) , 𝑠𝑝 𝑠 𝑠 𝑝 𝑝 𝑝∈𝑉 𝑝∈𝑆

where 𝑣 = |𝑉| is the number of vertices. Hence (3.42)

𝜒orb (𝑋) = 𝜒(𝑋) − ∑ (1 − 𝑝∈𝑆

1 ). 𝑠𝑝

The formula for 𝜒orb (𝑋) in Definition 3.78 has the following explanation. At any local chart (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ), every edge or face in 𝑈𝑝 gives 𝑠𝑝 edges or faces in 𝑈𝑝̃ . If we make the count of 𝑣, 𝑒, 𝑓 at 𝑈𝑝̃ , we have to divide by 𝑠𝑝 . But then the central point, which is a vertex, should contribute by 1/𝑠𝑝 . An orbifold cover 𝑓 ∶ 𝑋 ′ → 𝑋 between oriented orbisurfaces 𝑋 ′ , 𝑋 is a continuous surjective map such that for each 𝑞 ∈ 𝑋 there is a neighbourhood 𝑈𝑞 with 𝑓−1 (𝑈𝑞 ) = ⨆𝑖∈𝐼 𝑉 𝑖 , and orbifolds charts fitting into a commutative diagram (3.43)

𝑉 ̃ 𝑖 = 𝐵𝜖 (0)

Id

𝜑𝑝

 𝑉 𝑖 = 𝐵𝜖 (0)/Γ𝑝

/ 𝑈𝑞̃ = 𝐵𝜖 (0) 𝜑𝑞

𝑓

 / 𝑈𝑞 = 𝐵𝜖 (0)/Γ𝑞

where {𝑝} = 𝑓−1 (𝑞) ∩ 𝑉 𝑖 , Γ𝑝 = ⟨𝜌2𝜋/𝑠′𝑝 ⟩ and Γ𝑞 = ⟨𝜌2𝜋/𝑠𝑞 ⟩. Using the homeomorphisms 𝜛𝑠′𝑝 ∶ 𝐵𝜖 (0)/Γ𝑝 → 𝐵𝜖 (0) and 𝜛𝑠𝑞 ∶ 𝐵𝜖 (0)/Γ𝑞 → 𝐵𝜖 (0) of Lemma 3.77, 𝑓 becomes 𝑓′ = 𝑠

𝜛𝑠𝑞 ∘ 𝑓 ∘ (𝜛𝑠′𝑝 )−1 , given by 𝑓′ (𝑟𝑒i𝜃 ) = 𝑟𝑒i𝑚𝑝 𝜃 , where 𝑚𝑝 = 𝑠′𝑞 . In particular, 𝑓 is a 𝑝 ramified cover (Definition 2.60), with ramification locus inside the singular locus. We declare the degree of 𝑓 as its degree as ramified cover. Proposition 3.79. Let 𝑓 ∶ 𝑋 ′ → 𝑋 be an orbifold cover between compact oriented orbisurfaces of degree 𝑑. Then 𝜒orb (𝑋 ′ ) = 𝑑 𝜒orb (𝑋). Proof. By (2.4), we have 𝜒(𝑋 ′ ) = 𝑑𝜒(𝑋) − ∑𝑝∈𝑅 (𝑚𝑝 − 1), where 𝑅 ⊂ 𝑋 ′ is the set of ramification points. We can add points to 𝑅 since non-ramification points contribute with 0, so we can take 𝑅 = 𝑓−1 (𝑆 ′ ∪ 𝑓(𝑆)), where 𝑆 ⊂ 𝑋, 𝑆 ′ ⊂ 𝑋 ′ are the singular points

3.2. Riemannian surfaces and the Gauss-Bonnet theorem

of both orbifolds. For any point 𝑞 ∈ 𝑓(𝑅), we have that ∑𝑓(𝑝)=𝑞 𝑚𝑝 = ∑𝑓(𝑝)=𝑞 Then 𝜒orb (𝑋 ′ ) = 𝜒(𝑋 ′ ) − ∑ (1 − 𝑝∈𝑅′

= 𝑑𝜒(𝑋) − ∑

𝑠𝑞 𝑠′𝑝

= 𝑑.

1 1 ) = 𝑑𝜒(𝑋) − ∑ (𝑚𝑝 − 1) − ∑ (1 − ′ ) 𝑠𝑝′ 𝑠 𝑝 ′ ′ 𝑝∈𝑅 𝑝∈𝑅 (

𝑝∈𝑅′ ,𝑓(𝑝)=𝑞

179

𝑠𝑞 𝑠𝑞 1 1 ∑ ′ (1 − ) ′ − ′ ) = 𝑑𝜒(𝑋) − ∑ 𝑠 𝑠𝑝 𝑠𝑝 𝑠 𝑝 𝑞 𝑞∈𝑓(𝑅) 𝑓(𝑝)=𝑞

= 𝑑𝜒(𝑋) − ∑ 𝑑 (1 − 𝑞∈𝑓(𝑅)

1 ) = 𝑑 𝜒orb (𝑋) . 𝑠𝑞

□

Remark 3.80. Let 𝑆 be a compact surface. In particular 𝑆 is an orbisurface with no singular points and 𝜒orb (𝑆) = 𝜒(𝑆). Let Γ be a finite group acting continuously on 𝑆 1 with locally linearizable action. Then 𝑋 = 𝑆/Γ is an orbifold and 𝜒orb (𝑋) = 𝑑 𝜒(𝑆), where 𝑑 = |Γ|. Riemannian orbisurfaces. Let 𝑋 be a differentiable orbifold. A Riemannian orbifold metric consists of giving a metric g𝑝 on 𝑈𝑝̃ which is Γ𝑝 -invariant, for each chart (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ). Let (𝑋, g) be a Riemannian oriented orbisurface, i.e., an oriented differentiable orbisurface with an orbifold Riemannian metric. For a singular point 𝑝 ∈ 𝑆 with 𝑠 = 𝑠𝑝 , take a circle 𝐶𝑟 (𝑝) ⊂ 𝑈𝑝 of small radius 𝑟 > 0 centered at 𝑝. This is the image of a (geodesic) circle 𝐶𝑟̃ (0) ⊂ 𝑈𝑝̃ = 𝐵𝜖 (0) under the quotient map 𝜑𝑝 ∶ 𝑈𝑝̃ = 𝐵𝜖 (0) → 𝐵𝜖 (0)/Γ𝑝 , where Γ𝑝 = ⟨𝜌2𝜋/𝑠 ⟩. As the length ℓ(𝐶𝑟̃ (0)) ≈ 2𝜋𝑟, and 𝜋 is a degree 𝑠 map, we have that ℓ(𝐶𝑟 (𝑝)) ≈ 2𝜋𝑟/𝑠. In this way we see geometrically the index of the singular point. Let (𝑆, g) be an oriented Riemannian surface and let Γ < Isom(𝑆, g) be a finite group preserving the orientation of 𝑆. For each 𝑝 ∈ 𝑆, we consider Γ𝑝 = {𝑓 ∈ Γ|𝑓(𝑝) = 𝑝} and 𝑠𝑝 = |Γ𝑝 |. For 𝑓 ∈ Γ𝑝 , there is a commutative diagram (see (3.45)) 𝐵𝜀 (0) ⊂ 𝑇𝑝 𝑆

𝑑𝑝 𝑓

exp𝑝

 𝐵𝜀𝑑 (𝑝)

/ 𝐵𝜀 (0) ⊂ 𝑇𝑝 𝑆 exp𝑝

𝑓

 / 𝐵𝜀𝑑 (𝑝).

For 𝜀 > 0 small, exp𝑝 is a diffeomorphism, and 𝑑𝑝 𝑓 is a linear isometry. So 𝑓 is linearizable at 𝑝. This implies that the action is locally linearizable, and hence 𝑋 = 𝑆/Γ is an oriented Riemannian orbisurface (cf. Remark 3.80). Remark 3.81. The above is true if Γ is infinite but acts properly, since in this case the isotropy groups are finite (see Remark 2.65). For a Riemannian oriented orbisurface there is an orbifold volume form 𝜈 (Exercise 3.27) that is written in orbifold oriented charts by the same local formula as for surfaces (3.16). If the orbisurface is compact, then there is a well defined integral of orbifold functions by integrating outside the singular points, since these are of measure zero. Alternatively, it can be defined at every chart (𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 ) by doing the integral on

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3. Riemannian geometry

𝑈𝑝̃ and dividing by |Γ𝑝 |, i.e., ∫ 𝑓= 𝑈𝑝

1 ∫ (𝑓 ∘ 𝜑𝑝 )𝜈. |Γ𝑝 | 𝑈̃ 𝑝

Theorem 3.82 (Gauss-Bonnet, orbifold case). Let (𝑋, g) be a compact Riemannian oriented orbisurface. Then ∫𝑋 𝜅𝑋 = 2𝜋𝜒orb (𝑋), where 𝜅𝑋 is the Gaussian (orbifold) curvature of 𝑋. Proof. Take a singular point 𝑝 ∈ 𝑆 of index 𝑠 = 𝑠𝑝 , and take a small geodesic circle 𝐶𝑟 (𝑝) around it. As 𝐶𝑟 (𝑝) = 𝜑𝑝 (𝐶𝑟̃ (0)), where 𝐶𝑟̃ (0) ⊂ 𝑈𝑝̃ is a circle of radius 𝑟 > 0, we can apply the local Gauss-Bonnet theorem for 𝑅̃ = 𝐵𝑟 (0) ⊂ 𝑈𝑝̃ , ∫ 𝜅𝑋 + ∫ 𝑅̃

𝐶𝑟̃ (0)

𝑘𝑔 = 2𝜋 .

Here the Gaussian curvature 𝜅𝑋 , which is an orbifold function, becomes a Γ𝑝 -invariant smooth function on 𝑅.̃ 𝑟→0 2𝜋

1

The integral ∫𝑅̃ 𝜅𝑋 is 𝑂(𝑟2 ) for 𝑟 small, hence ∫𝐶𝑟 (𝑝) 𝑘𝑔 = 𝑠 ∫𝐶𝑟̃ (0) 𝑘𝑔 ⟶ 𝑠 . Let 𝑋𝑟𝑜 = 𝑋−⨆𝑝∈𝑆 𝐵𝑟𝑑 (𝑝), whose Euler-Poincaré characteristic is 𝜒(𝑋𝑟𝑜 ) = 𝜒(𝑋)−|𝑆|. Using the Gauss-Bonnet theorem for 𝑋𝑟𝑜 , and taking into account that the induced orientation on 𝐶𝑟 (𝑝) is negative, we obtain ∫ 𝜅𝑋 = lim ∫ 𝜅𝑋 = lim (2𝜋𝜒(𝑋𝑟𝑜 ) + ∑ ∫ 𝑋

𝑟→0

𝑋𝑟𝑜

𝑟→0

𝑘𝑔 )

𝑝∈𝑆 𝐶𝑟 (𝑝)

= 2𝜋𝜒(𝑋) − 2𝜋|𝑆| + ∑ 𝑝∈𝑆

2𝜋 = 2𝜋𝜒orb (𝑋), 𝑠𝑝 □

using (3.42). Remark 3.83. We can rewrite Theorem 3.82 as ∫𝑋 𝜅𝑋 = 2𝜋𝜒(𝑋) − 2𝜋 ∑𝑝∈𝑋 (1 − Take the distribution 𝜅𝑋̂ = 𝜅𝑋 + 2𝜋 ∑𝑝∈𝑋 (1 −

1 )𝛿 , 𝑠𝑝 𝑝

1 ). 𝑠𝑝

where 𝛿𝑝 is the Dirac delta at 𝑝,

then ∫𝑋 𝜅𝑋̂ = 2𝜋𝜒(𝑋). So one may interpret a Riemannian orbisurface as a surface whose curvature concentrates at the singular points. Example 3.84. The pillowcase 𝑋 = 𝑇 2 /ℤ2 of Example 2.66 is an orbifold with four 1 1 points of index 2. By Remark 3.80, 𝜒orb (𝑋) = 2 𝜒(𝑇 2 ) = 0, so 𝜒(𝑋) = 4(1 − 2 ) = 2. Hence 𝑆 is topologically a sphere. This can also be computed observing that the torus 𝑇 2 = ℝ2 /ℤ2 has a metric of curvature 𝜅𝑇 2 ≡ 0 (this is the metric g = 𝑑𝑥2 + 𝑑𝑦2 in coordinates (𝑥, 𝑦) for ℝ2 /ℤ2 , see page 222). The group ℤ2 = ⟨𝑎⟩, 𝑎(𝑥, 𝑦) = (−𝑥, −𝑦), acts on 𝑇 2 by orientation preserving isometries with four fixed points 𝑝, 𝑞, 𝑟, 𝑠. Hence 𝑋 has an orbifold metric with 𝜅𝑋 ≡ 0, and 𝜒orb (𝑋) = 0 by Theorem 3.82. The distributional curvature is 𝜅𝑋̂ = 𝜋(𝛿𝑝 + 𝛿𝑞 + 𝛿𝑟 + 𝛿𝑠 ), and ∫𝑋 𝜅𝑋̂ = 4𝜋.

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181

3.3. Isotropic, symmetric, and homogeneous manifolds The classification of general Riemannian manifolds up to isometry would give a too𝑐𝑜 broad list 𝕃𝐑𝐢𝐞𝐦𝑛 . On the one hand, we proved in Proposition 3.22 that there are plenty of metrics on a differentiable manifold (an infinite dimensional space). On the other hand, there are very few isometries between two Riemannian connected manifolds which are diffeomorphic (Exercise 3.39). This means that the set of Riemannian structures on a given smooth manifold, i.e., the fibers of the map (3.18), is very large. This leads us naturally to restrict the study from general Riemannian metrics to a subset of some fewer metrics satisfying some property, to make the question more manageable, whereas maintaining enough objects. In some situations we will keep the surjectivity of (3.18), such as in the case of constant curvature metrics for surfaces (Theorem 4.82). But this may not happen in some other situations of interest (Remark 3.110). We will restrict ourselves to studying Riemannian manifolds which “look similar” at any point in the manifold. This is relevant from our two main points of view in differential geometry. In physics, it is natural to study spaces in which physics laws are independent of the point of space, thus leading naturally to spaces which have isometries taking any point to any other. In geometry, since the early steps of Euclidean geometry, it is natural to deal with spaces in which “figures” can be moved (i.e., transported isometrically) from one place of space to another. Definition 3.85. Let 𝑀 be a Riemannian manifold. We say that 𝑀 is (1) Homogeneous, if for every 𝑝, 𝑞 ∈ 𝑀, there is an isometry 𝑓 ∶ 𝑀 → 𝑀 such that 𝑓(𝑝) = 𝑞. (2) Symmetric, if for every 𝑝 ∈ 𝑀, there exists an isometry 𝑓 ∶ 𝑀 → 𝑀 such that 𝑓(𝑝) = 𝑝 and 𝑑𝑝 𝑓 = − Id. (3) Isotropic, if for every 𝑝 ∈ 𝑀 and any linear isometry 𝐿 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝑀, there exists an isometry 𝑓 ∶ 𝑀 → 𝑀 such that 𝑓(𝑝) = 𝑝 and 𝑑𝑝 𝑓 = 𝐿. (4) Locally homogeneous, if for every 𝑝, 𝑞 ∈ 𝑀, there exist neighbourhoods 𝑈 𝑝 , 𝑉 𝑞 of 𝑝, 𝑞, respectively, and an isometry 𝑓 ∶ 𝑈 → 𝑉 such that 𝑓(𝑝) = 𝑞. (5) Locally symmetric, if for every 𝑝 ∈ 𝑀, there exists a neighbourhood 𝑈 𝑝 of 𝑝 and an isometry 𝑓 ∶ 𝑈 → 𝑈 such that 𝑓(𝑝) = 𝑝 and 𝑑𝑝 𝑓 = − Id. (6) Locally isotropic, if for every 𝑝 ∈ 𝑀 and any linear isometry 𝐿 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝑀, there exists a neighbourhood 𝑈 𝑝 of 𝑝 and an isometry 𝑓 ∶ 𝑈 → 𝑈 such that 𝑓(𝑝) = 𝑝 and 𝑑𝑝 𝑓 = 𝐿. Remark 3.86. (1) Homogeneous manifolds are spaces which look similar at every point, and they permit the transport of figures from one place to another. (2) Isotropic manifolds are similar when we look in any direction, so the figures can be rotated. We shall see that they are also homogeneous (Corollary 3.89).

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(3) Symmetric manifolds admit central reflections at every point. They are geometrically interesting although not so physically relevant. However, mathematically they are of high importance, since the condition of being symmetric can be expressed in terms of the curvature tensor (see Exercise 3.35). (4) Suppose that 𝑀 is not connected with connected components 𝑀 = ⨆𝜆∈Λ 𝑀𝜆 . Then 𝑀 is homogeneous if and only if all the components 𝑀𝜆 are homogeneous and isometric. On the other hand, 𝑀 is symmetric (resp., isotropic) if and only if all the 𝑀𝜆 are symmetric (resp., isotropic). However, no requirement of the components being isometric is needed. The reason is that we take the isometry on the connected component of 𝑝 with the desired property, and the identity on the other components to construct a global isometry. Also, 𝑀 is locally homogeneous if and only if all the components 𝑀𝜆 are locally homogeneous and the different components admit local isometries, and 𝑀 is locally symmetric (resp., locally isotropic) if and only if all the 𝑀𝜆 are locally symmetric (resp., locally isotropic). (5) The local counterparts that are homogeneous, isotropic, and symmetric may look awkward, but in some sense they are more natural from the physical point of view. We only have physical access to a (small) neighbourhood around us. However, a global isometry 𝑓 ∶ 𝑀 → 𝑀 such that 𝑓(𝑝) = 𝑞 sends a geodesic 𝛾𝑝,𝑣 (𝑡) to a geodesic 𝛾𝑞,𝑣′ (𝑡), 𝑣′ = 𝑑𝑝 𝑓(𝑣), for all 𝑡, thus implying that physical information of any point is detected at 𝑝, regardless of how far away it is. (6) From the geometrical point of view, the local conditions are also natural. Axioms like the fifth postulate of Euclid (which will be addressed in section 4.4) talk about what happens too far away (whether two lines do not intersect is something that has to be checked on the whole extension of both lines). So they are in fact very strong global conditions. This is why in modern geometry, a geometry is defined as a locally homogeneous space (see Remark 4.87). (7) There is also a mathematical reason for dealing with the local versions in Definition 3.85. A proper (connected) open subset 𝑈 ⊊ 𝑀 of a homogeneous (resp., symmetric, isotropic) manifold 𝑀 is only locally homogeneous (resp., locally symmetric, locally isotropic). Also, for a Riemannian cover 𝜋 ∶ 𝑀 ′ → 𝑀 with 𝑀 ′ homogeneous (resp., symmetric, isotropic), the manifold 𝑀 is only locally homogeneous (resp., locally symmetric, locally isotropic). In short, the local conditions work well with local isometries, which are the morphisms 𝑛 of 𝐑𝐢𝐞𝐦 . (8) And last but not least, there are too few isotropic manifolds. In the origins of general relativity, the field equations of Einstein (see Remark 6.58(5)) were applied to the whole universe, where they successfully predicted the big bang. Nowadays, the assumptions include the local isotropy of the universe (which is equivalent to an equidistribution of the galaxies in rough terms). But the first cosmologists assumed connectedness and isotropy and were led to only three possibilities: 3-sphere 𝑆 3 , Euclidean space ℝ3 , or the hyperbolic 3-space

3.3. Isotropic, symmetric, and homogeneous manifolds

183

ℍ3 (they missed ℝ𝑃3 , by the way, see Exercise 4.4). Since the 1980s, cosmologists have returned to the local isotropic condition, which allows quotients of these manifolds, and hence universe may have a non-trivial fundamental group. Here topology enters! 3.3.1. Relation between isotropic, symmetric, and homogeneous properties. The objective of this section is to show that these concepts are strongly related. Actually, we have the following diagram of implications for a connected Riemannian manifold 𝑀. (3.44) + complete, simply connected

qy

+3 Locally isotropic ks

Isotropic

+3 Constant sectional curvature

+ complete, simply connected



 +3 Locally symmetric ks

qy

Symmetric

+3 ∇𝑅 = 0

+ complete, simply connected

 +3 Locally homogeneous

qy



Homogeneous



Complete

Proposition 3.87. 𝑀 homogeneous ⟹ 𝑀 complete. Proof. Let 𝛾 ∶ 𝐼 → 𝑀 be a unitary geodesic, where 𝐼 is the maximal interval of definition for 𝛾. We want to show that 𝐼 = ℝ. Suppose otherwise that 𝐼 is bounded above or below. We examine the case where 𝐼 is bounded above, the other one being analogous. Hence 𝐼 = (𝑎, 𝑏), with 𝑏 < ∞. Fix an accessory point 𝑝0 ∈ 𝑀 and a geodesic ball 𝐵𝜖𝑑 (𝑝0 ) around it. Then the unitary geodesics from 𝑝0 live at least for time 𝜖. Take 1 1 𝑞 = 𝛾(𝑏 − 2 𝜖) and 𝑣 𝑞 = 𝛾′ (𝑏 − 2 𝜖). By the definition of a homogeneous manifold, there exists an isometry 𝑓 ∶ 𝑀 → 𝑀 with 𝑓(𝑝0 ) = 𝑞. Let 𝑣𝑝0 = (𝑑𝑝0 𝑓)−1 (𝑣 𝑞 ) and let 𝛾(𝑡) = 𝑓(exp𝑝0 (𝑡𝑣𝑝0 )) for −𝜖 < 𝑡 < 𝜖. Observe that we have 𝛾(0) ̄ = 𝑓(𝑝0 ) = 𝑞 and 𝛾′̄ (0) = 𝑑𝑝0 𝑓(𝑣𝑝0 ) = 𝑣 𝑞 so, by uniqueness of geodesics, we have that 𝛾(𝑡) ̃ ={

𝛾(𝑡), 1 𝛾(𝑡 ̄ − (𝑏 − 2 𝜖)),

𝑡 ∈ (𝑎, 𝑏), 3 1 𝑡 ∈ (𝑏 − 2 𝜖, 𝑏 + 2 𝜖), 1

is a geodesic extending 𝛾 on the interval (𝑎, 𝑏 + 2 𝜖) ⊋ 𝐼, contradicting that 𝐼 is maximal. □ In the previous proof we do not need 𝑀 to be globally homogeneous. Since the geodesic always lies in a connected component, it is enough to suppose that all the connected components of 𝑀 are homogeneous. For the implication symmetric ⇒ homogeneous, we first need to check a weaker implication.

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Lemma 3.88. 𝑀 symmetric ⟹ 𝑀 complete. Proof. The key point to the proof is that, in a symmetric space, the geodesics can be reflected at any point, giving two geodesic segments of equal length at each side of the centre of reflection. Actually if 𝑝 ∈ 𝑀 and 𝑓 ∶ 𝑀 → 𝑀 is an isometry with 𝑓(𝑝) = 𝑝, 𝑑𝑝 𝑓 = − Id, then 𝑓(𝛾𝑝,𝑣 (𝑡)) = 𝛾𝑝,−𝑣 (𝑡), 𝑣 ∈ 𝑇𝑝 𝑀. Therefore 𝑓(𝛾(𝑡)) = 𝛾(−𝑡), for any geodesic with 𝛾(0) = 𝑝. Let 𝛾 ∶ 𝐼 → 𝑀 be a unitary geodesic, where 𝐼 is the maximal interval of definition. Suppose that 𝐼 ≠ ℝ. As before, we suppose that 𝐼 = (𝑎, 𝑏) with 𝑏 < ∞. Let us take 1 (𝑎 + 𝑏) < 𝑐 < 𝑏 (it also works if 𝑎 = −∞). Set 𝑝 = 𝛾(𝑐), and let 𝑓 ∶ 𝑀 → 𝑀 be an 2 isometry with 𝑓(𝑝) = 𝑝 and 𝑑𝑝 𝑓 = − Id. Let 𝛾(𝑡) ̄ = 𝑓(𝛾(2𝑐 − 𝑡)) for 2𝑐 − 𝑏 < 𝑡 < 2𝑐 − 𝑎. Observe that 𝛾(𝑐) ̄ = 𝑓(𝛾(𝑐)) = 𝛾(𝑐) and 𝛾′̄ (𝑐) = −𝑑𝑝 𝑓(𝛾′ (𝑐)) = 𝛾′ (𝑐), so 𝛾(𝑡) ̃ ={

𝛾(𝑡), 𝛾(𝑡), ̄

𝑡 ∈ (𝑎, 𝑏), 𝑡 ∈ (2𝑐 − 𝑏, 2𝑐 − 𝑎),

is a geodesic extending 𝛾 to the interval (𝑎, 2𝑐 − 𝑎), which is strictly bigger than 𝐼, since 2𝑐 − 𝑎 > 𝑏. This shows that 𝐼 is not maximal and gives a contradiction. □ Corollary 3.89. 𝑀 is symmetric, and all its connected components are isometric ⟹ 𝑀 homogeneous. Proof. Observe that it is enough to prove that each connected component of 𝑀 is homogeneous. Indeed, if 𝑝, 𝑞 ∈ 𝑀 are points in different connected components, let ℎ ∶ 𝑀 → 𝑀 be an isometry with ℎ(𝑝) in the same component as 𝑞 (e.g., take ℎ an isometry between the connected components of 𝑝 and 𝑞 and extend it with the identity to the other components). Hence, if 𝑓 is an isometry with 𝑓(ℎ(𝑝)) = 𝑞, then 𝑓 ∘ ℎ is the desired isometry. For this reason, we can suppose that 𝑀 is connected. Let us pick points 𝑝, 𝑞 ∈ 𝑀 in the same connected component. By Lemma 3.88, 𝑀 is complete, so by Theorem 3.35 there exists a unitary geodesic 𝛾 ∶ [0, 𝑙] → 𝑀 with 𝛾(0) = 𝑝 and 𝛾(𝑙) = 𝑞, where 𝑙 = 𝑑(𝑝, 𝑞). Set 𝑠 = 𝛾(𝑙/2). Since 𝑀 is symmetric, there exists an isometry 𝑓 ∶ 𝑀 → 𝑀 such that 𝑓(𝑠) = 𝑠 and 𝑑𝑠 𝑓 = − Id. By the argument of Lemma 3.88, we have that 𝑓(𝛾(𝑡)) = 𝛾(𝑙 − 𝑡). In particular, 𝑓(𝑝) = 𝑓(𝛾(0)) = 𝛾(𝑙) = 𝑞, as required. □ Remark 3.90. • The implications 𝑀 (locally) isotropic ⟹ 𝑀 (locally) symmetric are trivial since the symmetry property is just the isotropy property with 𝐿 = − Id. • Since the local versions of isotropy, symmetry, and homogeneity are just restrictions of the global versions, it is immediate that the global version implies the local one. • Suppose that M is connected. Since isotropy, symmetry, or homogeneity implies that 𝑀 is complete, we have that if 𝑈 ⊊ 𝑀 is a proper open subset and 𝑀 is isotropic (resp., symmetric, homogeneous), then 𝑈 is locally isotropic but not isotropic (resp., locally symmetric but not symmetric, locally homogeneous but not homogeneous), since 𝑈 cannot be complete (see Remark 3.36).

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185

An argument similar to that of Corollary 3.89 proves the local version. Proposition 3.91. 𝑀 is locally symmetric, and all its connected components are locally isometric (at one point on each component) ⟹ 𝑀 locally homogeneous. Proof. By the asme argument as in Corollary 3.89, we can suppose that 𝑀 is connected. Let us fix 𝑝 ∈ 𝑀, and let 𝐴𝑝 be the subset of the points 𝑞 ∈ 𝑀 such that there exists a local isometry 𝑓 ∶ 𝑈 → 𝑉, with 𝑈 and 𝑉 neighbourhoods of 𝑝 and 𝑞, respectively, such that 𝑓(𝑝) = 𝑞. Let us see that 𝐴𝑝 is open. Take 𝑞 ∈ 𝐴𝑝 and 𝑓 ∶ 𝐵𝜖 (𝑝) → 𝐵𝜖 (𝑞) a local isometry, for some small 𝜖 > 0. For 𝑟 ∈ 𝐵2𝜖/3 (𝑞), take the (unique) geodesic from 𝑞 to 𝑟 and let 𝑠 be its midpoint. Then 𝑑(𝑞, 𝑠) < 𝜖/3 and hence 𝐵2𝜖/3 (𝑠) ⊂ 𝐵𝜖 (𝑞), so the local symmetry with center on 𝑠, as in Corollary 3.89, is defined in such a ball, 𝑔 ∶ 𝐵2𝜖/3 (𝑠) → 𝐵2𝜖/3 (𝑠), and it takes 𝑔(𝑞) = 𝑟. Moreover 𝐵𝜖/3 (𝑞), 𝐵𝜖/3 (𝑟) ⊂ 𝐵2𝜖/3 (𝑠), so 𝑔 ∶ 𝐵𝜖/3 (𝑞) → 𝐵𝜖/3 (𝑟), and hence 𝑟 ∈ 𝐴𝑝 . This proves that 𝐴𝑝 is open. Observe that if 𝐴𝑝 ∩ 𝐴𝑝′ ≠ ∅ for 𝑝, 𝑝′ ∈ 𝑀, then 𝐴𝑝 = 𝐴𝑝′ . So the sets 𝐴𝑝 form a disjoint open covering of 𝑀. As 𝑀 is connected, there can only be one of them. Therefore 𝐴𝑝 = 𝑀, and 𝑀 is locally homogeneous. □ Proposition 3.92. 𝑀 locally isotropic ⟹ 𝑀 has constant sectional curvature on each connected component. Proof. Before starting the proof, recall that given an isometry 𝑓 ∶ 𝑀 → 𝑀 ′ , with 𝑀, 𝑀 ′ Riemannian manifolds, 𝑝 ∈ 𝑀 and 𝜎 ⊂ 𝑇𝑝 𝑀 a plane, we have 𝐾𝑝 (𝜎) = 𝐾𝑓(𝑝) (𝑑𝑝 𝑓(𝜎)). Since the statement is about constant curvature on each connected component, without loss of generality we can suppose that 𝑀 is connected. Fix a point 𝑝 ∈ 𝑀 and let 𝜎1 , 𝜎2 ⊂ 𝑇𝑝 𝑀 be two planes, say 𝜎1 = ⟨𝑢1 , 𝑢2 ⟩ and 𝜎2 = ⟨𝑣 1 , 𝑣 2 ⟩, where (𝑢1 , 𝑢2 ) and (𝑣 1 , 𝑣 2 ) are both orthonormal. Complete them to orthonormal bases (𝑢1 , 𝑢2 , . . . , 𝑢𝑛 ) and (𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 ) of 𝑇𝑝 𝑀, and let 𝐿 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝑀 be the linear isometry with 𝐿(𝑢𝑖 ) = 𝑣 𝑖 for 1 ≤ 𝑖 ≤ 𝑛. Since 𝑀 is locally isotropic, there exist a neighbourhood 𝑈 𝑝 of 𝑝 and an isometry 𝑓 ∶ 𝑈 → 𝑈 with 𝑓(𝑝) = 𝑝 and 𝑑𝑝 𝑓 = 𝐿. Since 𝐿(𝜎1 ) = 𝜎2 , we have 𝐾𝑝 (𝜎1 ) = 𝐾𝑓(𝑝) (𝑑𝑝 𝑓(𝜎1 )) = 𝐾𝑝 (𝐿(𝜎1 )) = 𝐾𝑝 (𝜎2 ). Therefore, the sectional curvatures at a point 𝑝 ∈ 𝑀 are independent of the plane in 𝑇𝑝 𝑀. Denote this value by 𝑘𝑝 ∈ ℝ. It remains to see that 𝑘𝑝 does not depend on 𝑝 ∈ 𝑀. As 𝑀 is locally symmetric and we are assuming that it is connected, by Proposition 3.91, 𝑀 is also locally homogeneous. Thus, given any 𝑝, 𝑞 ∈ 𝑀, there exists an isometry 𝑓 ∶ 𝑈 𝑝 → 𝑉 𝑞 with 𝑓(𝑝) = 𝑞. Then, 𝑘𝑝 = 𝐾𝑝 (𝜎) = 𝐾𝑓(𝑝) (𝑑𝑝 𝑓(𝜎)) = 𝐾𝑞 (𝜎′ ) = 𝑘𝑞 , taking any plane 𝜎 ⊂ 𝑇𝑝 𝑀 and 𝜎′ = 𝑑𝑝 𝑓(𝜎). Hence 𝐾𝑝 (𝜎) = 𝑘0 , for all 𝑝 ∈ 𝑀 and planes 𝜎 ⊂ 𝑇𝑝 𝑀, for a constant 𝑘0 ∈ ℝ. □ In principle, the sectional curvature on a locally isotropic manifold 𝑀 may vary between connected components. However, if we suppose that the connected components of 𝑀 are locally isometric, then the sectional curvature is forced to be globally constant. 3.3.2. Cartan lemma. To proceed further, for the remaining implications in (3.44), we need to construct isometries by starting with a linear isometry at the tangent space. First, we shall construct an isometry on a small (geodesic) ball. For the local analysis

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of an isometry, we start by noticing that, if an isometry exists, then it is determined by the information on a point. To be precise, let 𝑁, 𝑁 ′ be two Riemannian manifolds of the same dimension (they may be the same), and 𝑝 ∈ 𝑁, 𝑝′ ∈ 𝑁 ′ such that there are geodesic neighbourhoods 𝑈 = 𝐵𝜖𝑑 (𝑝), 𝑉 = 𝐵𝜖𝑑 (𝑝′ ), respectively, and an isometry 𝑓 ∶ 𝑈 → 𝑉 with 𝑝′ = 𝑓(𝑝). Then as 𝑓 carries geodesics to geodesics, it must be 𝑓(𝛾𝑝,𝑣 (𝑡)) = 𝛾𝑝′ ,𝑣′ (𝑡), with 𝑣′ = 𝑑𝑝 𝑓(𝑣). That is, 𝑓(exp𝑝 (𝑡𝑣)) = exp𝑝′ (𝑡 𝑑𝑝 𝑓(𝑣)). This means that there is a commutative diagram, (3.45)

𝐵𝜖 (0) ⊂ 𝑇𝑝 𝑁

𝑑𝑝 𝑓

/ 𝐵𝜖 (0) ⊂ 𝑇𝑝′ 𝑁 ′

exp𝑝

 𝐵𝜖𝑑 (𝑝)

exp𝑝′

𝑓

 / 𝐵𝜖𝑑 (𝑝′ )

or, equivalently, 𝑓|𝐵𝜖 (𝑝) = exp𝑝′ ∘ 𝑑𝑝 𝑓 ∘ exp−1 𝑝 .

(3.46)

In particular, 𝑓 is locally determined by the image 𝑓(𝑝) and the differential 𝑑𝑝 𝑓. From a different point of view, given points 𝑝 ∈ 𝑁, 𝑝′ ∈ 𝑁 ′ and a linear isometry 𝐿 ∶ 𝑇𝑝 𝑁 → 𝑇𝑝′ 𝑁 ′ , the only possible candidate to be an isometry 𝑓 ∶ 𝐵𝜖𝑑 (𝑝) → 𝐵𝜖𝑑 (𝑝′ ) with 𝑓(𝑝) = 𝑝′ and 𝑑𝑝 𝑓 = 𝐿 is 𝑓 = exp𝑝′ ∘𝐿 ∘ exp−1 𝑝 , where 𝜖 > 0 is chosen small enough so that 𝐵𝜖𝑑 (𝑝), 𝐵𝜖𝑑 (𝑝′ ) are geodesic balls. However, this does not guarantee that 𝑓 is actually an isometry. Sufficient conditions for 𝑓 to be an isometry are given by the Cartan lemma, whose proof can be found in [DC2]. In order to state it, we introduce some notation. Let 𝑞 = exp𝑝 (𝑣) ∈ 𝐵𝜖𝑑 (𝑝) and its image 𝑞′ = 𝑓(𝑞) = exp𝑝 (𝐿(𝑣)). Let 𝛾(𝑡) = exp𝑝 (𝑡𝑣), 𝛾′ (𝑡) = exp𝑝′ (𝑡 𝐿(𝑣)) = 𝑓(𝛾(𝑡)) be the geodesics from the center 𝑝 to 𝑞 in 𝑁, and from the center 𝑝′ to 𝑞′ in 𝑁 ′ . We define the map (−)′ ∶ 𝑇𝑞 𝑁 (3.47)

𝑥

p

⟶ ↦

𝑇𝑞′ 𝑁 ′ 1,0 𝑥′ = 𝑃𝛾0,1 (𝑥))) ′ (𝐿(𝑃𝛾

p0

γ

q

γ0

q0

L(Pγ1;0 (x))

Pγ1;0 (x) x

1;0 x0 = Pγ0;1 (x))) 0 (L(Pγ

Theorem 3.93 (Cartan lemma). Let (𝑁, g), (𝑁 ′ , g′ ) be Riemannian manifolds of the same dimension, let 𝑝 ∈ 𝑁, 𝑝′ ∈ 𝑁 ′ and 𝐿 ∶ 𝑇𝑝 𝑁 → 𝑇𝑝′ 𝑁 ′ be a linear isometry. 𝑑 𝑑 𝑑 ′ Define 𝑓 = exp𝑝′ ∘𝐿 ∘ exp−1 𝑝 on a geodesic ball 𝐵𝜖 (𝑝). Then 𝑓 ∶ 𝐵𝜖 (𝑝) → 𝐵𝜖 (𝑝 ) is

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187

an isometry if and only if 𝑅𝑞 (𝑥, 𝑦, 𝑧, 𝑤) = 𝑅′𝑞′ (𝑥′ , 𝑦′ , 𝑧′ , 𝑤′ ), for any 𝑞 ∈ 𝐵𝜖𝑑 (𝑝) and any 𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝑇𝑞 𝑁, where 𝑅, 𝑅′ denote the curvature tensors of 𝑁, 𝑁 ′ , respectively. Remark 3.94. • In Theorem 3.93 it is enough to check 𝐾𝑞 (⟨𝑥, 𝑦⟩) = 𝐾𝑞′ (⟨𝑥′ , 𝑦′ ⟩) for all orthonormal pairs {𝑥, 𝑦}. This is due to the fact that the sectional curvature determines uniquely the curvature tensor 𝑅. • The hypothesis of Theorem 3.93 cannot be substituted with the hypothesis that 𝑅𝑞 (𝑥, 𝑦, 𝑧, 𝑤) = 𝑅𝑞′ (𝑓∗ 𝑥, 𝑓∗ 𝑦, 𝑓∗ 𝑧, 𝑓∗ 𝑤), for 𝑞 ∈ 𝐵𝜖𝑑 (𝑝), 𝑥, 𝑦, 𝑧, 𝑤 ∈ 𝑇𝑞 𝑀. For instance, if 𝑀 = (ℝ𝑛 , g𝑠𝑡𝑑 ), then any differentiable map 𝑓 ∶ ℝ𝑛 → ℝ𝑛 satisfies such a hypothesis, although it may be not an isometry. • Otherwise said, the implication 𝑓∗ g′ = g ⟹ 𝑓∗ 𝑅′ = 𝑅 holds, but the converse does not. • Note that the map (3.47) is always a linear isometry. If the hypothesis of Theorem 3.93 holds, then 𝑓 is an isometry and, a fortiori, 𝑓∗ 𝑥 = 𝑥′ , for all vectors 𝑥. Corollary 3.95. 𝑀 has constant sectional curvature on each connected component ⟹ 𝑀 locally isotropic. Proof. Let 𝑝 ∈ 𝑀 and let 𝐿 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝑀 be a linear isometry. Let 𝐵𝜖𝑑 (𝑝) be a geo𝑑 𝑑 desic neighbourhood of 𝑝, and define 𝑓 = exp𝑝 ∘𝐿 ∘ exp−1 𝑝 ∶ 𝐵𝜖 (𝑝) → 𝐵𝜖 (𝑝). Clearly 𝑓(𝑝) = 𝑝 and 𝑑𝑝 𝑓 = 𝐿. To see that 𝑓 is an isometry, we check that it satisfies the hypothesis of Theorem 3.93. Take 𝑞 ∈ 𝐵𝜖𝑑 (𝑝) and set 𝑞′ = 𝑓(𝑞). Given 𝑢, 𝑣 ∈ 𝑇𝑞 𝑀 orthonormal, let 𝑢′ , 𝑣′ ∈ 𝑇𝑞′ 𝑀 be their images under (3.47), which are also orthonormal. Then 𝐾𝑝′ (⟨𝑢, 𝑣⟩) = 𝐾𝑞′ (⟨𝑢′ , 𝑣′ ⟩), since the sectional curvature is constant. Thus Theorem 3.93 applies and 𝑓 is an isometry. □ The same argument gives the following improvement. Corollary 3.96. Let 𝑀1 , 𝑀2 be two Riemannian manifolds, both with the same constant sectional curvature. Take 𝑝1 ∈ 𝑀1 , 𝑝2 ∈ 𝑀2 , and let 𝐿 ∶ 𝑇𝑝1 𝑀1 → 𝑇𝑝2 𝑀2 be a linear isometry. Then there is some 𝜖 > 0 and an isometry 𝑓 ∶ 𝐵𝜖𝑑 (𝑝1 ) → 𝐵𝜖𝑑 (𝑝2 ), with 𝑓(𝑝1 ) = 𝑝2 , 𝑑𝑝1 𝑓 = 𝐿. Remark 3.97. Note that in Corollary 3.96 it is necessary that 𝐵𝜖𝑑 (𝑝1 ) be a geodesic ball (so that exp𝑝1 has an inverse there), but it is only necessary that exp𝑝2 be defined on 𝐵𝜖 (0) (for instance, this is automatic if 𝑀2 is complete). It is a consequence that 𝐵𝜖𝑑 (𝑝2 ) is a geodesic ball. Now we move to the issue of constructing isometries 𝑓 ∶ 𝑁 → 𝑁 ′ globally. We start with a result on uniqueness. Lemma 3.98. Let 𝑁, 𝑁 ′ be Riemannian manifolds, with 𝑁 connected, and let 𝑓1 , 𝑓2 ∶ 𝑁 → 𝑁 ′ be two local isometries. If there exists 𝑝 ∈ 𝑁 such that 𝑓1 (𝑝) = 𝑓2 (𝑝) and 𝑑𝑝 𝑓1 = 𝑑𝑝 𝑓2 , then 𝑓1 = 𝑓2 .

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Proof. Let 𝐴 = {𝑞 ∈ 𝑁 | 𝑓1 (𝑞) = 𝑓2 (𝑞), 𝑑𝑞 𝑓1 = 𝑑𝑞 𝑓2 }. By hypothesis, 𝐴 is non-empty since 𝑝 ∈ 𝐴. Moreover, 𝐴 is a closed set by definition. On the other hand, given 𝑞 ∈ 𝐴 if we take 𝐵𝜖 (𝑞) a geodesic neighbourhood of 𝑞, we know that 𝑓1 |𝐵𝜖 (𝑞) = exp𝑓1 (𝑞) ∘ 𝑑𝑞 𝑓1 ∘ −1 𝑑 exp−1 𝑞 = exp𝑓2 (𝑞) ∘ 𝑑𝑞 𝑓2 ∘ exp𝑞 = 𝑓2 |𝐵𝜖𝑑 (𝑞) by (3.46). So 𝐵𝜖 (𝑞) ⊂ 𝐴, proving that 𝐴 is open. As 𝑁 is connected, we have that 𝐴 = 𝑁, finishing the proof. □ Theorem 3.99. Let 𝑀, 𝑁 be connected Riemannian 𝑛-manifolds with the same constant sectional curvature, and suppose that 𝑀 is simply connected and 𝑁 is complete. Fix 𝑝0 ∈ 𝑀 and 𝑞0 ∈ 𝑁 and a linear isometry 𝐿0 ∶ 𝑇𝑝0 𝑀 → 𝑇𝑞0 𝑁. Then there exists a unique local isometry 𝑓 ∶ 𝑀 → 𝑁 with 𝑓(𝑝0 ) = 𝑞0 and 𝑑𝑝0 𝑓 = 𝐿0 . Furthermore if 𝑀 is complete, then 𝑓 is a Riemannian cover. Proof. Let us define 𝑆 = {(𝑝, 𝑞, 𝐿) | 𝑝 ∈ 𝑀, 𝑞 ∈ 𝑁, 𝐿 ∶ 𝑇𝑝 𝑀 → 𝑇𝑞 𝑁 linear isometry}. Endow 𝑆 with the topology generated by the following sets: let 𝑔 ∶ 𝑈 → 𝑉 be an isometry with 𝑈 ⊂ 𝑀 and 𝑉 ⊂ 𝑁 open sets, then we define 𝑊𝑔 = {(𝑥, 𝑔(𝑥), 𝑑𝑥 𝑔) | 𝑥 ∈ 𝑈}. Let us see that {𝑊𝑔 } is a basis of a topology on 𝑆. If 𝑔1 ∶ 𝑈1 → 𝑉1 and 𝑔2 ∶ 𝑈2 → 𝑉2 satisfy that 𝑊𝑔1 ∩ 𝑊𝑔2 ≠ ∅, then by Lemma 3.98, 𝑔1 |𝑈1 ∩𝑈2 = 𝑔2 |𝑈1 ∩𝑈2 so 𝑊𝑔1 ∩ 𝑊𝑔2 = 𝑊𝑔 , with 𝑔 = 𝑔1 |𝑈1 ∩𝑈2 ∶ 𝑈1 ∩ 𝑈2 → 𝑉1 ∩ 𝑉2 . On the other hand, by Corollary 3.96, for any (𝑝, 𝑞, 𝐿), there are 𝑈 𝑝 , 𝑉 𝑞 and an isometry 𝑔 ∶ 𝑈 → 𝑉 with 𝑔(𝑝) = 𝑞 and 𝑑𝑝 𝑔 = 𝐿, so {𝑊𝑔 } cover 𝑆. This proves that {𝑊𝑔 } is a basis for a topology for 𝑆. Next observe that the map 𝜋1 ∶ 𝑆 → 𝑀, 𝜋1 (𝑝, 𝑞, 𝐿) = 𝑝, is a cover. Certainly, for 𝑝 ∈ 𝑀, take a geodesic ball 𝐵𝜖𝑑 (𝑝). Then 𝜋−1 (𝐵𝜖𝑑 (𝑝)) = ⨆𝑔 𝑊𝑔 , where the union is taken over the isometries 𝑔 ∶ 𝐵𝜖𝑑 (𝑝) → 𝐵𝜖𝑑 (𝑞) for 𝑞 ∈ 𝑁. Here we use that all such isometries 𝑔 are defined over the 𝜖-ball for every 𝑞 (by Remark 3.97) because 𝑁 is complete. Let 𝑆 0 be the connected component containing (𝑝0 , 𝑞0 , 𝐿0 ) in 𝑆. Then 𝜋1 |𝑆0 ∶ 𝑆 0 → 𝑀 is a cover between connected spaces. As 𝑀 is simply connected, we have that 𝜋1 |𝑆0 ∶ 𝑆 0 → 𝑀 is a homeomorphism. We give 𝑆 0 the induced differentiable structure. This produces a map 𝑓 = 𝜋2 ∘ (𝜋1 |𝑆0 )−1 ∶ 𝑀 → 𝑁, where 𝜋2 ∶ 𝑆 → 𝑁, 𝜋2 (𝑝, 𝑞, 𝐿) = 𝑞. The map 𝑓 is a local isometry with 𝑓(𝑝0 ) = 𝑞0 and 𝑑𝑝0 𝑓 = 𝐿0 . If 𝑀 is complete, Exercise 3.32 implies that 𝑓 is a Riemannian cover.

□

Corollary 3.100. Under the hypothesis of Theorem 3.99, if 𝑀 is complete and 𝑁 is also simply connected, then 𝑓 is an isometry. □ Corollary 3.101. 𝑀 is locally isotropic, simply connected, and complete ⟹ 𝑀 is isotropic. Proof. Take 𝑀 = 𝑁 and 𝑝 = 𝑞 in Corollary 3.100.

□

Remark 3.102. • The fact that 𝑀 is locally symmetric ⟺ ∇𝑅 = 0 is in Exercise 3.35. • If 𝑀 is locally symmetric, complete, and simply connected, then 𝑀 is symmetric; see Exercise 3.36.

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189

• It is also true that if 𝑀 is locally homogeneous, complete, and simply connected, then 𝑀 is homogeneous,2 but we shall not prove it here. This completes the proof of (3.44). Now fix a connected Riemannian 𝑛-manifold 𝑀 and denote Isom(𝑀) the group of isometries of 𝑀. We want to analyse Isom(𝑀) in detail. Fix a point 𝑝0 ∈ 𝑀 and identify 𝑇𝑝0 𝑀 = ℝ𝑛 . Define the orthogonal frame bundle 𝑂𝑀 = {(𝑝, 𝐿) | 𝑝 ∈ 𝑀, 𝐿 ∈ O(ℝ𝑛 , 𝑇𝑝 𝑀)} . Let 𝜋 ∶ 𝑂𝑀 → 𝑀 be the projection 𝜋(𝑝, 𝐿) = 𝑝. Its fiber is 𝑂𝑝 𝑀 = O(ℝ𝑛 , 𝑇𝑝 𝑀), the space of linear isometries 𝐿 ∶ ℝ𝑛 → 𝑇𝑝 𝑀, also called frames of 𝑇𝑝 𝑀. This acquires the structure of a smooth manifold in a similar way as the tangent bundle (see page 153). If 𝑈 ⊂ 𝑀 is a coordinate chart, then there is an identification 𝜋−1 (𝑈) ≅ 𝑈 × O(𝑛), where O(𝑛) is the set of linear isometries of ℝ𝑛 , usually called the orthogonal group (see section 4.1.1 for a thorough discussion). Endow 𝑂𝑀 with the topology that makes all these identifications homeomorphisms. With this topology, 𝑂𝑀 is a differentiable 𝑛(𝑛−1) manifold. As dim O(𝑛) = 2 (see (4.2)), we have that 𝑛(𝑛 + 1) . 2 We define the map Ψ ∶ Isom(𝑀) → 𝑂𝑀, Ψ(𝑓) = (𝑓(𝑝0 ), 𝑑𝑝0 𝑓). By Lemma 3.98, this map is injective. It is a fact (that we shall not prove here) that Isom(𝑀) is a Lie group (that is, it is a differentiable manifold and a group, such that the product in the group and the inverse are both differentiable maps). Therefore, Ψ is an immersion, and thus dim 𝑂𝑀 = dim 𝑀 + dim O(𝑛) =

dim Isom(𝑀) ≤

𝑛(𝑛 + 1) . 2

Remark 3.103. • If 𝑀 is a connected and isotropic manifold, then Ψ ∶ Isom(𝑀) → 𝑂𝑀 is a 𝑛(𝑛+1) diffeomorphism and dim Isom(𝑀) = . To see that, let (𝑝, 𝐿) ∈ 𝑂𝑀. 2 Since 𝑀 is in particular homogeneous, there exists an isometry 𝑔 ∶ 𝑀 → 𝑀 with 𝑔(𝑝0 ) = 𝑝. Using that 𝑀 is isotropic, there exists an isometry ℎ ∶ 𝑀 → 𝑀 with ℎ(𝑝0 ) = 𝑝0 and 𝑑𝑝0 ℎ = (𝑑𝑝 𝑔)−1 ∘ 𝐿. Taking 𝑓 = 𝑔 ∘ ℎ, we have that Ψ(𝑓) = (𝑝, 𝐿). • The space 𝑂𝑀 has at most two connected components (since 𝑀 is connected and O(𝑛) has two connected components). Actually, it has one if 𝑀 is nonorientable, and it has two if 𝑀 is orientable. Therefore for an isotropic manifold 𝑀, Isom(𝑀) has two connected components if 𝑀 is orientable, and one if 𝑀 is non-orientable. • It cannot happen that 𝑂𝑀 has two connected components and Isom(𝑀) is exactly (under Ψ) one of them (see Exercise 4.3). • If 𝑀 is a homogeneous manifold, then the composition 𝜋 ∘ Ψ ∶ Isom(𝑀) → 𝑂𝑀 → 𝑀 is surjective, so dim Isom(𝑀) ≥ 𝑛. 2

We thank Marco Castrillón (Universidad Complutense de Madrid) for sharing this information with us.

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Remark 3.104. If 𝑀 is a connected orientable Riemannian manifold, then we can decompose Isom(𝑀) = Isom+ (𝑀) ⊔ Isom− (𝑀), where Isom+ (𝑀) and Isom− (𝑀) are the sets of isometries that preserve and reverse orientation, respectively. As in statement (b) on page 78, if Isom− (𝑀) ≠ ∅, then there is a bijection Isom+ (𝑀) → Isom− (𝑀), defined by composition with any fixed orientation reversing isometry. Remark 3.105. In general, given a Riemannian manifold 𝑀, we have an action of Isom(𝑀) on 𝑀 by 𝑓 ⋅ 𝑝 = 𝑓(𝑝) for 𝑓 ∈ Isom(𝑀) and 𝑝 ∈ 𝑀. In this setting, being a homogeneous Riemannian manifold means that Isom(𝑀) acts transitively. In general, a homogeneous manifold is defined as a manifold 𝑀 with a transitive action of some Lie group 𝐺 on it. See [War] for notions on Lie groups and Lie algebras. In that case, fix 𝑝0 ∈ 𝑀, and let 𝐻 be the isotropy group of 𝑝0 , i.e., 𝐻 = {𝑔 ∈ 𝐺 | 𝑔 ⋅ 𝑝0 = 𝑝0 }. This is a closed Lie subgroup of 𝐺. The map 𝐺 → 𝑀, 𝑔 ↦ 𝑔 ⋅ 𝑝0 is surjective, so it induces a bijection 𝐺/𝐻 ≅ 𝑀. There is a geometric construction of a differentiable structure for the quotient 𝐺/𝐻 (for any Lie group 𝐺 and closed subgroup 𝐻), which can be found in [Hel], and the map 𝐺/𝐻 ≅ 𝑀 turns out to be a diffeomorphism. This is another point of view for homogeneous manifolds coming from the theory of Lie groups. Symmetric spaces can also be regarded from this point of view, since they are homogeneous manifolds 𝑀 which have an isometry 𝑓 ∶ 𝑀 → 𝑀, with 𝑓(𝑝0 ) = 𝑝0 , 𝑑𝑝0 𝑓 = − Id (the central reflections at other points are obtained by the homogeneity of 𝑀). If we write 𝑀 = 𝐺/𝐻 and consider the Lie algebras 𝔤, 𝔥 of 𝐺, 𝐻, respectively, then we have a decomposition 𝔤 = 𝔥 ⊕ 𝔪 with 𝔪 = 𝔤/𝔥 ≅ 𝑇𝑝0 𝑀 in a natural way, and 𝑓 induces a map 𝜎 ∶ 𝔤 → 𝔤 with 𝜎|𝔥 = Id𝔥 and 𝜎|𝔪 = − Id𝔪 . 3.3.3. Space forms. To finish this chapter, let us introduce some terminology previous to the detailed study of the Riemannian manifolds with constant sectional curvature that we shall carry in Chapter 4. Definition 3.106. A space form is a complete connected Riemannian manifold with constant sectional curvature. Remark 3.107. • The terminology comes from cosmology. In the early times of general relativity, it was assumed in cosmological models that the universe must be a complete, locally isotropic, connected Riemannian manifold. For this reason, these spaces received the name of space forms, as they were the possible forms of “the space” (the universe). See Remark 3.86(8). • If the universe is assumed to be isotropic, then (with one exception) it is a simply connected space form (Exercise 4.4). Theorem 3.108. Let 𝑀1 and 𝑀2 be simply connected space forms of the dimension and the same constant sectional curvature. Then 𝑀1 and 𝑀2 are isometric. Proof. Pick any points 𝑝 ∈ 𝑀1 and 𝑞 ∈ 𝑀2 and any linear isometry 𝐿 ∶ 𝑇𝑝 𝑀1 → 𝑇𝑞 𝑀2 . Then apply Corollary 3.100. □

3.3. Isotropic, symmetric, and homogeneous manifolds

191

This proposition implies that, given 𝑘0 ∈ ℝ, there exists, up to isometry, at most one 𝑛-dimensional simply connected space form with curvature 𝑘0 , which we will denote by 𝐸𝑘𝑛0 . In Chapter 4 we will construct such space forms for every 𝑘0 ∈ ℝ, proving that actually there exists one and only one. Theorem 3.109. There exists a unique (up to isometry) simply connected space form of dimension 𝑛 and constant curvature 𝑘0 , for 𝑛 ≥ 2, 𝑘0 ∈ ℝ. We shall also see that we may rescale the metric so that we get spaces with a metric of curvature 𝑘0 ∈ {1, 0, −1}. These simply connected space forms receive the following names: • For 𝑘0 = 0, Euclidean space. • For 𝑘0 = 1, sphere (it is the only compact, simply connected space form). • For 𝑘0 = −1, hyperbolic space. The reason behind these names hinges also on the fact that the classical Euclidean, spherical (or projective), and hyperbolic geometries are defined on these space forms. We shall see this in detail in Chapter 4. Now, let (𝑀, g) be any complete connected Riemannian 𝑛-manifold of constant sectional curvature 𝑘0 . Let 𝜋 ∶ 𝑀̃ → 𝑀 be its universal cover. Endow 𝑀̃ with the pull-back metric 𝜋∗ g, so that 𝜋 is a Riemannian cover. By Exercise 3.31, 𝑀̃ is also complete. So 𝑀̃ is a complete, simply connected Riemannian manifold of constant sectional curvature 𝑘0 . By Theorem 3.108, 𝑀̃ is isometric to 𝐸𝑘𝑛0 . Thus we have a cover 𝜋 ∶ 𝐸𝑘𝑛0 → 𝑀 so 𝑀 = 𝐸𝑘𝑛0 /Γ for some group Γ = Deck(𝜋) acting freely and properly. By Exercise 3.31, Γ < Isom(𝐸𝑘𝑛0 ). Conversely, if a subgroup Γ of a Lie group acts freely and properly, then it is discrete (Exercise 3.40). Thus, 𝑀 = 𝐸𝑘𝑛0 /Γ for some discrete subgroup Γ < Isom(𝐸𝑘𝑛0 ). Therefore, in order to classify the complete connected Riemannian 𝑛-manifolds of constant sectional curvature 𝑘0 , we have to find the discrete subgroups Γ < Isom(𝐸𝑘𝑛0 ) and determine the quotient spaces 𝐸𝑘𝑛0 /Γ. We shall undertake this task for the case of surfaces (𝑛 = 2) in the next chapter. In particular, we shall see that any compact connected surface admits at least one metric of constant curvature. In the case of the sphere and ℝ𝑃2 , it will have 𝑘0 = 1. In the case of the torus and the Klein bottle, it will have 𝑘0 = 0. And in the case of Σ𝑔 and 𝑋𝑘 , for 𝑔 ≥ 2 and 𝑘 ≥ 3, it will have 𝑘0 = −1. Remark 3.110. As a final comment, this task in dimension 3 is much more difficult and strongly links with the so-called Thurston geometrization program, which is used in the origins of the proof by Perelman of the Poincaré conjecture (Remark 2.31). The fact is that not every 3-dimensional connected manifold admits a metric of constant sectional curvature. Note that if any manifold admitted such metric, then a compact simply connected one would be a compact simply connected space form, and hence it would be diffeomorphic to the 3-sphere, proving Poincaré conjecture. Actually, Thurston found that there are eight possible homogeneous geometries (Remark 4.87), three of which are the ones with constant sectional curvature, and the other five are not locally isotropic (they have directions with different curvatures). The Thurston geometrization program says that any connected compact 3-manifold can be

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split into pieces, in a precise geometric way, such that every piece has one of these geometries. Philosophically this is similar to the case of surfaces, but in a highly more involved way.

Problems In these problems, all manifolds are assumed to be smooth. Exercise 3.1. Prove that a map 𝑇 ∶ Ω1 (𝑀)𝑟 × 𝔛(𝑀)𝑠 → 𝐶 ∞ (𝑀) defines a tensor if and only if it is 𝐶 ∞ (𝑀)-multilinear. Exercise 3.2. Let 𝑋 be a vector field in an 𝑛-manifold 𝑀. Show that there is an open subset 𝒱 ⊂ 𝑋 × ℝ containing 𝑋 × {0} and a smooth map Φ ∶ 𝒱 → 𝑋, such that the curves 𝑐𝑝 (𝑡) = Φ(𝑝, 𝑡) are integral curves of 𝑋, that is, 𝑐𝑝′ (𝑡) = 𝑋(𝑐𝑝 (𝑡)) for all 𝑡. This is called the flow of 𝑋, and it is denoted 𝜙𝑡 = Φ(−, 𝑡). Exercise 3.3. Let 𝑋 be a vector field in an 𝑛-manifold with 𝑋(𝑝) ≠ 0. Show that there are 𝜕 coordinates (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) in a neighbourhood of 𝑝 such that 𝑋 = 𝜕𝑥 . 1

Exercise 3.4. Let 𝑋, 𝑌 be two vector fields in an 𝑛-manifold, and let 𝜙𝑡 , 𝜓𝑠 be the (local) flows of 𝑋, 𝑌 around a point 𝑝 ∈ 𝑀. Show that, in coordinates, we have the formula [𝑋, 𝑌 ]𝑝 = lim

𝑡,𝑠→0

𝜓𝑠 (𝜙𝑡 (𝑝)) − 𝜙𝑡 (𝜓𝑠 (𝑝)) . 𝑡𝑠

Exercise 3.5. Let 𝑋, 𝑌 be two vector fields in an 𝑛-manifold with {𝑋(𝑝), 𝑌 (𝑝)} linearly independent. Show that if [𝑋, 𝑌 ] = 0, then there exist coordinates (𝑥1 , . . . , 𝑥𝑛 ) in a neighbourhood of 𝑝 𝜕 𝜕 with 𝑋 = 𝜕𝑥 , 𝑌 = 𝜕𝑥 . 1

2

Exercise 3.6. Let 𝑋, 𝑌 be two vector fields linearly independent on a surface. Show that locally there exist two non-zero functions 𝑓, 𝑔 such that [𝑓 𝑋, 𝑔 𝑌 ] = 0. Exercise 3.7. Show that 𝐿𝑋 on page 143 can be extended to a derivation of tensors satisfying (1), (3), (4), and (5) of Definition 3.6. If 𝑋 is a vector field and 𝜙𝑡 is the flow of 𝑋, prove that for any tensor 𝑇, 𝜙∗ 𝑇 − 𝑇 𝑑 𝐿𝑋 (𝑇) = || (𝜙∗𝑡 𝑇) = lim 𝑡 . 𝑑𝑡 𝑡=0 𝑡 𝑡→0 Exercise 3.8. Let 𝐻 ⊂ 𝑀 be a submanifold, and let 𝑇 ∈ 𝒯 𝑟,𝑠 (𝐻) be any tensor. Prove that there is an open subset 𝑈 ⊃ 𝐻 and 𝑇 ′ ∈ 𝒯 𝑟,𝑠 (𝑈) such that 𝑇 ′ |𝐻 = 𝑇. If 𝐻 is closed, then we can take 𝑇 ′ ∈ 𝒯 𝑟,𝑠 (𝑀). Exercise 3.9. If ∇ is a metric connection on a Riemannian manifold, then we have ∇(↓𝑇) =↓∇𝑇, for any tensor 𝑇 and any lowering of index. Prove the analagous statement for the raising index operators. Exercise 3.10. Show that if a connection ∇ has zero torsion, then for any 𝑘-form 𝛼 we have 𝑑𝛼 = 𝒜(∇𝛼), where 𝒜 is the antisymmetrization operator on (0, 𝑘 + 1)-tensors. That is, the map 1 such that 𝒜(𝑑𝑥𝑖1 ⊗ ⋯ ⊗ 𝑑𝑥𝑖𝑘+1 ) = (𝑘+1)! 𝑑𝑥𝑖1 ∧ ⋯ ∧ 𝑑𝑥𝑖𝑘+1 . Exercise 3.11. Prove that in geodesic coordinates around 𝑝, we have Γ𝑖𝑗𝑘 (𝑝) = 0. Exercise 3.12. Let ∇ be a connection, and 𝑇 be a tensor. Prove that the following are equivalent: • ∇𝑇 = 0. 𝑡 ,𝑡

• 𝑃𝑐 0 1 𝑇𝑐(𝑡0 ) = 𝑇𝑐(𝑡1 ) for any curve 𝑐 ∶ [𝑡0 , 𝑡1 ] → 𝑀. 𝑝

• 𝑇𝑝 is fixed by Hol∇ , for 𝑝 ∈ 𝑀.

Problems

193

This result is usually known as the holonomy principle. 𝑐𝑜

Exercise 3.13. Determine 𝕃𝐑𝐢𝐞𝐦1 . Exercise 3.14. Prove that any smooth manifold admits a geodesically complete metric. 𝑘 Exercise 3.15. Let 𝑀 be a compact manifold. Prove that 𝐻𝑑𝑅 (𝑀) is finite dimensional (use a covering by geodesically convex balls).

Exercise 3.16. Prove that a compact smooth surface admits a triangulation. Exercise 3.17. Let (𝑀, g) be a Riemannian manifold with coordinates (𝑥1 , . . . , 𝑥𝑛 ) and 𝑝 = (0, 0, . . . , 0). Assume that 𝑔𝑖𝑗 (𝑝) = 𝛿𝑖𝑗 . Consider a square with edges: (a) (𝑡, 0, . . . , 0), 𝑡 ∈ [0, 𝜖]; (b) (𝜖, 𝑡, . . . , 0), 𝑡 ∈ [0, 𝜖]; (c) (0, 𝑡, . . . , 0), 𝑡 ∈ [0, 𝜖]; (d) (𝑡, 𝜖, . . . , 0), 𝑡 ∈ [0, 𝜖]. Let 𝑣 ∈ 𝑇𝑝 𝑀 and we parallel transport it along the edges of the square. Show that 𝑅(𝜕𝑥1 , 𝜕𝑥2 )𝑣 = lim 𝜖→0

𝑃𝑏0,𝜀 (𝑃𝑎0,𝜀 (𝑣)) − 𝑃𝑑0,𝜀 (𝑃𝑐0,𝜀 (𝑣)) . 𝜖2

Exercise 3.18. Determine how many independent coefficients the curvature tensor of an 𝑛manifold depends on at a point. Exercise 3.19. Let 𝑀 be an 𝑛-dimensional Riemannian manifold, 𝑝 ∈ 𝑀, and let 𝑆𝑟 (𝑝) be the geodesic (𝑛 − 1)-sphere of radius 𝑟 around 𝑝, for 𝑟 > 0 small. Then prove that 1 Vol(𝑆𝑟 (𝑝)) = 𝜔𝑛−1 𝑟𝑛−1 − 𝜔𝑛−1 Scal(𝑝)𝑟𝑛+1 + 𝑂(𝑟𝑛+2 ), 6 where Scal(𝑝) is the scalar curvature at 𝑝, 𝜔𝑛−1 = Vol(𝕊𝑛−1 ), for the standard Euclidean (𝑛 − 1)sphere of radius 1. Exercise 3.20. Let 𝑆 be a Riemannian surface, and let 𝐶 ⊂ 𝑆 be a regular curve. Show that there are (local) orthogonal coordinates (𝑢, 𝑣) such that the curve is 𝐶 = {𝑣 = 0}. Exercise 3.21. Consider a surface of revolution 𝑆 ⊂ ℝ3 which is compact, connected and without boundary. Compute its Gaussian curvature and test the formula of Gauss-Bonnet. Do the same for the case that 𝑆 has boundary. Exercise 3.22. Show that if 𝑆 ⊂ ℝ3 is a compact connected surface not homeomorphic to 𝑆2 , then 𝑆 has points with 𝐾 > 0, points with 𝐾 = 0, and points with 𝐾 < 0. Exercise 3.23. Prove, using the Preissman theorem (Remark 3.71), that if 𝑀 is a compact connected Riemannian manifold with 𝐾 < 0, then 𝜋1 (𝑀) cannot be Abelian. Exercise 3.24. Let 𝑀 be a smooth manifold, and let Γ be a finite group acting by diffeomorphisms. Prove that the action is locally linearizable. Exercise 3.25. Make the 𝑛-orbifolds into a category. Exercise 3.26. Prove that a differentiable orbifold 𝑋 admits orbifold partitions of unity. Use it to prove that 𝑋 always admits orbifold Riemannian metrics and the space of orbifold metrics is convex. Exercise 3.27. Prove that a differentiable oriented orbifold 𝑋 admits an orbifold volume form (i.e., non-vanishing orbifold 𝑛-forms). Prove that this is equivalent to 𝑋 having an atlas whose changes of charts are orientation preserving. Exercise 3.28. Let 𝑓 ∶ 𝑋 ′ → 𝑋 be an orbifold cover of compact oriented orbisurfaces, of degree 𝑑. Take an orbifold Riemannian metric g on 𝑋. Prove that 𝑓∗ g is an orbifold Riemannian metric. Using Gauss-Bonnet Theorem 3.82, prove that 𝜒orb (𝑋 ′ ) = 𝑑𝜒orb (𝑋). Exercise 3.29. Let 𝑋 be a compact oriented orbisurface, and let 𝑝 ∈ 𝑋 be a smooth point. Define the category 𝐂𝐨𝐯(𝑋, 𝑝) of orbifold covers of (𝑋, 𝑝), and prove a classification result as in (2.2).

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Exercise 3.30. Let 𝑃 ⊂ ℝ3 be a polyhedron, and consider the topological surface 𝑆 = 𝜕𝑃, which is homeomorphic to a 2-sphere. On each face 𝑃𝛼2 ⊂ 𝑆, put the Euclidean metric g𝛼 = 𝑖∗ g𝑠𝑡𝑑 . Prove that these glue nicely to give a smooth and Riemannian struture at 𝑆 − 𝑉, where 𝑉 is the set of vertices. For each vertex 𝑣 ∈ 𝑉, let 𝜃𝑣 be the total angle around 𝑣. Prove that ∑𝑣∈𝑉 (2𝜋−𝜃𝑣 ) = 2𝜋 using the Gauss-Bonnet theorem. Can you find a combinatorial proof? Exercise 3.31. Let 𝜋 ∶ 𝑀 ′ → 𝑀 be a cover space and suppose that (𝑀, g) is a Riemannian manifold. Show that g′ = 𝜋∗ g is a Riemannian metric on 𝑀 ′ such that the deck transformations are isometries. Prove that if 𝑀 is complete, then 𝑀 ′ is also complete. Conversely, suppose that (𝑀 ′ , g′ ) is a Riemannian manifold and Γ < Isom(𝑀 ′ , g′ ) is a group acting freely and properly, then show that 𝑀 = 𝑀 ′ /Γ is a Riemannian manifold and 𝜋 ∶ 𝑀 ′ → 𝑀 is a Riemannian cover. Exercise 3.32. Let 𝑀, 𝑁 be connected Riemannian manifolds, 𝑀 complete, and let 𝜋 ∶ 𝑀 → 𝑁 be a local isometry. Show that 𝜋 is a cover and that 𝑁 is complete. Exercise 3.33. Find a (non-compact) connected Riemannian manifold 𝑀 and a local isometry 𝑓 ∶ 𝑀 → 𝑀 which is not an isometry. Prove that for a compact manifold, a local isometry is always an isometry. Exercise 3.34. Prove that there are inextensible connected Riemannian manifolds which are not complete. Exercise 3.35. Show that a Riemannian manifold (𝑀, g) is locally symmetric if and only if its curvature tensor 𝑅 satisfies ∇𝑅 = 0. Exercise 3.36. Prove that if 𝑀 is locally symmetric, complete, and simply connected, then it is symmetric. Exercise 3.37. Show that for a surface, locally homogeneous is equivalent to locally isotropic. ̃ be a space form (complete, connected, of constant curvature), Exercise 3.38. Let 𝑀 = 𝑀/Γ ̃ Show that 𝑀 is isotropic if and where 𝑀̃ is the simply connected space form and Γ < Isom(𝑀). ̃ ̃ only if Γ is normal in Isom(𝑀), and that in this case Isom(𝑀) ≅ Isom(𝑀)/Γ. Exercise 3.39. Let 𝑀 be a smooth connected manifold and let g1 , g2 be two Riemannian metrics 1 on 𝑀. Prove that Isom((𝑀, g1 ), (𝑀, g2 )) is at most of dimension 2 𝑛(𝑛 + 1). Conclude that for general metrics g1 , g2 , (𝑀, g1 ) and (𝑀, g2 ) are not isometric. Exercise 3.40. Let 𝑀 be a differentiable manifold, and let 𝐺 be a Lie group acting on 𝑀 freely, properly, and smoothly (i.e., 𝐺 × 𝑀 → 𝑀, (𝑔, 𝑥) ↦ 𝑔 𝑥 is smooth). Prove that 𝐺 is discrete (i.e., its topology is discrete). The converse is not true: take 𝑆1 ≅ ℝ/ℤ and 𝐺 = ⟨𝑓⟩ ≅ ℤ, with 𝑓(𝑥) = 𝑥 + 𝛼, where 𝛼 > 0 is an irrational number. Show that the action of 𝐺 on 𝑆1 is not proper.

References and extra material Basic reading. A general introduction to differentiable manifolds can be found in [Boo], and for a more specific study of Riemannian geometry, we recommend [DC2] or [Sp2]. The theory of surfaces in Euclidean space is extensively treated in [DC1] and [Sp3]. [Boo] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Pure and Applied Mathematics, Vol. 120, 2nd Edition, Academic Press, 2002. [DC1] M.P. Do Carmo, Differential Geometry of Curves and Surfaces, 2nd Edition, Dover Publications Inc., 2017. [DC2] M.P. Do Carmo, Riemannian Geometry, Mathematics: Theory and Applications, Birkhäuser, 1992.

References and extra material

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[Sp2] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition, Publish-or-Perish, 1999. [Sp3] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition, Publish-or-Perish, 1999.

Further reading. The following topics are of interest in the area of Riemannian geometry. • Geodesic completeness. To develop the necessary tools to prove Hopf-Rinow Theorem 3.35, see [DC2]. • Curvature and topology. We propose giving the proof of the Bonnet-Myers theorem or the Hadamard theorem following [DC2]. It is necessary to develop the theory of variation of geodesics and Jacobi fields. • Semi-Riemannian geometry. To develop the theory of semi-Riemannian manifolds, and to give applications to relativity, we can follow the book: [O’N] B. O’Neill, Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, Academic Press, 1983. • Lie groups. For this important topic, we recommend: [War] F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, Vol. 94, 1983. • Symmetric and homogeneous manifolds. We recommend the classic book: [Hel] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, 1978. • Orbifolds. One can follow Chapter 13 of the book: [Thu] W. Thurston, The Geometry and Topology of Three-Manifolds, Princeton University lecture notes, Princeton University Press, 2002. This can include the theory of orbifold covers and orbifold fundamental groups. • Riemannian holonomy. The Berger theorem determines the possible holonomy groups of Riemannian manifolds. [Ber] M. Berger, Sur les groupes d’holonomie homogènes des variétés a connexion affines et des variétés Riemanniennes, Bull. Soc. Math. France, 83, 279-330, 1953. • Fiber bundles. Unfortunately, we have not introduced or used the concept of fiber bundle in this book. This unifies important notions in differential geometry, starting from the tangent bundle. [Hus] D. Husemöller, Fibre Bundles, Graduate Texts in Mathematics, Vol. 20, Springer, 1993. • Connections and curvature on vector bundles. The natural setting for defining connection and curvature is that of vector bundles. A classical reference is Chapter 4 of the book: [Tau] C.H. Taubes, Differential Geometry: Bundles, Connections, Metrics and Curvature, Oxford Graduate Texts in Mathematics, Vol. 23, Oxford University Press, 2011. • Characteristic classes. Characteristic classes are cohomology classes associated to vector bundles. This can be found in: [B-T] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer, 1982. [M-S] J.W. Milnor, J.D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, Vol. 76, Princeton University Press, 1974.

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3. Riemannian geometry

References. [Do2] S. Donaldson, An application of gauge theory to four-dimensional topology, J. Diff. Geom. 18, 279-315, 1983. [Gra] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd Ed., CRC Press, 504-507, 1997. [Jo1] J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2008. [Joy] D. Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press, 2000. [K-N] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. 1, Wiley Classics Library, Wiley Interscience, 1963. [Sat] I. Satake, On a generalization of the notion of manifold, Proc. Natl. Acad. Sci. U.S.A. 42, 359-363, 1956. [Str] S. Sternberg, Curvature in Mathematics and Physics, Dover Books on Mathematics, Dover Publications Inc, 2012.

Chapter 4

Constant curvature

In this chapter we enter into a detailed study of compact connected Riemannian manifolds of constant curvature. As we have seen in Chapter 3, these are the locally isotropic manifolds and all of them are quotients of the corresponding simply connected space form by a group of isometries acting freely and properly. Therefore, we shall first construct the simply connected space forms for all dimensions and curvatures and proceed to the study of the group of isometries. For the classification of the compact connected manifolds of constant curvature, we shall focus on the case of surfaces. Historically, the appearance of hyperbolic geometry was a result of a long controversy with respect to the axiomatic definition of “geometry” dating back to Euclid. In this context, the spaces of constant curvature appeared as models of the different classical geometries. We shall give a brief discussion that links classical geometry to the modern point of view of differential geometry at the end of the chapter.

4.1. Positive constant curvature We start by studying complete connected manifolds with constant positive curvature 𝑘0 > 0. By rescaling the metric (Remark 4.9), we can arrange that the curvature is 𝑘0 = 1. So we focus on constructing the 𝑛-dimensional simply connected space form with 𝑘0 = 1, which is the 𝑛-sphere. 4.1.1. The orthogonal group. Let 𝑛 ≥ 1, and consider the Euclidean vector space ℝ𝑛 with scalar product ⟨𝑥, 𝑦⟩ = 𝑥𝑡 𝑦, where the vectors 𝑥 ∈ ℝ𝑛 are written as columns, 𝑥1 𝑥 = ( ⋮ ). The orthogonal group is the group of matrices 𝑥𝑛 O(𝑛) = {𝐴 ∈ 𝑀𝑛×𝑛 (ℝ) | 𝐴𝑡 𝐴 = Id}. The matrix 𝐴𝑡 denotes the transpose matrix of 𝐴, and 𝑀𝑛×𝑛 (ℝ) is the set of 𝑛×𝑛 matrices with entries in ℝ. The group O(𝑛) corresponds to the set of linear maps 𝜑𝐴 ∶ ℝ𝑛 → ℝ𝑛 , 197

4. Constant curvature

198

𝜑𝐴 (𝑥) = 𝐴𝑥, which are linear isometries, since ⟨𝐴𝑥, 𝐴𝑦⟩ = (𝐴𝑥)𝑡 (𝐴𝑦) = 𝑥𝑡 𝐴𝑡 𝐴𝑦 = 𝑥𝑡 𝑦 = ⟨𝑥, 𝑦⟩, for all 𝑥, 𝑦 ∈ ℝ𝑛 , is equivalent to 𝐴𝑡 𝐴 = Id. Let 𝐴 ∈ O(𝑛), and let (𝑒 1 , . . . , 𝑒 𝑛 ) be the canonical basis of ℝ𝑛 . Then (𝐴𝑒 1 , . . . , 𝐴𝑒 𝑛 ) is an orthonormal basis, and since 𝐴𝑒𝑗 is the 𝑗th column of 𝐴, we have that the columns of 𝑎 ⎛ 11 𝑎21 𝐴=⎜ ⎜ ⋮ ⎝ 𝑎𝑛1

(4.1)

𝑎12 𝑎22 ⋮ 𝑎𝑛2

... ... ⋱ ...

𝑎1𝑛 𝑎2𝑛 ⋮ 𝑎𝑛𝑛

⎞ ⎟ ⎟ ⎠

form an orthonormal basis of ℝ𝑛 . From this, we can compute the dimension of O(𝑛). The first vector 𝑣 1 = 𝐴𝑒 1 is unitary, so 𝑣 1 ∈ 𝑆 𝑛−1 ⊂ ℝ𝑛 . The second vector 𝑣 2 = 𝐴𝑒 2 is unitary and 𝑣 2 ⟂ 𝑣 1 , so 𝑣 2 ∈ 𝑆 𝑛−1 ∩ ⟨𝑣 1 ⟩⟂ ≅ 𝑆 𝑛−2 . Repeating the argument 𝑣 3 = 𝐴𝑒 3 ∈ 𝑆 𝑛−1 ∩ ⟨𝑣 1 , 𝑣 2 ⟩⟂ ≅ 𝑆 𝑛−3 , . . . , 𝑣 𝑛 ∈ 𝑆 0 . Hence (4.2)

dim O(𝑛) = (𝑛 − 1) + (𝑛 − 2) + ⋯ + 1 + 0 =

𝑛(𝑛 − 1) . 2

Remark 4.1. Actually O(𝑛) is a Lie group, that is, a group which is also a smooth manifold, and such that the multiplication and inverse maps are smooth maps [War]. In order to study O(𝑛), recall the following result from linear algebra. Lemma 4.2. If 𝐴 ∈ O(𝑛), then there exists an orthonormal basis of ℝ𝑛 so that the expression of 𝐴 with respect to this basis is

(4.3)

cos 𝜃1 ⎛ − sin 𝜃1 ⎜ ⋮ ⎜ ⎜ ⎜ ⎜ ⎜ ⋮ ⎜ ⎜ ⎜ ⋮ ⎜ 0 ⎝

sin 𝜃1 cos 𝜃1

⋯

⋯

⋯

⋱ cos 𝜃𝑝 − sin 𝜃𝑝

sin 𝜃𝑝 cos 𝜃𝑝 −1 ⋱ −1 1 ⋱

⋯

0

⎞ ⎟ ⋮ ⎟ ⎟ ⎟ ⎟ ⋮ ⎟ ⎟ ⎟ ⎟ ⎟ 1 ⎠

where 0 < 𝜃𝑖 < 𝜋, for 1 ≤ 𝑖 ≤ 𝑝. Proof. This has an easy proof in linear algebra as follows. Seeing 𝐴 as an automorphism of ℂ𝑛 , 𝐴 has an eigenvalue 𝜆 ∈ ℂ. Then |𝜆| = 1 since if 𝑣 ∈ ℂ𝑛 is an eigenvector, then we have (𝐴𝑣)𝑡 𝐴𝑣 = 𝑣𝑡 𝐴𝑡 𝐴𝑣 = 𝑣𝑡 𝑣 = ||𝑣||2 and (𝐴𝑣)𝑡 𝐴𝑣 = (𝜆𝑣)𝑡 𝜆 𝑣 = |𝜆|2 𝑣𝑡 𝑣 = |𝜆|2 ||𝑣||2 , with ||𝑣|| ≠ 0. There are two possibilities. If 𝜆 is real, then 𝜆 = ±1, and we take the eigenvector as a unitary vector 𝑣 1 = 𝑣 ∈ ℝ𝑛 . Consider the hyperplane 𝐻 = ⟨𝑣 1 ⟩⟂ and observe that 𝐴(𝐻) ⊂ 𝐻 since if 𝑤 ∈ 𝐻, then ⟨𝑣 1 , 𝐴𝑤⟩ = ±⟨𝐴𝑣 1 , 𝐴𝑤⟩ = ±⟨𝑣 1 , 𝑤⟩ = 0, hence 𝐴𝑤 ∈ 𝐻. So 𝐴|𝐻 ∶ 𝐻 → 𝐻, and by induction there is a basis (𝑣 2 , . . . , 𝑣 𝑛 ) of 𝐻 such that 𝐴|𝐻 has

4.1. Positive constant curvature

199

the required form. The basis (𝑣 1 , 𝑣 2 , . . . , 𝑣 𝑛 ) is orthonormal and 𝐴 has the form given in the statement. If 𝜆 = 𝑒i𝜃 is not real, neither is 𝑣 ∈ ℂ𝑛 a real vector. Changing 𝜆 by its conjugate if necessary, we can assume 0 < 𝜃 < 𝜋. Write 𝑣 = 𝑣 1 + i𝑣 2 and normalize the length so that ||𝑣 1 || = 1. Then 𝐴𝑣 = 𝜆𝑣 is rewritten as 𝐴(𝑣 1 + i𝑣 2 ) = (cos 𝜃 + i sin 𝜃)(𝑣 1 + i𝑣 2 ). So 𝐴𝑣 1 = cos 𝜃 𝑣 1 − sin 𝜃 𝑣 2 , 𝐴𝑣 2 = sin 𝜃 𝑣 1 + cos 𝜃 𝑣 2 . Also 𝜆2 𝑣𝑡 𝑣 = (𝐴𝑣)𝑡 𝐴𝑣 = 𝑣𝑡 𝐴𝑡 𝐴𝑣 = 𝑣𝑡 𝑣 and 𝜆2 ≠ 1 imply that 0 = 𝑣𝑡 𝑣 = (𝑣 1 + i𝑣 2 )𝑡 (𝑣 1 + i𝑣 2 ) = ||𝑣 1 ||2 − ||𝑣 2 ||2 + 2i⟨𝑣 1 , 𝑣 2 ⟩. Hence ⟨𝑣 1 , 𝑣 2 ⟩ = 0 and ||𝑣 2 || = ||𝑣 1 || = 1. Take now 𝐻 = ⟨𝑣 1 , 𝑣 2 ⟩⟂ . Arguing as before, 𝐴|𝐻 ∶ 𝐻 → 𝐻. The result follows then by induction. □ With this result at hand, let us prove that O(𝑛) has two connected components. First if 𝐴𝑡 𝐴 = Id, taking determinants, we have det(𝐴)2 = 1, and hence det(𝐴) = ±1. + The set O (𝑛) = {𝐴 ∈ O(𝑛) | det(𝐴) = 1} consists of linear isometries preserving the − orientation, and the set O (𝑛) = {𝐴 ∈ O(𝑛) | det(𝐴) = −1} consists of linear isometries reversing the orientation. Both are closed, disjoint and non-empty. We prove that both + − are connected. Note that O (𝑛) ≅ O (𝑛), by the map 𝐴 ↦ 𝐴0 𝐴, where 𝐴0 is a fixed −1 . . . 0 ⎛ ⎞ ⋮ 1 ⋮ − ⎟. Hence it is enough to see that matrix in O (𝑛), for instance 𝐴0 = ⎜ ⋱ ⎜ ⎟ 1 ⎠ ⎝ 0 ⋯ + O (𝑛) is connected. This subgroup is called the special orthogonal group, +

SO(𝑛) = O (𝑛) = {𝐴 ∈ O(𝑛) | det(𝐴) = 1}. The letter “S” stands for “special”, and it is the customary way of referring to matrices with determinant one, that is, volume preserving. To see that SO(𝑛) is path connected, take 𝐴 ∈ SO(𝑛). Then there is some matrix 𝑃 such that 𝐴 = 𝑃 −1 𝐵𝑃, where 𝐵 is as in (4.3). As det(𝐵) = 1, the number of entries −1 0 cos 𝜃 sin 𝜃 equal to −1 is even. A 2 × 2-block ( ) is of the form 𝐶 𝜃 = ( ) 0 −1 − sin 𝜃 cos 𝜃 for 𝜃 = 𝜋. So 𝐵 is composed by blocks of the type 𝐶 𝜃𝑖 , 0 < 𝜃𝑖 ≤ 𝜋, and possibly some entries equal to 1. We define 𝐵𝑡 by using blocks 𝐶𝑡𝜃𝑖 , 𝑡 ∈ [0, 1], and 𝐴𝑡 = 𝑃 −1 𝐵𝑡 𝑃. Then 𝐴0 = Id and 𝐴1 = 𝐴, giving a path joining 𝐴 with Id. 4.1.2. The 𝑛-sphere. In ℝ𝑛+1 consider the bilinear form given by the scalar product 𝐵(𝑥, 𝑦) = ⟨𝑥, 𝑦⟩ = 𝑥𝑡 𝑦 . It may seem a bit redundant to introduce new notation, but it is a useful point of view that will be mimicked in section 4.3. The 𝑛-sphere is the manifold described by the equation 𝐵(𝑥, 𝑥) = ||𝑥||2 = 1, 𝑆 𝑛 = {𝑥 ∈ ℝ𝑛+1 | 𝐵(𝑥, 𝑥) = 1} = {𝑥 = (𝑥0 , . . . , 𝑥𝑛 ) ∈ ℝ𝑛+1 | ||𝑥||2 = 𝑥02 + ⋯ + 𝑥𝑛2 = 1} ⊂ ℝ𝑛+1 . We consider ℝ𝑛+1 a Riemannian manifold with the standard metric g𝑠𝑡𝑑 = 𝑑𝑥12 + ⋯ + 𝑑𝑥𝑛2 , which is the scalar product 𝐵 on each tangent space g𝑠𝑡𝑑 (𝑢, 𝑣) = 𝐵(𝑢, 𝑣), for all 𝑝 ∈ ℝ𝑛+1 , 𝑢, 𝑣 ∈ 𝑇𝑝 ℝ𝑛+1 = ℝ𝑛+1 . We endow 𝑆 𝑛 with the induced metric g𝑆𝑛 = 𝑖∗ g𝑠𝑡𝑑 , where 𝑖 ∶ 𝑆 𝑛 ↪ ℝ𝑛+1 is the inclusion.

4. Constant curvature

200

Definition 4.3. The round 𝑛-sphere is the Riemannian 𝑛-manifold (𝑆 𝑛 , g𝑆𝑛 ). It will be denoted 𝕊𝑛 . As 𝕊𝑛 is compact, it is automatically complete. The function 𝐹(𝑥) = 𝐵(𝑥, 𝑥) = has differential 𝑑𝑥 𝐹 = 2𝑥0 𝑑𝑥0 + ⋯ + 2𝑥𝑛 𝑑𝑥𝑛 . Therefore the tangent space at a point 𝑥 ∈ 𝕊𝑛 is given as 𝑥02 + ⋯ + 𝑥𝑛2

𝑇𝑥 𝕊𝑛 = {𝑦 ∈ ℝ𝑛+1 | 𝑑𝑥 𝐹(𝑦) = 2𝐵(𝑥, 𝑦) = 0} = ⟨𝑥⟩⟂ . This means that the normal vector 𝐍(𝑥) = 𝑥 equals the position vector. Consider 𝐴 ∈ O(𝑛 + 1) and 𝜑𝐴 ∶ ℝ𝑛+1 → ℝ𝑛+1 the associated linear isometry, 𝜑𝐴 (𝑥) = 𝐴𝑥 for 𝑥 ∈ ℝ𝑛+1 . Then 𝜑𝐴 ∶ (ℝ𝑛+1 , g𝑠𝑡𝑑 ) → (ℝ𝑛+1 , g𝑠𝑡𝑑 ) is an isometry as a Riemannian manifold (this is an easy calculation that is done in section 4.2.1). As ||𝜑𝐴 (𝑥)|| = ||𝑥||, we have that 𝜑𝐴 ∶ 𝑆 𝑛 → 𝑆 𝑛 . The metric of 𝕊𝑛 is the one induced from ℝ𝑛+1 , hence 𝜑𝐴 ∶ 𝕊𝑛 → 𝕊𝑛 is an isometry. So we conclude that O(𝑛 + 1) ≅ {𝜑𝐴 | 𝐴 ∈ O(𝑛 + 1)} < Isom(𝕊𝑛 ). Let us show that, in fact, both groups are equal. Proposition 4.4. The sphere 𝕊𝑛 is isotropic. Moreover, we have the equality Isom(𝕊𝑛 ) = O(𝑛 + 1). Proof. Let (𝑒 0 , 𝑒 1 , . . . , 𝑒 𝑛 ) be the standard basis of ℝ𝑛+1 . Then we can fix 𝑝0 = 𝑒 0 ∈ 𝕊𝑛 and 𝐵0 = (𝑒 1 , . . . , 𝑒 𝑛 ) which is an orthonormal basis of 𝑇𝑝0 𝕊𝑛 . Take another point 𝑝 ∈ 𝕊𝑛 and any orthonormal basis (𝑢1 , . . . , 𝑢𝑛 ) of 𝑇𝑝 𝕊𝑛 . As 𝑇𝑝 𝕊𝑛 = ⟨𝑝⟩⟂ , we have that ⟨𝑝, 𝑢𝑗 ⟩ = 0, 1 ≤ 𝑗 ≤ 𝑛. Also ||𝑝|| = 1 so denoting 𝑢0 = 𝑝 ∈ ℝ𝑛+1 , we have that (𝑢0 , 𝑢1 , . . . , 𝑢𝑛 ) is an orthonormal basis of ℝ𝑛+1 . There is a linear isometry 𝜑𝐴 ∶ ℝ𝑛+1 → ℝ𝑛+1 such that 𝜑𝐴 (𝑒𝑗 ) = 𝑢𝑗 , 𝑗 = 0, 1, . . . , 𝑛. This is given by an orthogonal matrix 𝐴 whose columns are 𝑝 = 𝑢0 , 𝑢1 , . . . , 𝑢𝑛 . Then 𝜑𝐴 (𝑝0 ) = 𝑝. Also 𝑑𝑝0 𝜑𝐴 = 𝐴 ∶ 𝑇𝑝0 ℝ𝑛+1 → 𝑇𝑝 ℝ𝑛+1 , since for 𝑣 ∈ 𝑇𝑝0 ℝ𝑛+1 we take the curve 𝑐(𝑡) = 𝑝0 + 𝑡𝑣 with 𝑑 𝑑 𝑐(0) = 𝑝0 , 𝑐′ (0) = 𝑣, and then 𝑑𝑝0 𝜑𝐴 (𝑣) = 𝑑𝑡 |𝑡=0 𝜑𝐴 (𝑐(𝑡)) = 𝑑𝑡 |𝑡=0 𝐴(𝑝0 + 𝑡𝑣) = 𝐴𝑣. By restriction, 𝑑𝑝0 𝜑𝐴 = 𝐴 ∶ 𝑇𝑝0 𝕊𝑛 → 𝑇𝑝 𝕊𝑛 . Therefore 𝑑𝑝0 𝜑𝐴 (𝑒𝑗 ) = 𝑢𝑗 , 1 ≤ 𝑗 ≤ 𝑛. So 𝜑𝐴 is an isometry of 𝕊𝑛 sending 𝑝0 to 𝑝 and the basis 𝐵0 = (𝑒 1 , . . . , 𝑒 𝑛 ) of 𝑇𝑝0 𝕊𝑛 to the basis 𝐵 = (𝑢1 , . . . , 𝑢𝑛 ) of 𝑇𝑝 𝕊𝑛 . This implies that 𝕊𝑛 is isotropic: if 𝑝1 , 𝑝2 ∈ 𝕊𝑛 and 𝐵1 , 𝐵2 are orthonormal bases of 𝑇𝑝1 𝕊𝑛 and 𝑇𝑝2 𝕊𝑛 , respectively, take 𝜑𝐴𝑗 , 𝑗 = 1, 2, isometries such that 𝜑𝐴𝑖 (𝑝0 ) = 𝑝𝑗 , −1 and 𝑑𝑝0 𝜑𝐴𝑖 sends 𝐵0 to 𝐵𝑖 , 𝑖 = 1, 2. Then 𝜑𝐴2 ∘ 𝜑𝐴 sends 𝑝1 to 𝑝2 , and its differential 1 −1 𝑑𝑝1 (𝜑𝐴2 ∘ 𝜑𝐴1 ) sends 𝐵1 to 𝐵2 . Now, given any isometry 𝜑 ∈ Isom(𝕊𝑛 ), by the result above we can find 𝐴 ∈ O(𝑛 + 1) so that 𝜑(𝑝0 ) = 𝜑𝐴 (𝑝0 ) and 𝑑𝑝0 𝜑(𝐵0 ) = 𝑑𝑝0 𝜑𝐴 (𝐵0 ), which yields that 𝜑 = 𝜑𝐴 by Lemma 3.98. □ Remark 4.5. • Note that dim Isom(𝕊𝑛 ) = dim O(𝑛+1) = manifold (Remark 3.103).

(𝑛+1)𝑛 , as expected for an isotropic 2

• The orientation preserving isometries of the sphere are Isom+ (𝕊𝑛 ) = SO(𝑛 + 1), and the orientation reversing isometries are Isom− (𝕊𝑛 ) = − O (𝑛 + 1). These are all the connected components of Isom(𝕊𝑛 ).

4.1. Positive constant curvature

201

Now let us describe the geodesics of 𝕊𝑛 . These are the maximal circles of 𝕊𝑛 , i.e., circles obtained by intersecting 𝕊𝑛 with vectorial planes of ℝ𝑛+1 . Proposition 4.6. The geodesics of 𝕊𝑛 are the curves of the form 𝛾𝑞,ᵆ (𝑡) = (cos 𝑡)𝑞 + ′ (sin 𝑡)𝑢 for 𝛾𝑞,ᵆ (0) = 𝑞 ∈ 𝕊𝑛 and 𝛾𝑞,ᵆ (0) = 𝑢 ∈ 𝑇𝑞 𝕊𝑛 a unitary vector. The images of the geodesics are the maximal circles of 𝕊𝑛 . Proof. We begin with the maximal circle 𝛾(𝑡) = (cos 𝑡, sin 𝑡, 0, . . . , 0), which is the intersection of the plane ⟨𝑒 0 , 𝑒 1 ⟩ with 𝕊𝑛 . A direct computation shows that 𝛾″ (𝑡) = −𝛾(𝑡). Since 𝑇𝛾(𝑡) 𝕊𝑛 = ⟨𝛾(𝑡)⟩⟂ , we conclude that 𝛾″ (𝑡) ⟂ 𝑇𝛾(𝑡) 𝕊𝑛 and therefore 0 (see (3.24)), hence 𝛾 is a geodesic.

𝐷𝛾′ 𝑑𝑡

=(

𝑇 𝑑𝛾′ ) 𝑑𝑡

=

We can also write 𝛾(𝑡) = (cos 𝑡)𝑝0 + (sin 𝑡)𝑒 1 , with 𝑝0 = 𝑒 0 ∈ 𝕊𝑛 . For any 𝑞 ∈ 𝕊𝑛 and unitary vector 𝑢 ∈ 𝑇𝑞 𝕊𝑛 , we consider an isometry 𝜑𝐴 of 𝕊𝑛 so that 𝜑𝐴 (𝑝0 ) = 𝑞 and 𝑑𝑝0 𝜑𝐴 (𝑒 1 ) = 𝑢. For this it is enough to complete to an orthonormal basis (𝑢0 = 𝑞, 𝑢1 = 𝑢, 𝑢2 , . . . , 𝑢𝑛 ) and take the matrix 𝐴 ∈ O(𝑛 + 1) such that 𝐴𝑒 𝑖 = 𝑢𝑖 , 0 ≤ 𝑖 ≤ 𝑛. We have that 𝛾𝑞,ᵆ (𝑡) = 𝜑𝐴 (𝛾(𝑡)) = 𝐴((cos 𝑡)𝑝0 + (sin 𝑡)𝑒 1 ) = (cos 𝑡)𝑞 + (sin 𝑡)𝑢 is the geodesic from 𝑞 with initial vector 𝑢. This is also a maximal circle because 𝜑𝐴 transforms planes into planes, so 𝜑𝐴 (𝛾(ℝ)) = 𝕊𝑛 ∩ 𝜑𝐴 (⟨𝑝0 , 𝑒 1 ⟩) = 𝕊𝑛 ∩ ⟨𝑞, 𝑢⟩. All geodesics are obtained in this way. □ Remark 4.7. There is a constructive proof of Proposition 4.6. Take 𝛾(𝑡) = (𝑥0 (𝑡), 𝐷𝛾′ 𝑥1 (𝑡), . . . , 𝑥𝑛 (𝑡)) a unitary geodesic in 𝕊𝑛 . Then 𝑑𝑡 = 𝛾″ (𝑡)𝑇 = 0 implies that 𝛾″ (𝑡) = 𝜆(𝑡)𝛾(𝑡), for some smooth function 𝜆(𝑡). We differentiate ⟨𝛾, 𝛾′ ⟩ ≡ 0 to get ⟨𝛾′ , 𝛾′ ⟩ + ⟨𝛾, 𝛾″ ⟩ ≡ 0, which is rewritten as 1 + 𝜆 ≡ 0, using that ||𝛾′ || = 1. Then 𝜆(𝑡) = −1 and hence 𝛾″ (𝑡) = −𝛾(𝑡), for all 𝑡 ∈ ℝ. Each coordinate satisfies 𝑥𝑗″ = −𝑥𝑗 , and hence 𝑥𝑖 (𝑡) = 𝑝 𝑖 cos 𝑡 + 𝑢𝑖 sin 𝑡, for suitable real numbers 𝑝 𝑖 , 𝑢𝑖 , 0 ≤ 𝑖 ≤ 𝑛. This is rewritten as 𝛾(𝑡) = (cos 𝑡)𝑝 + (sin 𝑡)𝑢, for 𝑝 = (𝑝0 , . . . , 𝑝𝑛 ) and 𝑢 = (𝑢0 , . . . , 𝑢𝑛 ). For 𝛾(𝑡) to be an unitary geodesic in 𝕊𝑛 , it is necessary that 𝑝 = 𝛾(0) ∈ 𝕊𝑛 and 𝑢 = 𝛾′ (0) is a unitary vector in 𝑇𝑝 𝕊𝑛 . From this description of the geodesics, we can easily get the exponential map at the north pole 𝑝0 = 𝑒 0 ∈ 𝕊𝑛 , as exp𝑝0 ∶ 𝑇𝑝0 𝕊𝑛 → 𝕊𝑛 given by (4.4)

exp𝑝0 (𝑢) = 𝛾𝑝0 ,

𝑢 ||𝑢||

(||𝑢||) = (cos ||𝑢||)𝑝0 + (sin ||𝑢||)

𝑢 . ||𝑢||

In coordinates, take 𝑢 = (0, 𝑡1 , . . . , 𝑡𝑛 ) ∈ 𝑇𝑝0 𝕊𝑛 , then exp𝑝0 (𝑢) = (cos ||𝑢||,

sin ||𝑢|| sin ||𝑢|| 𝑡 , ... , 𝑡 ). ||𝑢|| 1 ||𝑢|| 𝑛 sin ||ᵆ||

By expanding in Taylor series, we see that both cos ||𝑢|| and ||ᵆ|| are analytic functions (in particular smooth) in the variables 𝑡1 , . . . , 𝑡𝑛 , since only even terms appear in the Taylor expansion, and ||𝑢||2 = 𝑡12 + ⋯ + 𝑡𝑛2 . The exponential map (4.4) is a diffeomorphism from 𝐵𝜋 (0) ⊂ 𝑇𝑝0 𝕊𝑛 to 𝕊𝑛 − {−𝑝0 }, and it collapses {||𝑢|| = 𝜋} to the south pole {−𝑝0 }. So the injectivity radius is 𝜋 (Remark 3.33). This happens for all points 𝑝 ∈ 𝕊𝑛 , by the isotropy. Now let us compute the curvature of 𝕊𝑛 . As it is isotropic, the sectional curvature is constant, so it is enough to compute 𝐾𝑝0 (𝜎) for the point 𝑝0 = 𝑒 0 and the plane

4. Constant curvature

202

𝜎 = ⟨𝑒 1 , 𝑒 2 ⟩. By Proposition 3.45 and Theorem 3.51, we have 𝐾𝑝0 (𝜎) = 𝜅𝑆 (𝑝0 ), where 𝑆 is the surface given as 𝑆 = exp𝑝0 (𝜎). By the description of the exponential map, we have 𝑆 = {(𝑥0 , 𝑥1 , 𝑥2 , 0, . . . , 0) ∈ 𝕊𝑛 } ≅ 𝑆 2 , and the metric is the one induced by the embedding 𝑆 2 ⊂ ℝ3 ⊂ ℝ𝑛+1 , that is the metric g𝑆2 . This means that we only need to compute the Gaussian curvature of the round sphere 𝕊2 . We compute the curvature of 𝕊2 in several ways: (1) Consider the geodesic circle 𝐶𝑟 (𝑝0 ) of radius 𝑟 from the north pole, that is 𝛼𝑟 (𝜃) = exp𝑝0 (𝑟 cos 𝜃, 𝑟 sin 𝜃) = (cos 𝑟, sin 𝑟 cos 𝜃, sin 𝑟 sin 𝜃). Therefore 𝛼′𝑟 (𝜃) = (0, − sin 𝜃 sin 𝑟, cos 𝜃 sin 𝑟) and ||𝛼′𝑟 (𝜃)|| = sin 𝑟. Hence the Taylor series expansion 2𝜋

ℓ(𝐶𝑟 (𝑝0 )) = ∫

||𝛼′𝑟 (𝜃)||𝑑𝜃 = 2𝜋 sin 𝑟

0

= 2𝜋𝑟 − 2𝜋

𝑟3 2𝜋 3 + ⋯ = 2𝜋𝑟 − 𝑟 +⋯ 3! 6

says that 𝜅𝕊2 (𝑝0 ) = 1, by comparing it with the formula in Theorem 3.58. (2) Consider the (geodesic) triangle 𝑇 in 𝕊2 with sides given by two meridians from the north pole to the equator separated by an angle 𝛼, and the arc of 𝜋 𝜋 the equator joining them. Therefore the angles of 𝑇 are 𝛼, 2 , 2 . We have 𝜋 𝜋 by (3.38) that 𝛼 + 2 + 2 = 𝜋 + ∫𝑇 𝜅𝕊2 = 𝜋 + 𝑘0 area(𝑇), where 𝑘0 = 𝜅𝕊2 . Now the total area of 𝕊2 is 4𝜋 (see Remark 4.10), so the area of the triangle is 𝛼 4𝜋 area(𝑇) = 2𝜋 2 = 𝛼. Hence 𝛼 + 𝜋 = 𝜋 + 𝑘0 𝛼, which implies that 𝑘0 = 1. (3) Let us perform the parallel transport of a vector along the previous triangle from the north pole to compute the holonomy. Take a vector 𝑣 0 tangent to the first meridian. We parallel transport it downwards to the equator, and it remains tangent to the meridian (because the meridian is a geodesic). Now we parallel transport it along the equator, and it remains perpendicular to the equator (again the tangent direction to the equator is parallel, and so is the vector perpendicular to it). Finally we parallel transport it upwards along the second meridian, remaining tangent to the meridian until we reach the north pole again, obtaining a final vector 𝑣 1 , as in Figure 4.1. As the angle between the two meridians is 𝛼, Theorem 3.64 says that 𝛼 = ∠(𝑣 0 , 𝑣 1 ) = ∫𝑇 𝜅𝕊2 = 𝑘0 area(𝑇). As area(𝑇) = 𝛼, we have that the curvature is 𝑘0 = 1. v1

v0 r

α Cr

Figure 4.1. Computing the curvature of 𝕊2 .

α

4.1. Positive constant curvature

203

(4) By Gauss-Bonnet Theorem 3.68, ∫𝕊2 𝜅𝕊2 = 2𝜋𝜒(𝕊2 ). So 𝑘0 area(𝕊2 ) = 4𝜋𝑘0 = 2𝜋𝜒(𝑆2 ) = 4𝜋. Therefore 𝑘0 = 1. (5) We can also compute 𝜅𝕊2 with the definition (3.29). Parametrize 𝕊2 with spherical coordinates, P(𝑢, 𝑣) = (sin 𝑢 cos 𝑣, sin 𝑢 sin 𝑣, cos 𝑢). Then the first and second fundamental forms are 𝐈=(

1 0

0 ) sin2 𝑢

and

𝐈𝐈 = (

−1 0

0 ). − sin2 𝑢

So 𝜅𝕊2 = 1. (6) We can also use Proposition 3.54 for the curvature in orthogonal coordinates, with the spherical coordinates above. We will also compute 𝜅𝕊2 again in (4.8) using conformal coordinates. We have completed the proof of the following. Theorem 4.8. The simply connected space form 𝐸1𝑛 of curvature 𝑘0 = 1 is the round 𝑛-sphere 𝕊𝑛 . Remark 4.9. • To modify the value of the curvature, we use the following fact. Two metrics g, g′ are called homothetic if g′ = 𝜆2 g, with 𝜆 ∈ ℝ>0 . First, ||𝑢||′ = 𝜆||𝑢||, where || − || denotes the norm for (𝑀, g) and || − ||′ is the norm for (𝑀, g′ ). A computation using (3.19) shows that the Christoffel symbols g and g′ , which 1 we denote Γ𝑖𝑗𝑘 and Γ′𝑘𝑖𝑗 , satisfy Γ′𝑘𝑖𝑗 = Γ𝑖𝑗𝑘 , using that 𝑔′𝑖𝑗 = 𝜆2 𝑔𝑖𝑗 . From (3.27), the curvature tensors verify 𝑅′𝑙𝑖𝑗𝑘 = 𝑅𝑙𝑖𝑗𝑘 . The sectional curvature is given as g′ (𝑅′ (𝑢, 𝑣)𝑣, 𝑢) ||𝑢||′2 ||𝑣||′2 − g′ (𝑢, 𝑣)2 𝜆2 g(𝑅(𝑢, 𝑣)𝑣, 𝑢) 1 = 4 = 2 𝐾𝑝 (⟨𝑢, 𝑣⟩), 𝜆 𝜆 (||𝑢||2 ||𝑣||2 − g(𝑢, 𝑣)2 )

𝐾𝑝′ (⟨𝑢, 𝑣⟩) =

where 𝐾 ′ is the sectional curvature of (𝑀, g′ ) and 𝐾 is the sectional curvature 1 of (𝑀, g). In particular, (𝑆𝑛 , 𝜆2 g𝑆𝑛 ) has constant curvature 𝑘0 = 𝜆2 , so it is the 𝑛 space form 𝐸1/𝜆2 . • Alternatively, consider the sphere of radius 𝑅, given by the space 𝑆 𝑛𝑅 = {𝑥 ∈ ℝ𝑛+1 | ||𝑥|| = 𝑅}, with the metric given by the inclusion 𝑆 𝑛𝑅 ⊂ ℝ𝑛+1 . 1 We can directly compute that 𝑆 𝑛𝑅 has constant curvature 𝑘0 = 𝑅2 as before: first we reduce to 𝑆 2𝑅 , and then apply Gauss-Bonnet to get ∫𝑆2 𝜅𝑆𝑅2 = 4𝜋𝑅2 𝑘0 = 2𝜋𝜒(𝑆2𝑅 ) = 4𝜋, which implies 𝑘0 =

1 . 𝑅2

𝑅

• As required by Theorem 3.108, there is an isometry 𝑓 ∶ (𝑆 𝑛 , 𝑅2 g𝑆𝑛 ) → (𝑆 𝑛𝑅 , g𝑆𝑛 ), 𝑅 given by 𝑓(𝑥) = 𝑅𝑥, between the two previous models of constant curvature 1 𝑘0 = 𝑅2 . Stereographic projection. We will consider the coordinates on 𝕊2 given by the stereographic projection (see Figure 4.2), 𝑥1 𝑥2 (4.5) 𝜑 ∶ 𝕊2 − {𝑁} → ℝ2 , (𝑥0 , 𝑥1 , 𝑥2 ) ↦ (𝑢, 𝑣) = ( , ), 1 − 𝑥0 1 − 𝑥0

4. Constant curvature

204

N p

φ(p)

Figure 4.2. Stereographic projection.

from the north pole 𝑒 0 = (1, 0, 0). Its inverse is the parametrization P = 𝜑−1 ∶ ℝ2 → 𝕊2 − {𝑒 0 }, (4.6)

(𝑢, 𝑣) ↦ (𝑥0 , 𝑥1 , 𝑥2 ) = (

−1 + 𝑢2 + 𝑣2 2𝑢 2𝑣 , , ). 1 + 𝑢2 + 𝑣 2 1 + 𝑢2 + 𝑣 2 1 + 𝑢2 + 𝑣 2

Therefore Pᵆ = (

4𝑢 2(1 − 𝑢2 + 𝑣2 ) −4𝑢𝑣 , , ), (1 + 𝑢2 + 𝑣2 )2 (1 + 𝑢2 + 𝑣2 )2 (1 + 𝑢2 + 𝑣2 )2

P𝑣 = (

4𝑣 −4𝑢𝑣 2(1 + 𝑢2 − 𝑣2 ) , , ), (1 + 𝑢2 + 𝑣2 )2 (1 + 𝑢2 + 𝑣2 )2 (1 + 𝑢2 + 𝑣2 )2

from where we get 𝐸 = 𝐺 = (4.7)

g𝑆2 =

4 , (1+ᵆ2 +𝑣2 )2

𝐹 = 0. Hence

4 4 1 0 (𝑑𝑢2 + 𝑑𝑣2 ) = ( ). (1 + 𝑢2 + 𝑣2 )2 (1 + 𝑢2 + 𝑣2 )2 0 1

In these coordinates, we can compute the curvature directly using Corollary 3.56. Writ1 ing 𝐸 = 𝑒2𝑓 with 𝑓 = 2 log 𝐸 = log √𝐸 = log 2 − log(1 + 𝑢2 + 𝑣2 ), we have (4.8)

𝜅 = −𝑒−2𝑓 Δ𝑓 = −

(1 + 𝑢2 + 𝑣2 )2 −4 = 1. 4 (1 + 𝑢2 + 𝑣2 )2

Remark 4.10. The volume element is given by √𝐸𝐺 − 𝐹 2 𝑑𝑢 ∧ 𝑑𝑣 = 𝐸𝑑𝑢 ∧ 𝑑𝑣, so the area of 𝕊2 is 2

area(𝕊 ) = ∫ ℝ2

2𝜋

4 (1 + 𝑢2 + 𝑣2 )2

∞

𝑑𝑢𝑑𝑣 = ∫

∫

0

0

4𝜌 𝑑𝜌𝑑𝜃 (1 + 𝜌2 )2

∞

−2 = 2𝜋 [ ] = 4𝜋. 1 + 𝜌2 0 Stereographic coordinates can be written for 𝕊𝑛 in the same vein.

4.1. Positive constant curvature

205

Conformal maps. Conformal geometry will be treated in section 6.1. However, some definitions on conformal maps will be useful in dealing with the stereographic projection. Definition 4.11. Let (𝑀1 , g1 ) and (𝑀2 , g2 ) be two Riemannian manifolds of the same dimension. A smooth map 𝜑 ∶ 𝑀1 → 𝑀2 is locally conformal if 𝜑∗ g2 = 𝜇 g1 for some smooth map 𝜇 ∶ 𝑀1 → ℝ>0 . We say that 𝜑 is conformal if it is locally conformal and a diffeomorphism. Remark 4.12. (1) Definition 4.11 says that for 𝑝 ∈ 𝑀1 and 𝑢, 𝑣 ∈ 𝑇𝑝 𝑀1 , we have 𝜇(𝑝)g1 (𝑢, 𝑣)𝑝 = g2 (𝑑𝑝 𝜑(𝑢), 𝑑𝑝 𝜑(𝑣))𝜑(𝑝) . This means that 𝑑𝑝 𝜑 ∶ 𝑇𝑝 𝑀1 → 𝑇𝜑(𝑝) 𝑀2 is a dilation, that is, a linear map such that ||𝑑𝑝 𝜑(𝑢)|| = 𝜆||𝑢||, for all 𝑢 ∈ 𝑇𝑝 𝑀1 , where 𝜆 = √𝜇(𝑝) > 0. (2) In particular 𝜑 is a local diffeomorphism. Also, if (𝑒 1 , . . . , 𝑒 𝑛 ) form an orthonormal basis of 𝑇𝑝 𝑀1 , then (𝑑𝑝 𝜑(𝑒 1 ), . . . , 𝑑𝑝 𝜑(𝑒 𝑛 )) is an orthogonal basis for 𝑇𝜑(𝑝) 𝑀2 whose vectors are of norm 𝜆(𝑝). (3) It is typical to write 𝜑∗ g2 = 𝜆2 g1 to make explicit the dilation factor 𝜆 > 0. Also taking 𝑓 = log 𝜆, we can write 𝜑∗ g2 = 𝑒2𝑓 g1 , where 𝑓 ∈ 𝐶 ∞ (𝑀1 ). (4) The composition of locally conformal maps is a locally conformal map. If 𝜑 is conformal, then it admits and inverse 𝜑−1 which is also a conformal map. (5) If the dilation factor 𝜆 is a constant function, then 𝜑 is called a homothety. Lemma 4.13. Let 𝐿 ∶ 𝑉1 → 𝑉2 be a linear map between Euclidean vector spaces. The following are equivalent. (a) There exists 𝜆 > 0 so that ||𝐿(𝑢)|| = 𝜆||𝑢|| for all 𝑢 ∈ 𝑉1 (𝐿 is a dilation). (b) For every 𝑢, 𝑣 ∈ 𝑉1 we have ⟨𝐿(𝑢), 𝐿(𝑣)⟩ = 𝜆2 ⟨𝑢, 𝑣⟩. (c) The map 𝐿 preserves angles, i.e., ∠(𝑢, 𝑣) = ∠(𝐿(𝑢), 𝐿(𝑣)) for all non-zero 𝑢, 𝑣 ∈ 𝑉1 . (d) If 𝑢 ⟂ 𝑣, then 𝐿(𝑢) ⟂ 𝐿(𝑣). Proof. To see (a) ⇒ (b), we use the formula to recover the scalar product from the norm, 1 ⟨𝐿(𝑢), 𝐿(𝑣)⟩ = (||𝐿(𝑢) + 𝐿(𝑣)||2 − ||𝐿(𝑢)||2 − ||𝐿(𝑣)||2 ) 2 1 = 𝜆2 (||𝑢 + 𝑣||2 − ||𝑢||2 − ||𝑣||2 ) = 𝜆2 ⟨𝑢, 𝑣⟩. 2 To see (b) ⇒ (c), we use the formula which relates the angles and the scalar product. cos ∠(𝐿(𝑢), 𝐿(𝑣)) =

⟨𝐿(𝑢), 𝐿(𝑣)⟩ 𝜆2 ⟨𝑢, 𝑣⟩ = 2 = cos ∠(𝑢, 𝑣), 𝜆 ||𝑢|| ||𝑣|| ||𝐿(𝑢)|| ||𝐿(𝑣)||

so ∠(𝐿(𝑢), 𝐿(𝑣)) = ∠(𝑢, 𝑣). Clearly (c) ⇒ (d). Finally let us see (d) ⇒ (a). Take (𝑒 1 , . . . , 𝑒 𝑛 ) an orthonormal basis of 𝑉1 . By (d), the basis (𝐿(𝑒 1 ), . . . , 𝐿(𝑒 𝑛 )) is orthogonal. Let us see that all the vectors 𝐿(𝑒 𝑖 ) have the same norm for all 𝑖. It is clear that

4. Constant curvature

206

⟨𝑒 𝑖 +𝑒𝑗 , 𝑒 𝑖 −𝑒𝑗 ⟩ = 0 for all 𝑖 ≠ 𝑗, so by (d) we have 𝐿(𝑒 𝑖 )+𝐿(𝑒𝑗 ) = 𝐿(𝑒 𝑖 +𝑒𝑗 ) ⟂ 𝐿(𝑒 𝑖 −𝑒𝑗 ) = 𝐿(𝑒 𝑖 ) − 𝐿(𝑒𝑗 ). This tells us that 0 = ⟨𝐿(𝑒 𝑖 ) + 𝐿(𝑒𝑗 ), 𝐿(𝑒 𝑖 ) − 𝐿(𝑒𝑗 )⟩ = ||𝐿(𝑒 𝑖 )||2 − ||𝐿(𝑒𝑗 )||2 for all 𝑖 ≠ 𝑗. Let us call 𝜆 = ||𝐿(𝑒 𝑖 )||. Then for any 𝑢 = ∑ 𝑥𝑖 𝑒 𝑖 ∈ 𝑉1 , we have that ||𝐿(𝑢)||2 = ∑ 𝑥𝑖2 ||𝐿(𝑒 𝑖 )||2 = 𝜆2 ||𝑢||2 , as required. □ Therefore, by Lemma 4.13, a map 𝜑 ∶ (𝑀1 , g1 ) → (𝑀2 , g2 ) is conformal if and only if it preserves the angles, or if and only if it preserves the orthogonality. Definition 4.14. Let (𝑀, g) be a Riemannian manifold. A coordinate chart 𝜑 ∶ 𝑈 ⊂ 𝑀 → 𝜑(𝑈) ⊂ ℝ𝑛 are conformal coordinates if it is a conformal map as a map 𝜑 ∶ (𝑈, g) → (ℝ𝑛 , g𝑠𝑡𝑑 ). Remark 4.15. • A conformal chart is a chart for which angles measured in the manifold coincide with angles measured in ℝ𝑛 (with the standard metric). Therefore the sizes of figures are distorted by 𝜑, but not their shape. • Also a chart is conformal if g|𝑈 = 𝑒2𝑓 g𝑠𝑡𝑑 = 𝑒2𝑓 (𝑑𝑥12 + ⋯ + 𝑑𝑥𝑛2 ), for some smooth function 𝑓. This means that the matrix (𝑔𝑖𝑗 ) is a multiple of the identity. • Conformal coordinates always exist for 𝑛 = 2. This is no longer true for 𝑛 ≥ 3 (Remark 6.10). The stereographic coordinates of 𝕊2 are conformal, by (4.7). So the angles of two curves in 𝕊2 coincide with the angles of the projected curves in ℝ2 . The geodesics in the chart (see Figure 4.3) are the images of the maximal circles. The image of a circle 𝐶 in 𝕊2 is a circle (if 𝑒 0 ∉ 𝐶) or a line (if 𝑒 0 ∈ 𝐶) in ℝ2 , being 𝑒 0 = (1, 0, 0) the north pole (Exercise 4.7). Therefore the geodesics in the chart are the following. • Lines through the origin, which are the images of meridians. • The unit circle 𝑆 1 , which is the image of the equator. • Circles passing through antipodal points of 𝑆 1 .

Figure 4.3. Geodesics in the stereographic chart.

Let us describe now the isometries in the stereographic coordinates (𝑢, 𝑣). Let 𝑓 ∈ Isom(𝕊2 ) be an isometry and assume for a while that it does not fix the north pole 𝑒 0 . Denote 𝑞 = 𝜑 ∘ 𝑓−1 (𝑒 0 ), 𝑝 = 𝜑 ∘ 𝑓(𝑒 0 ). Then 𝑓 is, in the stereographic coordinates, an

4.1. Positive constant curvature

207

isometry 𝑓 ∶ (ℝ2 − {𝑞}, g𝑆2 ) → (ℝ2 − {𝑝}, g𝑆2 ). Since Id ∶ (ℝ2 , g𝑆2 ) → (ℝ2 , g𝑠𝑡𝑑 ) is a conformal map and the composition of conformal maps is also conformal, we have that 𝑓 ∶ (ℝ2 − {𝑞}, g𝑠𝑡𝑑 ) → (ℝ2 − {𝑝}, g𝑠𝑡𝑑 ) is a conformal map with respect to the standard metric of ℝ2 . If 𝑓 fixes the norh pole, then 𝑝 and 𝑞 do not exist and we have a conformal map on the whole of ℝ2 . Conformal maps in ℝ2 = ℂ (with the standard metric) are closely related to holomorphic maps (see section 5.1.1 for the definition and properties of holomorphic maps). We have the following useful equivalence (see item (7) on page 267). Lemma 4.16. Let 𝑈 ⊂ ℂ be an open set, and let 𝑓 ∶ 𝑈 → ℂ be a smooth map. If we interpret 𝑓 as a map 𝑓 ∶ 𝑈 ⊂ ℝ2 → ℝ2 , then the following are equivalent: • 𝑓 is locally conformal and preserves the orientation. • 𝑓 is holomorphic and 𝑓′ (𝑧) ≠ 0, for all 𝑧 ∈ 𝑈. There is a natural compactification of ℂ by adding one point, denoted ∞. Topologically this is the one point compactification, also called Alexandroff compactification. We write (4.9)

ℂ = ℂ ∪ {∞}

and endow it with the usual topology of ℂ, and a basis of the neighbourhoods of ∞ are the sets (ℂ − 𝐵𝑅̄ (0)) ∪ {∞}, for 𝑅 large. The stereographic map (4.5) extends naturally to a homeomorphism 𝜑̄ ∶ 𝕊2 → ℂ,

(4.10)

̄ with 𝜑(𝑁) = ∞. The space ℂ is called the Riemann sphere. There is a natural description of ℂ as the complex projective line ℂ𝑃1 . Recall that ℂ𝑃 = (ℂ2 − {(0, 0)})/ℂ∗ , where ℂ∗ = ℂ − {0} acts on ℂ2 − {(0, 0)} by 𝜆 ⋅ (𝑧0 , 𝑧1 ) = (𝜆𝑧0 , 𝜆𝑧1 ) (see section 5.2.3 for more details). The space ℂ𝑃 1 is a complex manifold (Definition 5.3). Let [𝑧0 , 𝑧1 ] be the homogeneous coordinates of ℂ𝑃1 . There is a natural bijection 1

𝜛 ∶ ℂ𝑃 1 ⟶ ℂ,

(4.11) given as 𝑧 = 𝜛([𝑧0 , 𝑧1 ]) = 1

𝑧1 𝑧0

for 𝑧0 ≠ 0, and 𝜛([0, 1]) = ∞ (we can write 𝑧 =

𝑧1 𝑧0

always, by agreeing that 0 = ∞). The inverse is given by 𝜛−1 (𝑧) = [1, 𝑧] for 𝑧 ∈ ℂ, and 𝜛−1 (∞) = [0, 1]. The differentiable structure of ℂ𝑃1 is given by the following two charts (see section 5.2.3): 𝑈0 = {[𝑧0 , 𝑧1 ] | 𝑧0 ≠ 0}, with 𝜙0 ∶ 𝑈0 → ℂ, 𝑧 = 𝜙0 ([𝑧0 , 𝑧1 ]) = 𝑧1 𝑧 ; and 𝑈1 = {[𝑧0 , 𝑧1 ] | 𝑧1 ≠ 0}, with 𝜙1 ∶ 𝑈1 → ℂ, 𝑧′ = 𝜙1 ([𝑧0 , 𝑧1 ]) = 𝑧0 . The change 𝑧 0

1

1

of charts is given by 𝑧′ = 𝑧 , which is obviously smooth (it is actually a biholomorphism, i.e., a holomorphic bijective map with holomorphic inverse). This gives ℂ the structure of a complex manifold. With this differentiable structure, we see that (4.10) is actually a diffeomorphism. For the chart 𝑈0 , we have that 𝜑 = 𝜙0 ∘ 𝜛−1 ∘ 𝜑̄ ∶ 𝕊2 − {𝑒 0 } → ℂ is given by 𝑧 = 𝑥1 +i𝑥2 ̄ 0 , 𝑥1 , 𝑥2 ) = 1−𝑥 𝜑(𝑥 , by (4.5). For the chart 𝑈1 , we have 𝜑′ = 𝜙1 ∘ 𝜛−1 ∘ 𝜑̄ ∶ 𝕊2 − 0

{−𝑒 0 } → ℂ, given by 𝑧′ =

1 𝑧

=

1−𝑥0 𝑥1 +i𝑥2

=

(1−𝑥0 )(𝑥1 −i𝑥2 ) 𝑥21 +𝑥22

=

(1−𝑥0 )(𝑥1 −i𝑥2 ) 1−𝑥02

=

𝑥1 −i𝑥2 , 1+𝑥0

where

4. Constant curvature

208

−𝑒 0 = (−1, 0, 0) is the south pole. The inverse of the chart 𝜑′ is the stereographic projection from the south pole composed with a reflection: (4.12)

(𝜑′ )−1 (𝑢, 𝑣) = (

1 − 𝑢2 − 𝑣 2 2𝑢 2𝑣 , ,− ), 1 + 𝑢2 + 𝑣 2 1 + 𝑢2 + 𝑣 2 1 + 𝑢2 + 𝑣 2

which again define conformal coordinates. Now consider 𝑓 ∈ Isom+ (𝕊2 ), and let 𝑝 = 𝑓−1 (𝑒 0 ), 𝑞 = 𝑓(𝑒 0 ), as before. If we express 𝑓 in the stereographic coordinates, we have a smooth map 𝑓 ∶ ℂ−{𝑝} → ℂ−{𝑞} which is conformal and preserves the orientation. By Lemma 4.16, 𝑓 is holomorphic in ℂ − {𝑝}. We extend 𝑓 naturally to 𝑓 ∶ ℂ → ℂ by setting 𝑓(𝑝) = ∞ and 𝑓(∞) = 𝑞. Using the charts (4.6) and (4.12), we see that 𝑓 is actually holomorphic on the whole of ℂ. As 𝑓−1 is also holomorphic, we have that 𝑓 is a biholomorphism. Therefore, in order to understand the isometries of 𝕊2 , we have to study the biholomorphisms of ℂ. We will see that they are the so-called Möbius maps. Remark 4.17. The same argument as above says that an orientation preserving conformal map of 𝕊2 is a biholomorphism of ℂ, and conversely (see Remark 6.14(7)). 4.1.3. Möbius maps. Recall that given a complex vector space 𝑉, the group GL(𝑉) is the group of complex linear automorphisms of 𝑉, and we denote GL(𝑛, ℂ) = GL(ℂ𝑛 ), which consists of 𝑛 × 𝑛 matrices 𝐴 ∈ 𝑀𝑛×𝑛 (ℂ) with complex coefficients and non-zero determinant. The special complex linear group is SL(𝑛, ℂ) = {𝐴 ∈ GL(𝑛, ℂ)| det 𝐴 = 1}. A projective matrix [𝐴] is the class of a non-zero matrix in (𝑀𝑛×𝑛 (ℂ) − {0})/ℂ∗ , where the action of ℂ∗ is given by scalar multiplication. And the projective (complex) linear group is the group PGL(𝑛, ℂ) consisting of projective matrices [𝐴] where 𝐴 ∈ GL(𝑛, ℂ). Such [𝐴] induces a projective isomorphism 𝑓[𝐴] ∶ ℂ𝑃 𝑛−1 → ℂ𝑃 𝑛−1 given by 𝑓[𝐴] ([𝑧0 , . . . , 𝑧𝑛−1 ]) = [𝐴(𝑧0 , . . . , 𝑧𝑛−1 )𝑡 ]. Definition 4.18. We define the group Mob(ℂ) of Möbius maps as the set of maps 𝑓 ∶ 𝑎𝑧+𝑏 ℂ → ℂ of the form 𝑓(𝑧) = 𝑐𝑧+𝑑 , for 𝑧 ∈ ℂ, where 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℂ with 𝑎𝑑 − 𝑏𝑐 ≠ 0. Here we understand that 𝑓(

−𝑑 ) 𝑐

= ∞ and 𝑓(∞) =

𝑎 . 𝑐

If 𝑐 = 0, then 𝑓(∞) = ∞.

The group Mob(ℂ) can be described as the group PGL(2, ℂ) = {[𝐴] | 𝐴 ∈ GL(2, ℂ)} of 𝑑 𝑐 projective isomorphisms of ℂ𝑃 1 . Indeed, for a matrix 𝐴 = ( ) ∈ PGL(2, ℂ), we 𝑏 𝑎 have the projective isomorphism 𝑓[𝐴] ∶ ℂ𝑃 1 → ℂ𝑃 1 , 𝑧=[

𝑧0 𝑑 ] ↦ [( 𝑧1 𝑏

𝑎𝑧 + 𝑏𝑧0 𝑎𝑧 + 𝑏 𝑐 𝑧 𝑑𝑧0 + 𝑐𝑧1 = , ]= 1 ) ( 0 )] = [ 𝑎 𝑧1 𝑏𝑧0 + 𝑎𝑧1 𝑐𝑧1 + 𝑑𝑧0 𝑐𝑧 + 𝑑

following the standard identification 𝑧 =

𝑧1 𝑧0

∈ ℂ. It is a basic result that, given three

distinct points in ℂ, there is a unique Möbius map that sends them to {0, 1, ∞}.

4.1. Positive constant curvature

209

Remark 4.19. We enlarge the Möbius group Mob(ℂ) by adding the conjugation 𝑟(𝑧) = 𝑧. This gives the extended Möbius group, − ˆ Mob(ℂ) = Mob(ℂ) ⊔ Mob (ℂ), −

Mob (ℂ) = {𝑓(𝑧) =

𝑎𝑧 + 𝑏 | 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℂ, 𝑎𝑑 − 𝑏𝑐 ≠ 0} . 𝑐𝑧 + 𝑑 |

−

The transformations of Mob (ℂ) are of the form 𝑓 ∘ 𝑟, where 𝑓 ∈ Mob(ℂ). There is a classical characterization in projective geometry for the Möbius transformations as those preserving the set of lines and circles in ℂ. Lemma 4.20. Let ℛ be the set consisting of generalized circles on ℂ, that is circles of ℂ and unions of straight lines with ∞. Then a Möbius map 𝑓 ∈ Mob(ℂ) sends ℛ to itself. Proof. We shall give two proofs. For the first one, recall that the Möbius maps are the projective transformations of ℂ𝑃 1 , and those maps preserve the double ratio, defined 𝑑−𝑎 𝑐−𝑏 as [𝑎 ∶ 𝑏 ∶ 𝑐 ∶ 𝑑] = 𝑑−𝑏 ⋅ 𝑐−𝑎 , for 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℂ (here no three points are equal, and a ∞ ∞−𝑎 𝑧−𝑎 quotient ∞ is interpreted as a limit, so ∞−𝑏 = lim 𝑧−𝑏 = 1, if 𝑎, 𝑏 ∈ ℂ). As we shall 𝑧→∞

see shortly, given three different points 𝑎, 𝑏, 𝑐 ∈ ℂ, the set 𝑅 = {𝑧 ∈ ℂ | [𝑎 ∶ 𝑏 ∶ 𝑐 ∶ 𝑧] ∈ ℝ ∪ {∞}} is the (unique) generalized circle passing through 𝑎, 𝑏, 𝑐, thus 𝑅 ∈ ℛ. A Möbius map preserves the condition [𝑎 ∶ 𝑏 ∶ 𝑐 ∶ 𝑧] being real, hence it sends elements of ℛ to elements of ℛ. To see that 𝑅 ∈ ℛ, let us rewrite the condition of [𝑎 ∶ 𝑏 ∶ 𝑐 ∶ 𝑧] = 𝑧−𝑎 𝑐−𝑏 𝑧−𝑎 ⋅ 𝑐−𝑎 ∈ ℝ ∪ {∞}. If 𝑏 = ∞, then it is 𝑐−𝑎 ∈ ℝ, and writing 𝑐 − 𝑎 = 𝜌𝑒i𝜃 in 𝑧−𝑏 polar coordinates, we get the line defined as 𝑧 = 𝑎 + 𝜆𝑒i𝜃 , 𝜆 ∈ ℝ ∪ {∞}. If 𝑎 = ∞, 𝑐−𝑏 the argument is similar. If 𝑎, 𝑏 ≠ ∞, then we write 𝑐−𝑎 = 𝜌𝑒i𝜃 . The defining equation 𝑧−𝑎

for 𝑅 is 𝑒i𝜃 𝑧−𝑏 ∈ ℝ ∪ {∞}, or equivalently 𝑒i𝜃 (𝑧 − 𝑎)(𝑧 − 𝑏) ∈ ℝ ∪ {∞}, which after expanding is 𝑒i𝜃 (|𝑧|2 − 𝑎𝑧 − 𝑏𝑧 + 𝑎𝑏) ∈ ℝ ∪ {∞}. This gives the equation Im(𝑒i𝜃 (|𝑧|2 − 𝑎𝑧 − 𝑏𝑧 + 𝑎𝑏)) = (sin 𝜃)(𝑥2 + 𝑦2 ) + 𝑎1 𝑥 + 𝑎2 𝑦 + 𝑎0 = 0, for 𝑧 = 𝑥 + i𝑦, and for some real numbers 𝑎0 , 𝑎1 , 𝑎2 . If sin 𝜃 ≠ 0, then this is a circle, and if sin 𝜃 = 0, then it is a line (and by a continuity argument, it contains ∞, so it is a generalized circle). For the second proof, we characterize the Möbius transformations more explicitly. Consider the following maps: • 𝜏(𝑧) = 𝑧 + 𝑎, the translation of vector 𝑎 ∈ ℂ. We denote 𝒯 the set of translations. • 𝜌(𝑧) = 𝑒i𝜃 (𝑧 − 𝑎) + 𝑎, the rotation of center 𝑎 ∈ ℂ and angle 𝜃 ∈ [0, 2𝜋). We denote 𝒫 the set of rotations. • ℎ(𝑧) = 𝜆(𝑧 − 𝑎) + 𝑎, the homothety of center 𝑎 ∈ ℂ and dilation factor 𝜆 ∈ ℝ∗ = ℝ − {0}. We denote ℋ the set of (classical) homotheties. Note that our definition of homotheties for Riemannian manifolds in Remark 4.12(5) is wider. So we will call it classical homothety to ℎ(𝑧).

4. Constant curvature

210

• 𝑟(𝑧) = 𝑒i𝜃 𝑒−i𝜃 (𝑧 − 𝑎)+𝑎, the reflection with respect to the line through 𝑎 ∈ ℂ with angle 𝜃 ∈ [0, 2𝜋) with the horizontal. We denote 𝒮 the set of reflections. (𝑧−𝑎)

• 𝐼(𝑧) = 𝑅2 |𝑧−𝑎|2 + 𝑎 =

𝑅2 𝑧−𝑎

+ 𝑎, the inversion of center 𝑎 ∈ ℂ with respect to

the circle |𝑧 − 𝑎| = 𝑅 of radius 𝑅 > 0, that is, 𝐼(𝑎 + 𝜌𝑒i𝜃 ) = 𝑎 + denote ℐ the set of inversions.

𝑅2 i𝜃 𝑒 . 𝜌

We

The set 𝒯 ∪ 𝒫 ∪ ℋ is the set of orientation preserving homotheties of the plane ℝ2 (cf. Remark 4.12(5)), and the set 𝒯 ∪ 𝒫 ∪ ℋ ∪ (ℋ ∘ 𝒮) is the set of all homotheties of ℝ2 = ℂ. These coincide with the (extended) Möbius maps which send ∞ to ∞. −

For the inversions, clearly ℐ ⊂ Mob (ℂ). The composition of an inversion and a reflection is then a Möbius map. Actually, an extended Möbius map is a composition 𝑎𝑧+𝑏 of maps in 𝒯, 𝒫, ℋ, 𝒮, and ℐ. If 𝑓(𝑧) = 𝑐𝑧+𝑑 , then we can express it as (4.13)

𝑓(𝑧) =

𝑎 𝑏𝑐 − 𝑎𝑑 1 + , 𝑐 𝑐2 𝑧 + 𝑑/𝑐 𝑑

1

the composition of the translation 𝑧 ↦ 𝑧 + 𝑐 , the inversion with reflection 𝑧 ↦ 𝑧 , and the classical homothety 𝑧 ↦ with the reflection 𝑧 ↦ 𝑧.

𝑎 𝑐

+

𝑏𝑐−𝑎𝑑 𝑧. 𝑐2

If 𝑓(𝑧) =

𝑎𝑧+𝑏 , 𝑐𝑧+𝑑

then we compose the above

ˆ Now the preservation of ℛ by the maps in Mob(ℂ) comes down to checking it for each of the above types of maps. It is well known that all homotheties preserve lines 1 and circles, so we only need to check it for the inversion 𝐼(𝑧) = 𝑧 with respect to the unit circle. This is a basic fact in plane geometry: inversions send lines through the center to themselves; lines not passing through the center to circles through the center, and conversely; and circles not through the center to other circles not through the center. □ Remark 4.21. When viewed in 𝕊2 , Möbius maps send circles in 𝕊2 to circles. By Exercise 4.7, the generalized circles of ℂ correspond to circles in 𝕊2 , and the Möbius maps preserve them by Lemma 4.20. Proposition 4.22. Any biholomorphism of ℂ is a Möbius map. Proof. Suppose that 𝑓 ∶ ℂ → ℂ is biholomorphic. Choose 𝑔 ∈ Mob(ℂ) so that 𝑔(𝑞) = ∞, where 𝑞 = 𝑓(∞) ∈ ℂ, and consider the map ℎ = 𝑔 ∘ 𝑓. Then ℎ is a biholomorphism and ℎ(∞) = ∞. It is enough to check that ℎ is a Möbius transformation, since then 𝑓 = 𝑔−1 ∘ ℎ is a Möbius transformation as well. As ℎ(∞) = ∞, we need to see that ℎ(𝑧) = 𝑎𝑧 + 𝑏 for suitable 𝑎, 𝑏 ∈ ℂ, 𝑎 ≠ 0. The map ℎ restricts to a holomorphic map ℎ ∶ ℂ → ℂ, 𝑤 = ℎ(𝑧). We look at the map ℎ on the charts around ∞ ∈ ℂ in the origin and ∞ ∈ ℂ in the target. These are 1 1 given by coordinates 𝑧′ = 𝑧 , 𝑤′ = 𝑤 , respectively, and the map 𝑤 = ℎ(𝑧) is written as ̂ ′ ), where 𝑤′ = ℎ(𝑧 ̂ ′ ) = 𝑤′ = 1 = 1 = 1 . ℎ(𝑧 1 𝑤 ℎ(𝑧) ℎ( 𝑧′ ) ̂ ̂ ′ ) is a diffeomorphism, a standard arguAs ℎ(0) = 0 and 𝑟 = |ℎ′̂ (0)| > 0, because ℎ(𝑧 ment using the Taylor expansion of ℎ ̂ near 0 implies that for 𝜀 > 0 small enough we

4.1. Positive constant curvature

211

1 ̂ ′ )| ≤ 3 𝑟|𝑧′ | if |𝑧′ | < 𝜀. Substituting back 𝑧′ = have that 2 𝑟|𝑧′ | ≤ |ℎ(𝑧 2

we have

𝑟 1 2 |𝑧|

≤

1 |ℎ(𝑧)|

≤

3𝑟 1 , 2 |𝑧|

1 𝑧

̂ ′) = and ℎ(𝑧

1 , ℎ(𝑧)

1

for |𝑧| > 𝜀 . This yields 2 2 |𝑧| ≤ |ℎ(𝑧)| ≤ |𝑧| 3𝑟 𝑟

1

on 𝑈 = {|𝑧| > 𝜀 }, i.e., for 𝑧 on a neighbourhood of ∞. Since ℂ − 𝑈 is compact, ℎ(𝑧) is bounded there. So, summing up, there exists 𝐴, 𝐵 > 0 such that |ℎ(𝑧)| ≤ 𝐴|𝑧| + 𝐵, for all 𝑧 ∈ ℂ. By the Liouville theorem (Exercise 5.1), we conclude that ℎ(𝑧) = 𝑎 + 𝑏𝑧, for some 𝑎, 𝑏 ∈ ℂ, as required. □ Corollary 4.23. If we regard the sphere 𝕊2 as ℂ = ℂ ∪ {∞}, Isom+ (𝕊2 ) < Mob(ℂ) ≅ ˆ PGL(2, ℂ) is a subgroup of the Möbius group, and Isom− (𝕊2 ) < Mob(ℂ) is a subgroup ˆ of the extended Möbius group. The group Mob(ℂ) is the group of conformal maps of 𝕊2 . Proof. Any orientation preserving conformal map of 𝕊2 is biholomorphic, by Remark 4.17. By Proposition 4.22 it is a Möbius map. Regarding orientation reversing conformal maps of 𝕊2 , if 𝑔 ∶ 𝕊2 → 𝕊2 is orientation reversing and conformal, then 𝑔 ∘ 𝑟 is orientation preserving and conformal, where 𝑟(𝑧) = 𝑧. By the above, 𝑔 ∘ 𝑟 = 𝑓 ∈ Mob(ℂ) − and thus 𝑔 = 𝑓 ∘ 𝑟 ∈ Mob (ℂ), by Remark 4.19. □ Remark 4.24. We can give an alternative proof of the description of the geodesics of 𝕊2 in stereographic coordinates using Corollary 4.23 (see Figure 4.3). Take the standard geodesic 𝑆 1 ⊂ ℂ, which is the image of the equator. Applying the isometries of 𝕊2 , we get generalized circles, because isometries are Möbius maps and we use Lemma 4.20. If the geodesic is a line, then it passes through ∞, which corresponds to the north pole, hence it is a meridian, i.e., a line through the origin. If the geodesic is a circle, it corresponds to a maximal circle in 𝕊2 , which intersect the equator in two antipodal points. However, the group of isometries of 𝕊2 is a proper subgroup of the Möbius group since dim Isom+ (𝕊2 ) = 3 < dim Mob(ℂ) = 6. The latter dimension agrees with the fact that a Möbius transformation is determined by three points (each of them accounts for two dimensions). Let us see exactly which subgroup it is. Definition 4.25. We define the unitary group as U(𝑛) = {𝐴 ∈ GL(𝑛, ℂ)|𝐴∗ 𝐴 = Id}, where 𝐴∗ = 𝐴𝑡 is the adjoint matrix (the conjugate transpose). We define the special unitary group as SU(𝑛) = {𝐴 ∈ U(𝑛)| det 𝐴 = 1}. The groups U(𝑛) can be interpreted as the complex linear transformations of ℂ𝑛 preserving a Hermitian product, which is the natural analogue of scalar product in a complex vector space (cf. Definition 5.18 and Remark 5.35). If 𝐴 ∈ U(𝑛), then 1 = det(𝐴∗ 𝐴) = det 𝐴 det 𝐴 = | det 𝐴|2 . Hence | det 𝐴| = 1. Proposition 4.26. The group U(2) consists of the matrices of the form U(2) = {(

𝑎 −𝜆𝑏 | 2 ) | |𝑎| + |𝑏|2 = 1, |𝜆| = 1} . 𝑏 𝜆𝑎

4. Constant curvature

212

The group SU(2) consists of the matrices of the form SU(2) = {(

Proof. Write 𝐴 = ( (

𝑎 −𝑏 | 2 ) | |𝑎| + |𝑏|2 = 1} . 𝑏 𝑎

𝑎 𝑐 ). The condition 𝐴∗ 𝐴 = Id gives the equations 𝑏 𝑑

𝑏 𝑎 𝑐 |𝑎|2 + |𝑏|2 )( )=( 𝑏 𝑑 𝑑 𝑎𝑐 + 𝑏𝑑

𝑎 𝑐

𝑎𝑐 + 𝑏𝑑 1 )=( 2 2 0 |𝑐| + |𝑑|

0 ). 1

Therefore we have 𝑎𝑐 + 𝑏𝑑 = 0, so 𝑐 = −𝜆𝑏 and 𝑑 = 𝜆𝑎, for some 𝜆 ∈ ℂ. The other equations give |𝑎|2 + |𝑏|2 = 1 and 1 = |𝑐|2 + |𝑑|2 = |𝜆|2 (|𝑏|2 + |𝑎|2 ) = |𝜆|2 , which implies |𝜆| = 1. This gives that the matrix 𝐴 has the desired aspect. In the case of 𝐴 ∈ SU(2), we have that det 𝐴 = 𝜆(|𝑎|2 + |𝑏|2 ) = 𝜆 = 1. □ For projective matrices, we are allowed to change the determinant. First, we introduce the following notation. For a subgroup G < GL(𝑛, ℂ), it is customary to denote PG = 𝜋(G) < PGL(𝑛, ℂ) under the quotient map 𝜋 ∶ GL(𝑛, ℂ) → PGL(𝑛, ℂ). Now if [𝐴] ∈ PGL(2, ℂ), then the matrix 𝐴′ = (det 𝐴)−1/2 𝐴 is another representative with [𝐴′ ] = [𝐴] and has det 𝐴′ = 1. So PGL(2, ℂ) = GL(2, ℂ)/ℂ∗ ≅ SL(2, ℂ)/ℤ2 = PSL(2, ℂ),

(4.14)

since for matrices 𝐴, 𝐴′ ∈ SL(2, ℂ). If 𝐴′ = 𝜆𝐴, 𝜆 ∈ ℂ∗ = ℂ − {0}, then 1 = det 𝐴′ = 𝜆2 det 𝐴 = 𝜆2 implies that 𝜆 = ±1, hence 𝐴′ = ±𝐴. The same argument works for a matrix [𝐴] ∈ PU(2), so PU(2) = PSU(2) = SU(2)/ℤ2 . Proposition 4.27. The group Isom+ (𝕊2 ) coincides with the subgroup of Möbius transformations PSU(2) < PGL(2, ℂ). Proof. First, we recall the form of the metric g𝑆2 given in (4.7). We can abbreviate the expression using complex numbers. As 𝑧 = 𝑢 + i𝑣, 𝑑𝑧 = 𝑑𝑢 + i𝑑𝑣. Also |𝑧|2 = 𝑢2 + 𝑣2 and |𝑑𝑧|2 = 𝑑𝑢2 + 𝑑𝑣2 . Then 4|𝑑𝑧|2 g𝑆2 = . (1 + |𝑧|2 )2 𝑎 𝑏

−𝑏 ) ∈ SU(2) whose corresponding Möbius map 𝑓[𝐴] ∶ ℂ → ℂ, 𝑎

𝑏+𝑎𝑧

. Let us see that 𝑓[𝐴] is an isometry of 𝕊2 . Using that 𝑤 = − +

Now take 𝐴 = ( 𝑤 = 𝑓[𝐴] (𝑧) = 1

1

𝑏 𝑎−𝑏𝑧

𝑎−𝑏𝑧

, we have 𝑑𝑤 =

𝑎

−1

𝑏

𝑑𝑧. So 2

(𝑎−𝑏𝑧)

4 |𝑑𝑤|2 ∗ 𝑓[𝐴] (g𝑆2 ) = = (1 + |𝑤|2 )2 =

4

|𝑑𝑧|2 |𝑎−𝑏𝑧|4

(1 +

|𝑏+𝑎𝑧|2 |𝑎−𝑏𝑧|2

4|𝑑𝑧|2 (|𝑎 − 𝑏𝑧|2 + |𝑏 + 𝑎𝑧|2 )

2

=

2

) 4 |𝑑𝑧|2 = g𝑆2 , (|𝑧|2 + 1)2

4.1. Positive constant curvature

213

since |𝑎−𝑏𝑧|2 +|𝑏+𝑎𝑧|2 = |𝑎|2 −𝑎𝑏𝑧−𝑎 𝑏𝑧+|𝑏|2 |𝑧|2 +|𝑏|2 +𝑏𝑎𝑧+𝑏 𝑎𝑧+|𝑎|2 |𝑧|2 = |𝑧|2 +1. We conclude that PSU(2) < Isom+ (𝕊2 ) = SO(3). Since both groups are connected and have the same dimension, we conclude that both groups are equal (Exercise 4.5), so Isom+ (𝕊2 ) ≅ PSU(2). □ Remark 4.28. Composing the isometries of Proposition 4.27 with 𝑟(𝑧) = 𝑧, we get the orientation reversing isometries Isom− (𝕊2 ) = {𝑓[𝐴] ∘ 𝑟 | [𝐴] ∈ PSU(2)} = {𝑓(𝑧) =

𝑏 + 𝑎𝑧 | 2 2 | |𝑎| + |𝑏| = 1} . 𝑎 − 𝑏𝑧

Remark 4.29. The Lie group SU(2) is diffeomorphic to 𝑆 3 since a matrix 𝐴 ∈ SU(2) is determined by (𝑎, 𝑏) ∈ ℂ2 = ℝ4 satisfying |𝑎|2 + |𝑏|2 = 1, that is (𝑎, 𝑏) ∈ 𝑆 3 ⊂ ℂ2 . On the other hand, we already know that Isom+ (𝕊2 ) = SO(3). By Proposition 4.27, we have that SO(3) ≅ PSU(2) = SU(2)/ℤ2 , where ℤ2 acts on SU(2) as 𝐴 ∼ −𝐴, that is (𝑎, 𝑏) ∼ (−𝑎, −𝑏). This means that SO(3) is diffeomorphic to the manifold ℝ𝑃 3 = 𝑆 3 /ℤ2 . The isomorphism of Proposition 4.27, 𝐹 ∶ PSU(2) → SO(3), is given by 𝐹([𝐴]) = 𝜑 ∘𝑓[𝐴] ∘𝜑, for [𝐴] ∈ PSU(2), where 𝜑 is the stereographic chart in (4.5). In particular, the map 𝜋 ∶ SU(2) ⟶ PSU(2) ≅ SO(3), 𝜋(𝐴) = 𝐹([𝐴]), −1

is the degree 2 cover 𝜋 ∶ 𝑆 3 → ℝ𝑃 3 . As 𝑆 3 is simply connected, we have that 𝜋 is the universal cover and 𝜋1 (SO(3)) ≅ ℤ2 . In Exercise 4.12, we ask for the explicit equations of 𝜋. This is a very important cover in theoretical physics called the spin cover. Remark 4.30. Another interesting consequence comes from the description of 𝕊2 as a homogeneous space (Remark 3.105). Consider the group 𝐺 = SU(2) acting on ℂ by isometries, and take the point 𝑝0 = 0 = [1, 0] ∈ ℂ = ℂ𝑃 1 . The isotropy is given by 𝐻 = {𝐴 ∈ SU(2)|𝑓[𝐴] (𝑝0 ) = 𝑝0 } = {(

𝑒i𝜃 0

0 )} ≅ 𝑆 1 < SU(2). 𝑒−i𝜃

Then we have a homeomorphism 𝐺/𝐻 ≅ 𝑆 2 , given by [𝐴] ↦ 𝑓[𝐴] (𝑝0 ). This gives the interesting equality 𝑆 2 ≅ SU(2)/𝑆 1 ≅ 𝑆 3 /𝑆 1 . Alternatively, we have a map, called the Hopf map, ℎ ∶ 𝑆3 ⟶ 𝑆2 , whose fibers ℎ−1 (𝑞) are all (homeomorphic to) circles. An explicit description is given in Exercise 4.13. The map ℎ can be interpreted as ℎ ∶ 𝑆 3 ⊂ ℂ2 → ℂ𝑃 1 , ℎ(𝑎, 𝑏) = [𝑎, 𝑏]. A fact in homotopy theory is that the Hopf map ℎ gives a non-trivial element (actually a generator) of 𝜋3 (𝑆 2 ) ≅ ℤ (see Remark 2.27). 4.1.4. Real projective plane. The standard Riemannian metric on ℝ𝑃2 is obtained from the one on 𝕊2 . Note that ℝ𝑃2 = 𝕊2 /ℤ2 , where ℤ2 = ⟨𝑎⟩, 𝑎 ∶ 𝕊2 → 𝕊2 , 𝑎(𝑥, 𝑦, 𝑧) = (−𝑥, −𝑦, −𝑧), is the antipodal map. Clearly the group ℤ2 = ⟨𝑎⟩ acts on 𝕊2 by isometries, hence the quotient map 𝜋 ∶ 𝕊2 → ℝ𝑃 2 is a local isometry, where we endow ℝ𝑃 2 with the metric induced by 𝕊2 .

4. Constant curvature

214

We use the coordinates in ℝ𝑃2 given by the affine chart (Example 1.33(5)) 𝑥 𝑥 𝜑0 ∶ 𝑈0 → ℝ2 , 𝜑0 ([𝑥0 , 𝑥1 , 𝑥2 ]) = ( 𝑥1 , 𝑥2 ), where 𝑈0 = {[𝑥0 , 𝑥1 , 𝑥2 ] | 𝑥0 ≠ 0}. Let 0 0 𝑆 2+ = {(𝑥0 , 𝑥1 , 𝑥2 ) ∈ 𝑆 2 |𝑥0 > 0}. Then 𝜋 ∶ 𝑆 2+ → 𝑈0 is a diffeomorphism, and hence an isometry. The inverse of 𝜑0 ∘ 𝜋 ∶ 𝑆 2+ → ℝ2 is given by (𝑢, 𝑣) ↦ [1, 𝑢, 𝑣] followed by [1, 𝑢, 𝑣] ↦ (𝑥0 , 𝑥1 , 𝑥2 ) = (

1 √1 +

𝑢2

+

𝑣2

,

𝑢 √1 +

𝑢2

+

𝑣2

,

𝑣 √1 + 𝑢 2 + 𝑣 2

),

since it consists of taking the unitary vector in the direction of (1, 𝑢, 𝑣). From this expression, we get 𝑑𝑥0 =

−𝑢 𝑑𝑢 − 𝑣 𝑑𝑣 , (1 + 𝑢2 + 𝑣2 )3/2

𝑑𝑥1 =

(1 + 𝑣2 )𝑑𝑢 − 𝑢𝑣 𝑑𝑣 , (1 + 𝑢2 + 𝑣2 )3/2

𝑑𝑥2 =

(1 + 𝑢2 )𝑑𝑣 − 𝑢𝑣 𝑑𝑢 . (1 + 𝑢2 + 𝑣2 )3/2

The metric is then (1 + 𝑣2 )𝑑𝑢2 − 2𝑢𝑣 𝑑𝑢𝑑𝑣 + (1 + 𝑢2 )𝑑𝑣2 (1 + 𝑢2 + 𝑣2 )2 1 1 + 𝑣2 −𝑢𝑣 = ( ), 2 2 2 −𝑢𝑣 1 + 𝑢2 (1 + 𝑢 + 𝑣 )

gℝ𝑃 2 = 𝑑𝑥02 + 𝑑𝑥12 + 𝑑𝑥22 =

which, in contrast with the stereographic projection, is not conformal. However we can characterize orthogonality in easy terms (see Remark 4.31). The volume element is given by √𝐸𝐺 − 𝐹 2 = (1 + 𝑢2 + 𝑣2 )−3/2 , and hence the area of ℝ𝑃2 is 2𝜋

1 area(ℝ𝑃 ) = ∫ 𝑑𝑢𝑑𝑣 = ∫ ∫ 2 + 𝑣2 )3/2 (1 + 𝑢 ℝ2 0 0 2

∞

𝜌 𝑑𝜌 𝑑𝜃 (1 + 𝜌2 )3/2

∞

= 2𝜋 [

−1 1 ] = 2𝜋 = area(𝕊2 ), 2 1/2 2 (1 + 𝜌 ) 0

which should come as no surprise since 𝕊2 is a cover of ℝ𝑃2 of degree 2. By Proposition 4.6, the (unitary) geodesics of ℝ𝑃 2 are of the form 𝛾(𝑡) = [𝑝0 cos 𝑡 + 𝑣 0 sin 𝑡, 𝑝1 cos 𝑡 + 𝑣 1 sin 𝑡, 𝑝2 cos 𝑡 + 𝑣 2 sin 𝑡], where (𝑝0 , 𝑝1 , 𝑝2 ), (𝑣 0 , 𝑣 1 , 𝑣 2 ) are unitary and orthogonal. These are the straight (projective) lines in ℝ𝑃 2 , but parametrized in a way such that they are closed with length 𝜋. As an example, the geodesic 𝛾(𝑡) = [cos 𝑡, sin 𝑡, 0] is, in the affine chart, of the form 𝜋 𝜋 (𝑢(𝑡), 𝑣(𝑡)) = (tan 𝑡, 0), for − 2 < 𝑡 < 2 . The isometries of ℝ𝑃 2 are obtained from the isometries of 𝕊2 . First recall that PGL(𝑛, ℝ) = GL(𝑛, ℝ)/ℝ∗ is the projective (real) linear group consisting of projective matrices [𝐴], where 𝐴 ∈ GL(𝑛, ℝ), modulo multiplication by non-zero real numbers. A projective matrix [𝐴] ∈ PGL(𝑛, ℝ) induces a projective isomorphism 𝑓[𝐴] ∶ ℝ𝑃 𝑛−1 → ℝ𝑃 𝑛−1 , 𝑓[𝐴] ([𝑥0 , . . . , 𝑥𝑛−1 ]) = [𝐴(𝑥0 , . . . , 𝑥𝑛−1 )𝑡 ]. The linear maps in O(3) < GL(3, ℝ) induce a subset of projective maps of ℝ𝑃2 , denoted PO(3) < PGL(3, ℝ). If 𝐴, 𝐴′ ∈ O(3) satisfy that 𝐴′ = 𝜆𝐴, then 𝜆 = ±1, and hence 𝐴′ = ±𝐴. Note that O(3) = SO(3) ⊔ − − O (3), and the map 𝐴 ↦ −𝐴 interchanges SO(3) and O (3), therefore PO(3) ≅ SO(3). 2 Let us see that Isom(ℝ𝑃 ) = PO(3). If 𝑓 ∈ PO(3), 𝑓 ∶ ℝ𝑃 2 → ℝ𝑃 2 , consider the corresponding map 𝑓 ̃ ∈ SO(3), 𝑓 ̃ ∶ 𝕊2 → 𝕊2 . Then 𝜋 ∘ 𝑓 ̃ = 𝑓 ∘ 𝜋, where 𝜋 ∶ 𝕊2 → ℝ𝑃 2

4.1. Positive constant curvature

215

is the projection. As 𝑓 ̃ is an isometry, 𝑓 is a local isometry, and as 𝑓 is a diffeomorphism, it is an isometry. Conversely, if 𝑓 ∶ ℝ𝑃 2 → ℝ𝑃 2 is an isometry, lift the map to the universal cover 𝑓 ̃ ∶ 𝕊2 → 𝕊2 . This is again an isometry, so 𝑓 is induced by a map 𝑓 ̃ ∈ O(3), and hence 𝑓 ∈ PO(3). Note that ℝ𝑃2 is isotropic. Remark 4.31. Consider the imaginary quadric 𝑄 = {𝑥02 + 𝑥12 + 𝑥22 = 0} of ℝ𝑃 2 . If [𝐴] ∈ PGL(3, ℝ), then the induced map 𝑓[𝐴] ∶ ℝ𝑃 2 → ℝ𝑃 2 is an isometry if and only if it preserves the quadric 𝑄 (that is, 𝐴𝑡 𝐴 = Id, or equivalently [𝐴] ∈ PO(3)). For any point 𝑝 = [𝑎1 , 𝑎2 , 𝑎3 ], the polar line of 𝑝 respect to the quadric 𝑄 is given by 𝑟𝑝 = {𝑎0 𝑥0 + 𝑎1 𝑥1 + 𝑎2 𝑥2 = 0} ⊂ ℝ𝑃 2 . We also say that 𝑝 is the pole of 𝑟𝑝 . We will write 𝑝 = 𝑝𝑟 if 𝑝 is the pole of a projective line 𝑟. Then we have the following: two lines 𝑟, 𝑠 are perpendicular if the pole of 𝑟 belongs to 𝑠. Actually if the pole of 𝑟 is [𝑎0 , 𝑎1 , 𝑎2 ] and the pole of 𝑠 is [𝑏0 , 𝑏1 , 𝑏2 ], then 𝑝 ∈ 𝑟𝑝 when 𝑎0 𝑏0 + 𝑎1 𝑏1 + 𝑎2 𝑏2 = 0, which happens when 𝑟, 𝑠 are orthogonal. This is clear since the maximal circles in 𝕊2 defined by 𝑟 and 𝑠, that is 𝑎0 𝑥0 + 𝑎1 𝑥1 + 𝑎2 𝑥2 = 0 and 𝑏0 𝑥0 + 𝑏1 𝑥1 + 𝑏2 𝑥2 = 0, are perpendicular. 4.1.5. Manifolds with constant curvature 1. Let (𝑀, g) be a complete connected 𝑛-manifold with constant sectional curvature 𝐾𝑀 ≡ 1. Its universal cover is 𝕊𝑛 so 𝑀 is isometric to 𝕊𝑛 /Γ for some Γ < Isom(𝕊𝑛 ) = O(𝑛 + 1) acting freely and properly on 𝕊𝑛 . Observe that 𝑀 is compact being the image of a compact space by the quotient map 𝕊𝑛 → 𝑀 = 𝕊𝑛 /Γ. Since 𝕊𝑛 is compact, we deduce also that Γ must be a finite group. Proposition 4.32. Let Γ < O(𝑛 + 1) be a finite group acting freely and properly on 𝕊𝑛 . Then (1) If 𝑛 is even, either Γ = {Id} or Γ = {± Id} ≅ ℤ2 . (2) If 𝑛 is odd, Γ < Isom+ (𝕊𝑛 ) = SO(𝑛 + 1). Proof. (1) Suppose that 𝑛 is even and take 𝜑 ∈ Γ. By Lemma 4.2, 𝜑 has a matrix as in (4.3) on a suitable orthonormal basis. If 𝜑 ≠ Id, then the eigenvalue +1 cannot appear since otherwise 𝜑 would have a fixed point, which is ruled out by hypothesis. As 𝑛 + 1 is odd, then there must be at least one −1 in the diagonal of (4.3). This implies that 𝜑2 has at least one +1 in the diagonal, so it must be 𝜑2 = Id. The conclusion is that 𝜑 has only the eigenvalues ±1. But the eigenvalues +1 are ruled out, thus 𝜑 = − Id. (2) Suppose now that 𝑛 is odd, so 𝑛 + 1 is even, and let 𝜑 ∈ Γ, 𝜑 ≠ Id. On a suitable orthonormal basis, 𝜑 has matrix as in (4.3). As it cannot have any +1 as eigenvalue, then it only has (possibly) some eigenvalues −1 and some 2 × 2 matrices (representing rotations in the corresponding planes) in the diagonal. As 𝑛 + 1 is even, the number of entries −1 in the diagonal has to be even, so det(𝜑) = 1. This proves that Γ < Isom+ (𝕊𝑛 ), as required. □ Corollary 4.33. (1) If 𝑛 is even, then the list of compact, connected Riemannian 𝑛-manifolds with 𝑐𝑜 constant curvature 𝑘0 ≡ 1 is given by 𝕃𝐑𝐢𝐞𝐦𝑛 = {𝕊𝑛 , ℝ𝑃 𝑛 }. In particular, for 𝑐𝑜

surfaces, 𝕃𝐑𝐢𝐞𝐦2

𝐾≡1

= {𝕊2 , ℝ𝑃 2 }.

𝐾≡1

4. Constant curvature

216

(2) If 𝑛 is odd, then every compact Riemannian manifold of dimension 𝑛 and constant curvature 𝑘0 ≡ 1 is orientable. 𝑛

Here we denote 𝐑𝐢𝐞𝐦𝐾≡𝑘0 the category of complete Riemannian 𝑛-manifolds of constant curvature 𝐾 ≡ 𝑘0 . Example 4.34. Let us show some non-trivial 3-dimensional manifolds with constant curvature 𝑘0 = 1. (1) We have ℝ𝑃3 = 𝕊3 /ℤ2 , which is orientable. (2) Consider the group Γ = ⟨𝜑⟩ ≅ ℤ𝑝 < SO(4) with 2𝜋

cos 𝑝 ⎛ 2𝜋 ⎜ sin 𝑝 𝜑=⎜ 0 ⎜ 0 ⎝

2𝜋 𝑝 2𝜋 𝑝

− sin

0

0

cos

0

0

0

cos

0

sin

2𝜋𝑞 𝑝 2𝜋𝑞 𝑝

2𝜋𝑞 𝑝 2𝜋𝑞 𝑝

− sin cos

⎞ ⎟ ⎟. ⎟ ⎠

Here 𝑝 is a positive integer number and 𝑞 is another integer with 0 < 𝑞 < 𝑝 and gcd(𝑝, 𝑞) = 1. The last condition ensures that no power 𝜑𝑘 , 0 < 𝑘 < 𝑝, has a fixed point. The group Γ ≅ ℤ𝑝 is a finite cyclic group acting freely and properly in 𝕊3 (actually, any finite cyclic subgroup of SO(4) is conjugate to one of this form). Therefore the quotient 𝐿(𝑝, 𝑞) = 𝕊3 /Γ is a Riemannian manifold with constant curvature 𝑘0 = 1. The manifold 𝐿(𝑝, 𝑞) is called lens space with coefficients 𝑝, 𝑞. Observe that 𝜋1 (𝐿(𝑝, 𝑞)) ≅ ℤ𝑝 . The lens spaces 𝐿(𝑝, 𝑞1 ), 𝐿(𝑝, 𝑞2 ) are homotopy equivalent if and only if 𝑞1 𝑞2 ≡ ±𝑛2 (mod 𝑝) for some 𝑛 ∈ ℤ. They are homeomorphic if and only if 𝑞1 ≡ ±𝑞2 (mod 𝑝) or 𝑞1 𝑞2 ≡ ±1 (mod 𝑝). This shows that the homotopy and the topological classifications of 𝑛-manifolds differ for 𝑛 ≥ 3 (Remark 2.7). (3) The Poincaré dodecahedral manifold. This is a 3-manifold of the form 𝑀 = 𝕊3 /Γ, where Γ < SO(4) is a perfect group, that is, a group whose commutator subgroup [Γ, Γ] = Γ. Therefore 𝜋1 (𝑀) = Γ is non-trivial, but 𝐻1 (𝑀) = Ab(𝜋1 (𝑀)) = Γ/[Γ, Γ] = 0. The importance of this manifold comes from the fact that the initial question of Poincaré was whether a compact 3-manifold 𝑀 with 𝐻• (𝑀) ≅ 𝐻• (𝑆3 ) is homeomorphic to 𝑆3 . Poincaré himself produced this counterexample and corrected the question to whether a compact simply connected 3-manifold is homeomorphic to 𝑆 3 (see [Poi]). Note that by Exercise 2.25, this is rephrased as 𝑀 ∼ 𝑆 3 ⇒ 𝑀 ≅ 𝑆 3 . This is the way in which the Poincaré conjecture is stated in Remark 2.31. (4) The group Γ of item (3) is constructed as follows (we just sketch the steps). Take a standard dodecahedron 𝐷 ⊂ ℝ3 . Consider the group 𝐺 𝐷 < SO(3) of isometries preserving 𝐷. This is isomorphic to the alternating group 𝐴5 , which is seen as follows: the dodecahedron 𝐷 contains five cubes 𝑄1 , . . . , 𝑄5 whose sides are diagonals of 𝐷. The isometries of 𝐺 𝐷 permute the cubes defining a permutation of 𝑆 5 , which can be seen to be an even permutation. So 𝐺 𝐷 ≅ 𝐴5 , which is a simple group of order |𝐴5 | = 60. As it is simple

4.2. Vanishing constant curvature

217

(i.e., it has no non-trivial normal subgroups), [𝐴5 , 𝐴5 ] = 𝐴5 . Use the double cover 𝜋 ∶ SU(2) → SO(3) and its preimage Γ = 𝜋−1 (𝐺 𝐷 ). Using that SU(2) < SO(4), via ℂ2 = ℝ4 , we have Γ < SO(4), which is of order |Γ| = 120. As 𝐺 𝐷 = Γ/ℤ2 , the commutator [Γ, Γ] maps surjectively to [𝐺 𝐷 , 𝐺 𝐷 ] = 𝐺 𝐷 . If [Γ, Γ] were a proper subgroup, then it has to be of index 2, and this gives an isomorphism Γ ≅ 𝐺 𝐷 × ℤ2 . It can be proved that this would imply that SU(2) ≅ SO(3) × ℤ2 , which is false. So [Γ, Γ] = Γ, thus Γ is perfect, and Γ < SO(4) acts on 𝑆 3 . (5) There is a list of all finite groups Γ < SO(4) although it is relatively involved [C-S]. With this at hand, we can classify all compact connected 3-manifolds with constant curvature 𝑘0 = 1.

4.2. Vanishing constant curvature 4.2.1. The Euclidean space. To construct the simply connected space form of constant sectional curvature zero, we take ℝ𝑛 as underlying differentiable manifold, use the global canonical coordinates (𝑥1 , . . . , 𝑥𝑛 ), and consider the standard Riemannian metric g𝑠𝑡𝑑 = 𝑑𝑥12 + ⋯ + 𝑑𝑥𝑛2 . Observe that, since the Christoffel symbols are given as derivatives of the components of the Riemannian metric and those are constant for g𝑠𝑡𝑑 , we have Γ𝑖𝑗𝑘 = 0 for 1 ≤ 𝑖, 𝑗, 𝑘 ≤ 𝑛. This implies that the curvature tensor vanishes, 𝑅 ≡ 0, and so the sectional curvature vanishes too. Again due to the vanishing of the Christoffel symbols, the associated Levi-Civita connection is just the trivial connection ∇𝑋 𝑌 = ∑𝑖 𝑋(𝑌 𝑖 )𝜕𝑥𝑖 , for 𝑌 = ∑𝑖 𝑌 𝑖 𝜕𝑥𝑖 and 𝑋 ∈ 𝔛(ℝ𝑛 ) (cf. Example 3.17). In particular, the covariant derivative along a curve 𝐷𝑋 𝛾 ∶ 𝐼 → ℝ𝑛 is just the usual derivative, 𝑑𝑡 = 𝑋 ′ (𝑡), for 𝑋 ∈ 𝔛(𝛾). A vector field is parallel if and only if its coordinate functions are constant, 𝑋(𝑡) = ∑ 𝑎𝑖 𝜕𝑥𝑖 . Therefore 𝑡 ,𝑡 the parallel transport is the identity 𝑃𝛾 0 1 (𝑣) = 𝑣, identifying 𝑇𝑝 ℝ𝑛 ≅ ℝ𝑛 , for all 𝑝 ∈ ℝ𝑛 . Hence the holonomy Hol∇ = {Id}. This again proves that the curvature vanishes by Remark 3.65. Remark 4.35. When the curvature of a connection vanishes, we say that the connection is flat. A Riemannian manifold is flat if its sectional curvature vanishes, that is the Levi-Civita connection is flat. This implies that the parallel transport between two points does not depend on the choice of path joining them for homotopic paths, see Exercise 4.17. A curve 𝛾 ∶ 𝐼 ⊂ ℝ → ℝ𝑛 is a geodesic if and only if 𝐷𝛾′ (𝑡) = 𝛾″ (𝑡). 𝑑𝑡 This means that 𝛾(𝑡) = (𝑎1 𝑡 + 𝑏1 , . . . , 𝑎𝑛 𝑡 + 𝑏𝑛 ), for some constants 𝑎𝑖 , 𝑏𝑖 ∈ ℝ, and 𝑡 ∈ ℝ, i.e., the geodesics are the straight lines of ℝ𝑛 . In fact, the exponential map is given by exp𝑝 ∶ 𝑇𝑝 ℝ𝑛 ≅ ℝ𝑛 ⟶ ℝ𝑛 , exp𝑝 (𝑣) = 𝑝 + 𝑣, 0=

4. Constant curvature

218

for any 𝑝 ∈ ℝ𝑛 . This implies that ℝ𝑛 is complete. The injectivity radius is infinite, as exp𝑝 is a global diffeomorphism. Thus (ℝ𝑛 , g𝑠𝑡𝑑 ) is a simply connected complete Riemannian manifold of vanishing sectional curvature, so it is the simply connected space form of curvature zero. It is called the Euclidean space, and will be denoted 𝔼𝑛 = (ℝ𝑛 , g𝑠𝑡𝑑 ). Suppose that 𝜑 ∶ 𝔼𝑛 → 𝔼𝑛 is an isometry. Let 𝑝 = 𝜑(0) be the image of the origin. Then 𝜑(exp0 (𝑣)) = exp𝑝 (𝑑0 𝜑(𝑣)), for any 𝑣 ∈ ℝ𝑛 . Then 𝜑(𝑣) = 𝑝 + 𝐿(𝑣), where 𝐿 = 𝑑0 𝜑 is a linear isometry. A linear isometry is given by an orthogonal matrix 𝐴 ∈ O(𝑛) with matrix given by (4.1). Hence 𝜑 is the affine transformation given as (𝑦1 , . . . , 𝑦𝑛 ) = 𝜑(𝑥1 , . . . , 𝑥𝑛 ), 𝑦1 𝑝1 𝑎11 (⋮)=(⋮)+( ⋮ 𝑦𝑛 𝑝𝑛 𝑎𝑛1

... ⋱ ...

𝑎1𝑛 𝑥1 ⋮ )( ⋮ ), 𝑎𝑛𝑛 𝑥𝑛

or 1 ⎛ ⎞ ⎛ 𝑦1 ⎜ ⎟=⎜ ⎜⋮⎟ ⎜ ⎝𝑦𝑛 ⎠ ⎝

1 𝑝1 ⋮ 𝑝𝑛

0 𝑎11 ⋮ 𝑎𝑛1

... ... ⋱ ...

0 𝑎1𝑛 ⋮ 𝑎𝑛𝑛

1 ⎞⎛ ⎞ 𝑥1 ⎟⎜ ⎟, ⎟⎜ ⋮ ⎟ ⎠ ⎝𝑥𝑛 ⎠

where 𝑝 = (𝑝1 , . . . , 𝑝𝑛 ). Conversely, given 𝐴 ∈ O(𝑛) and a point 𝑝 ∈ ℝ𝑛 , the map 𝜑(𝑥) = 𝑝 + 𝐴𝑥 is an isometry. Indeed, we have that 𝑑𝑞 𝜑 = 𝐴 for all 𝑞 ∈ ℝ𝑛 . To check this, take 𝑣 ∈ 𝑇𝑞 ℝ𝑛 and the curve 𝑐(𝑡) = 𝑞 + 𝑡𝑣 with 𝑐(0) = 𝑞, 𝑐′ (0) = 𝑣. Then 𝑑 𝑑 𝑑𝑞 𝜑(𝑣) = 𝑑𝑡 |𝑡=0 𝜑(𝑐(𝑡)) = 𝑑𝑡 |𝑡=0 (𝑝 + 𝐴(𝑞 + 𝑡𝑣)) = 𝐴𝑣. So 𝑑𝑞 𝜑 = 𝐴 is a linear isometry for all 𝑞 ∈ ℝ𝑛 , hence 𝜑 ∶ (ℝ𝑛 , g𝑠𝑡𝑑 ) → (ℝ𝑛 , g𝑠𝑡𝑑 ) is an isometry. We define the affine orthogonal group as AO(𝑛) = {(

1 𝑝𝑡

0 ) | 𝑝 ∈ ℝ𝑛 , 𝐴 ∈ O(𝑛)} , 𝐴

and by the discussion above, Isom (𝔼𝑛 ) ≅ AO(𝑛). Remark 4.36. • The previous computation actually shows that 𝔼𝑛 is isotropic. Indeed, given two points 𝑝, 𝑞 ∈ ℝ𝑛 , and a linear isometry 𝐿 ∶ 𝑇𝑝 ℝ𝑛 → 𝑇𝑞 ℝ𝑛 , given by 𝐿(𝑣) = 𝐴𝑣, 𝐴 ∈ O(𝑛), we have that 𝜑(𝑥) = (𝑞 − 𝐴𝑝) + 𝐴𝑥 is the desired global isometry with 𝜑(𝑝) = 𝑞 and 𝑑𝑝 𝜑 = 𝐴. • From the formula of the dimension of O(𝑛) in (4.2), we have that dim Isom(𝔼𝑛 ) = dim ℝ𝑛 + dim O(𝑛) = 𝑛 +

𝑛(𝑛 − 1) (𝑛 + 1)𝑛 = , 2 2

as expected for an isotropic space. • Given 𝜑 ∈ AO(𝑛), say 𝜑(𝑥) = 𝑝 + 𝐴𝑥, we have 𝑑𝑞 𝜑 = 𝐴, so 𝜑 preserves orientation if and only if det 𝐴 = 1. As the group O(n) has two connected components (section 4.1.1), AO(n) has also two connected components. One

4.2. Vanishing constant curvature

219

consists of the isometries that preserve orientation and the other of the isometries that reverse orientation. Isom+ (𝔼𝑛 ) = ASO(𝑛) = {( −

Isom− (𝔼𝑛 ) = AO (𝑛) = {(

1 𝑝𝑡

0 | ) 𝑝 ∈ ℝ𝑛 , 𝐴 ∈ SO(𝑛)} , 𝐴 |

1 𝑝𝑡

0 | ) 𝑝 ∈ ℝ𝑛 , 𝐴 ∈ O(𝑛), det 𝐴 = −1} . 𝐴 |

Proposition 4.37. Let 𝜑 ∈ AO(𝑛). Then there is an orthonormal affine reference in which 𝜑 has matrix of one of the following forms:

(4.15)

⎛ ⎜ ⎜ (a) ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ (b) ⎜ ⎜ ⎜ ⎜ ⎝

1 𝑙 0 0 ⋮

1 0 0 cos 𝜃1 0 − sin 𝜃1 ⋮

⋯ sin 𝜃1 cos 𝜃1

0

⋱ −1 ⋱

0

0 0 1 0 0

0 0 cos 𝜃1 − sin 𝜃1

⋯ ⋯ ⋯ sin 𝜃1 cos 𝜃1 ⋱ −1 ⋱

0

0

⋯

⎞ ⎟ ⎟ ⋮ ⎟, ⎟ ⎟ ⎟ 1 ⎠ 0 ⎞ 0 ⎟ ⎟ ⎟ , ⋮ ⎟ ⎟ ⎟ ⎟ 1 ⎠

where 0 < 𝜃𝑗 < 𝜋, and 𝑙 > 0. Proof. The differential 𝐴 = 𝑑𝑝 𝜑 is an orthogonal matrix, so by Lemma 4.2, there is an orthonormal basis (𝑣 1 , . . . , 𝑣 𝑛 ) such that 𝐴 has matrix (4.3) with respect to this basis. If 𝜑 has a fixed point 𝑝0 , then, with respect to the orthonormal affine reference {𝑝0 ; (𝑣 1 , . . . , 𝑣 𝑛 )}, 𝜑 has matrix (a). If 𝜑 does not have a fixed point, then note that there must be a vector in the gen1 0 eralized eigenspace associated to the eigenvalue 𝜆 = 1 of the matrix 𝐴′ = ( 𝑡 ) 𝑝 𝐴 𝑛 associated to 𝜑, which is not in {0} × ℝ . This is because the eigenvalue 𝜆 = 1 has multiplicity one unit more for 𝐴′ than for 𝐴. Let 𝑎 = (𝑎0 , 𝑎1 , . . . , 𝑎𝑛 ) with 𝑎0 ≠ 0, be an element in such generalized eigenspace, given as ker(𝐴′ −Id)𝑟 for 𝑟 large enough. Dividing by 𝑎0 , we can assume that 𝑎0 = 1, so 𝑎 corresponds to a point 𝑝0 = (𝑎1 , . . . , 𝑎𝑛 ) ∈ ℝ𝑛 . Write 𝜑(𝑝0 ) = 𝑝0 + 𝑣, for some vector 𝑣 ≠ 0. Then 𝑣 = (𝐴′ − Id)(𝑎), so 𝑣 lives in the eigenspace of eigenvalue 1 of the matrix 𝐴, hence it must be a fixed vector of 𝐴. 𝑣 We can choose the basis (𝑣 1 , . . . , 𝑣 𝑛 ) of Lemma 4.2 in such a way that 𝑣 1 = ||𝑣|| . Letting 𝑙 = ||𝑣|| > 0, then we have that 𝜑 has matrix (b) with respect to the orthonormal reference {𝑝0 ; (𝑣 1 , . . . , 𝑣 𝑛 )}. □

4. Constant curvature

220

Flat surfaces. Let 𝑀 be a complete connected Riemannian 𝑛-manifold with curvature 𝐾 ≡ 0. Then 𝑀 = 𝔼𝑛 /Γ, where Γ < Isom(𝔼𝑛 ) = AO(𝑛) acts freely and properly. In particular, any element of Γ different from the identity cannot have a fixed point. So it is of the form (b) in (4.15). Hence we are led to study groups Γ in which its non-trivial elements are exclusively of type (b). In the case of surfaces, the elements of type (b) are of two kinds: 1 0 0 1 0 0 (4.16) (i) ( 𝑙 1 0 ) or (ii) ( 𝑙 1 0 ) , 0 0 1 0 0 −1 where 𝑙 > 0. The first one is orientation preserving and the second one is orientation reversing. Thus the fixed point free isometries of (ℝ2 , g𝑠𝑡𝑑 ) are the following: (i) 𝜏𝑣 (𝑥) = 𝑥 + 𝑣, for a non-zero vector 𝑣 ∈ ℝ2 . These are translations, which are the orientation preserving fixed point free isometries. They are of type (i) in (4.16). (ii) 𝜎ℓ,𝑣 (𝑥) = 𝑠ℓ (𝑥) + 𝑣, where ℓ ⊂ ℝ2 is a straight line, 𝑠ℓ is the reflection with axis ℓ, and 𝑣 ∈ ℝ2 is a non-zero vector in the direction of ℓ. These are the so-called glide reflections and are the orientation reversing fixed point free isometries. They are of type (ii) in (4.16). 4.2.2. Classification of orientable surfaces with vanishing curvature. Suppose now that 𝑆 = 𝔼2 /Γ is a flat complete connected surface which is orientable. Then Γ < Isom+ (𝔼2 ) acts on ℝ2 freely and properly by orientation preserving isometries. By the previous discussion, Γ is a subgroup of the group of translations. The group of translations is Abelian, so Γ is an Abelian group. Let ΛΓ ⊂ ℝ2 be the set of vectors of the translations in Γ, that is ΛΓ = {𝑣 ∈ ℝ2 | 𝜏𝑣 ∈ Γ} . When Γ is understood from the context, we will just write Λ. We have an isomorphism of groups Γ ≅ (Λ, +), 𝜏𝑣 ↦ 𝑣. Definition 4.38. Let Λ ⊂ ℝ𝑛 be an additive subgroup. We say that Λ is a lattice if Λ is generated by some 𝑣 1 , . . . , 𝑣 𝑟 ∈ ℝ𝑛 linearly independent (over ℝ). In this case, Λ = ℤ⟨𝑣 1 , . . . , 𝑣 𝑟 ⟩ ≅ ℤ𝑟 as an Abelian group, so 𝑟 is the rank of Λ (Remark 2.106(6)). Example 4.39. Being a lattice is strictly stronger than being a finitely generated Abelian subgroup Λ ⊂ ℝ𝑛 . For example, take 𝑛 = 1 and Λ = ℤ⟨1, 𝜋⟩ ⊂ ℝ. Then Λ is finitely generated, but it is not a lattice. Actually, as the previous example shows, the condition of being a lattice can be formulated in topological terms. Lemma 4.40. Λ ⊂ ℝ𝑛 is a lattice if and only if it is a discrete additive subgroup of ℝ𝑛 . Proof. Assume that Λ is a lattice, say Λ = ℤ⟨𝑣 1 , . . . , 𝑣 𝑟 ⟩. Complete 𝑣 1 , . . . , 𝑣 𝑟 to a basis 𝑣 1 , . . . , 𝑣 𝑛 of ℝ𝑛 and let 𝑓 ∶ ℝ𝑛 → ℝ𝑛 be the linear map with 𝑓(𝑣 𝑖 ) = 𝑒 𝑖 where 𝑒 1 , . . . , 𝑒 𝑛 is the standard basis of ℝ𝑛 . Since 𝑓 is a homeomorphism, Λ is discrete if and only if 𝑛−𝑟 𝑓(Λ) is. But this is obvious since 𝑓(Λ) = ℤ𝑟 × {0} .

4.2. Vanishing constant curvature

221

On the other hand, suppose that Λ ⊂ ℝ𝑛 is a discrete set. Let 𝑙 = inf{||𝑤|| | 𝑤 ∈ Λ − {0}} . Observe that 𝑙 is a minimum since Λ ∩ 𝐵̄𝑙+1 (0) is a discrete compact set, hence finite. Thus, there exists 𝑣 ∈ Λ − {0} with |𝑣| = 𝑙 > 0. We claim that Λ ∩ ℝ⟨𝑣⟩ = ℤ⟨𝑣⟩. To see that, let 𝑤 ∈ Λ ∩ ℝ⟨𝑣⟩, say 𝑤 = 𝜆𝑣 with 𝜆 ∈ ℝ. We can write 𝜆 = 𝑚 + 𝛼 with 𝑚 ∈ ℤ and 0 ≤ 𝛼 < 1 so 𝑤 − 𝑚𝑣 = 𝛼𝑣 ∈ Λ. Since ||𝛼𝑣|| = 𝛼𝑙 < 𝑙, by the minimality of ||𝑣|| we must have that 𝛼 = 0, which implies that 𝑤 = 𝑚𝑣 ∈ ℤ⟨𝑣⟩. Now, let us prove that Λ ⊂ ℝ𝑛 is a lattice by induction in 𝑛. For 𝑛 = 1 the result follows from the previous argument. For 𝑛 > 1, let 𝑣 with ||𝑣|| = 𝑙 as above, and take the orthogonal decomposition ℝ𝑛 = ⟨𝑣⟩ ⊕ 𝑊. Let 𝜋 ∶ ℝ𝑛 → 𝑊 be the orthogonal projection, and let us see that the image Λ′ = 𝜋 (Λ) is a discrete additive subgroup of 𝑊 ≅ ℝ𝑛−1 . Suppose that there is a convergent sequence 𝑥𝑛′ ∈ Λ′ of distinct points. ′ Then 𝑦′𝑛 = 𝑥𝑛+1 − 𝑥𝑛′ ∈ Λ′ = 𝜋(Λ) and 𝑦′𝑛 → 0. Write 𝑦′𝑛 = 𝜋(𝑦𝑛 ), where 𝑦𝑛 = ′ 𝑦𝑛 + 𝜆𝑛 𝑣 ∈ Λ, for some 𝜆𝑛 ∈ ℝ. We write 𝜆𝑛 = 𝑚𝑛 + 𝛼𝑛 with 𝑚𝑛 ∈ ℤ and 0 ≤ 𝛼𝑛 < 1. Then 𝑦𝑛̂ = 𝑦𝑛 − 𝑚𝑛 𝑣 = 𝑦′𝑛 + 𝛼𝑛 𝑣 ∈ Λ and ||𝑦𝑛̂ || ≤ ||𝑦′𝑛 || + ||𝑣||. So 𝑦𝑛̂ is a bounded sequence of distinct points in Λ. This contradicts that Λ is discrete. By the induction hypothesis, Λ′ is a lattice, say Λ′ = ℤ⟨𝑤′1 , . . . , 𝑤𝑠′ ⟩ with 𝑤′1 , . . . , 𝑤𝑠′ ∈ Λ linearly independent. Pick vectors 𝑤 1 , . . . , 𝑤𝑠 ∈ Λ with 𝜋(𝑤 𝑖 ) = 𝑤′𝑖 , for 1 ≤ 𝑖 ≤ 𝑠. It is clear that 𝑣, 𝑤 1 , . . . , 𝑤𝑠 are linearly independent. It remains to see that Λ = ℤ⟨𝑣, 𝑤 1 , . . . , 𝑤𝑠 ⟩. As 𝑣, 𝑤 1 , . . . , 𝑤𝑠 ∈ Λ, we have ℤ⟨𝑣, 𝑤 1 , . . . , 𝑤𝑠 ⟩ ⊂ Λ. For the other inclusion, let 𝑤 ∈ Λ and let 𝑤′ = 𝜋(𝑤) ∈ Λ′ . Then we have 𝑤′ = 𝑚1 𝑤′1 + ⋯ + 𝑚𝑠 𝑤𝑠′ for some 𝑚1 , . . . , 𝑚𝑠 ∈ ℤ, which implies that 𝜋 (𝑤 − ∑ 𝑚𝑖 𝑤 𝑖 ) = 0. Thus 𝑤 −∑𝑖 𝑚𝑖 𝑤 𝑖 = 𝜆𝑣 for some 𝜆 ∈ ℝ. So 𝜆𝑣 ∈ Λ, which implies that 𝜆 ∈ ℤ, since 𝑣 has minimal length in Λ. Therefore 𝑤 = 𝜆𝑣 + ∑𝑖 𝑚𝑖 𝑤 𝑖 ∈ ℤ⟨𝑣, 𝑤 1 , . . . , 𝑤𝑠 ⟩, as we wanted to prove. □ ′

Proposition 4.41. Let Γ be a subgroup of the group of translations of ℝ𝑛 . Then Γ acts freely and properly on ℝ𝑛 if and only if ΛΓ is a lattice. Proof. Suppose that Γ acts freely and properly on ℝ𝑛 . Then, there exists an open neighbourhood 𝑈 of 0 such that 𝜏𝑣 𝑈 ∩ 𝑈 = ∅ for all 𝑣 ∈ ΛΓ − {0}. Since 𝜏𝑣 𝑈 is an open neighbourhood of 𝑣, we have that 𝑈 ∩ΛΓ = {0}, which implies that 0 is not a limit point of ΛΓ . Since translations are homeomorphisms of ℝ𝑛 , we see that 𝜏𝑣 𝑈 ∩ ΛΓ = {𝑣} for all 𝑣 ∈ ΛΓ . This proves that ΛΓ is a discrete additive subgroup of ℝ𝑛 , hence a lattice by Lemma 4.40 (see also Exercise 3.40). Suppose that ΛΓ is a lattice. By Lemma 4.40, ΛΓ is a discrete additive subgroup of ℝ𝑛 . So there exists 𝜖 > 0 such that 𝐵𝜖 (0) ∩ ΛΓ = {0}. In that case, we have that 𝜏𝑣 𝐵𝜖/2 (0) ∩ 𝐵𝜖/2 (0) = 𝐵𝜖/2 (𝑣) ∩ 𝐵𝜖/2 (0) = ∅, for all 𝑣 ∈ ΛΓ − {0}. Since translations are homeomorphisms, the same holds at any 𝑥 ∈ ℝ𝑛 . This proves that Γ acts freely and properly on ℝ𝑛 . □ Let us return to the study of orientable surfaces of zero curvature. Let 𝑆 be an orientable complete connected Riemannian surface of zero curvature. Then 𝑆 = 𝔼2 /Γ, where Γ is a group of translations acting freely and properly. By Proposition 4.41, Λ = ΛΓ ⊂ ℝ2 is a lattice, so we can write 𝑆 = 𝔼2 /Λ. The lattice Λ is generated by 𝑟 linearly

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independent vectors, for some 0 ≤ 𝑟 ≤ 2. We have the following possibilities: (1) If 𝑟 = 0, then Λ = {0} so 𝑆 = 𝔼2 . (2) If 𝑟 = 1, then Λ = ℤ⟨𝑣⟩ for some 𝑣 ≠ 0. After rotating ℝ2 , we can suppose that 𝑣 = (𝑑, 0) where 𝑑 = |𝑣| > 0. Then, we have that 𝑆 = 𝔼2 /Λ is isometric to ([0, 𝑑] × ℝ)/ ∼ , where (0, 𝑦) ∼ (𝑑, 𝑦). Equivalently, 𝑆 ≅ 𝑆 1𝑑/2𝜋 × ℝ, where 𝑆 1𝑑/2𝜋 = ℝ/ℤ⟨𝑑⟩ is the circle of length 𝑑 (i.e., of radius 𝑑/2𝜋). This space Cyl𝑑 = 𝑆 1𝑑/2𝜋 × ℝ is the infinite cylinder of perimeter 𝑑 > 0. These are nonisometric spaces for different 𝑑 > 0, since 𝑑 can be recovered as the length of the shortest closed geodesic in 𝑆. (3) If 𝑟 = 2, then Λ = ⟨𝑣, 𝑤⟩, with 𝑣, 𝑤 linearly independent. Thus, ℝ2 /Λ is diffeomorphic to a torus 𝑇 2 . Indeed, the linear map 𝑓 ∶ ℝ2 → ℝ2 that sends (𝑢, 𝑣) to the standard basis (𝑒 1 , 𝑒 2 ), takes Λ to ℤ2 , so it descends to a diffeomorphism 𝑓 ̄ ∶ ℝ2 /Λ → ℝ2 /ℤ2 = 𝑇 2 . This map is not an isometry in general. As we will see, the metric structure of the flat torus depends on the lattice Λ. A Riemannian torus of this kind is called a flat torus. The fundamental domain for 𝑆 = ℝ2 /Λ is the parallelogram 𝑃 = {𝑡1 𝑣 + 𝑡2 𝑤 | 𝑡1 , 𝑡2 ∈ [0, 1]}. The universal cover 𝜋 ∶ ℝ2 → 𝑆 implies that there is a natural isomorphism 𝜋1 (𝑆, 𝑝0 ) ≅ 𝜋−1 (𝑝0 ) = Λ, where 𝑝0 = 𝜋(0, 0). So 𝐻1 (𝑆) ≅ 𝜋1 (𝑆, 𝑝0 ) ≅ Λ, in a natural way. So the lattice can be recovered topologically. Also from Example 2.159, the generators of 𝐻1 (𝑆) are the paths 𝑎1 (𝑡) = 𝑡𝑣, 𝑎2 (𝑡) = 𝑡𝑤, for 𝑡 ∈ [0, 1], the sides of the parallelogram 𝑃. Proposition 4.42. Let 𝑆 = 𝔼2 /Λ and 𝑆 ′ = 𝔼2 /Λ′ be two flat tori, with Λ, Λ′ ⊂ ℝ2 lattices. Then 𝑆 is isometric to 𝑆 ′ if and only if there exists a linear isometry 𝐿 ∈ O(2) such that 𝐿(Λ) = Λ′ . Proof. If 𝐿 ∶ 𝔼2 → 𝔼2 is a linear isometry with 𝐿(Λ) = Λ′ , then 𝐿 descends to the quotient to a diffeomorphism 𝐿 ̄ ∶ 𝑆 = 𝔼2 /Λ → 𝑆 ′ = 𝔼2 /Λ′ . Since the projections onto the quotient space are local isometries, we have that 𝐿 ̄ is an isometry. On the other hand, if 𝜑 ∶ 𝑆 → 𝑆 ′ is an isometry, let 𝜑̃ ∶ 𝔼2 → 𝔼2 be any lift of 𝜑 to ̃ the universal covers of 𝑆 and 𝑆 ′ . Then 𝜑̃ is an isometry of 𝔼2 so 𝜑(𝑥) = 𝑝0 + 𝐿(𝑥), for ̃ = 𝜏𝐿(𝑤) for any 𝑤 ∈ ℝ2 . In some 𝑝0 ∈ 𝔼2 and 𝐿 ∈ O(2). This implies that 𝜑̃ ∘ 𝜏𝑤 ∘ 𝜑−1 particular, letting 𝜋 ∶ 𝔼2 → 𝑆 and 𝜋′ ∶ 𝔼2 → 𝑆 ′ be the quotient maps, for any 𝑣 ∈ Λ, we have ̃ = 𝜑 ∘ 𝜋 ∘ 𝜏𝑣 ∘ 𝜑−1 ̃ = 𝜑 ∘ 𝜋 ∘ 𝜑−1 ̃ = 𝜑 ∘ 𝜑−1 ∘ 𝜋′ = 𝜋′ , 𝜋′ ∘ 𝜏𝐿(𝑣) = 𝜋′ ∘ 𝜑̃ ∘ 𝜏𝑣 ∘ 𝜑−1 so 𝐿(𝑣) ∈ Λ′ . Therefore 𝐿(Λ) ⊂ Λ′ . Repeating the argument with 𝜑−1 , we obtain the reverse inclusion. □ 4.2.3. Moduli of flat tori. As seen in the previous section, the flat structures on the torus are in one-to-one correspondence with the set of possible rank 2 lattices up to linear isometry. Let us denote by ℛ the set of all rank 2 lattices in ℝ2 . There is an action of O(2) on ℛ, given by (𝐴, Λ) ↦ 𝜑𝐴 (Λ), for 𝐴 ∈ O(2) and Λ ∈ ℛ (this action is on the left, see Remark 4.43). By Proposition 4.42, the flat tori are parametrized by the space ℳ𝑇 2 = O(2)\ℛ.

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Remark 4.43. So far we have written 𝑋/𝐺 for the quotient of a space 𝑋 by the action of a group 𝐺. But now we will need a more specific notation. When a group 𝐺 acts on a space 𝑋 on the left, i.e., (ℎ, 𝑥) ↦ ℎ ⋅ 𝑥, satisfying 𝑔 ⋅ (𝑔′ ⋅ 𝑥) = (𝑔𝑔′ ) ⋅ 𝑥, for 𝑔, 𝑔′ ∈ 𝐺 and 𝑥 ∈ 𝑋, it is customary to write 𝐺\𝑋 for the quotient space. If a group 𝐺 acts on a space 𝑋 on the right, i.e., (𝑔, 𝑥) ↦ 𝑥 ⋅ 𝑔, satisfying (𝑥 ⋅ 𝑔′ ) ⋅ 𝑔 = 𝑥 ⋅ (𝑔′ 𝑔), for 𝑔, 𝑔′ ∈ 𝐺 and 𝑥 ∈ 𝑋, then the quotient is denoted 𝑋/𝐺. This notation is used generally for the situations in which one cares about specifying the actions that are on the left or on the right, and it is important when there are simultaneously actions of both types. Note that the actions previously used (Definition 2.42) were always on the left. However, although uncomfortable, this is not a crucial issue, see Exercise 4.6. We are going to show that there is a natural way of putting a topology on ℳ𝑇 2 reflecting some kind of continuity notion for Riemannian structures. In this sense, the flat structures on the torus are parametrized by the topological space ℳ𝑇 2 . Such parametrizing spaces are usually called moduli spaces (since “modulus” is the Latin word for parameter). The notion of moduli spaces is very important in modern geometry. For a historical introduction to moduli spaces, see [Gro]. Let us start by describing ℛ. For a lattice Λ ∈ ℛ, we pick a basis 𝑣 1 , 𝑣 2 so that Λ = ℤ⟨𝑣 1 , 𝑣 2 ⟩. If we write 𝑣 1 = (𝑣 11 , 𝑣 21 ), 𝑣 2 = (𝑣 12 , 𝑣 22 ) ∈ ℝ2 , then this basis is given by the matrix (4.17)

𝑀=(

𝑣 11 𝑣 21

𝑣 12 ) ∈ GL(2, ℝ), 𝑣 22

whose columns are the vectors 𝑣 1 , 𝑣 2 , respectively. Note that the action of 𝐴 ∈ O(2) on 𝑀 is given by the matrix multiplication 𝐴 𝑀. However, Λ can be given by different matrices. Let 𝑣′1 , 𝑣′2 be another basis, Λ = ℤ⟨𝑣′1 , 𝑣′2 ⟩, with associated matrix 𝑀 ′ ∈ GL(2, ℝ). Then there must be integers 𝑛11 , 𝑛12 , 𝑛21 , 𝑛22 such that (4.18)

𝑣′1 = 𝑛11 𝑣 1 + 𝑛21 𝑣 2 , 𝑣′2 = 𝑛12 𝑣 1 + 𝑛22 𝑣 2 .

𝑛 𝑛12 Writing 𝑁 = ( 11 ), this is rewritten as 𝑀 ′ = 𝑀𝑁. Note that 𝑁 ∈ GL(2, ℤ), the 𝑛21 𝑛22 set of 2 × 2 matrices with integer coefficients that have an inverse also with integer coefficients. Certainly, there must be another matrix with integer coefficients 𝑁 ′ such that 𝑀 = 𝑀 ′ 𝑁 ′ . Hence 𝑀 = 𝑀𝑁𝑁 ′ , so that 𝑁𝑁 ′ = Id. This means that 𝑁 is invertible over the integers. Remark 4.44. A matrix 𝑁 ∈ 𝑀2×2 (ℤ) is invertible over the integers if and only if det 𝑁 is a unit in ℤ, that is if det 𝑁 = ±1. Indeed if 𝑁 is invertible, then there is an integer matrix 𝑁 ′ with 𝑁𝑁 ′ = Id. Taking determinants, (det 𝑁)(det 𝑁 ′ ) = 1, so det 𝑁 = ±1. Conversely, if det 𝑁 = ±1, then 𝑁 −1 is computed by taking the matrix of the adjoints of 𝑁 and dividing by det 𝑁. The adjoints are obviously integer numbers. As det 𝑁 is a unit, we get that 𝑁 −1 is a matrix with integer coefficients. The action of GL(2, ℤ) on GL(2, ℝ) is on the right, since (𝑁, 𝑀) ↦ 𝑀𝑁. The discussion above means that (4.19)

ℛ = GL(2, ℝ)/ GL(2, ℤ).

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Therefore the moduli space ℳ𝑇 2 is the double coset space (4.20)

ℳ𝑇 2 = O(2)\ GL(2, ℝ)/ GL(2, ℤ).

Note that this is actually an action of O(2) × GL(2, ℤ), since the actions of the two groups commute (Exercise 4.6). To topologize ℳ𝑇 2 , we consider the usual topology on GL(2, ℝ) and give ℳ𝑇 2 the quotient topology. To find an explicit description of (4.20), we need a fundamental domain for the action. Theorem 4.45. A fundamental domain for the action of O(2) × GL(2, ℤ) on GL(2, ℝ) is given by the matrices, 𝒟 = {(

𝑑 0

𝛼 | ) 𝑑 > 0, 𝛽 > 0, 𝛼2 + 𝛽 2 ≥ 𝑑 2 , 0 ≤ 𝛼 ≤ 𝑑/2} ⊂ GL(2, ℝ). 𝛽 |

Moreover, the space ℳ𝑇 2 is homeomorphic to 𝒟 ≅ {(𝑑, 𝛼, 𝛽) ∈ ℝ3 |𝑑 > 0, 𝛽 > 0, 𝛼2 +𝛽 2 ≥ 𝑑 2 , 0 ≤ 𝛼 ≤ 𝑑/2} (see Figure 4.4). Proof. Given a lattice Λ, we are going to find a special basis of it, which is essentially unique. First, let (4.21)

𝑑 = min{||𝑤|| | 𝑤 ∈ Λ − {0} } > 0,

and take a vector 𝑣 1 ∈ Λ − {0} with ||𝑣 1 || = 𝑑. Then Λ ∩ ℝ⟨𝑣 1 ⟩ = ℤ⟨𝑣 1 ⟩. Consider now (4.22)

𝑙 = min{||𝑤|| | 𝑤 ∈ Λ − ℤ⟨𝑣 1 ⟩ } ≥ 𝑑,

and take a vector 𝑣 2 ∈ Λ − {0} with ||𝑣 2 || = 𝑙. Changing 𝑣 2 by −𝑣 2 if necessary, we can assume that the angle 𝜃 = ∠(𝑣 1 , 𝑣 2 ) is acute or right, i.e., ⟨𝑣 1 , 𝑣 2 ⟩ ≥ 0. Let us write an associated matrix to the basis (𝑣 1 , 𝑣 2 ). Take an orthonormal basis of ℝ2 (which corresponds to acting by an isometry of O(2) on ℝ2 ) in the following way: 𝑣 let 𝑒 1 = ||𝑣1 || , and let 𝑒 2 be a unitary vector perpendicular to 𝑒 1 and in the direction of 1 𝑣 2 , i.e., ⟨𝑒 2 , 𝑣 2 ⟩ ≥ 0. In these coordinates, we have 𝑣 1 = (𝑑, 0) and 𝑣 2 = (𝛼, 𝛽), with 𝛽 > 0 and 𝑙 = √𝛼2 + 𝛽 2 . As ⟨𝑣 1 , 𝑣 2 ⟩ ≥ 0, we have 𝛼 ≥ 0. Moreover, 𝛼 ∈ [0, 𝑑/2], since if 𝛼 > 𝑑/2, then we make the division 𝛼 = 𝑛𝑑 + 𝑎, with 𝑛 ∈ ℤ, −𝑑/2 ≤ 𝑎 ≤ 𝑑/2, and so 𝑣 2 − 𝑛𝑣 1 = (𝑎, 𝛽) ∈ Λ and ||𝑣 2 − 𝑛𝑣 1 ||2 ≤ (𝑑/2)2 + 𝛽 2 < ||𝑣 2 ||2 , which is a contradiction with the choice of 𝑣 2 . Summarizing, we get a matrix 𝑀 ∈ 𝒟. Note that we can parametrize (𝑑, 𝛼, 𝛽) by (𝑑, 𝑙, 𝜃), where 𝛼 = 𝑙 cos 𝜃, 𝛽 = 𝑙 sin 𝜃. To guarantee that the vectors 𝑣 1 , 𝑣 2 are a basis of Λ, we check that the vectors are chosen by the procedure followed in the proof of Lemma 4.40. This is clear for 𝑣 1 . In the coordinates above, the orthogonal decomposition is ℝ2 = ⟨𝑣 1 ⟩ ⊕ 𝑊 = ⟨𝑒 1 ⟩ ⊕ ⟨𝑒 2 ⟩, and the orthogonal projection 𝜋 ∶ ℝ2 → 𝑊 ≅ ℝ is 𝜋(𝑥, 𝑦) = 𝑦. We only have to see that 𝜋(𝑣 2 ) = 𝛽 generates Λ′ = 𝜋(Λ) ⊂ ℝ. Indeed if there is some 0 < 𝛽 ′ < 𝛽 with 𝛽 ′ ∈ Λ′ , then 𝛽 = 𝑘𝛽 ′ for some integer 𝑘 ≥ 2. There is some 𝑤 = (𝛼′ , 𝛽 ′ ) ∈ Λ, and we can arrange that 𝛼′ ∈ [−𝑑/2, 𝑑/2] by adding a multiple of 𝑣 1 = (𝑑, 0) as before. So ||𝑤||2 = (𝛼′ )2 + (𝛽 ′ )2 ≤ (𝑑/2)2 + (𝛽/2)2 ≤ 𝑑 2 /4 + 𝑙2 /4 < 𝑙2 /2, which is a contradiction, since 𝑣 2 was chosen to be of lowest norm on Λ − ℤ⟨𝑣 1 ⟩. The previous discussion says that the quotient map GL(2, ℝ) → ℳ𝑇 2 restricts to a surjection 𝒟 → ℳ𝑇 2 . We want to see that the map 𝒟 → ℳ𝑇 2 is also injective. This

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0

d=2

d

Figure 4.4. Moduli of flat tori. The point (𝛼, 𝛽) lies in the shadowed region.

means that the parameters (𝑑, 𝛼, 𝛽) are uniquely determined by Λ. For this we have to study the indeterminacy in the choice of the basis of Λ. First note that the parameters (4.21) and (4.22) are well defined. We have the following cases: (1) If 𝑙 > 𝑑, 0 < 𝛼 < 𝑑/2, the choice of 𝑣 1 is unique up to sign, and the choice of 𝑣 2 is unique up to sign. As we require that ⟨𝑣 1 , 𝑣 2 ⟩ ≥ 0, we are left with two possible bases (𝑣 1 , 𝑣 2 ) and (−𝑣 1 , −𝑣 2 ). Such bases are always equivalent via − Id ∈ O(2). So the parameters (𝑑, 𝛼, 𝛽) are well defined. (2) If 𝑙 > 𝑑, 𝛼 = 0, similarly as in (1), 𝑣 1 , 𝑣 2 are defined uniquely up to sign. Now we are left with four possible bases ±(𝑣 1 , 𝑣 2 ), ±(𝑣 1 , −𝑣 2 ). All of them have the same parameters (𝑑, 𝑙, 𝜃 = 𝜋/2). (3) If 𝑙 > 𝑑, 𝛼 = 𝑑/2, then 𝑣 1 is defined up to sign. But once fixed 𝑣 1 , 𝑣 2 , we have that ||𝑣 2 − 𝑣 1 || = ||𝑣 2 || (in the coordinates above, 𝑣 2 = (𝑑/2, 𝛽) and 𝑣 2 − 𝑣 1 = (−𝑑/2, 𝛽)), so the vectors with length 𝑙 are ±𝑣 2 , ±(𝑣 2 − 𝑣 1 ). The condition that the angle is acute leaves us with the following bases: ±(𝑣 1 , 𝑣 2 ), ±(𝑣 1 , −(𝑣 2 − 𝑣 1 )). All of them have the same parameters (𝑑, 𝛼, 𝛽). (4) If 𝑙 = 𝑑, 0 < 𝛼 < 𝑑/2, then the choice of the shortest vector is not unique, actually ±𝑣 1 , ±𝑣 2 are the vectors of length 𝑑. The second shortest vector has to be the one that we do not chose as first vector. The condition of the angle being acute yields the bases ±(𝑣 1 , 𝑣 2 ), ±(𝑣 2 , 𝑣 1 ). Again the parameters (𝑑, 𝑙 = 𝑑, 𝜃) are well defined. (5) If 𝑙 = 𝑑, 𝛼 = 0, then the argument in item (4) gives the eight bases (±𝑣 1 , ±𝑣 2 ), (±𝑣 2 , ±𝑣 1 ). The parameters (𝑑, 𝑙 = 𝑑, 𝜃 = 𝜋/2) are well defined. (6) If 𝑙 = 𝑑, 𝛼 = 𝑑/2. There are six shortest vectors: ±𝑣 1 , ±𝑣 2 , ±(𝑣 2 − 𝑣 1 ). This yields 12 different bases: ±(𝑣 1 , 𝑣 2 ), ±(𝑣 1 , 𝑣 1 − 𝑣 2 ), ±(𝑣 2 , 𝑣 1 ), ±(𝑣 2 , 𝑣 2 − 𝑣 1 ), ±(𝑣 1 − 𝑣 2 , 𝑣 1 ), ±(𝑣 2 − 𝑣 1 , 𝑣 2 ). All of them have the same parameters (𝑑, 𝑙 = 𝑑, 𝜃 = 𝜋/3).

4. Constant curvature

226

It remains to be seen that 𝜋 ∶ 𝒟 → ℳ𝑇 2 is a homeomorphism. It is obviously continuous and bijective, so we need to check that it is a closed map. Suppose that 𝐶 ⊂ 𝒟 is a closed set, and consider 𝜋(𝐶) ⊂ ℳ𝑇 2 . To see that it is closed, we need to check that its preimage in ℛ is closed. Take a point in the adherence of this preimage, that is a basis (𝑣, 𝑤) ∈ ℛ such that there is a sequence in ℛ with (𝑣 𝑛 , 𝑤 𝑛 ) → (𝑣, 𝑤), and Λ𝑛 = ℤ⟨𝑣 𝑛 , 𝑤 𝑛 ⟩ has parameters (𝑑𝑛 , 𝛼𝑛 , 𝛽𝑛 ) ∈ 𝐶. Let (𝑑, 𝛼, 𝛽) be the parameters of Λ = ℤ⟨𝑣, 𝑤⟩. By geometric construction, it is easy to see that 𝑑𝑛 → 𝑑, 𝑙𝑛 → 𝑙, and 𝜃𝑛 → 𝜃, i.e., (𝑑𝑛 , 𝛼𝑛 , 𝛽𝑛 ) → (𝑑, 𝛼, 𝛽). As 𝐶 is closed, (𝑑, 𝛼, 𝛽) ∈ 𝐶 and hence Λ ∈ 𝜋(𝐶), as we wanted to prove. □ Remark 4.46. Given a flat torus (𝑆, g), we can find geometrically the parameters (𝑑, 𝛼, 𝛽) which determine its Riemannian structure. The parameter 𝑑 is the length of the shortest closed geodesic, 𝑙 is the length of the shortest closed geodesic with homology class linearly independent with the first one, and 0 < 𝜃 ≤ 𝜋/2 is the (acute or right) angle between them. Then we recover (𝑑, 𝛼, 𝛽) = (𝑑, 𝑙 cos 𝜃, 𝑙 sin 𝜃). Intuitively, if we were living in a “universe” (which is a flat torus), then we can recover its geometrical structure by the light rays that we receive (light rays follow geodesics), and from the time they take to reach us, we recover the distance travelled (the length of closed geodesics).

l θ d For other ways to recover the geometry of the flat torus from the closed geodesics, see Exercise 4.15 and Exercise 4.16. Other geometrical properties follow from these invariants. For instance, area(𝑆) = 𝑑 𝑙 sin 𝜃, which is the area of the fundamental parallelogram. Remark 4.47. The indeterminacy in the choice of basis of Λ is related to the existence of self-isometries of a flat torus 𝑆 = 𝔼2 /Λ. The group Isom(𝑆) contains all translations 𝜏𝑤 (𝑥) = 𝑥 + 𝑤, 𝑤 ∈ Λ, but it also contains the elements 𝐴 ∈ O(2) such that 𝐴(Λ) = Λ. If (𝑣 1 , 𝑣 2 ) is a “special” basis for Λ, then (𝐴𝑣 1 , 𝐴𝑣 2 ) is another special basis for Λ. So we have the following: (1) For a general lattice 𝑙 > 𝑑, 0 < 𝛼 < 𝑑/2, we only have 𝐴 = ± Id. (2) For a lattice with 𝑙 > 𝑑, 𝛼 = 𝑑/2, we have ± Id and 𝑟(𝑥, 𝑦) = (𝑥, −𝑦). There is a fundamental domain for 𝑆 defined by the vectors (−𝑑/2, 𝛽), (𝑑/2, 𝛽) which is a rhombus.

4.2. Vanishing constant curvature

227

(3) For a lattice with 𝑙 > 𝑑, 𝛼 = 0, the fundamental domain is a rectangle. We have ± Id and 𝑟(𝑥, 𝑦) = (𝑥, −𝑦). (4) For a lattice with 𝑙 = 𝑑, 𝛼 < 0 < 𝑑/2, the fundamental domain is a rhombus. We have ± Id and the reflection with 𝑟(𝑣 1 ) = 𝑣 2 , 𝑟(𝑣 2 ) = 𝑣 1 . (5) For a lattice with 𝑙 = 𝑑, 𝛼 = 0, its fundamental domain is a square. Hence we have ± Id, 𝑟1 (𝑥, 𝑦) = (𝑥, −𝑦), 𝑟2 (𝑥, 𝑦) = (−𝑥, 𝑦), and the rotation of angle 𝜋/2. This produces the dihedral group 𝐷4 of symmetries of the square. (6) For a lattice with 𝑙 = 𝑑, 𝛼 = 𝑑/2, we have a hexagonal lattice (whose points are at the vertices of the hexagon). We have ± Id, 𝑟1 (𝑥, 𝑦) = (𝑥, −𝑦), 𝑟2 (𝑥, 𝑦) = (−𝑥, 𝑦), the reflection with 𝑟(𝑣 1 ) = 𝑣 2 and 𝑟(𝑣 2 ) = 𝑣 1 , and the rotation of angle 𝜋/3. This produces the dihedral group 𝐷6 of symmetries of the hexagon. Remark 4.48. The result of Remark 4.47 produces some well known results, such as the classification of the groups of plane tessellations. These are discrete subgroups of AO(2) which contain a lattice of translations (they correspond to the symmetry group of a tessellation by figures with two independent translations of symmetry). Therefore any group of plane tessellations is a subgroup of the group of isometries of a flat torus, extended with the translations of the lattice. Accounting for the possible subgroups of the automorphism group in each of the cases (1)–(6), and removing repetitions, we get finally the amusing number of 17 groups (Exercise 4.18). Under the identification of Theorem 4.45, ℳ𝑇 2 becomes a topological space. Actually it is a 3-dimensional smooth manifold with corners (see Remark 2.122). This completes the classification of complete connected oriented Riemannian surfaces of zero curvature. They are 𝑐𝑜

𝕃𝐑𝐢𝐞𝐦2

𝐾≡0

2 = {𝔼2 , 𝑆 1𝑑/2𝜋 × ℝ, 𝑇𝑑,𝛼,𝛽 | 𝑑 > 0, (𝑑, 𝛼, 𝛽) ∈ ℳ𝑇 2 } ,

2 where 𝑆 1𝑑/2𝜋 is the circle of length 𝑑, and 𝑇𝑑,𝛼,𝛽 is the flat torus with parameters (𝑑, 𝛼, 𝛽). 1 The family of cylinders Cyl𝑑 = 𝑆 𝑑/2𝜋 ×ℝ form also a (topological) moduli space ℳ𝑐𝑦𝑙 = (0, ∞) parametrized by 𝑑 > 0. So we can write 𝑐𝑜

𝕃𝐑𝐢𝐞𝐦2

𝐾≡0

= {ℝ2 } ⊔ ℳ𝑐𝑦𝑙 ⊔ ℳ𝑇 2 .

It is natural to topologize the three pieces into a single topological space. First when 𝑑 → +∞ in ℳ𝑐𝑦𝑙 = (0, ∞), the cylinder Cyl𝑑 becomes the flat plane 𝔼2 . Similarly, when 2 𝛽 → +∞ in ℳ𝑇 2 and 𝑑 stays fixed, then 𝑇𝑑,𝛼,𝛽 tends to the cylinder Cyl𝑑 = 𝑆 1𝑑/2𝜋 × ℝ. 2 When simultaneously, 𝑑, 𝛽 → +∞, then 𝑇𝑑,𝛼,𝛽 tends to the Euclidean plane 𝔼2 . This 𝑐𝑜 makes 𝕃 = 𝕃𝐑𝐢𝐞𝐦2 into a topological space. Note that for fixed 𝑑 > 0, the space 𝐾≡0

{(𝛼, 𝛽) ∈ ℝ2 |𝛼 ∈ [0, 𝑑/2], 𝛽 > 0, 𝛼2 + 𝛽 2 ≥ 𝑑 2 } ⊔ {Cyl𝑑 } is topologically a triangle 𝒯. So 𝕃 = 𝒯 × (0, ∞) ⊔ {𝔼2 }, where the extra point {𝔼2 } is the limit for 𝑑 → ∞. That is, 𝕃 is topologically a cone on a triangle. Remark 4.49. We can do a similar discussion to study Riemannian flat tori of dimension 𝑛 ≥ 2. The space that parametrizes them is the double coset ℳ𝑇 𝑛 = O(𝑛)\ GL(𝑛, ℝ)/ GL(𝑛, ℤ),

4. Constant curvature

228

1

1

which is a topological space of dimension 𝑛2 − 2 𝑛(𝑛 − 1) = 2 𝑛(𝑛 + 1). It can be seen that it is a manifold with corners. Moduli of oriented flat tori. We want to classify now oriented flat tori, that is flat torus with a fixed orientation, up to orientation preserving isometries. Any such 𝑆 is given as a quotient 𝑆 = 𝔼2 /Λ, where the orientation of 𝑆 is that induced by the canonical orientation of ℝ2 . We have a result similar to Proposition 4.42. Proposition 4.50. Let 𝑆 = 𝔼2 /Λ and 𝑆 ′ = 𝔼2 /Λ′ be two oriented flat tori, with Λ, Λ′ ⊂ ℝ2 lattices. Then 𝑆 is oriented isometric to 𝑆 ′ if and only if there exists an orientation preserving linear isometry 𝐿 ∈ SO(2) such that 𝐿(Λ) = Λ′ . From this proposition, as in the previous section, we have that the oriented flat tori are parametrized by the space ℳ𝑇𝑜𝑟2 = SO(2)\ℛ = SO(2)\ GL(2, ℝ)/ GL(2, ℤ). There is a map 𝜛 ∶ ℳ𝑇𝑜𝑟2 → ℳ𝑇 2 which assigns to each oriented torus (𝑆, 𝑜) the torus 𝑆. This is generically two-to-one since the preimage of 𝑆 is 𝜛−1 (𝑆) = {(𝑆, 𝑜), (𝑆, −𝑜)}, where 𝑜 is one orientation and −𝑜 is the opposite orientation. However, there are tori which admit an orientation reversing isometry (cases (2)–(6) in Remark 4.47). For them, there is an oriented isometry 𝑓 ∶ (𝑆, 𝑜) → (𝑆, −𝑜), and hence the preimage 𝜛−1 (𝑆) consists only of one point. For an oriented torus 𝑆 = 𝔼2 /Λ, we can choose a basis (𝑣 1 , 𝑣 2 ) of Λ which is oriented. Then the matrix 𝑀 in (4.17) satisfies det 𝑀 > 0. The changes of generators in (4.18) have to be positive since they preserve the orientation. So the matrix 𝑁 in (4.18) satisfies det 𝑁 = 1 and hence 𝑁 ∈ SL(2, ℤ). This means that we also have a description + ℛ = GL (2, ℝ)/ SL(2, ℤ) and (4.23)

+

ℳ𝑇𝑜𝑟2 = SO(2)\ℛ = SO(2)\ GL (2, ℝ)/ SL(2, ℤ).

The analogue of Theorem 4.45 is the following. +

Theorem 4.51. A fundamental domain for the action of SO(2) × SL(2, ℤ) on GL (2, ℝ) + is given by the matrices 𝒟̂ ⊂ GL (2, ℝ) given by 𝑑 𝒟̂ = {( 0

𝛼 | ) 𝑑 > 0, 𝛽 > 0, 𝛼2 + 𝛽 2 ≥ 𝑑 2 , −𝑑/2 ≤ 𝛼 ≤ 𝑑/2} . 𝛽 |

̂ ∼ ≅ {(𝑑, 𝛼, 𝛽) ∈ ℝ3 |𝑑 > 0, 𝛽 > Moreover, the space ℳ𝑇𝑜𝑟2 is homeomorphic to 𝒟/ 2 2 2 0, 𝛼 + 𝛽 ≥ 𝑑 , −𝑑/2 ≤ 𝛼 ≤ 𝑑/2}/ ∼ with the identification on the boundary given by (𝑑, −𝑑/2, 𝑦) ∼ (𝑑, 𝑑/2, 𝑦) for 𝑦 ≥ 𝑑√3/2, and (𝑑, 𝑥, 𝑦) ∼ (𝑑, −𝑥, 𝑦) for 𝑥2 + 𝑦2 = 𝑑 2 . Under this identification, the map 𝜛 ∶ ℳ𝑇𝑜𝑟2 → ℳ𝑇 2 is given by 𝜛(𝑑, 𝛼, 𝛽) = (𝑑, |𝛼|, 𝛽). Proof. The same proof as in Theorem 4.45 works here, with the difference that we use the oriented angle 𝜃 between 𝑣 1 and 𝑣 2 , where (𝑣 1 , 𝑣 2 ) is a positive basis of Λ. We cannot require the condition for 𝜃 to be acute or right, since we cannot change 𝑣 2 by −𝑣 2 . So we get that −𝑑/2 ≤ 𝛼 ≤ 𝑑/2. The argument of Theorem 4.45 gives now a surjective map 𝒟̂ → ℳ𝑇𝑜𝑟2 . The map 𝜛 is clearly given by 𝜛(𝑑, 𝛼, 𝛽) = (𝑑, |𝛼|, 𝛽). Therefore, if two different points (𝑑1 , 𝛼1 , 𝛽1 ), (𝑑2 , 𝛼2 , 𝛽2 ) define the same oriented torus, in particular

4.2. Vanishing constant curvature

229

they have the same image through 𝜛, so 𝑑1 = 𝑑2 , 𝛽1 = 𝛽2 and 𝛼2 = −𝛼1 . Hence the corresponding torus 𝑇𝑑21 ,𝛼1 ,𝛽1 must have an orientation reversing isometry. This happens only for the flat tori in the cases (2)–(6) in Remark 4.47, which corresponds to (𝑑𝑖 , 𝛼𝑖 , 𝛽 𝑖 ), 𝑖 = 1, 2, being at the boundary of 𝒟.̂ This describes the identification ∼ in the statement. □

a

a

b

-d

-d=2

b

0

d=2

d

Figure 4.5. The moduli space ℳ𝑇𝑜𝑟2 . The boundary is identified according to the letters.

Theorem 4.51 gives a topological description of ℳ𝑇𝑜𝑟2 . Let ℳ𝑇𝑜𝑟,𝑑 be the space con2 sisting of those lattices with fixed 𝑑 (i.e., fixing the length of the shortest closed geodesic), so ℳ𝑇𝑜𝑟2 ≅ ℳ𝑇𝑜𝑟,1 × (0, ∞). The space ℳ𝑇𝑜𝑟,𝑑 is topologically a square with 2 2 one vertex removed and with the identification along the boundary given by the word 𝑎𝑏𝑏−1 𝑎−1 (see Figure 4.5). This corresponds to a sphere with a point removed. Therefore ℳ𝑇𝑜𝑟,𝑑 ≅ 𝑆 2 − {𝑒 0 } ≅ ℝ2 is a topological manifold, where 𝑒 0 ∈ 𝑆 2 is the north pole. 2 Note that adding the cylinder Cyl𝑑 to the moduli space ℳ𝑇𝑜𝑟,𝑑 corresponds to adding 2 the missing point to the sphere. Thus ℳ𝑇𝑜𝑟,𝑑 ⊔ {Cyl𝑑 } ≅ 𝑆 2 , 2

(4.24)

which implies that the moduli space of (complete, connected) oriented surfaces of constant vanishing curvature, 𝑐𝑜

𝕃𝐑𝐢𝐞𝐦2

𝐾≡0,𝑜𝑟

= {𝔼2 } ⊔ ℳ𝑐𝑦𝑙 ⊔ ℳ𝑇𝑜𝑟2 ,

is homeomorphic to 𝑆 2 × (0, ∞) with a point added corresponding to the limit 𝑑 → ∞. This is homeomorphic to an open 3-dimensional ball 𝐵 3 . The manifold ℳ𝑇𝑜𝑟,𝑑 has actually the structure of a (differentiable) orbifold with 2 two orbifold points of indices 2 and 3 (section 3.2.4). Let 𝜋 ∶ 𝒟̂ 𝑑 = {(𝛼, 𝛽)| − 𝑑/2 ≤ 𝛼 ≤ 𝑑/2, 𝛽 > 0, 𝛼2 + 𝛽 2 ≥ 𝑑 2 } → ℳ𝑇𝑜𝑟,𝑑 2 . For a point in the vertical boundary 𝑝 = 𝜋(−𝑑/2, 𝑦) = 𝜋(𝑑/2, 𝑦), 𝑦 > 𝑑√3/2, a smooth chart for ℳ𝑇𝑜𝑟,𝑑 is constructed by gluing two semiballs around (−𝑑/2, 𝑦) and (𝑑/2, 𝑦), 2 respectively. The same can be done for a point in the circular arcs 𝑝 = 𝜋(𝑥, 𝑦) = 𝜋(−𝑥, 𝑦), 𝑥2 + 𝑦2 = 𝑑 2 , 0 < 𝑥 < 𝑑/2. The point 𝑝 = 𝜋(0, 𝑑) is an orbifold point of

4. Constant curvature

230

order 2 with the chart being given by a semiball 𝐵𝜀 (0, 𝑑) ∩ {𝑥2 + 𝑦2 ≥ 𝑑 2 }, which gives a cone point with angle 𝜋. Finally, the point 𝑝 = 𝜋(𝑑/2, 𝑑√3/2) = 𝜋(−𝑑/2, 𝑑√3/2) has a chart constructed by gluing 𝐵𝜀 (𝑑/2, 𝑑√3/2) ∩ 𝒟̂ 𝑑 and 𝐵𝜀 (−𝑑/2, 𝑑√3/2) ∩ 𝒟̂ 𝑑 , to get a cone point of angle 2𝜋/3. Actually, ℳ𝑇𝑜𝑟,𝑑 can be given the structure of a Riemannian orbisurface (section 2 3.2.4), but the metric is hyperbolic (it has constant curvature −1), see Remark 4.78. This is a sample of the common fact that a moduli space parametrizing some class of geometric structures has in general some geometric structure of similar or related type. 4.2.4. Classification of non-orientable surfaces with vanishing curvature. Let 𝑆 be a non-orientable complete connected surface of zero curvature. In this case, 𝑆 = 𝔼2 /Γ for some Γ ⊂ Isom(𝔼2 ), acting freely and properly, with Γ ∩ Isom− (𝔼2 ) ≠ ∅. Let us write Γ+ = Γ ∩ Isom+ (𝔼2 ) and Γ− = Γ ∩ Isom− (𝔼2 ) so that Γ = Γ+ ⊔ Γ− . Observe that the quotient 𝑆 ̂ = 𝔼2 /Γ+ is the (Riemannian) oriented cover of 𝑆. By the discussion in section 4.2.2, Γ+ is a subgroup of the group of translations given by some lattice Λ ⊂ ℝ2 . On the other hand, all elements in Γ− are glide reflections 𝜎ℓ,ᵆ = 𝜏ᵆ ∘ 𝑠ℓ , a composition of a reflection with respect to a line ℓ with the translation of some vector 𝑢 ≠ 0 parallel to ℓ. Given any 𝜑 = 𝜎ℓ,ᵆ ∈ Γ− , we have 𝜑2 = 𝜏2ᵆ , so 2𝑢 ∈ Λ. Take a vector 𝑣 ∈ Λ ∩ ℝ⟨𝑢⟩ of minimum length. Then 2𝑢 = 𝑛𝑣, for some integer 𝑛. It cannot be 𝑢 ∈ Λ, since otherwise the reflection 𝑠ℓ = 𝜏−ᵆ ∘ 𝜎ℓ,ᵆ ∈ Γ. Hence 𝑛 is odd, say 𝑛 = 2𝑘 + 1, 𝑘 ∈ ℤ. Then 𝜑0 = 𝜎ℓ,𝑣/2 = 𝜏−𝑘𝑣 ∘ 𝜎ℓ,ᵆ ∈ Γ− , where 𝜑20 = 𝜏𝑣 . This implies that all other elements of Γ− are of the form 𝜏𝑤 ∘ 𝜑0 = 𝜏𝑤+𝑣/2 ∘ 𝑠ℓ , for 𝑤 ∈ Λ. In particular, all of them are glide reflections along lines parallel to ℓ. We have two cases: • If rank Λ = 1, then Λ = ℤ⟨𝑣⟩. Thus Γ is generated by the glide reflection 𝜑0 = 𝜎ℓ,𝑣/2 . Taking coordinates so that 𝑣 = (𝑑, 0), 𝑑 > 0, we have 𝜑0 (𝑥, 𝑦) = (𝑥 + 𝑑/2, −𝑦). Hence 𝑆 is diffeomorphic to a Möbius strip. Metrically, it is characterized by the length of the shortest closed geodesic, which is 𝑑/2. Note that this geodesic is unique. We obtain a 1-parameter family of Möbius stripes Mob𝑑 , with 𝑑 ∈ (0, ∞) the length of the shortest closed geodesic. The oriented cover of Mob𝑑 is the cylinder Cyl2𝑑 . • If rank Λ = 2, then Λ = ℤ⟨𝑣, 𝑤⟩. The oriented cover of 𝑆 is the flat torus 𝑆 ̂ = 𝔼2 /Λ. Actually, we have that Γ = ⟨𝜎ℓ,𝑣/2 , 𝜏𝑤 ⟩, so the surface 𝑆 is diffeomorphic to a Klein bottle. The moduli of flat Klein bottles will be denoted ℳKl . It is characterised next. Theorem 4.52. The lattice Λ = ℤ⟨𝑣, 𝑤⟩ associated to a flat Klein bottle 𝑆 is rectangular, that is, 𝑤 ⟂ 𝑣. The moduli of Klein bottles is ℳKl ≅ {(𝑑, 𝑙) | 𝑑, 𝑙 > 0} = (0, ∞)2 , where 𝑑 = ||𝑣||, 𝑙 = ||𝑤||. Proof. We can take coordinates so that 𝑣 = (𝑑, 0) and 𝑤 = (𝛼, 𝛽) with 𝛽 > 0. Substituting 𝑤 by 𝑤 − 𝑛𝑣, with a suitable 𝑛 ∈ ℤ, we can arrange that −𝑑/2 ≤ 𝛼 ≤ 𝑑/2. As before 𝜑0 ∈ Γ− , where 𝜑0 (𝑥, 𝑦) = (𝑥 + 𝑑/2, −𝑦). The other elements of Γ− are of the

4.2. Vanishing constant curvature

231

form 𝜑 = 𝜏𝑛1 𝑣+𝑛2 𝑤 ∘ 𝜑0 . So 𝜑(𝑥, 𝑦) = (𝑥 + (𝑛1 + 1/2)𝑑 + 𝑛2 𝛼, 𝑛2 𝛽 − 𝑦), 1

which is the composition of the reflection with respect to the line 𝑦 = 2 𝑛2 𝛽 with the translation of vector 𝑢 = ((𝑛1 + 1/2)𝑑 + 𝑛2 𝛼, 0). As 𝜑2 ∈ Γ+ , we have that 2𝑢 = ((2𝑛1 + 1)𝑑 + 2𝑛2 𝛼, 0) ∈ Λ, that is 2𝑢 = 𝑚𝑣, for an integer 𝑚. As before, 𝑚 has to be odd, since otherwise 𝑢 ∈ Λ and 𝜏−ᵆ ∘ 𝜑 would be a reflection in Γ− . Therefore 𝑚 = 2𝑘 + 1, 𝑘 ∈ ℤ, and hence (2𝑛1 + 1)𝑑 + 2𝑛2 𝛼 = (2𝑘 + 1)𝑑, for all 𝑛1 , 𝑛2 ∈ ℤ. Choosing 𝑛1 = 0, 𝑛2 = 1, we get 2𝛼 = 2𝑘𝑑 for some 𝑘. But |𝛼| ≤ 𝑑/2, hence 𝑘 = 0 and 𝛼 = 0. This concludes that 𝑤 = (0, 𝛽). Thus 𝑙 = ||𝑤|| = 𝛽, so 𝑤 = (0, 𝑙) and 𝑤 ⟂ 𝑣, as required. We denote Kl 𝑑,𝑙 the Klein bottle with Γ = ⟨𝜑0 , 𝜏𝑤 ⟩, where 𝜑0 (𝑥, 𝑦) = (𝑥 + 𝑑/2, −𝑦) and 𝑤 = (0, 𝑙), 𝑑, 𝑙 > 0. The parameters 𝑑, 𝑙 are detected geometrically, as 𝑑/2 is the length of the shortest closed geodesic that reverses the orientation, and 𝑙 is the length of the shortest closed geodesic perpendicular (or just linearly independent in homology) to the first one. Therefore ℳKl ≅ (0, ∞)2 . □ The moduli space ℳKl is a 2-dimensional manifold. Moreover, the oriented cover 2 2 of Kl 𝑑,𝑙 is the torus 𝑇𝑑,0,𝑙 if 𝑙 ≥ 𝑑, or the torus 𝑇𝑙,0,𝑑 if 𝑑 ≥ 𝑙. Therefore the map (4.25)

ℳKl ⟶ ℳ𝑇 2 ,

𝑆 ↦ 𝑆̂

has image {(𝑑, 0, 𝑙)|𝑙 ≥ 𝑑}, that corresponds to the vertical axis in Figure 4.5. Note that the map (4.25) is two-to-one except for the “square” Klein bottle (𝑑 = 𝑙). Example 4.53. In dimensions 𝑛 ≥ 3, there is more phenomenology for complete connected manifolds of zero curvature, since there are orientation preserving isometries with no fixed points which are not translations. For 𝑛 = 3, these have standard form of the type 1 ⎛ 1 𝜓𝜃 = ⎜ ⎜ 0 ⎝ 0

0 1 0 0

0 0 cos 𝜃 − sin 𝜃

0 0 sin 𝜃 cos 𝜃

⎞ ⎟, ⎟ ⎠

with 𝜃 ∈ (0, 𝜋]. For 𝜃 ∈ (0, 𝜋) these are helicoidal movements. Some examples are the following. • A non-compact example is 𝑀 = 𝔼3 /⟨𝜓𝜃 ⟩. It is always homeomorphic to 𝑆 1 × ℝ2 , but the angle 𝜃 is detected metrically. • 𝑀 = 𝔼3 /Γ with Γ = ⟨𝜓𝜋/2 , 𝜏𝑒2 , 𝜏𝑒3 ⟩, where (𝑒 1 , 𝑒 2 , 𝑒 3 ) denotes the canonical basis. The fundamental domain is the cube [0, 1]3 , but the left and right faces are identified by a rotation of angle 𝜋/2. The group Γ is not Abelian, so 𝑀 is not homeomorphic to a torus. • There are also new non-orientable manifolds, as 𝑀 = 𝔼3 /Γ′ where Γ′ = ⟨𝜓𝜋/2 , 𝜏𝑒2 , 𝜑⟩, 𝜑(𝑥, 𝑦, 𝑧) = (𝑥, 1 − 𝑦, 𝑧 + 1). The fundamental domain is the cube [0, 1]3 , the left and right faces are identified by a rotation of angle 𝜋/2, and the bottom and top faces are identified by a reflection. Remark 4.54. The compact, connected manifolds of zero curvature are called Bieberbach manifolds. If 𝑀 = 𝔼𝑛 /Γ is such manifold, with Γ < AO(𝑛), then it has a finite

4. Constant curvature

232

cover which is a flat torus, that is 𝜋 ∶ 𝑀 ′ = 𝔼𝑛 /Λ → 𝑀 = 𝔼𝑛 /Γ of some degree 𝑑 > 0. This corresponds to a subgroup Λ < Γ of index 𝑑, which is a lattice (i.e., a discrete group of translations). This subgroup is normal and 𝐹 = Γ/Λ is a finite group acting freely on the torus 𝑀 ′ , so that 𝑀 = 𝑀 ′ /𝐹. The action of 𝐹 on the homology of 𝑀 ′ , 𝐻1 (𝑀 ′ , ℤ) = Λ ≅ ℤ𝑛 , gives a homomorphism 𝐹 → GL(𝑛, ℤ), which characterizes 𝑀 topologically.

4.3. Negative constant curvature Hyperbolic manifolds are complete connected Riemannian manifolds with constant sectional curvature equal to −1. As shown in section 3.3.3, these are quotients of the 𝑛 simply connected space form 𝐸−1 by a group of isometries acting freely and properly. We start by proving the existence of this simply connected space form, usually called the hyperbolic 𝑛-space and denoted ℍ𝑛 . This will complete the proof of Theorem 3.109.

4.3.1. The hyperbolic space. To construct ℍ𝑛 , we will try to mimic as much as possible the construction of 𝕊𝑛 in section 4.1.2, but starting with an indefinite quadratic form. Given a smooth 𝑛-manifold 𝑀, a (0, 2)-tensor g is a semi-Riemannian metric if, for all 𝑝 ∈ 𝑀, g𝑝 ∶ 𝑇𝑝 𝑀 × 𝑇𝑝 𝑀 → ℝ is a non-degenerate symmetric bilinear form. If g𝑝 has signature (𝑟, 𝑠) for all 𝑝 ∈ 𝑀 (that is, g𝑝 has 𝑟 positive eigenvalues and 𝑠 negative eigenvalues, with 𝑛 = 𝑟 + 𝑠), then g is said to have signature (𝑟, 𝑠) (cf. Exercise 4.19). In the particular case where g has signature (𝑛 − 1, 1), it is customary to call g a Lorentz metric, and the pair (𝑀, g) a Lorentz manifold. We will not enter into the thorough study of semi-Riemannian manifolds, just let us mention that many results of Riemannian geometry have their counterpart in the semi-Riemannian world [O’N]. Remark 4.55. Whereas Riemannian manifolds serve as the natural ground for studying physical problems in (curved) spaces, Lorentz manifolds are the natural place to develop the theory of general relativity of Einstein. A Lorentz manifold is a spacetime, in which space and time are interlaced. The directions in which the semi-Riemannian metric is positive and negative serve to distinguish between space and time directions. For our purposes, we consider the bilinear form in ℝ𝑛+1 given by 𝐵(𝑥, 𝑦) = 𝑥𝑡 𝑄𝑦, for 𝑥, 𝑦 ∈ ℝ𝑛+1 , where −1 ⎛ 0 𝑄=⎜ ⎜ ⋮ ⎝ 0

0 1 0

... ⋱ ...

0 0 ⋮ 1

⎞ ⎟. ⎟ ⎠

This is a bilinear form of signature (𝑛, 1), so using the identification 𝑇𝑝 ℝ𝑛+1 ≅ ℝ𝑛+1 , for 𝑝 ∈ ℝ𝑛+1 , we have that 𝐵 defines the Lorentz metric in ℝ𝑛+1 denoted g𝐿 = −𝑑𝑥02 + 𝑑𝑥12 + ⋯ + 𝑑𝑥𝑛2 . The space (ℝ𝑛+1 , g𝐿 ) is usually called the Minkowski space. A remarkable fact is that the restriction of g𝐿 to a suitable hypersurface of (ℝ𝑛+1 , g𝐿 ) is actually Riemannian.

4.3. Negative constant curvature

233

Proposition 4.56. Consider the hypersurface 𝑀 = {𝑥 = (𝑥0 , . . . , 𝑥𝑛 ) ∈ ℝ𝑛+1 |𝐵(𝑥, 𝑥) = −𝑥02 + 𝑥12 + ⋯ + 𝑥𝑛2 = −1}. Then g𝐿 |𝑀 is a Riemannian metric. Proof. The function 𝐹(𝑥) = 𝐵(𝑥, 𝑥) = 𝑥𝑡 𝑄𝑥 has differential 𝑑𝑥 𝐹(𝑣) = 2𝑥𝑡 𝑄𝑣 = 2𝐵(𝑥, 𝑣), so for a point 𝑥 ∈ 𝑀, the tangent space is 𝑇𝑥 𝑀 = {𝑣 ∈ ℝ𝑛+1 |𝐵(𝑥, 𝑣) = 0}. This is the 𝐵-orthogonal space to the line ⟨𝑥⟩. As 𝐵(𝑥, 𝑥) < 0, we have a decomposition ℝ𝑛+1 = ⟨𝑥⟩⊕⟨𝑥⟩⟂𝐵 , where ⟂𝐵 means “orthogonal with respect to 𝐵”, and 𝑇𝑥 𝑀 = ⟨𝑥⟩⟂𝐵 . In this decomposition, 𝐵 restricted to the line ⟨𝑥⟩ has signature (0, 1), hence 𝐵 restricted to 𝑇𝑥 𝑀 has signature (𝑛, 0), that is g𝐿 |𝑇𝑥 𝑀 is positive definite. □ Observe that 𝑀 = {𝑥 ∈ ℝ𝑛+1 |𝐵(𝑥, 𝑥) = −1} is an analogue of the 𝑛-sphere, where we use 𝐵 instead of the standard scalar product. The hypersurface 𝑀 is a hyperboloid of two sheets, each of them being the graph 𝑥0 = ±√𝑥12 + . . . + 𝑥𝑛2 + 1. Note that the other “𝐵-sphere” is defined by the equation 𝐵(𝑥, 𝑥) = 1 which is the hyperboloid of one sheet, but this does not inherit a Riemannian metric from the Minkowski space. We denote 𝑀+ = 𝑀 ∩ {𝑥0 > 0} and 𝑀− = 𝑀 ∩ {𝑥0 < 0}, for each of the two sheets of 𝑀 (see Figure 4.6).

Figure 4.6. The cone 𝐵(𝑥, 𝑥) = 0, the hyperboloid of one sheet 𝐵(𝑥, 𝑥) = 1, and the hyperboloid of two sheets 𝐵(𝑥, 𝑥) = −1.

Definition 4.57. The hyperbolic 𝑛-space is the Riemannian 𝑛-manifold ℍ𝑛 = (𝑀+ , gℍ𝑛 = g𝐿 |𝑀+ ). Clearly, ℍ𝑛 is homeomorphic to ℝ𝑛 , hence it is simply connected. The group O(𝑛, 1). In order to study the isometries of ℍ𝑛 , we need to introduce the group O(𝑛, 1) = {𝐴 ∈ GL(𝑛 + 1, ℝ)|𝐴𝑡 𝑄𝐴 = 𝑄}

4. Constant curvature

234

of linear maps 𝜑𝐴 ∶ ℝ𝑛+1 → ℝ𝑛+1 , 𝜑𝐴 (𝑥) = 𝐴𝑥, which preserve the quadratic form 𝐵. This holds since for 𝐴 ∈ O(𝑛, 1), 𝐵(𝜑𝐴 (𝑥), 𝜑𝐴 (𝑦)) = 𝐵(𝐴𝑥, 𝐴𝑦) = (𝐴𝑥)𝑡 𝑄𝐴𝑦 = 𝑥𝑡 𝐴𝑡 𝑄𝐴𝑦 = 𝑥𝑡 𝑄𝑦 = 𝐵(𝑥, 𝑦). Let

𝑎 ⎛ 00 𝑎 𝐴 = ⎜ 10 ⎜ ⋮ ⎝ 𝑎𝑛0

𝑎01 𝑎11 ⋮ 𝑎𝑛2

... ... ⋱ ...

𝑎0𝑛 𝑎1𝑛 ⋮ 𝑎𝑛𝑛

⎞ ⎟ ∈ O(𝑛, 1), ⎟ ⎠

and let 𝑎𝑖 = (𝑎0𝑖 , 𝑎1𝑖 , . . . , 𝑎𝑛𝑖 ) be the 𝑖th column of 𝐴. So 𝑎𝑖 = 𝐴𝑒 𝑖 , where (𝑒 0 , 𝑒 1 , . . . , 𝑒 𝑛 ) is the canonical basis of ℝ𝑛+1 . As 𝐵(𝑒 0 , 𝑒 0 ) = −1, 𝐵(𝑒 𝑖 , 𝑒 𝑖 ) = 1 for 1 ≤ 𝑖 ≤ 𝑛, and 𝐵(𝑒 𝑖 , 𝑒𝑗 ) = 0 for 𝑖 ≠ 𝑗, then 𝐴 ∈ O(𝑛, 1) if and only if the basis (𝑎0 , 𝑎1 , . . . , 𝑎𝑛 ) is 𝐵orthonormal, that is 𝐵(𝑎0 , 𝑎0 ) = −1, 𝐵(𝑎𝑖 , 𝑎𝑖 ) = 1 for 1 ≤ 𝑖 ≤ 𝑛, and 𝐵(𝑎𝑖 , 𝑎𝑗 ) = 0 for 𝑖 ≠ 𝑗. This means that ℝ𝑛+1 = ⟨𝑎0 ⟩ ⊕ 𝑉, where 𝑉 = ⟨𝑎0 ⟩⟂𝐵 = ⟨𝑎1 , . . . , 𝑎𝑛 ⟩, and (𝑎1 , . . . , 𝑎𝑛 ) is an orthonormal basis of 𝑉. This implies that a matrix 𝐴 ∈ O(𝑛, 1) is determined by a point 𝑎0 ∈ 𝑀 and an orthonormal basis of the 𝑛-dimensional Riemannian vector space 1 1 (𝑇𝑎0 𝑀, g𝐿 |𝑀 ), hence dim O(𝑛, 1) = dim 𝑀 +dim O(𝑛) = (𝑛+1)+ 2 𝑛(𝑛−1) = 2 𝑛(𝑛+1). The above description also says that O(𝑛, 1) has four connected components, since 𝑀 has two connected components (𝑎0 ∈ 𝑀+ if 𝑎00 > 0, and 𝑎0 ∈ 𝑀− if 𝑎00 < 0), and O(𝑛) has two connected components (which depend on the orientation of the basis (𝑎1 , . . . , 𝑎𝑛 )). We denote these components as follows: +

O> (𝑛, 1) = {𝐴 ∈ O(𝑛, 1)| 𝑎00 > 0, det(𝐴) = 1}, −

O> (𝑛, 1) = {𝐴 ∈ O(𝑛, 1)| 𝑎00 > 0, det(𝐴) = −1}, +

O< (𝑛, 1) = {𝐴 ∈ O(𝑛, 1)| 𝑎00 < 0, det(𝐴) = 1}, −

O< (𝑛, 1) = {𝐴 ∈ O(𝑛, 1)| 𝑎00 < 0, det(𝐴) = −1}. ±

+

−

We shall also shorten O> (𝑛, 1) = O> (𝑛, 1) ⊔ O> (𝑛, 1). Note that from 𝐴𝑡 𝑄𝐴 = 𝑄 it follows that det(𝐴) = ±1. The sign of det 𝐴 says whether 𝜑𝐴 preserves or reverses the orientation of ℝ𝑛+1 . By construction, for 𝐴 ∈ O(𝑛, 1), the map 𝜑𝐴 ∶ (ℝ𝑛+1 , g𝐿 ) → (ℝ𝑛+1 , g𝐿 ), 𝜑𝐴 (𝑥) = ∗ 𝐴𝑥, preserves the Minkowski metric, i.e., 𝜑𝐴 g𝐿 = g𝐿 . For 𝑥 ∈ 𝑀, 𝐵(𝑥, 𝑥) = −1 implies ± 𝐵(𝜑𝐴 (𝑥), 𝜑𝐴 (𝑥)) = −1. This means 𝜑𝐴 ∶ 𝑀 → 𝑀. For 𝐴 ∈ O> (𝑛, 1), 𝜑𝐴 preserves the two sheets of 𝐻, since it sends 𝑒 0 ∈ 𝑀+ to 𝑎0 ∈ 𝑀+ . On the other hand, for ± ± 𝐴 ∈ O< (𝑛, 1), 𝜑𝐴 swaps the two sheets of 𝑀. Therefore if 𝐴 ∈ O> (𝑛, 1), then 𝜑𝐴 ∶ ℍ𝑛 = (𝑀+ , gℍ𝑛 ) → ℍ𝑛 = (𝑀+ , gℍ𝑛 ) +

−

is an isometry. Thus O> (𝑛, 1) ⊔ O> (𝑛, 1) < Isom(ℍ𝑛 ).

4.3. Negative constant curvature

235

Proposition 4.58. The hyperbolic 𝑛-space ℍ𝑛 is isotropic. + − Isom+ (ℍ𝑛 ) = O> (𝑛, 1) and Isom− (ℍ𝑛 ) = O> (𝑛, 1).

Moreover we have

Proof. Take 𝑝0 = 𝑒 0 = (1, 0, . . . , 0) ∈ ℍ𝑛 and consider the orthonormal basis (𝑒 1 , . . . , 𝑒 𝑛 ) of 𝑇𝑝0 ℍ𝑛 . Take also another point 𝑝 = 𝑎0 ∈ ℍ𝑛 and an orthonormal basis (𝑎1 , . . . , 𝑎𝑛 ) ± of 𝑇𝑝 ℍ𝑛 = ⟨𝑎0 ⟩⟂𝐵 . Then the matrix 𝐴 = (𝑎0 , 𝑎1 , . . . , 𝑎𝑛 ) ∈ O> (𝑛, 1). Moreover we have 𝑛 𝑛 𝜑𝐴 (𝑝0 ) = 𝐴𝑒 0 = 𝑎0 = 𝑝, and 𝑑𝑝0 𝜑𝐴 ∶ 𝑇𝑝0 ℍ → 𝑇𝑝 ℍ satisfies 𝑑𝑝0 𝜑𝐴 (𝑒𝑗 ) = 𝐴𝑒𝑗 = 𝑎𝑗 , 1 ≤ 𝑗 ≤ 𝑛. This proves the assertion. □ Now let us study the geodesics of ℍ𝑛 . We will not develop the theory of the LeviCivita connections for a semi-Riemannian metric, but it is similar to the Riemannian case. For us, it suffices to say that the trivial connection ∇̄ on ℝ𝑛+1 is torsion free and ̄ = 0, since the Lorentz metric g has constant coefficients. For 𝑝 ∈ ℍ𝑛 , satisfies ∇g 𝐿 𝐿 we have the 𝐵-orthogonal splitting 𝑇𝑝 ℝ𝑛+1 = 𝑇𝑝 ℍ𝑛 ⊕ (𝑇𝑝 ℍ𝑛 )⟂𝐵 , where (𝑇𝑝 ℍ𝑛 )⟂𝐵 = ⟨𝑝⟩. Analogously to the Riemannian case (3.22), there are projections (−)𝑇𝐵 ∶ 𝑇𝑝 ℝ𝑛+1 → 𝑇𝑝 ℍ𝑛 and (−)𝑁𝐵 ∶ 𝑇𝑝 ℝ𝑛+1 → ⟨𝑝⟩. The same proof as in Proposition 3.37 shows that ∇𝑋 𝑌 = (∇̄ 𝑋 𝑌 )𝑇𝐵 , for 𝑋, 𝑌 ∈ 𝔛(ℍ𝑛 ), defines a connection which is torsion free and ∇gℍ𝑛 = 0. Hence ∇ is the Levi-Civita connection for ℍ𝑛 = (𝑀+ , gℍ𝑛 ). The covariant derivative of a vector 𝐷𝑋 field 𝑋 ∈ 𝔛(ℍ𝑛 ) along a curve 𝛾 ∶ [0, 1] → ℍ𝑛 is given by 𝑑𝑡 = (∇̄ 𝛾′ 𝑋)𝑇𝐵 = (𝑋 ′ (𝑡))𝑇𝐵 . A curve 𝛾 ∶ [0, 1] → ℍ𝑛 parametrized by arc length is a geodesic if it satisfies that 𝐷𝛾′ = (𝛾″ (𝑡))𝑇𝐵 = 0, which happens if and only if there exists a function 𝜆 such that 𝑑𝑡 ″ 𝛾 (𝑡) = 𝜆(𝑡)𝛾(𝑡). On the other hand we have 𝐵(𝛾, 𝛾) = −1, and taking derivatives we get 𝐵(𝛾, 𝛾′ ) = 0 and 0 = 𝐵(𝛾′ , 𝛾′ ) + 𝐵(𝛾, 𝛾″ ) = 1 − 𝜆(𝑡) which concludes that 𝛾″ = 𝛾. The solutions to this equation are of the form 𝛾𝑝,𝑣 (𝑡) = (cosh 𝑡)𝑝 + (sinh 𝑡)𝑣, for 𝑝 = 𝛾(0) ∈ ℍ𝑛 , 𝑣 = 𝛾′ (0) ∈ 𝑇𝑝 ℍ𝑛 unitary. Note that 𝐵(𝑝, 𝑝) = −1, 𝐵(𝑝, 𝑣) = 0 and 𝐵(𝑣, 𝑣) = 1, so 𝐵(𝛾(𝑡), 𝛾(𝑡)) = −1 for all 𝑡 ∈ ℝ, and then 𝛾(𝑡) is a curve in ℍ𝑛 . In particular, the trace of the geodesic is the intersection of a plane through the origin with ℍ𝑛 ⊂ ℝ𝑛+1 , namely ⟨𝑝, 𝑣⟩ ∩ ℍ𝑛 . From this it is clear that the geodesics are defined for all 𝑡 ∈ ℝ, and therefore ℍ𝑛 is complete. The exponential map at 𝑝0 = (1, 0, . . . , 0) ∈ ℍ𝑛 is exp𝑝0 ∶ 𝑇𝑝0 ℍ𝑛 → ℍ𝑛 , 𝑣 = (0, 𝑣 1 , . . . , 𝑣 𝑛 ) ↦ exp𝑝0 (𝑣) = (cosh ||𝑣||)𝑝0 +

sinh ||𝑣|| 𝑣. ||𝑣||

Clearly it is smooth, and it is a diffeomorphism onto its image. The inverse is given by sending 𝑥 = (𝑥0 , 𝑥1 , . . . , 𝑥𝑛 ) ∈ ℍ𝑛 to exp−1 𝑝0 (𝑥) = Therefore the injectivity radius is infinity.

−1

sinh (||𝑥′ ||) ′ 𝑥, ||𝑥′ ||

where 𝑥′ = (𝑥1 , . . . , 𝑥𝑛 ).

Theorem 4.59. ℍ𝑛 is the simply connected space form of curvature 𝑘0 = −1. Proof. We already know that ℍ𝑛 is simply connected and isotropic, hence complete and of constant curvature. It remains to compute the curvature. For this it is enough

4. Constant curvature

236

to compute 𝐾𝑝0 (𝜎) for 𝑝0 = (1, 0, . . . , 0) and the plane 𝜎 = ⟨𝑒 1 , 𝑒 2 ⟩ ⊂ 𝑇𝑝0 ℍ𝑛 . The image of 𝜎 under the exponential map is exp𝑝0 (𝜎) = { (cosh ||𝑣||,

sinh ||𝑣|| sinh ||𝑣|| 𝑣1 , 𝑣 2 , 0, . . . , 0) || ||𝑣|| ||𝑣|| 𝑣 = (0, 𝑣 1 , 𝑣 2 , 0, . . . , 0) ∈ 𝑇𝑝0 ℍ2 }

= {(𝑥0 , 𝑥1 , 𝑥2 , 0, . . . , 0)| − 𝑥02 + 𝑥12 + 𝑥22 = −1, 𝑥0 > 0} ≅ ℍ2 , which is the hyperbolic plane (of dimension 2). It has the induced hyperbolic metric g𝐿 |ℍ2 = gℍ2 . Hence it is enough to compute the Gaussian curvature of ℍ2 . For this, we will compute the length of a small circle in ℍ2 and use Theorem 3.58. Take 𝑝0 = (1, 0, 0) and consider the circle 𝐶𝑟 (𝑝0 ), which is parametrized by 𝛼𝑟 (𝜃) = exp𝑝0 (𝑟(cos 𝜃)𝑒 1 + 𝑟(sin 𝜃)𝑒 2 ) = (cosh 𝑟, cos 𝜃 sinh 𝑟, sin 𝜃 sinh 𝑟), where 𝜃 ∈ [0, 2𝜋]. Then the derivative is 𝛼′𝑟 (𝜃) = (0, − sin 𝜃 sinh 𝑟, cos 𝜃 sinh 𝑟) with norm ||𝛼′𝑟 (𝜃)|| = 𝐵(𝛼′𝑟 , 𝛼′𝑟 )1/2 = sinh 𝑟. Therefore the length is 2𝜋

||𝛼′𝑟 (𝜃)|| 𝑑𝜃 = 2𝜋 sinh 𝑟 = 2𝜋(𝑟 +

ℓ(𝐶𝑟 (𝑝0 )) = ∫ 0

𝑟3 + ⋯) 3!

𝜋 𝜅 2 (𝑝 )𝑟3 + 𝑂(𝑟4 ), 3 ℍ 0 according to Theorem 3.58. This implies that 𝜅ℍ2 (𝑝0 ) = −1, and we conclude that the sectional curvature of ℍ𝑛 is equal to 𝑘0 = −1. □ = 2𝜋𝑟 −

Remark 4.60. To obtain the simply connected space form of curvature 𝑘0 < 0, we proceed as explained in Remark 4.9. 1

• We can consider (ℍ𝑛 , 𝜆2 gℍ𝑛 ), with 𝜆 > 0, which has curvature 𝐾 ≡ 𝑘0 = − 𝜆2 . • We can consider ℍ𝑛𝑅 = {𝑥 ∈ ℝ𝑛+1 |𝐵(𝑥, 𝑥) = −𝑅2 , 𝑥0 > 0} with the Riemannian metric gℍ𝑛 = g𝐿 |ℍ𝑛𝑅 . It can easily be checked as above that 𝐾ℍ𝑛𝑅 ≡ 𝑘0 = 1

𝑅

− 𝑅2 . Observe that the map (ℍ𝑛𝑅 , gℍ𝑛 ) → (ℍ𝑛 , 𝑅2 gℍ𝑛 ), given by 𝑥 ↦ 𝑅

𝑥 , 𝑅

is an isometry.

There are other models of the hyperbolic space which have the advantage that they have natural global coordinates. We will focus on the case 𝑛 = 2 to simplify the notation, but the computations can be done in general (Remark 4.73). 4.3.2. Beltrami-Klein model. This model has the advantage of being natural from the point of view of projective geometry. The drawback is that it is not a conformal model; i.e., the angles seen in the chart in ℝ2 (with the standard metric) are different from the angles in ℍ2 with its hyperbolic metric. We consider the projection from the origin. Any point of ℍ2 belongs to a unique vector line, and no such line lies in the hyperplane {𝑥0 = 0}. Therefore we can see ℍ2 as a subset of the affine chart 𝑥0 ≠ 0 of the projective plane ℝ𝑃2 . This subset is easily seen to be the open unit disc 𝐵2 = {(𝑢, 𝑣) ∈ ℝ2 | 𝑢2 + 𝑣2 < 1}. More precisely, given (𝑥0 , 𝑥1 , 𝑥2 ) ∈ ℍ2 we write (𝑥0 , 𝑥1 , 𝑥2 ) = (𝜆, 𝜆𝑢, 𝜆𝑣) for some 𝜆 ≥ 1 and (𝑢, 𝑣) ∈ ℝ2 .

4.3. Negative constant curvature

Then 𝑢2 +𝑣2 = or equivalently, 𝑥0 =

𝜆2 −1 𝜆2

< 1, so (𝑢, 𝑣) ∈ 𝐵 2 . Therefore 𝑢 = 1

√1 −

237

𝑢2

−

𝑣2

,

𝑥1 =

𝑢 √1 −

𝑢2

−

𝑣2

𝑥1 ,𝑣 𝑥0

,

=

𝑥2 ,𝜆 𝑥0

𝑥2 =

= (1−𝑢2 −𝑣2 )−1/2 , 𝑣

√1 − 𝑢2 − 𝑣2

.

From the above formulae, we obtain 𝑑𝑥0 =

𝑢 𝑑𝑢 + 𝑣 𝑑𝑣 , (1 − 𝑢2 − 𝑣2 )3/2

𝑑𝑥1 =

(1 − 𝑣2 )𝑑𝑢 + 𝑢𝑣𝑑𝑣 , (1 − 𝑢2 − 𝑣2 )3/2

𝑑𝑥2 =

𝑢𝑣𝑑𝑢 + (1 − 𝑢2 )𝑑𝑣 . (1 − 𝑢2 − 𝑣2 )3/2

Using this, we compute the expression of the metric in the chart gℍ2 = − 𝑑𝑥02 + 𝑑𝑥12 + 𝑑𝑥22 =

1

((−𝑢2 + 1 − 2𝑣2 + 𝑣4 + 𝑢2 𝑣2 )𝑑𝑢2 (1 − − 𝑣2 )3 + (−2𝑢𝑣 + 2(1 − 𝑣2 )𝑢𝑣 + 2𝑢𝑣(1 − 𝑢2 ))𝑑𝑢𝑑𝑣+ 𝑢2

+ (−𝑣2 + 𝑢2 𝑣2 + 1 + 𝑢4 − 2𝑢2 )𝑑𝑣2 ) =

1 ((1 − 𝑣2 )𝑑𝑢2 + 2𝑢𝑣 𝑑𝑢𝑑𝑣 + (1 − 𝑢2 )𝑑𝑣2 ). (1 − 𝑢2 − 𝑣2 )2

Definition 4.61. We define the Beltrami-Klein model for the hyperbolic plane as ℍ2𝐵𝐾 = (𝐵 2 , g𝐵𝐾 ), where 𝐵 2 is the open disc and the metric is g𝐵𝐾 =

1 ((1 − 𝑣2 )𝑑𝑢2 + 2𝑢𝑣 𝑑𝑢𝑑𝑣 + (1 − 𝑢2 )𝑑𝑣2 ). (1 − 𝑢2 − 𝑣2 )2

In this model, the volume element is given by √𝐸𝐺 − 𝐹 2 𝑑𝑢 ∧ 𝑑𝑣 =

1 (1 −

𝑢2

− 𝑣 2 )2

√(1 − 𝑣2 )(1 − 𝑢2 ) − 𝑢2 𝑣2 𝑑𝑢 ∧ 𝑑𝑣 3

= (1 − 𝑢2 − 𝑣2 )− 2 𝑑𝑢 ∧ 𝑑𝑣,

(x; y; z)

(u; v)

Figure 4.7. Beltrami-Klein model of the hyperbolic plane, and the image of a geodesic.

4. Constant curvature

238

so the area of ℍ2 is given by 2𝜋 2

area(ℍ ) = ∫ 0

1

1

𝜌 −1 ∫ 𝑑𝜌𝑑𝜃 = 2𝜋 [ ] = ∞. 1 2 )3/2 (1 − 𝜌 0 (1 − 𝜌2 ) 2 0

As we showed above, a geodesic in ℍ2 is the intersection of a plane through the origin with ℍ2 ⊂ ℝ3 . By projecting from the origin, the geodesic transforms into the intersection of the projective line defined by the plane with ℍ2𝐵𝐾 ⊂ ℝ𝑃 2 , which is a segment in the disc ℍ2𝐵𝐾 (Figure 4.7). In coordinates, a geodesic in ℍ2 ⊂ ℝ3 is of the form 𝛾(𝑡) = (cosh 𝑡)𝑝 + (sinh 𝑡)𝑣 for 𝑝 = (𝑎0 , 𝑎1 , 𝑎2 ) ∈ ℍ2 and 𝑣 = (𝑏0 , 𝑏1 , 𝑏2 ) ∈ 𝑇𝑝 ℍ2 unitary. Hence, the geodesic in the Beltrami-Klein model has coordinates 𝛾(𝑡) = (

𝑎1 cosh 𝑡 + 𝑏1 sinh 𝑡 𝑎2 cosh 𝑡 + 𝑏2 sinh 𝑡 , ). 𝑎0 cosh 𝑡 + 𝑏0 sinh 𝑡 𝑎0 cosh 𝑡 + 𝑏0 sinh 𝑡 sinh 𝑡

The image of 𝛾 is a segment in the disc. For instance 𝛾(𝑡) = ( cosh 𝑡 , 0) = (tanh 𝑡, 0), 𝑡 ∈ ℝ, is the geodesic that defines the horizontal line through the origin in ℍ2𝐵𝐾 . By Proposition 4.58, the isometries of ℍ2 are given by the matrices of O(2, 1) which preserve the upper sheet of the hyperboloid. Since ℍ2𝐵𝐾 ⊂ ℝ𝑃 2 is obtained by projec± tivizing ℍ2 , the isometries of ℍ2𝐵𝐾 are obtained by projectivization of O> (2, 1). Note that, changing the sign of a matrix 𝐴 = (𝑎𝑖𝑗 ) if necessary, we can arrange that 𝑎00 > 0. Hence the above coincides with the projectivization of O(2, 1), so (4.26)

Isom(ℍ2𝐵𝐾 ) = PO(2, 1) = {[𝐴] | 𝐴 ∈ O(2, 1)} < PGL(3, ℝ).

The subgroup PO(2, 1) < PGL(3, ℝ) consists of those projective maps 𝜑𝐴 ∶ ℝ𝑃 2 → ℝ𝑃 2 such that 𝜑𝐴 (ℍ2𝐵𝐾 ) = ℍ2𝐵𝐾 . To check it, observe that 𝜑𝐴 has to send 𝜕ℍ2𝐵𝐾 to itself. This boundary is the projective conic −𝑥02 + 𝑥12 + 𝑥22 = 0 in ℝ𝑃2 , so 𝐴𝑡 𝑄𝐴 = 𝜆𝑄, for some 𝜆 ∈ ℝ∗ . This 𝜆 > 0 since the signature of 𝐴𝑡 𝑄𝐴 is equal to the signature of 𝑄. Taking determinants, det(𝐴)2 = 𝜆2 . Hence 𝐴′ = 𝜆−1/2 𝐴 ∈ O(2, 1) and it defines the same projective map [𝐴′ ] = [𝐴] ∈ PO(2, 1). Thus Isom(ℍ2𝐵𝐾 ) = {𝜑𝐴 ∶ ℝ𝑃 2 → ℝ𝑃 2 projective map| 𝜑𝐴 (ℍ2𝐵𝐾 ) = ℍ2𝐵𝐾 }. Note that projective maps send (projective) lines to lines. So an element 𝜑𝐴 ∶ ℍ2𝐵𝐾 → ℍ2𝐵𝐾 sends segments of ℍ2𝐵𝐾 to segments of ℍ2𝐵𝐾 . This gives a geometric way to send one geodesic to another in the Beltrami-Klein model via isometries. Remark 4.62. Let 𝑟, 𝑠 be two geodesics (segments) of ℍ2𝐵𝐾 . They are perpendicular if and only if the pole 𝑝𝑟 of 𝑟 with respect to the conic 𝑄 that defines 𝜕ℍ2𝐵𝐾 is in (the prolongation of) 𝑠 (cf. Remark 4.31). Indeed, let 𝑟,̄ 𝑠 ̄ be the projective lines so that 𝑟 = 𝑟∩ℍ ̄ 2𝐵𝐾 and 𝑠 = 𝑠∩ℍ ̄ 2𝐵𝐾 . In projective coordinates, 𝑟 ̄ = {[𝑥] = [𝑥0 , 𝑥1 , 𝑥2 ]|𝐵(𝑥, 𝑝𝑟 ) = 0}, where 𝑝𝑟 is the pole of 𝑟. Note that 𝐵(𝑝𝑟 , 𝑝𝑟 ) > 0, so 𝑝𝑟 lies outside ℍ2𝐵𝐾 . Let 𝑝 = 𝑟∩𝑠 ∈ ℍ2𝐵𝐾 . Take a 𝐵-orthogonal basis (𝑢0 , 𝑢1 , 𝑢2 ) of ℝ3 , where 𝑝 = [𝑢0 ] ∈ ℍ2𝐵𝐾 ⊂ ℝ𝑃 2 , 𝑢1 ∈ 𝑇ᵆ0 ℍ2 , 𝑟 ̄ = 𝑃(⟨𝑢0 , 𝑢1 ⟩) ⊂ ℝ𝑃 2 (the projective line generated by the plane ⟨𝑢0 , 𝑢1 ⟩), and 𝑝𝑟 = [𝑢2 ]. Then 𝑟, 𝑠 are 𝐵-orthogonal if and only if 𝑠 ̄ = 𝑃(⟨𝑢0 , 𝑢2 ⟩) ⊂ ℝ𝑃 2 , that is 𝑝𝑟 ∈ 𝑠.̄ This gives a geometric way to see the orthogonality in ℍ2𝐵𝐾 .

4.3. Negative constant curvature

239

r

p

Figure 4.8. Parallels and limiting parallels (dashed) to 𝑟 through the point 𝑝 in ℍ2𝐵𝐾 .

Remark 4.63. We say that two geodesics are parallel if they do not intersect. In the hyperbolic plane, given a geodesic 𝑟 and a point 𝑝 ∉ 𝑟, there are infinitely many parallels to 𝑟 through 𝑝. Such parallels form a circular sector, and the boundary geodesics 𝑠, 𝑠′ are called limiting parallel to 𝑟. In the Beltrami-Klein model, two limiting parallel lines 𝑟, 𝑠 intersect at a point in 𝜕ℍ2𝐵𝐾 . The points of 𝜕ℍ2𝐵𝐾 are called ideal points, since they are not points of the hyperbolic space. In this sense, limiting parallel lines approach indefinitely as we move in one direction, intersecting at an ideal point (see Figure 4.8). 4.3.3. Poincaré disc model. The next model is obtained from the Beltrami-Klein model as follows. Locate the Beltrami-Klein disc in ℝ3 at height 𝑥0 = 1. For a point (𝑢, 𝑣) = (1, 𝑢, 𝑣) ∈ ℍ2𝐵𝐾 = 𝐵 2 ⊂ ℝ3 , project it downwards in the vertical direction to a point in the upper hemisphere 𝑆 2+ of 𝑆 2 . Composing this map with the stereographic projection from the south pole of 𝑆 2 , we obtain a diffeomorphism 𝜙 ∶ 𝐵 2 ⟶ 𝑆 2+ ⟶ 𝐵 2 (𝑢, 𝑣) ↦ (𝑢, 𝑣, √1 − 𝑢2 − 𝑣2 ) ↦ (𝑥, 𝑦) = (

𝑢 1 + √1 −

𝑢2

−

𝑣2

,

𝑣 1 + √1 − 𝑢2 − 𝑣2

).

The Poincaré metric is given by g𝑃 = 𝜙∗ (g𝐵𝐾 ). From the definition of 𝜙 we have 2 𝑢 = (1 + √1 − 𝑢2 − 𝑣2 )𝑥, 𝑣 = (1 + √1 − 𝑢2 − 𝑣2 )𝑦, hence 1 + √1 − 𝑢2 − 𝑣2 = 2 2, 1+𝑥 +𝑦

2𝑥

2𝑦

which yields (𝑢, 𝑣) = ( 1+𝑥2 +𝑦2 , 1+𝑥2 +𝑦2 ). From this it is easy to compute that (4.27)

g𝑃 =

4 (𝑑𝑥2 + 𝑑𝑦2 ). (1 − 𝑥2 − 𝑦2 )2

Definition 4.64. We define the Poincaré disc model for the hyperbolic plane as ℍ2𝑃𝐷 = 4 (𝐵 2 , g𝑃 ), where 𝐵 2 is the open disc and the metric is g𝑃 = (1−𝑥2 −𝑦2 )2 (𝑑𝑥2 + 𝑑𝑦2 ). The Poincaré metric is conformal (Definition 4.14), since g𝑃 = 𝐸 g𝑠𝑡𝑑 with 𝐸 = 4 . This means that the angles in the chart, measured with the standard metric (1−𝑥2 −𝑦2 )2

4. Constant curvature

240

Figure 4.9. The Poincaré disc model. The image of a geodesic of ℍ2𝐵𝐾 is a geodesic in ℍ2𝑃𝐷 .

of ℝ2 , coincide with the angles in ℍ2𝑃𝐷 , that is, measured with respect to the Poincaré metric. Writing 𝑒2𝑓 = 𝐸, then 𝑓 = log 2 − log(1 − 𝑥2 − 𝑦2 ). Using Corollary 3.56, we can compute the curvature as 𝜅 = −𝑒−2𝑓 Δ𝑓 = −

(1 − 𝑥2 − 𝑦2 )2 4 = −1, 4 (1 − 𝑥2 − 𝑦2 )2

giving another proof that the sectional curvature of the hyperbolic plane is −1. The image of the geodesic 𝛾(𝑡) = (𝑢(𝑡), 𝑣(𝑡)) = (tanh 𝑡, 0) in ℍ2𝐵𝐾 under 𝜙 ∶ ℍ2𝐵𝐾 → ℍ2𝑃𝐷 is 𝜙(𝛾(𝑡)) = (𝑥(𝑡), 𝑦(𝑡)) = (tanh(𝑡/2), 0) for 𝑡 ∈ ℝ, since 𝑥(𝑡) = =

𝑢(𝑡) 1 + √1 − 2

𝑢2 (𝑡)

𝑣2 (𝑡)

=

tanh 𝑡 sinh 𝑡 = cosh 𝑡 + 1 1 + (cosh 𝑡)−1

− 2 sinh(𝑡/2) cosh(𝑡/2) 2

2

2

cosh (𝑡/2) + sinh (𝑡/2) + cosh (𝑡/2) − sinh (𝑡/2)

= tanh(𝑡/2),

by the usual formulae of hyperbolic trigonometry. In particular, a diameter through the origin of ℍ2𝑃𝐷 is a geodesic. Other geodesics can be described geometrically by looking at Figure 4.9 (see also Proposition 4.69). Consider any segment in ℍ2𝐵𝐾 and project it down to 𝑆 2+ . This gives half circles in 𝑆 2+ intersecting orthogonally the equator. By stereographic projection from the south pole and using Exercise 4.7, we have that the images are (portions of) circles or lines (diameters) intersecting orthogonally 𝑆 1 = 𝜕ℍ2𝑃𝐷 . Let us describe the isometries of the Poincaré disc ℍ2𝑃𝐷 . Instead of translating the isometries of ℍ2𝐵𝐾 = (𝐵 2 , g𝐵𝐾 ) through 𝜙, we shall make use of the conformality of the Poincaré metric. If 𝜑 ∈ Isom(ℍ2𝑃𝐷 ), then it is in particular conformal with respect to g𝑃 . As the metric g𝑃 is conformal to the standard metric g𝑠𝑡𝑑 of ℝ2 , 𝜑 is a conformal map with respect to g𝑠𝑡𝑑 . By Lemma 4.16, the conformal and orientation preserving maps 𝜑 ∶ 𝐵 2 ⊂ ℂ → 𝐵 2 ⊂ ℂ are biholomorphisms of 𝐵 2 . These maps are determined in the theory of complex variables by a result of Schwarz. We introduce groups of complex 2×2 matrices that are useful to describe the isometries of ℍ2𝑃𝐷 .

4.3. Negative constant curvature

241

Definition 4.65. Let us denote 𝑄 = (

1 0 ). We define the groups 0 −1

U(1, 1) = {𝐴 ∈ GL(2, ℂ)|𝐴𝑡 𝑄𝐴 = 𝑄} < GL(2, ℂ), SU(1, 1) = U(1, 1) ∩ SL(2, ℂ). Observe that this notation is justified by the fact that the matrices of U(1, 1) are 𝑡 the linear maps of ℂ2 which preserve the indefinite Hermitian metric ℎ(𝑧, 𝑤) = 𝑧 𝑄𝑤, for 𝑧, 𝑤 ∈ ℂ2 , or equivalently, the indefinite Hermitian norm ℎ(𝑧, 𝑧) = |𝑧0 |2 − |𝑧1 |2 , for 𝑧 = (𝑧0 , 𝑧1 ) ∈ ℂ2 . A computation similar to that carried out in Proposition 4.26 yields 𝑏 𝜆𝑐 that the matrices 𝐴 ∈ U(1, 1) are of the form 𝐴 = ( ), for some 𝑏, 𝑐, 𝜆 ∈ ℂ with 𝑐 𝜆𝑏 𝑏 𝑐 |𝑏|2 − |𝑐|2 = 1 and |𝜆| = 1. The matrices 𝐴 ∈ SU(1, 1) are of the form 𝐴 = ( ), 𝑐 𝑏 for some 𝑏, 𝑐 ∈ ℂ with |𝑏|2 − |𝑐|2 = 1. Writing (𝑏, 𝑐) = (𝑏1 , 𝑏2 , 𝑐 1 , 𝑐 2 ) ∈ ℂ2 = ℝ4 , the defining equation is 𝑏21 + 𝑏22 − 𝑐21 − 𝑐22 = 1, which is a hyperboloid of one sheet. Thus SU(1, 1) is a 3-dimensional connected Lie group. According to (4.14), we have isomorphisms PU(1, 1) = U(1, 1)/𝑆 1 ≅ SU(1, 1)/ℤ2 = PSU(1, 1), where the action of 𝜆 ∈ 𝑆 1 on 𝐴 ∈ U(1, 1) is given by (𝜆, 𝐴) ↦ 𝜆𝐴. For notational purposes, given 𝐴 ⊂ ℂ, we denote by Mob(𝐴) the set of 𝜑 ∈ Mob(ℂ) such that 𝜑(𝐴) = 𝐴. Analogously, Bihol(𝐴) denotes the set of biholomorphic maps 𝜑 ∶ 𝐴 → 𝐴, for 𝐴 open. Theorem 4.66. An orientation preserving isometry 𝜑 ∶ ℍ2𝑃𝐷 → ℍ2𝑃𝐷 has the form (4.28)

𝜑(𝑧) = 𝑒i𝜃

𝑧+𝑎 , 1 + 𝑎𝑧

for some 𝑎 ∈ ℂ with |𝑎| < 1 and 𝜃 ∈ ℝ. Furthermore, Isom+ (ℍ2𝑃𝐷 ) ≅ PSU(1, 1) ≅ Bihol(𝐵 2 ) = Mob(𝐵 2 ). Proof. A result of complex variables states that the biholomorphisms of the unit disc are of the form (4.28); a proof can be found in [Co2]. As any orientation preserving isometry of ℍ2𝑃𝐷 is a biholomorphism of the unit disc, because the metric of ℍ2𝑃𝐷 is conformal, we conclude that 𝜑 is of the above form. Note that 𝑎 and 𝜃 are characterised by the image of the origin, as 𝜑(0) = 𝑎𝑒i𝜃 and 𝜑′ (0) = 𝑒i𝜃 . The map 𝜑 is a Möbius map 1 𝑎 (Definition 4.18) associated to the matrix ( i𝜃 i𝜃 ). If we multiply this matrix by 𝑎𝑒 𝑒 𝑏 𝑐 𝑟𝑒−i𝜃/2 , we get the matrix ( ), with 𝑏 = 𝑟𝑒−i𝜃/2 and 𝑐 = 𝑟𝑎𝑒i𝜃/2 . Choosing 𝑟 so 𝑐 𝑏 that |𝑏|2 − |𝑐|2 = 1, i.e., taking 𝑟2 − 𝑟2 |𝑎|2 = 1 which gives 𝑟 = (1 − |𝑎|2 )−1/2 , we get 𝑏 𝑐 a matrix in SU(1, 1). This process can be reversed since, for any matrix ( ) with 𝑐 𝑏 |𝑏|2 − |𝑐|2 = 1, we can divide it by 𝑏 and put 𝑎 = 𝑐/𝑏 and 𝑒i𝜃 = 𝑏/𝑏 to get a Möbius

4. Constant curvature

242

1 𝑎 transformation of the type ( i𝜃 i𝜃 ), which is a biholomorphism of the disc. This 𝑎𝑒 𝑒 proves that Isom+ (ℍ2𝑃𝐷 ) < Bihol(𝐵 2 ) = PSU(1, 1). It remains to see that any biholomorphism is in fact an isometry of ℍ2𝑃𝐷 . For that, just observe that PSU(1, 1) = SU(1, 1)/ℤ2 is connected since SU(1, 1) is so. By Remark 3.103, as ℍ2𝑃𝐷 is isotropic, dim Isom+ (ℍ2𝑃𝐷 ) = 3. So Isom+ (ℍ2𝑃𝐷 ) and PSU(1, 1) are 3-dimensional connected Lie groups and thus, by Exercise 4.5, they are equal. Thus Isom+ (ℍ2𝑃𝐷 ) = PSU(1, 1). For the final equality, observe that if 𝜑 ∈ Bihol(𝐵 2 ), then 𝜑 has the form (4.28), so in particular it is a Möbius map sending 𝐵2 to itself, thus 𝜑 ∈ Mob(𝐵 2 ). □ Remark 4.67. A map 𝜑 ∈ Mob(𝐵 2 ) is a Möbius transformation that sends 𝑆 1 to itself and |𝜑(0)| < 1. A Möbius transformation is determined by the image of three points, so 𝜑 ∈ Mob(ℂ) sends 𝑆 1 to itself if and only if it sends three points, say 1, i, −1 ∈ 𝑆 1 , to three points 𝑎, 𝑏, 𝑐 ∈ 𝑆 1 . But such 𝜑 may send 𝐵2 to either the interior or the exterior or 𝑆 1 , that is 𝜑(𝐵 2 ) = 𝐵 2 or 𝜑(𝐵 2 ) = ℂ − 𝐷2 . These two cases are determined by whether 𝜑 preserves or reverses the cyclic order of points of 𝑆 1 . The cyclic order of three points 𝑎, 𝑏, 𝑐 ∈ 𝑆 1 is the orientation determined by the path that describes the arc of 𝑆 1 that goes from 𝑎 to 𝑐 passing through 𝑏. Therefore 𝜑(𝐵 2 ) = 𝐵 2 if and only if 𝑎, 𝑏, 𝑐 have positive cyclic order in 𝑆 1 . Remark 4.68. (1) The set Isom− (ℍ2𝑃𝐷 ) of isometries which reverse the orientation are obtained by composing the orientation preserving isometries with a fixed 𝜙 ∈ Isom− (ℍ2𝑃𝐷 ). We can take 𝑟(𝑧) = 𝑧 which clearly preserves the metric g𝑃 . Therefore Isom− (ℍ2𝑃𝐷 ) = Mob(𝐵 2 ) ∘ 𝑟, i.e., maps of the form 𝑧 ↦ 𝑒i𝜃

𝑧+𝑎 . 1 + 𝑎𝑧

ˆ These are the maps in Mob(ℂ) that preserve 𝐵 2 . (2) If 𝜑 ∶ ℍ2 → ℍ2 is a (global) conformal map, then it is an isometry. By (1), it is enough to prove it for orientation preserving conformal maps. We use the Poincaré disc model, so 𝜑 ∶ ℍ2𝑃𝐷 → ℍ2𝑃𝐷 is a biholomorphism of 𝐵 2 , hence by Theorem 4.66, it is an isometry. (3) In ℍ2 there is no concept of similarity of geometric figures, since this concept requires the existence of homotheties (Remark 4.12(5)). So, if two triangles have the same angles, then they have the same area, and there is an isometry which maps one triangle onto the other. By Remark 3.67, the sum of the angles of a triangle 𝑇 in ℍ2 is 𝛽1 + 𝛽2 + 𝛽3 = 𝜋 − area(𝑇), so the area 𝑇 is determined by the angles. If there were a homothety 𝜙 ∶ 𝑈 → 𝑉 of dilation factor 𝜆 > 0, defined on some open sets 𝑈, 𝑉 ⊂ ℍ2 , then 𝜙∗ gℍ2 = 𝜆2 gℍ2 , which implies that the lengths of the edges of 𝑇 are multiplied by 𝜆 and the area of 𝑇 is multiplied by 𝜆2 . Since the angles are preserved by 𝜙, this is impossible unless 𝜆 = 1. We conclude that homotheties do not exist, not even locally.

4.3. Negative constant curvature

243

(4) On the other hand, locally conformal maps do exist. Just take any holomorphic map with non-vanishing derivative on an small open subset of ℍ2𝑃𝐷 . After this study of the isometries of ℍ2𝑃𝐷 , we can reprove the shape of the geodesics in ℍ2𝑃𝐷 . Proposition 4.69. The geodesics of ℍ2𝑃𝐷 are the arcs of circles which intersect 𝑆 1 = 𝜕ℍ2𝑃𝐷 orthogonally and the diameters. Proof. We already know that the curve 𝛾0 (𝑡) = (tanh(𝑡/2), 0) is a geodesic. Since ℍ2𝑃𝐷 is isotropic, any other (unitary) geodesic 𝛾 has to be the image of 𝛾0 by a certain isometry. So there exists a Möbius map 𝜑 of 𝐵 2 so that 𝜑(𝛾0 (𝑡)) = 𝛾(𝑡). Since Möbius maps transform lines into either circles or lines by Lemma 4.20, we conclude that 𝛾 has to be an arc of a circle or a segment in 𝐵 2 . Also, a Möbius transformation preserves the orthogonality (in ℝ2 with the Euclidean metric), and the diameter 𝛾0 is perpendicular to 𝑆 1 = 𝜕ℍ2𝑃𝐷 , so the arc or segment 𝛾 has to intersect 𝑆 1 orthogonally. This proves that any geodesic is as described. Conversely, let us see that any diameter or arc of a circle orthogonal to 𝑆 1 = 𝜕ℍ2𝑃𝐷 is a geodesic. It is clear for diameters. For an arc 𝛾, take its endpoints 𝑃, 𝑄 ∈ 𝑆 1 , and any other point 𝑅 ∈ 𝑆 1 so that 𝑃, 𝑅, 𝑄 have positive cyclic order. There is a Möbius map 𝜑 which sends 1, i, −1 to 𝑃, 𝑅, 𝑄, respectively. Then 𝜑 ∈ Isom(ℍ2𝑃𝐷 ), and 𝜑−1 (𝛾) must be the diameter through 1, −1, which is the geodesic 𝛾0 . Hence 𝛾 = 𝜑(𝛾0 ) is a geodesic. □ Remark 4.70. In analogy with the Beltrami-Klein model, the points of 𝑆 1 = 𝜕ℍ2𝑃𝐷 are called the ideal points of the Poincaré disc. If we take three different ideal points in 𝑆 1 , there exist unique circles orthogonal to 𝜕ℍ2𝑃𝐷 passing through any pair of them. These circles form what is called an ideal triangle, i.e., a triangle 𝑇 whose vertices are at infinity. It is not a triangle in the strict sense, but a kind of limit of triangles in ℍ2𝑃𝐷 by pushing the vertices to infinity. Each pair of edges of 𝑇 are limiting parallel lines because they intersect at infinity. Note that by Remark 4.68(3), any triangle has area strictly smaller that 𝜋. However, ideal triangles have area equal to 𝜋. This holds since the ideal angle formed by two limiting parallel lines is zero. 4.3.4. Poincaré half-plane model. We study next another well known model for the hyperbolic plane which is also conformal. For this, consider a Möbius map 𝜙 ∈ Mob(ℂ) that sends the unit disc to the upper half-plane 𝐻 = {𝑤 ∈ ℂ | Im 𝑤 > 0}. To write a concrete map, fix 𝜙(1) = 0, 𝜙(i) = 1 and 𝜙(−1) = ∞, so 𝜙 is given by 𝑧−1 (4.29) 𝑤 = 𝜙(𝑧) = −i . 𝑧+1 Automatically 𝜙(𝜕ℍ2𝑃𝐷 ) = ℝ = ℝ ∪ {∞} and 𝜙(0) = i, so 𝜙 maps ℍ2𝑃𝐷 to the upper halfplane 𝐻. Let us compute the induced metric g𝐻 = 𝜙∗ g𝑃 in the coordinate 𝑤. Using complex coordinates 𝑧 = 𝑥 + i𝑦 ∈ 𝐵 2 , 𝑤 = 𝑢 + i𝑣 ∈ 𝐻, as we did in Proposition 4.27, we write the metric g𝑃 given in Definition 4.64 as (4.30)

g𝑃 =

4|𝑑𝑧|2 . (1 − |𝑧|2 )2

4. Constant curvature

244

@H2P D

@H2P H Figure 4.10. An ideal triangle in ℍ2𝑃𝐷 and in ℍ2𝑃𝐻 .

From (4.29), we have 𝑧 = g𝐻 = 𝜙∗ g𝑃 =

=

1+i𝑤 1−i𝑤

and thus 𝑑𝑧 =

2i 𝑑𝑤 . (1−i𝑤)2

16|𝑑𝑤|2 2 2

=

1+i𝑤 |1 − i𝑤|4 (1 − || 1−i𝑤 || )

Therefore 16|𝑑𝑤|2 2

(|1 − i𝑤|2 − |1 + i𝑤|2 )

16|𝑑𝑤|2 16|𝑑𝑤|2 𝑑𝑢2 + 𝑑𝑣2 = = . 2 2 𝑣2 (4𝑣) (2i𝑤 − 2i𝑤)

Definition 4.71. We define the Poincaré half-plane model of the hyperbolic plane as 1 ℍ2𝑃𝐻 = (𝐻, 𝑔𝐻 ), where 𝐻 = {(𝑢, 𝑣) ∈ ℝ2 | 𝑣 > 0}, and the metric g𝐻 = 𝑣2 (𝑑𝑢2 + 𝑑𝑣2 ). The metric g𝐻 is again conformal, although this was clear since g𝑃 is conformal, and the transformation 𝑤 = 𝜙(𝑧) is a conformal map. Once again we may compute the curvature using Corollary 3.56. Write 𝐸 = 𝑣−2 = 𝑒2𝑓 , so 𝑓 = − log 𝑣 and Δ𝑓 = 𝑣−2 . Therefore 𝜅 = −𝑒−2𝑓 Δ𝑓 = −1. To determine Isom(ℍ2𝑃𝐻 ), it is enough to transform the isometries of Isom(ℍ2𝑃𝐷 ) via 𝜙. Hence, the orientation preserving isometries of ℍ2𝑃𝐻 are the maps of the form 𝑓 = 𝜙 ∘ 𝜑 ∘ 𝜙−1 , where 𝜑 ∈ Isom+ (ℍ2𝑃𝐷 ) = Mob(𝐵 2 ). As 𝜙 is a Möbius map, we have that 𝑓 ∈ Mob(𝐻). Thus Isom+ (ℍ2𝑃𝐻 ) = Mob(𝐻). The condition that 𝑓(𝐻) = 𝐻 can be restated as requiring that 𝑓 maps ℝ = ℝ ∪ {∞} ≅ ℝ𝑃 1 to itself and that 𝑓|ℝ ∶ ℝ → ℝ respects the natural orientation of ℝ, which in turn means that 𝑓 respects the cyclic order of points of ℝ ≅ ℝ𝑃 1 , which is topologically a circle (see Remark 4.67). Note that when 𝑓 respects the orientation, then the normal vector on the positive direction (pointing upwards) is sent to itself, so 𝑓 maps the upper half-plane to the upper halfplane. The Möbius maps 𝑓 ∈ Mob(ℝ𝑃 1 ) are actually the projective maps of ℝ𝑃1 , that is, the elements of PGL(2, ℝ) < PGL(2, ℂ). Therefore +

(4.31)

Isom+ (ℍ2𝑃𝐻 ) ≅ Mob(𝐻) ≅ PGL (2, ℝ) = PSL(2, ℝ) < PSL(2, ℂ) = Mob(ℂ),

4.3. Negative constant curvature

245

+

where PGL (2, ℝ) is the group of projective maps of ℝ𝑃1 represented by matrices with positive determinant (hence they respect the orientation). For such a projective map, there is a representative with determinant 1, unique up to sign, hence PGL(2, ℝ) = PSL(2, ℝ) ≅ SL(2, ℝ)/ℤ2 . 1

The map 𝑔(𝑧) = 𝑧 is an orientation reversing isometry of ℍ2𝑃𝐻 (since it is an orien− tation reversing Möbius map), hence Isom− (ℍ2𝑃𝐻 ) = Isom+ (ℍ2𝑃𝐻 ) ∘ 𝑔 ≅ Mob (𝐻) = − {𝑓 ∈ Mob (ℂ)|𝑓(𝐻) = 𝐻}. The geodesics in ℍ2𝑃𝐻 are the vertical half lines and half circles which are orthogonal to 𝜕𝐻 = ℝ. This holds because the geodesics in ℍ2𝑃𝐷 are (the intersection with 𝐵 2 of) lines and circles orthogonal to 𝜕𝐵 2 , and the Möbius map 𝜙 transforms them to (the intersection with 𝐻 of) lines and circles which are orthogonal to 𝜕𝐻 = ℝ. As an example, the geodesic 𝛾(𝑡) = (tanh(𝑡/2), 0) in ℍ2𝑃𝐷 is transformed into tanh(𝑡/2) − 1 = −i 𝜙(𝛾(𝑡)) = −i tanh(𝑡/2) + 1

𝑒𝑡/2 −𝑒−𝑡/2 𝑒𝑡/2 +𝑒−𝑡/2 𝑒𝑡/2 −𝑒−𝑡/2 𝑒𝑡/2 +𝑒−𝑡/2

−1 +1

= −i

−2𝑒−𝑡/2 = 𝑒−𝑡 i, 2𝑒𝑡/2

which is the vertical line through the point i ∈ 𝐻. Remark 4.72. The area of an ideal triangle can be computed in this model (see Remark 4.70). Let us take a triangle 𝑇 with edges given by the upper half of the unit circle and 1 the lines {𝑢 = 1}, {𝑢 = −1}, as in Figure 4.10. Since the volume element in 𝐻 is 𝐸 = 𝑣2 , we have 1

∞

area(𝑇) = ∫ ∫ −1 1

=∫ −1

√1−ᵆ2

1

1 −1 ∞ ∫ 𝑑𝑣𝑑𝑢 = 𝑑𝑢 [ ] 𝑣 √1−ᵆ2 𝑣2 −1

1 √1 − 𝑢 2

ᵆ=1

𝑑𝑢 = [ arcsin 𝑢]

ᵆ=−1

= 𝜋.

This happens for every ideal triangle, since for any two ideal triangles there is a Möbius map which maps one onto the other. Remark 4.73. For the hyperbolic 𝑛-space ℍ𝑛 with 𝑛 > 2 there is a Beltrami-Klein model, a Poincaré ball model, and a Poincaré half space model (Exercise 4.21). The last two are conformal, whereas the first is related to projective geometry. For instance, the Poincaré half space model is given as (𝐻 𝑛 , g𝐻 𝑛 ) with 𝐻 𝑛 = {(𝑥1 , . . . , 𝑥𝑛 )|𝑥𝑛 > 0} and 1 g𝐻 𝑛 = 𝑥2 (𝑑𝑥12 + ⋯ + 𝑑𝑥𝑛2 ). 𝑛

Remark 4.74. The different models of the hyperbolic plane and the description of their corresponding isometry groups in (4.26), (4.31), and Theorem 4.66 give the following + interesting isomorphisms of Lie groups: PO (2, 1) ≅ PSU(1, 1) ≅ PSL(2, ℝ). Classification of isometries of ℍ2 . In order to know what subgroups of isometries of ℍ2 can act freely and properly, we need to have a geometric classification of the isometries. We introduce the following transformations: (1) Given 𝑒i𝜃 ∈ 𝑆 1 consider the map 𝜌𝜃 (𝑧) = 𝑒i𝜃 𝑧 in the model ℍ2𝑃𝐷 . This is an isometry with a unique fixed point 𝑃 in the hyperbolic plane. It acts as a rotation in the tangent space 𝑇𝑃 ℍ2 , so it rotates the geodesics emanating from 𝑃. We call such maps central rotations.

4. Constant curvature

246

(2) Given 𝑎 ∈ ℝ, consider the map 𝜏𝑎 (𝑧) = 𝑎𝑧 defined in the model ℍ2𝑃𝐻 . As a Möbius map, 𝜏𝑎 leaves fixed two points, namely 0 and ∞. Therefore 𝜏𝑎 fixes two ideal points, and so it leaves invariant the geodesic 𝛾 which joins them. As any geodesic is determined by its ideal endpoints, this is the only invariant geodesic. In the Poincaré half-plane model 𝛾(𝑡) = 𝑒𝑡 i and 𝜏𝑎 (𝛾(𝑡)) = 𝑒𝑡+𝑘 i, with 𝑘 = log 𝑎 ∈ ℝ. So 𝜏𝑎 as a translation along the geodesic 𝛾. Note that the family of geodesics perpendicular to 𝛾 is translated as well. For this reason, we call such maps translations along a geodesic. (3) Given 𝑏 ∈ ℝ, consider the transformation 𝜛 𝑏 (𝑧) = 𝑧 + 𝑏, defined in Poincaré half-plane ℍ2𝑃𝐻 . The map 𝜛 𝑏 is an isometry. As a Möbius map, it only fixes ∞ ∈ ℂ, which is an ideal point, so 𝜛 𝑏 has no fixed points on ℍ2𝑃𝐻 . The map 𝜛 𝑏 transforms vertical lines to vertical lines. This means that it moves a collection of limiting parallels through one ideal point. Seen in the Poincaré disc ℍ2𝑃𝐷 , 𝜛 𝑏 can be understood as a rotation around an ideal point which remains fixed. For this, we call such maps ideal rotations. Note that we can arrange 𝑏 = 1 in (3) by the change of variables 𝑧′ = 𝑏𝑧. However the rotation angle 𝜃 in (1) and the translation length 𝑘 > 0 in (2) are intrinsically defined.

Figure 4.11. Orientation preserving isometries of ℍ2 .

Let us see that every isometry of Isom+ (ℍ2 ) is of one of the types described above. Theorem 4.75. Let 𝜑 ≠ Id be an orientation preserving isometry of ℍ2 . Then 𝜑 is either an ideal rotation, a central rotation, or a translation along a geodesic. That is, there exists 𝜓 ∈ Isom(ℍ2 ) such that 𝜓 ∘ 𝜑 ∘ 𝜓−1 is one of the isometries described in Figure 4.11. 𝑎𝑧+𝑏

Proof. Consider 𝜑 in the Poincaré half-plane model ℍ2𝑃𝐻 . By (4.31), 𝜑(𝑧) = 𝑐𝑧+𝑑 with 𝑎, 𝑏, 𝑐, 𝑑 ∈ ℝ, 𝑎𝑑 − 𝑏𝑐 = 1. Let us see how many fixed points 𝜑 has. If 𝑧 ∈ ℂ is a fixed point, then 𝜑(𝑧) = 𝑧, which is equivalent to 𝑎𝑧 + 𝑏 = 𝑧(𝑐𝑧 + 𝑑). So 𝑧 is a zero of the polynomial 𝑝(𝑧) = 𝑐𝑧2 + (𝑑 − 𝑎)𝑧 + 𝑏, which is of degree (at most) 2 and with real coefficients. We distinguish the following cases: (1) The polynomial 𝑝 has one double root. In this case, 𝑝(𝑧) = 𝑐(𝑧 − 𝜆)2 with 𝑐 ≠ 0, and 𝜆 ∈ ℝ is the only fixed point of 𝜑, which is an ideal point of ℍ2𝑃𝐻 . In order to have a better picture of how the isometry 𝜑 behaves, consider

4.3. Negative constant curvature

247

𝜓 ∈ Mob(𝐻) with 𝜓(𝜆) = ∞ and change 𝜑 to 𝜑̂ = 𝜓 ∘ 𝜑 ∘ 𝜓−1 , so that 𝜑̂ has ∞ as the only fixed point. Möbius maps with ∞ as its only fixed point are ̂ ′ ) = 𝑧′ + 𝑏′ for some 𝑏′ ∈ ℝ, so 𝜑 is an ideal translations of ℂ. Therefore 𝜑(𝑧 rotation. (2) The polynomial 𝑝 has two real roots. In this case 𝑝(𝑧) = 𝑐(𝑧 − 𝜆)(𝑧 − 𝜇) with 𝑐 ≠ 0 and 𝜆, 𝜇 ∈ ℝ the only fixed points of 𝜑, both of which are ideal points of ℍ2𝐻𝑃 = 𝐻. As before, let us consider a change of coordinates 𝜓 ∈ Mob(𝐻) so that 𝜓(𝜆) = 0 and 𝜓(𝜇) = ∞, and 𝜑̂ = 𝜓 ∘ 𝜑 ∘ 𝜓−1 . The only fixed points of 𝜑̂ ̂ ′ ) = 𝑎′ 𝑧′ , for some 𝑎′ ∈ ℝ. Hence 𝜑 is a translation are 0 and ∞, therefore 𝜑(𝑧 along a geodesic. (3) The polynomial 𝑝 has two complex conjugate roots. In this case, write 𝑝(𝑧) = 𝑐(𝑧 − 𝑤 0 )(𝑧 − 𝑤0 ) with 𝑐 ≠ 0 and 𝑤 0 , 𝑤0 ∈ ℂ − ℝ are the only fixed points of 𝜑, with 𝑤 0 ∈ 𝐻 and 𝑤0 ∉ 𝐻. In this case, the disc model is more useful, so consider a Möbius map 𝜓 ∶ 𝐻 → ℍ2𝑃𝐷 such that 𝜓(𝑤 0 ) = 0, and let 𝜑̂ = 𝜓 ∘ 𝜑 ∘ 𝜓−1 . Then 𝜑̂ is a Möbius map of the Poincaré disc model ℍ2𝑃𝐷 with ̂ = 0. By Theorem 4.66, 𝜑(𝑧 ̂ ′ ) = 𝑒i𝜃 𝑧′ , and we conclude that 𝜑 is a central 𝜑(0) rotation. (4) The polynomial 𝑝 has degree 1, i.e., 𝑐 = 0. Therefore 𝜑(𝑧) = 𝛼𝑧 + 𝛽 with 𝛼 = 𝑎/𝑑 and 𝛽 = 𝑏/𝑑. In this case, 𝜑 leaves ∞ fixed and it may have another fixed point depending on whether 𝛼 equals 1 or not. If 𝛼 = 1, then 𝜑 is an ideal rotation, and if 𝛼 ≠ 1, then 𝜑 is a translation along a geodesic which 𝛽 leaves fixed the ideal points ∞ and 1−𝛼 . □ The orientation reversing isometries of the hyperbolic plane are classified in Exercise 4.27. 4.3.5. Compact surfaces with constant curvature −1. Any complete connected surface with constant curvature equal to −1 can be expressed as a quotient 𝑆 = ℍ2 /Γ, for Γ < Isom(ℍ2 ) a subgroup of isometries which acts freely and properly on ℍ2 . We address here the case where 𝑆 is compact and oriented. In this case, Γ < Isom+ (ℍ2 ) and the action of Γ in ℍ2 has a compact fundamental domain. Observe that the case where 𝑆 is non-orientable and compact can be easily tackled from the orientable case by considering the oriented cover 𝑆.̂ However the case where 𝑆 is non-compact is more difficult. Theorem 4.76. If Γ < Isom+ (ℍ2 ) acts freely and properly and 𝑆 = ℍ2 /Γ is compact, then every 𝜑 ∈ Γ − {Id} is a translation along a geodesic. Proof. The group Γ cannot contain central rotations, since it acts freely. Let us suppose that it contains ideal rotations. Taking the model ℍ2𝑃𝐻 , we can assume that it contains ideal rotations with an ideal fixed point ∞. These are translations of the form 𝑧 ↦ 𝑧+𝑏, 𝑏 ∈ ℝ∗ . As the action of Γ is proper, then the subgroup Γ ∩ 𝒯 < Γ, where 𝒯 is the group of translations in ℝ, also acts properly. By Proposition 4.41, this is a rank 1 lattice Λ ⊂ ℝ, hence generated by some 𝑏 > 0. By the comment after (3) in section 4.3.4, we can assume that 𝑏 = 1, so 𝜛(𝑧) = 𝑧 + 1 is the generator of Γ ∩ 𝒯. The matrix

4. Constant curvature

248

1 corresponding to 𝜛 is 𝐴 = ( 1

0 ) ∈ SL(2, ℝ). If we had Γ = ⟨𝜛⟩, then ℍ2𝑃𝐻 /Γ would 1 𝑎𝑧+𝑏

be non-compact, so there exists 𝜓 ∈ Γ − ⟨𝜛⟩. Writing 𝜓(𝑧) = 𝑐𝑧+𝑑 , the associated 𝑑 𝑐 matrix is 𝐵 = ( ) ∈ SL(2, ℝ) with 𝑑𝑎 − 𝑐𝑏 = 1. With that said, let us distinguish 𝑏 𝑎 cases. 𝑑𝑛 0 ) 𝑏𝑛 𝑎𝑛 for some 𝑏𝑛 ∈ ℝ, which we do not write explicitly, and the maps 𝜓𝑛 = 𝜓−𝑛 ∘ 𝜛 ∘ 𝜓𝑛 ∈ Γ with matrices

• If 𝑐 = 0, then 𝑎𝑑 = 1. We consider the maps 𝜓𝑛 with matrices 𝐵𝑛 = (

𝐵 −𝑛 𝐴𝐵𝑛 = (

𝑎𝑛 −𝑏𝑛

0 1 0 𝑑𝑛 )( )( 𝑑 𝑛 1 1 𝑏𝑛

0 1 ) = ( 2𝑛 𝑎𝑛 𝑑

0 ). 1

So 𝜓𝑛 (𝑧) = 𝑧 + 𝑑 2𝑛 . Therefore the points 𝑝𝑛 = 𝜓𝑛 (i) = i + 𝑑 2𝑛 , 𝑛 ∈ ℤ, are equivalent. If |𝑑| < 1, then 𝑝𝑛 → i as 𝑛 → ∞, and if |𝑑| > 1, then 𝑝𝑛 → i as 𝑛 → −∞. In either case, the group Γ does not act properly. If 𝑑 = ±1, then, 1 0 maybe after changing the sign of 𝐵, we have that 𝐵 = ( ), for some 𝑏 ≠ 0, 𝑏 1 which means 𝜓 ∈ Γ ∩ 𝒯 = ⟨𝜛⟩, which is ruled out by assumption. −1 • If 0 < |𝑐| < 1, then we consider the maps 𝜓0 = 𝜓, and 𝜓𝑛 = 𝜓𝑛−1 ∘ 𝜛 ∘ 𝜓𝑛−1 ∈ 𝑎𝑛 𝑧+𝑏𝑛 Γ for 𝑛 ≥ 1. Writing 𝜓𝑛 (𝑧) = 𝑐 𝑧+𝑑 , with matrix 𝐵𝑛 , we have that 𝐵0 = 𝐵 𝑛

−1 and 𝐵𝑛 = 𝐵𝑛−1 𝐴𝐵𝑛−1 , that is

𝑑 ( 𝑛 𝑏𝑛

𝑐𝑛 𝑑 ) = ( 𝑛−1 𝑎𝑛 𝑏𝑛−1

𝑛

𝑐𝑛−1 1 )( 𝑎𝑛−1 1

𝑎 𝑐 +1 = ( 𝑛−1 2𝑛−1 𝑎𝑛−1

0 𝑎𝑛−1 )( 1 −𝑏𝑛−1

−𝑐𝑛−1 ) 𝑑𝑛−1

2 −𝑐𝑛−1 ). 1 − 𝑎𝑛−1 𝑐𝑛−1

𝑛

2 So 𝑐𝑛 = −𝑐𝑛−1 = ±𝑐2 . As |𝑐| < 1, then 𝑐𝑛 → 0. Next 𝑎𝑛 = 1 − 𝑎𝑛−1 𝑐𝑛−1 . As |𝑐𝑛 | ≤ |𝑐|, we get |𝑎𝑛 | ≤ 1 + |𝑎𝑛−1 | |𝑐|. Hence |𝑎𝑛 | ≤ 1 + |𝑐| + |𝑐|2 + ⋯ + 1 |𝑐|𝑛−1 + |𝑐|𝑛 |𝑎| ≤ 1−|𝑐| + |𝑎||𝑐|, in particular, it is bounded. Going back to 𝑎𝑛 =

1−𝑎𝑛−1 𝑐𝑛−1 , we get 𝑎𝑛 → 1. Also 𝑑𝑛 = 1+𝑎𝑛−1 𝑐𝑛−1 → 1 and 𝑏𝑛 = 𝑎2𝑛−1 → 1, 𝑎 i+𝑏 as 𝑛 → ∞. With this, we get a sequence of points 𝑝𝑛 = 𝜓𝑛 (i) = 𝑐 𝑛i+𝑑 𝑛 ∈ ℍ2 , 𝑛 𝑛 all of them equivalent under the action of Γ, and accumulating at 1+i = 𝜛(i), contradicting that the action is proper. • Suppose that all maps 𝜓 ∈ Γ−⟨𝜛⟩ have matrices with |𝑐| ≥ 1. As 𝑆 is compact, we may consider a fundamental domain 𝐷 ⊂ 𝐻 for the action of Γ which is compact (e.g., take a planar representation 𝑞 ∶ P → 𝑆, and lift the map to the universal cover 𝜋 ∶ 𝐻 → 𝑆, say 𝑞 ̃ ∶ P → 𝐻 , and take 𝐷 = 𝑞(P)). ̃ Let 1 𝑀 > 0 so that 𝑀 ≤ Im(𝑧) ≤ 𝑀 for all 𝑧 ∈ 𝐷. By formula (4.13), write 𝑑

1

any 𝜓 ∈ Γ − ⟨𝜛⟩ as 𝜓 = 𝜏2 ∘ ℎ ∘ 𝚤 ∘ 𝜏1 , where 𝜏1 (𝑧) = 𝑧 + 𝑐 , 𝚤(𝑧) = − 𝑧 , ℎ(𝑧) =

1 𝑧, 𝑐2

𝑎

and 𝜏2 (𝑧) = 𝑧 + 𝑐 . For any translation 𝜏𝑘 , Im(𝜏𝑘 (𝑧)) = Im(𝑧);

for the inversion 𝚤, Im(𝚤(𝑧)) = Im(ℎ(𝑧)) =

1 Im(𝑧) 𝑐2

Im(𝑧) |𝑧|2

≤

1 Im(𝑧)

≤ 𝑀; and for the homothety,

≤ 𝑀, since |𝑐| ≥ 1. Therefore Im(𝜓(𝑧)) ≤ 𝑀 for all

4.3. Negative constant curvature

249

𝜓 ∈ Γ and 𝑧 ∈ 𝐷. This means that the image of the fundamental domain 𝐷 is included in {𝑧 ∈ 𝐻|Im(𝑧) ≤ 𝑀}. So the Γ-translates of 𝐷 cannot cover the whole of 𝐻, contradicting that it is a fundamental domain. □ Remark 4.77. • We have proved above that a group Γ < PSL(2, ℝ) acts properly on ℍ2𝑃𝐻 if and only if Γ is a discrete subset of PSL(2, ℝ), with its natural topology as a Lie group (Exercise 4.29). And Γ < PSL(2, ℝ) acts freely if and only if it does not contain central rotations. This is the analogue of Proposition 4.41 in hyperbolic geometry. • If Γ acts properly but contains a central rotation 𝜌 ∈ Γ, then 𝑆 = 𝐻/Γ has an orbifold point. Certainly, let 𝑝0 ∈ ℍ2 be the center of the rotation, and let 𝜃 be its angle. If 𝜃 is not a rational multiple of 2𝜋 (that is, if 𝜌 has no finite order), then the multiples 𝜌𝑛 , 𝑛 ∈ ℤ have accumulation points in the set 𝑆 1 ⊂ PSL(2, ℝ) consisting of central rotations of center 𝑝0 , and then Γ would not be discrete. Therefore we can take a generator of the rotations in Γ with 2𝜋 center in 𝑝0 , of some order 𝑚 > 0, and with angle 𝜃 = 𝑚 . By Remark 3.81, the quotient ℍ2 /Γ has the structure of a (Riemannian, oriented) orbisurface with a point of index 𝑚 at 𝑝0 . • If Γ acts properly on ℍ2𝑃𝐻 = 𝐻 and it contains an ideal rotation of the form 𝜑(𝑧) = 𝑧 + 1, then the argument in the proof of Theorem 4.76 shows that a fundamental domain must contain a stripe of the form 𝑅 = {0 ≤ Re(𝑧) ≤ 1, Im(𝑧) > 𝑀}, where the vertical edges are identified. This means that the fundamental domain contains a region between two geodesics ending at the (ideal) center of rotation, which are identified by Γ. In the quotient 𝑆 = ℍ2𝑃𝐻 /Γ, 𝜋(𝑅) is called a cusp of 𝑆. It is a non-compact region, which is accessible by a geodesic of infinite length (defined by a vertical line), which gets thinner and thinner (since the length of the segment {(𝑥, 𝑦0 )| 0 ≤ 𝑥 ≤ 1} is 1/𝑦20 → 0 as 𝑦0 → ∞), and which has finite area (see Remark 4.72). • The discrete subgroups Γ < PSL(2, ℝ) are called Fuchsian groups. Remark 4.78. The group PSL(2, ℤ) is discrete in PSL(2, ℝ). So the quotient 𝑆 = 𝐻/ PSL(2, ℤ) is a complete Riemannian oriented orbisurface with a metric of constant curvature 𝜅 ≡ −1. Let us check that this is the moduli of flat oriented tori ℳ𝑇𝑜𝑟,𝑑 on page 2 + 𝑜𝑟 229. By (4.23), ℳ𝑇 2 = SO(2)\ GL (2, ℝ)/ SL(2, ℤ). We consider oriented tori with total area equal to 1 and denote its moduli as ℳ𝑇𝑜𝑟,area=1 . Note that ℳ𝑇𝑜𝑟2 ≅ ℳ𝑇𝑜𝑟,area=1 ×ℝ>0 , 2 2 −1/2 with the map 𝐴 → (det(𝐴) 𝐴, det(𝐴)), where the second factor parametrizes the total area, so ℳ𝑇𝑜𝑟,area=1 = SO(2)\ SL(2, ℝ)/ SL(2, ℤ). There is a natural bijection 2 𝑜𝑟,area=1 𝑜𝑟,1 ℳ𝑇 2 → ℳ𝑇 2 , given by 𝐴 ↦ 𝛽𝐴, where (𝑑, 𝛼, 𝛽) are the parameters for 𝐴 and det 𝐴 = 𝛽𝑑 = 1. Finally, we have that 𝐻 = SO(2)\ SL(2, ℝ), since the isotropy group for the action of Isom+ (ℍ2𝑃𝐻 ) = PSL(2, ℝ) at 𝑝0 = i ∈ 𝐻 is PSO(2) < PSL(2, ℝ). This allows us to express 𝐻 as a homogeneous space (see Remark 3.105) as the quotient 𝐻 = PSO(2)\ PSL(2, ℝ) = SO(2)\ SL(2, ℝ). Putting it all together, we have an isomorphism (4.32)

ℳ𝑇𝑜𝑟,1 ≅ 𝐻/ PSL(2, ℤ). 2

4. Constant curvature

250

Therefore ℳ𝑇𝑜𝑟,1 is a Riemannian orbisurface. The action has fundamental domain 𝒟̂ 2 given in Figure 4.5. Consider the maps 𝜑(𝑧) = 𝑧 + 1, 𝜓(𝑧) = −1/𝑧 and 𝜙 = 𝜑 ∘ 𝜓, 𝜙(𝑧) = (𝑧 − 1)/𝑧, with associated matrices 𝐴=(

1 0 ), 1 1

𝐵=(

0 1 ), −1 0

𝐶=(

0 1 ). −1 1

Then 𝜑, 𝜓 serve to define the identifications of 𝒟.̂ The argument in Example 2.53 shows that PSL(2, ℤ) = ⟨𝜑, 𝜓⟩ and SL(2, ℤ) = ⟨𝐴, 𝐵⟩. The map 𝜑 has order 2 and fixed point 𝑝1 = i in 𝐷, and 𝜓 has order 3 and fixed point 𝑝2 = 𝑒𝜋i/3 . Note also that 𝑝3 = 𝑒2𝜋i/3 is a fixed point of order 3 of 𝜓 ∘ 𝜑, but it is identified with 𝑝2 in the quotient. There are no other fixed points: if 𝑓 ∈ PSL(2, ℤ) is a central rotation with center 𝑧0 , we can assume that 𝑧0 ∈ 𝒟.̂ But then there must be two geodesics emanating from 𝑧0 which are identified by 𝑓, and this only can happen for 𝑝1 , 𝑝2 , 𝑝3 . Therefore the quotient 𝐻/ PSL(2, ℤ) has one cusp (corresponding to ∞) and two orbifold points of indices 2 and 3 (see Figure 4.12).

2

3

1

Figure 4.12. The moduli space 𝐻/ PSL(2, ℤ). The cusp has cone angle 0, so the index should be ∞.

Finally, note that after compatifying 𝑆 = 𝐻/ PSL(2, ℤ) by adding the cuspidal point, we have a sphere with a metric with curvature 𝜅 ≡ −1 and three singular points. Two are orbifold points with cone angles 𝜋 and 2𝜋/3, and the cusp has cone angle 0 (this can be properly defined by a limit argument). An easy extension of the formula 1 2 13 𝜋 in Remark 3.83 gives that ∫𝑆 𝜅𝑆 = 2𝜋𝜒(𝑆) − 2𝜋( 2 + 3 + 1) = 4𝜋 − 3 𝜋 = − 3 . This 𝜋 agrees with the fact that area(𝑆) = 3 , which can be computed with the angles of the triangle 𝒟,̂ as in Remark 4.72. At this point, we have the tools to characterize the compact connected surfaces that admit a metric of curvature −1 and to detail how to construct such a metric. We focus on the orientable case. Let 𝑆 = Σ𝑔 be a compact connected orientable surface of genus 𝑔. The first obstruction on 𝑆 for admitting a metric of curvature −1 arises from the Gauss-Bonnet formula (3.41), which implies that 𝑔 ≥ 2. We are going to show the opposite, that is a compact connected orientable surface of genus 𝑔 ≥ 2 always has a metric of constant curvature −1. Let us review how we constructed a metric of zero constant curvature on the torus 𝑆 = Σ1 starting from the fundamental domain. Let Λ = ℤ⟨𝑢, 𝑣⟩ be a lattice of ℝ2 . By the discussion on page 222, a fundamental domain of 𝑆 = 𝔼2 /Λ is 𝐷 = {𝜆𝑢 + 𝜇𝑣|𝜆, 𝜇 ∈ [0, 1]}, which is a parallelogram. Then the translates 𝜑(𝐷), for 𝜑 ∈ Λ, tessellate by parallelograms the space ℝ2 . Actually 𝜋 ∶ ℝ2 → 𝑆 is the universal cover, and the

4.3. Negative constant curvature

251

transformations of Λ are the deck transformations which act by isometries. The metric of 𝑆 can be recovered from the flat metric of 𝐷 ⊂ ℝ2 . The edges are straight lines (geodesics) which are glued by an isometry (translation). The most difficult point is the vertex, for which we have to glue (isometrically) sectors to form a metric ball. For this, it is necessary that the total angle (the sum of the interior angles of all the vertices of the polygon 𝐷) adds up to 2𝜋. If we want to construct a metric of constant curvature on Σ𝑔 , for 𝑔 ≥ 2, with a similar procedure, we have to take a 4𝑔-gon whose edges are geodesics and such that the sum of the interior angles at the vertices is ∑ 𝜃𝑖 = 2𝜋. By (3.37), a 4𝑔-gon 𝐷 with constant curvature 𝑘0 has ∑ 𝜃𝑖 = (4𝑔 − 2)𝜋 + ∫ 𝜅 = (4𝑔 − 2)𝜋 + 𝑘0 area(𝐷). 𝐷

If we want this to be equal to 2𝜋, then we have to take 𝑘0 = −1 and area(𝐷) = (4𝑔−4)𝜋. Let us show that there is a hyperbolic 4𝑔-gon satisfying such constraint. We start by giving a simple proof for a particular 4𝑔-gon, namely the regular one (with all sides and angles equal). Lemma 4.79. There is a regular hyperbolic 𝑙-gon with angles 𝜃𝑖 = 2𝜋/𝑙 for any integer 𝑙 > 4. Proof. We draw a regular 𝑙-gon in the model ℍ2𝑃𝐷 , with center at the origin. If we draw the segments from the origin to the vertices, we have an isosceles triangle 𝑇 with 2𝜋 𝜋 interior angle 𝛼 = 𝑙 and opposite angles 𝛽 = 𝑙 . Hence 𝐷 is reconstructed from 𝑇 by rotating 𝑙 times by a rotation of angle 2𝜋/𝑙.

β

α

β

Let us fix 𝛼, 𝛽 > 0 such that 𝛼 + 2𝛽 < 𝜋. We shall construct an isosceles triangle in ℍ2𝑃𝐷 with angles 𝛼, 𝛽, 𝛽. Take two radii 𝑟1 , 𝑟2 from the origin forming an angle 𝛼. For 𝑑 > 0, take two points 𝑥1 (𝑑) ∈ 𝑟1 and 𝑥2 (𝑑) ∈ 𝑟2 so that 𝑑(0, 𝑥1 (𝑑)) = 𝑑(0, 𝑥2 (𝑑)) = 𝑑,

4. Constant curvature

252

A x1 β

r1 t1

O

α s

C

r2

Figure 4.13. Construction of a hyperbolic isosceles triangle with given angles.

and let 𝛾 (𝑑) be the geodesic joining 𝑥1 (𝑑) and 𝑥2 (𝑑). Consider the isosceles triangle 𝑇𝑑 with vertices {0, 𝑥1 (𝑑), 𝑥2 (𝑑)}, edges {𝑟1 , 𝑟2 , 𝛾 (𝑑) }, and angles {𝛼, 𝛽(𝑑), 𝛽(𝑑)}. Applying (3.38) to 𝑇𝑑 , we get 𝛼 + 2𝛽(𝑑) = 𝜋 − area(𝑇𝑑 ). If 𝑑 → 0, then area(𝑇𝑑 ) → 0, so 1 𝛽(𝑑) → 2 (𝜋 − 𝛼). On the other hand, if 𝑑 → ∞, then 𝛽(𝑑) → 0 (Remark 4.70). The 1

function 𝛽 ∶ (0, ∞) → (0, 2 (𝜋 − 𝛼)) is continuous and decreasing, hence there exists a unique 𝑑0 > 0 such that 𝛽(𝑑0 ) = 𝛽. Thus 𝑇𝑑0 is the desired triangle. There is another more constructive way to obtain such a triangle, as shown in Figure 4.13. Take two radii 𝑟1 , 𝑟2 from the origin 𝑂 with angle 𝛼 as before, and let 𝑠 be the bisectrix. Pick any point 𝑥1 ∈ 𝑟1 and draw a circle 𝐴 passing through 𝑥1 , forming an angle 𝛽 with 𝑟1 , and perpendicular to 𝑠. For this it is enough to take a circle whose center 𝐶 is the intersection of 𝑠 with the perpendicular to the line 𝑡1 at 𝑥1 forming an angle ∠(𝑟1 , 𝑡1 ) = 𝛽. This is the desired triangle, but in a disc of radius different from 1. Its circle of ideal points is the circle centered at the origin and orthogonal to 𝐴 (in heavy black). It passes by the intersection of 𝐴 with the circle with diameter 𝑂𝐶. Now make a homothety of the whole figure to take this last circle to the unit circle. □

With this polygon, we are going to paste the edges as in Figure 1.9 to obtain the corresponding surface, but this time we shall do it metrically and not just topologically. Theorem 4.80. Let 𝐷 ⊂ ℍ2𝑃𝐷 be a 4𝑔-gon with geodesic edges that represent the fundamental polygon of a compact connected surface of genus 𝑔 ≥ 2 whose associated word is −1 ′ ″ ′ ″ −1 −1 𝑎1 𝑏1 𝑎−1 1 𝑏1 ⋯ 𝑎𝑔 𝑏𝑔 𝑎𝑔 𝑏𝑔 . Let us denote 𝑎𝑗 , 𝑎𝑗 the two edges with label 𝑎𝑗 , and 𝑏𝑗 , 𝑏𝑗

4.3. Negative constant curvature

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the two edges with label 𝑏𝑗 . Suppose that: 4𝑔

(1) ∑𝑖=1 𝜃𝑖 = 2𝜋, (2) ℓ(𝑎𝑗′ ) = ℓ(𝑎𝑗″ ) and ℓ(𝑏𝑗′ ) = ℓ(𝑏𝑗″ ), for 1 ≤ 𝑗 ≤ 2𝑔. Then the quotient 𝑆 = 𝐷/∼ inherits from ℍ2𝑃𝐷 a metric of constant curvature 𝜅 ≡ −1. Proof. Denote 𝜛 ∶ 𝐷 → 𝑆 = 𝐷/∼, and let 𝑝 ∈ 𝑆. We have three cases. (1) 𝑝 ∈ 𝜛(Int(𝐷)) is the image of an interior point 𝑥 ∈ Int(𝐷). In this case, there exists 𝜀 > 0 so that 𝐵𝜀 (𝑥) ⊂ Int(𝐷), and 𝜛 ∶ 𝐵𝜀 (𝑥) → 𝑈 𝑝 = 𝜛(𝐵𝜀 (𝑥)) is a diffeomorphism. We take g𝑈 = 𝜛∗ (gℍ2 |𝐵𝜀 (𝑥) ), which is a metric on 𝑈 of curvature −1. (2) 𝑝 = 𝜛(𝑥′ ) for 𝑥′ ∈ 𝜕𝐷 in the interior of an edge. Suppose that the edge is 𝑎1 , and let 𝑥′ ∈ 𝑎′1 , 𝑥″ ∈ 𝑎″1 be the two points in 𝜕𝐷 with 𝑝 = 𝜛(𝑥′ ) = 𝜛(𝑥″ ). There exists 𝜀 > 0 so that 𝐵𝜀 (𝑥′ )∩𝜕𝐷 ⊂ Int(𝑎′1 ) and also 𝐵𝜀 (𝑥″ )∩𝜕𝐷 ⊂ Int(𝑎″1 ), where the balls are taken with respect to the hyperbolic metric. Since ℓ(𝑎′1 ) = ℓ(𝑎″1 ), there exists a unique 𝜑 ∈ Isom+ (ℍ2 ) so that 𝜑(𝑎′1 ) = 𝑎″1 . We take the identification ∼ of 𝑎′1 and 𝑎″1 via 𝜑, so 𝑥″ = 𝜑(𝑥′ ). Call 𝐵𝜀+ (𝑥′ ) = 𝐵𝜀 (𝑥′ )∩𝐷, 𝐵𝜀− (𝑥′ ) = 𝐵𝜀 (𝑥′ )∩(ℍ2 −Int(𝐷)), and analogously + ″ 𝐵𝜀 (𝑥 ) = 𝐵𝜀 (𝑥″ ) ∩ 𝐷, 𝐵𝜀− (𝑥″ ) = 𝐵𝜀 (𝑥″ ) ∩ (ℍ2 − Int(𝐷)). Since 𝜑 preserves the orientation of ℍ2 , we have that 𝜑(𝐵𝜀+ (𝑥′ )) = 𝐵𝜀− (𝑥″ ) and 𝜑(𝐵𝜀− (𝑥′ )) = 𝐵𝜀+ (𝑥″ ). A neighbourhood of 𝑝 ∈ 𝑆 = 𝜛(𝐷) is given by 𝑈 𝑝 = 𝐵𝜀+ (𝑥′ )∪𝜑 𝐵𝜀+ (𝑥″ ), where 𝜑 identifies 𝐵𝜀+ (𝑥′ ) ∩ 𝜕𝐷 with 𝐵𝜀+ (𝑥″ ) ∩ 𝜕𝐷. We have a chart 𝜑̂ = (Id, 𝜑) ∶ 𝐵𝜀 (𝑥′ ) = 𝐵𝜀+ (𝑥′ ) ∪Id 𝐵𝜀− (𝑥′ ) → 𝑈 𝑝 = 𝐵𝜀+ (𝑥′ ) ∪𝜑 𝐵𝜀+ (𝑥″ ) ⊂ 𝑆 = 𝜛(𝐷). We define the metric on 𝑈 as g𝑈 = 𝜑∗̂ (gℍ2 ). This is compatible with the metric defined in case (1). (3) 𝑝 ∈ 𝑆 is the image by 𝜛 of a vertex, so 𝑝 = 𝜛(𝑣 1 ) = ⋯ = 𝜛(𝑣 4𝑔 ), where 𝑣 𝑖 , 1 ≤ 𝑖 ≤ 4𝑔, are the 4𝑔 vertices of 𝐷. Given 𝜀 > 0 small, let us consider 𝐴𝑖 = 𝐵𝜀 (𝑣 𝑖 ) ∩ 𝐷. We have to see that all the 𝐴𝑖 , 1 ≤ 𝑖 ≤ 4𝑔, assemble to form an open set isometric to a small ball of ℍ2 . Indeed, choose 𝜑𝑖 ∶ 𝐴𝑖 → 𝑆 𝑖 ⊂ 𝔻 an isometry to a circular sector 𝑆 𝑖 in the Poincaré disc model ℍ2𝑃𝐷 centered at the origin, and with angle 𝜃𝑖 . If we label the vertices of 𝐷 in the order that they are identified starting from a fixed initial vertex 𝑣 1 (see (2) in section 1.3.1), then we follow the same (circular) order to assemble the circular sectors 𝑆 1 , . . . , 𝑆 4𝑔 . In this way, 𝑆 𝑖 and 𝑆 𝑖−1 intersect in points that correspond to 4𝑔

points of 𝐴𝑖 and 𝐴𝑖−1 which are identified by 𝜛. Finally, since ∑𝑖=1 𝜃𝑖 = 2𝜋, the last sector 𝑆 4𝑔 glues to the first sector 𝑆 1 , that is 𝑆 4𝑔 intersects 𝑆 1 along points that correspond to points of 𝐴4𝑔 and 𝐴1 which are identified by 𝜛. We 4𝑔

conclude that 𝑈 = 𝜛(⨆𝑖=1 𝐴𝑖 ) is an open neighbourhood of 𝑝 in 𝑆, and there is a homeomorphism 𝜑 = ⨆ 𝜑𝑖 ∶ 𝑈 → 𝐵𝜀 (0) ⊂ ℍ2𝑃𝐷 . We define g𝑈 = 𝜑∗ (g𝑃 ), and then all the metrics defined in each case coincide on the intersections (since g𝑃 is radial). This gives a well defined metric on 𝑆 of constant curvature −1. □

4. Constant curvature

254

Remark 4.81. A similar construction to that of Theorem 4.80 works for the nonorientable surfaces 𝑋𝑘 , 𝑘 ≥ 3. Let 𝐷 ⊂ ℍ2𝑃𝐷 be a 2𝑘-gon with 2𝑘 geodesic edges that represent the fundamental polygon of 𝑋𝑘 with associated word 𝑎1 𝑎1 ⋯ 𝑎𝑘 𝑎𝑘 . If 𝑎𝑗′ , 𝑎𝑗″ 2𝑘

are the two edges with label 𝑎𝑗 , ∑𝑖=1 𝜃𝑖 = 2𝜋 and ℓ(𝑎𝑗′ ) = ℓ(𝑎𝑗″ ), 1 ≤ 𝑗 ≤ 𝑘, then the quotient 𝑆 = 𝐷/ ∼ inherits from ℍ2𝑃𝐷 a metric of curvature 𝜅 ≡ −1. This time, the isometries which identify the edges 𝑎𝑗′ , 𝑎𝑗″ are in Isom− (ℍ2𝑃𝐷 ). With Lemma 4.79 and Theorem 4.80, we get that any compact orientable surface Σ𝑔 , for 𝑔 ≥ 2, admits (at least one) metric of constant curvature −1. A similar result is true for the non-orientable surfaces 𝑋𝑘 , 𝑘 ≥ 3, by Remark 4.81. See also Remark 3.70(4). Theorem 4.82. Any compact, connected (topological) surface admits a metric of con2 stant curvature. That is, if 𝐑𝐢𝐞𝐦𝐾≡cst,c denotes the category of smooth compact surfaces 𝑐𝑜 𝑐𝑜 with a Riemannian metric of constant curvature, 𝕃𝐑𝐢𝐞𝐦2 → 𝕃𝐓𝐌𝐚𝐧2 is surjective. More precisely:

𝐾≡cst,c

𝑐

(1) 𝑆 2 and ℝ𝑃 2 admit a metric of positive constant curvature. (2) 𝑇 2 and Kl admit a metric of zero constant curvature. (3) Σ𝑔 for 𝑔 ≥ 2, and 𝑋𝑘 for 𝑘 ≥ 3 admit a metric of negative constant curvature. No surface admits metrics of constant curvature with different signs. This concludes (at last) the result on the homotopy groups of surfaces mentioned in Remark 2.54. Corollary 4.83. The universal cover of Σ𝑔 for 𝑔 ≥ 1 and 𝑋𝑘 for 𝑘 ≥ 2 is ℝ2 . In particular, 𝜋𝑛 (Σ𝑔 ) = 𝜋𝑛 (𝑋𝑘 ) = 0 for 𝑛 ≥ 2, 𝑔 ≥ 1, 𝑘 ≥ 2. Proof. Let 𝑆 be Σ𝑔 or 𝑋𝑘 , for 𝑔 ≥ 2, 𝑘 ≥ 3. Take a metric of constant curvature −1, and consider the Riemannian universal cover 𝑆 ̃ → 𝑆. Then 𝑆 ̃ is a simply connected space form of curvature −1, that is 𝑆 ̃ = ℍ2 , which is homeomorphic to ℝ2 . The universal cover of Σ1 = 𝑇 2 and of 𝑋2 = Kl is ℝ2 , by Examples 2.50 and 2.53. The fact about the homotopy groups follows from Corollary 2.37. □ Remark 4.84. For (compact, connected) manifolds of higher dimension 𝑛 ≥ 3, there are manifolds that do not admit metrics of constant sectional curvature, for instance 𝑆 2 × 𝑆 1 . Certainly, if 𝑆 2 × 𝑆 1 admitted a metric of constant curvature, its universal cover 𝑆2 × ℝ would be a simply connected space form, hence diffeomorphic to 𝑆 3 or ℝ3 , which it is not. However, if a compact connected 𝑛-manifold admits a metric of constant curvature, then it can be of only one sign. This follows from Exercise 4.34. For the scalar curvature, the situation is not so rigid. All compact manifolds with 𝑛 ≥ 3 admit metrics of constant scalar curvature, which moreover can be found conformally equivalent to a given metric by the solution to the Yamabe problem (cf. Remark 6.76(4)). Actually the scalar curvature can be prescribed to be any function with negative values, by a result of Kazdan and Warner [K-W]. There are topological obstructions

4.3. Negative constant curvature

255

for a compact manifold to admit metrics of positive constant scalar curvature, but there are many examples of such manifolds. Observe that in such a case, they admit metrics with constant scalar curvature of different signs. So far, we have built metrics of constant curvature −1 on Σ𝑔 , for 𝑔 ≥ 2, using fundamental polygons in hyperbolic space. Actually, all such metrics can be obtained by this method, as the following result shows. Proposition 4.85. Let 𝑆 be a compact connected orientable surface of genus 𝑔 with a Riemannian metric of constant curvature −1. Then 𝑆 is isometric to the quotient of a 4𝑔gon 𝐷 ⊂ ℍ2 satisfying (1) and (2) of Theorem 4.80, where the edges of 𝐷 are identified by isometries of ℍ2 . Proof. Fix a basepoint 𝑝0 ∈ 𝑆 and fix loops 𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 which only intersect at 𝑝0 and such that ∏[𝑎𝑖 , 𝑏𝑖 ] = 1 in 𝜋1 (𝑆, 𝑝0 ), so that if we cut 𝑆 along these loops, we get a planar representation of 𝑆 with a canonical word as given in Example 1.62. By a theorem of Fricke [Kee] we can, maybe after moving 𝑝0 to another point 𝑝0′ , find geodesic loops 𝛼𝑖 , 𝛽 𝑖 only intersecting at 𝑝0′ . These loops are homotopic to the previous ones via a homotopy (actually it can be an isotopy of 𝑆) that sends 𝑝0 to 𝑝0′ . The loops are characterised by being the shortest paths in the corresponding homotopy class of loops. Therefore the loops satisfy ∏[𝛼𝑖 , 𝛽 𝑖 ] = 1 in 𝜋1 (𝑆, 𝑝0′ ). If we cut 𝑆 along 𝛼𝑖 , 𝛽 𝑖 we get a 4𝑔-gon whose boundary has to be identified in the usual manner to recover 𝑔 𝑆. We denote 𝑈 = 𝑆 − ⋃𝑖=1 (𝛼𝑖 ∪ 𝛽 𝑖 ) an open subset which is the interior of the 4𝑔gon. Consider now the Riemannian universal cover 𝜋 ∶ (ℍ2 , 𝑝0̃ ) → (𝑆, 𝑝0′ ), and let Γ = Deck(𝜋) < Isom+ (ℍ2 ). The set 𝜋−1 (𝑝0′ ) consists of the Γ-translates of 𝑝0̃ . The preimages by 𝜋 of the loops 𝛼𝑖 , 𝛽 𝑖 are geodesic segments between points of 𝜋−1 (𝑝0′ ). They split ℍ2 into connected open sets, which are the Γ-translates of some 𝑈0 with 𝜋 ∶ 𝑈0 → 𝑈, a diffeomorphism. The closure 𝐷 = 𝑈 0 is a polygon whose edges are geodesics of ℍ2 , and it is a fundamental domain of the action. It must satisfy the requirements of Theorem 4.80. □ Example 4.86. Let us describe geometrically the tessellation of ℍ2 and how the group Γ acts on it. We will do it for the surface Σ2 , the general case 𝑔 ≥ 2 being analogous. Compare this with Figure 2.4 for the Klein bottle. Given a metric of constant curvature −1 on Σ2 , let 𝜋 ∶ ℍ2 → Σ2 be the universal cover, and let Γ = Deck(𝜋) < Isom+ (ℍ2 ) be the group of deck transformations, so that ℍ2 /Γ ≅ Σ2 . Fix a basepoint 𝑝0 ∈ Σ2 , and 𝑎1 , 𝑏1 , 𝑎2 , 𝑏2 generators of 𝜋1 (Σ2 , 𝑝0 ) such that [𝑎1 , 𝑏1 ][𝑎2 , 𝑏2 ] = 1. Let 𝐷 ⊂ ℍ2 be a fundamental domain with word −1 −1 −1 ′ ″ 𝑎1 𝑏1 𝑎−1 1 𝑏1 𝑎2 𝑏2 𝑎2 𝑏2 given by Proposition 4.85. We denote by 𝑎𝑗 , 𝑎𝑗 the two edges ′ ″ with label 𝑎𝑗 , and by 𝑏𝑗 , 𝑏𝑗 the two edges with label 𝑏𝑗 , and let 𝑣 1 , . . . , 𝑣 8 be the vertices of 𝐷, ordered in the way that they are identified by the gluing of the edges as in Figure 4.14. The deck transformations are generated by the gluing maps of the geodesic edges of 𝐷. Since an orientation preserving isometry of ℍ2 is determined by the image of a point and the image of a unitary vector at the point, the requirement that it sends

4. Constant curvature

256

v4 v1

b01

a01

b002

v3 a001 v2

v6 a002 v7

b001 a02

b02

v5

v8 Figure 4.14. Fundamental domain 𝐷 for Σ2 , and corresponding tessellation.

a geodesic segment to another geodesic segment (of the same length) determines the isometry. Hence, Γ is generated by the following isometries: • 𝜑𝑎1 ∈ Isom+ (ℍ2 ) such that 𝜑𝑎1 ∶ 𝑎′1 → 𝑎″1 , 𝜑𝑎1 (𝑣 1 ) = 𝑣 2 , 𝜑𝑎1 (𝑣 4 ) = 𝑣 3 , • 𝜑𝑏1 ∈ Isom+ (ℍ2 ) such that 𝜑𝑏1 ∶ 𝑏′1 → 𝑏″1 , 𝜑𝑏1 (𝑣 4 ) = 𝑣 5 , 𝜑𝑏1 (𝑣 3 ) = 𝑣 2 , • 𝜑𝑎2 ∈ Isom+ (ℍ2 ) such that 𝜑𝑎2 ∶ 𝑎′2 → 𝑎″2 , 𝜑𝑎2 (𝑣 5 ) = 𝑣 6 , 𝜑𝑎2 (𝑣 8 ) = 𝑣 7 , • 𝜑𝑏2 ∈ Isom+ (ℍ2 ) such that 𝜑𝑏2 ∶ 𝑏′2 → 𝑏″2 , 𝜑𝑏2 (𝑣 8 ) = 𝑣 1 , 𝜑𝑏2 (𝑣 7 ) = 𝑣 6 . By Theorem 4.76, each of 𝜑𝑎1 , 𝜑𝑏1 , 𝜑𝑎2 , 𝜑𝑏2 are translations along geodesics, since the action of Γ is free and proper. For instance, for the case of the regular octagon, it is clear that 𝜑𝑎1 is the translation along the geodesic through the midpoints of 𝑎′1 and 𝑎″1 , which is perpendicular to both edges. −1 −1 −1 Observe that 𝜑𝑏2 ∘ 𝜑−1 𝑎2 ∘ 𝜑 𝑏2 ∘ 𝜑 𝑎2 ∘ 𝜑 𝑏1 ∘ 𝜑 𝑎1 ∘ 𝜑 𝑏1 ∘ 𝜑 𝑎1 (𝑣 1 ) = 𝑣 1 . Since a deck −1 transformation fixing a point must be the identity, we obtain that [𝜑𝑏2 , 𝜑−1 𝑎2 ][𝜑 𝑏1 , 𝜑 𝑎1 ] = Id. By (2.3), there is a group isomorphism

𝑓 ∶ Γ → 𝜋1 (Σ𝑔 , 𝑝0 ) = ⟨𝑎1 , 𝑏1 , 𝑎2 , 𝑏2 | [𝑎1 , 𝑏1 ][𝑎2 , 𝑏2 ] = 1⟩, that assigns to every 𝜑 ∈ Γ the loop 𝜋(𝛾𝑣1 ,𝜑(𝑣1 ) ) where 𝛾𝑣1 ,𝜑(𝑣1 ) is a path from 𝑣 1 to 𝜑(𝑣 1 ). Here we choose the basepoint 𝑣 1 ∈ ℍ2 , with 𝜋(𝑣 1 ) = 𝑝0 . If 𝑣 is another vertex and 𝛾𝑣,𝜑(𝑣) is a path from 𝑣 to 𝜑(𝑣), then taking a path 𝛼𝑣1 ,𝑣 from 𝑣 1 to 𝑣, we have that ←

𝛼𝑣1 ,𝑣 ∗ 𝛾𝑣,𝜑(𝑣) ∗ 𝜑( 𝛼 𝑣1 ,𝑣 ) goes from 𝑣 1 to 𝜑(𝑣 1 ). Hence 𝑓(𝜑) = [𝜋(𝛼𝑣1 ,𝑣 )] ∗ [𝜋(𝛾𝑣,𝜑(𝑣) )] ∗ [𝜋(𝛼𝑣1 ,𝑣 )]−1 . With this formula, we have 𝑓(𝜑𝑎1 ) = 𝑎1 𝑏1 𝑎−1 1 ,

−1 −1 𝑓(𝜑𝑏1 ) = 𝑎1 (𝑏1 𝑎−1 1 𝑏1 )𝑎1 ,

−1 −1 −1 −1 −1 𝑓(𝜑𝑎2 ) = [𝑎1 , 𝑏1 ]𝑎2 𝑏2 𝑎−1 2 [𝑎1 , 𝑏1 ] , 𝑓(𝜑 𝑏2 ) = [𝑎1 , 𝑏1 ]𝑎2 (𝑏2 𝑎2 𝑏2 )𝑎2 [𝑎1 , 𝑏1 ] .

4.3. Negative constant curvature

257

−1 A short computation reveals that 𝑓([𝜑𝑏2 , 𝜑−1 𝑎2 ][𝜑 𝑏1 , 𝜑 𝑎1 ]) = [𝑎1 , 𝑏1 ][𝑎2 , 𝑏2 ], so we get that a presentation of Γ is −1 Γ = ⟨𝜑𝑎1 , 𝜑𝑏1 , 𝜑𝑎2 , 𝜑𝑏2 | [𝜑𝑏2 , 𝜑−1 𝑎2 ][𝜑 𝑏1 , 𝜑 𝑎1 ] = Id ⟩.

As a final remark, let 𝑈 = Int(𝐷) ⊂ ℍ2 . Since 𝜋 is an isometry on 𝑈, we have that 𝜋−1 (𝜋(𝑈)) ⊂ ℍ2 is an infinite collection of disjoint octagons whose boundaries are geodesic edges. This defines a hyperbolic tessellations on ℍ2 . In the case of the regular octagon, the tessellation is shown in Figure 4.14. Teichmüller space. Fix 𝑔 ≥ 2. As we did for the classification of flat structures on a torus in section 4.2.3, we can put on the “list” of Riemannian structures on Σ𝑔 of constant curvature −1, the structure of a moduli space, i.e., we can endow this space with a geometric structure. We do it as follows. Fix a point 𝑝0 ∈ Σ𝑔 and a collection of loops 𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 only intersecting at 𝑝0 such that ∏[𝑎𝑖 , 𝑏𝑖 ] = 1. By Proposition 4.85, the metrics of curvature −1 on Σ𝑔 are equivalent to hyperbolic 4𝑔-gons satisfying (1) and (2) of Theorem 4.80. Let 𝒫 denote the set of such polygons. To construct the moduli space, we have to quotient by the two choices: the basepoint and the choice of loops in Σ𝑔 . This is similar to the double quotient in (4.20) for the moduli of flat tori. The Teichmüller space 𝒯𝑔 is defined as the quotient of 𝒫 by the choice of the basepoint. Let us compute the dimension of 𝒯𝑔 . For this we parametrize the set of polygons 𝒫. Let P ∈ 𝒫, and label its vertices 𝑣 1 , . . . , 𝑣 4𝑔 in cyclic order. We draw the 4𝑔 − 3 diagonals 𝑣 1 𝑣 3 , . . . , 𝑣 1 𝑣 4𝑔−1 . The lengths of the edges 𝑣 1 𝑣 2 , 𝑣 2 𝑣 3 , . . . , 𝑣 4𝑔 𝑣 1 and the lengths of the diagonals determine the polygon P, since P is assembled out of triangles 𝑣 1 𝑣 2 𝑣 3 , 𝑣 1 𝑣 3 𝑣 4 , . . . , 𝑣 1 𝑣 4𝑔−1 𝑣 4𝑔 , and each triangle is uniquely determined by the lengths of its edges (see Exercise 4.33). In our case, we have that the lengths of the 4𝑔 edges are determined by 2𝑔 numbers (since these are equal in pairs). So the number of parameters is 2𝑔 + 4𝑔 − 3 = 6𝑔 − 3. Finally, note that we have an extra equation ∑ 𝜃𝑖 = 2𝜋, which is rewritten as area(P) = (𝑙 − 4)𝜋. Fixing the area reduces the number of parameters to 6𝑔 − 4. Finally, the chosen basepoint moves in a 2-dimensional space. Certainly, the polygon P depends on the point 𝑝0 . This point is not arbitrary, since we can only use points for which the shortest paths in the homotopy classes of 𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 do not intersect except at 𝑝0 . However, this is an open condition, so 𝑝0 moves in an open set of 𝑆. Therefore, the Teichmüller space has dimension 6𝑔 − 6. Actually, it holds that 𝒯𝑔 is a smooth (6𝑔 − 6)-dimensional manifold diffeomorphic to ℝ6𝑔−6 . For constructing the space 𝒯𝑔 , we have a standard collection of loops 𝑎1 , 𝑏1 , . . . , 𝑎𝑔 , 𝑏𝑔 . This corresponds to a map P0 → Σ𝑔 from a fixed 4𝑔-gon P0 to Σ𝑔 , which can be rephrased as a homeomorphism 𝑓 ∶ Σ𝑔 = P0 / ∼ → Σ𝑔 . This is called a marking and such marking is defined up to isotopy. Thus the choices of markings are given by the mapping class group of Σ𝑔 , MCG(Σ𝑔 ), defined on page 16. The quotient ℳ𝑔 = 𝒯𝑔 /MCG(Σ𝑔 ) is an orbifold of dimension 6𝑔 − 6, and it is the moduli space that parametrizes Riemannian metrics of curvature −1 on Σ𝑔 . This should be compared with the moduli space of oriented flat tori of fixed area ℳ𝑇𝑜𝑟,area=1 discussed in Remark 4.78. In (4.32), the space 𝐻 parametrizes (flat) paral2 lelograms of fixed area, and the quotient by PSL(2, ℤ) corresponds to changes of the

4. Constant curvature

258

basis of the lattice. This group coincides with the mapping class group of 𝑇 2 (cf. Exercise 2.35). So the Teichmüller space for flat tori is 𝐻. A final remark is that not fixing the area for tori has its counterpart in hyperbolic geometry by taking a constant curvature metric with 𝜅 ≡ 𝑘0 , allowing any 𝑘0 < 0 for Σ𝑔 , 𝑔 ≥ 2. For genus 𝑔 ≥ 2, the area and the curvature are related (see Remark 3.70(5)).

4.4. Classical geometries Classical geometry dates back to the ancient formulation of Euclid. In it, geometry is defined by a space containing points and figures which satisfy some very natural axioms and postulates. Euclid’s famous five postulates are, verbatim: (1) To draw a straight line from any point to any point. (2) To produce a finite straight line continuously in a straight line. (3) To describe a circle with any center and distance. (4) That all right angles are equal to one another. (5) That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less than two right angles. In modern terms, we would say that we have a Riemannian manifold (distances and angles can be measured), which is connected and complete (the “straight lines”, i.e., the geodesics, can be extended indefinitely), and distances and angles can be compared from one place to another. This is equivalent to the fact that figures can be moved or rotated, so that they remain equal (i.e., isometric). This translates to the Riemannian metric being isotropic. The fifth postulate, however, is rather cumbersome. In 1795, Playfair gave an alternative (equivalent) version of the fifth postulate: “For a given point 𝑃 not on a line 𝑟, there is one and only one line passing through 𝑃 which does not meet 𝑟.” Such a line is called parallel to 𝑟. This is more natural to assume, but still it says something about the behaviour of the geodesic on the long run, which is a rather drastic non-local assertion. There was a big controversy for centuries as to whether this postulate would follow from the axioms and other postulates. In particular, one can postulate non-Euclidean geometries, by changing the fifth postulate. This gives three possibilities: • Elliptic geometry: through a given point 𝑃 not on a line 𝑟, there is no line parallel to 𝑟 (that is, any line through 𝑃 meets 𝑟). This is shown to be equivalent to the fact that the sum of the angles of a triangle is 𝛼 + 𝛽 + 𝛾 > 𝜋. • Euclidean geometry: through a given point 𝑃 not on a line 𝑟, there is a unique line parallel to 𝑟. Note that the parallel line to 𝑟 through 𝑃 is the perpendicular to the line 𝑠 perpendicular from 𝑃 to 𝑟. This is the previous Euclid fifth postulate and is equivalent to the sum of the angles of a triangle being 𝛼+𝛽+𝛾 = 𝜋.

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259

• Hyperbolic geometry: through a given point 𝑃 not on a line 𝑟, there are several lines parallel to 𝑟. This is equivalent to the sum of the angles of a triangle being 𝛼 + 𝛽 + 𝛾 < 𝜋. The spaces of constant curvature 𝑘0 = 1 are models of elliptic geometry. By Exercise 4.4, the isotropic spaces are 𝕊𝑛 and ℝ𝑃𝑛 . Classical geometry in 𝕊𝑛 is usually called spherical geometry, and classical geometry in ℝ𝑃 𝑛 is called projective geometry. The difference between them is that in 𝕊𝑛 two lines (geodesics) intersect at two (antipodal) points, and ℝ𝑃𝑛 has the advantage that two lines intersect at one point but the disadvantage that it is not simply connected. Spherical geometry was studied since ancient times with a view towards astronomy, and projective geometry since the Renaissance. The fact that lines are of finite length did not fit morally with the vision of Euclid for a geometry, so it was not truly considered as a competitor of Euclidean geometry. Euclidean geometry is characterised by the fact that there are homotheties, i.e., maps 𝜑 such that 𝜑∗ g = 𝜆2 g, where 𝜆 > 0 is a constant (Remark 4.68(3)). Applying 𝜑 to a figure, we get a “similar” figure, all whose lengths are multiplied by a factor 𝜆, and areas multiplied by a factor 𝜆2 , but the angles are preserved. In particular, one can construct triangles of arbitrarily large area. We recall that in elliptic or hyperbolic geometries, the areas of the triangles are necessarily bounded (Remark 3.67). In hyperbolic geometry, given a line 𝑟 and a point 𝑃 not in it, we can draw the perpendicular 𝑠 from 𝑃 to 𝑟. The line 𝑡 perpendicular to 𝑠 at 𝑃 is parallel to 𝑟. But there is some angle 𝜃0 > 0 such that all lines 𝑡 𝜃 forming angle 𝜃 with 𝑡, 𝜃 ∈ [−𝜃0 , 𝜃0 ] are also parallel to 𝑟. In particular there are infinitely many and form a sector in the hyperbolic plane.

θ0

P

θ0

t

s

r The lines 𝑡±𝜃0 are called limiting parallel, and they intersect the line 𝑟 at an ideal point (Remark 4.63). The line 𝑟 has two ideal points. It is natural to say that the angle formed by two limiting parallel lines at the ideal point is zero. The discovery of hyperbolic geometry is due to Lobachevsky (and independently Bolyai, and also Gauss in unpublished work) in nineteenth century. This was done by substituting the fifth postulate of Euclid by the hyperbolic alternative, and developing geometry from scratch (as Euclid did) in a constructive way.

260

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The name of hyperbolic geometry was introduced by Klein, and it seems to be related to the fact that many hyperbolic functions appear in formulas for hyperbolic figures (Exercise 4.31). The name of elliptic geometry followed by contraposition. Remark 4.87. In more modern terms, it is natural to understand a geometry as a homogeneous space [Gol]. As such space figures can be moved isometrically, but possibly not rotated. The discussion on the eight geometries of Thurston in dimension 3 fits into this paradigm. Thurston proved that in dimension 3 there are eight possible connected homogeneous spaces: ℝ3 , 𝕊3 , ℍ3 , 𝕊2 × ℝ, ℍ2 × ℝ, and another three twisted geometries called Solv, Nil and the universal cover of SL(2, ℝ). See Remark 3.110. A further step consists of defining a space with a geometry as a manifold with charts giving homeomorphisms to open subsets of a homogeneous manifold 𝑋 = 𝐺/𝐻, and with changes of charts being given by elements of 𝐺, which is the selected group of isometries of the modelling geometry 𝑋. This is basically saying that the space is locally homogeneous, but we note that this can be done without involving a Riemannian metric at all. These are called Cartan geometries.

Problems Exercise 4.1. Determine the connected surfaces of revolution in ℝ3 with constant curvature. Exercise 4.2. Solve the equation −𝑒−2𝑓 Δ𝑓 = 𝑘0 for a function on ℝ2 that only depends on the radius, 𝑓 = 𝑓(𝑟), where 𝑘0 is a constant. Exercise 4.3. Let 𝑀 be a connected Riemannian 𝑛-manifold. Prove that if dim Isom(𝑀) = 1 𝑛(𝑛 + 1), then 𝑀 is isotropic. In particular, if 𝑀 is orientable, then it has orientation reversing 2 isometries. Exercise 4.4. Show that the only isotropic connected 𝑛-manifolds are 𝕊𝑛 , ℝ𝑃𝑛 , 𝔼𝑛 , and ℍ𝑛 . Exercise 4.5. Let 𝐺 be a connected Lie group, and let 𝐻 < 𝐺 be a Lie subgroup. If they have the same dimension, then 𝐻 = 𝐺. Exercise 4.6. Let 𝑋 be a space with an action of a group 𝐺 on 𝑋 on the right. Convert it to an action of 𝐺 on 𝑋 on the left. Prove also that if 𝐺 acts on 𝑋 on the right and 𝐻 acts on 𝑋 on the left simultaneously, then there is an action of 𝐺 × 𝐻 on 𝑋. Exercise 4.7. Prove geometrically that the stereographic projection from the sphere to the plane preserves angles, and that it transforms circles in 𝑆2 to circles or lines in ℝ2 . ˆ Exercise 4.8. Give a classification (up to projective isomorphism) of the maps in Mob(ℂ). Exercise 4.9. Let 𝑓 ∶ ℂ → ℂ be a bijection that preserves the set of generalized circles. Prove that 𝑓 is an extended Möbius map. Exercise 4.10. The Mercator projection is the projection of 𝕊2 from the origin of ℝ3 onto the cylinder {(𝑥, 𝑦, 𝑧)|𝑥2 +𝑦2 = 1}. Show that it is conformal, compute the first fundamental form, the curvature, and draw the geodesics and the loxodromes (these are the curves that have constant angle with the meridians). Exercise 4.11. The Archimedes projection is the radial projection of 𝕊2 from the axis 𝑥 = 𝑦 = 0 onto the cylinder {(𝑥, 𝑦, 𝑧)|𝑥2 + 𝑦2 = 1, −1 < 𝑧 < 1}. Compute the first fundamental form, show that it is iso-areal (it preserves areas) but not conformal. Compute the curvature and draw the geodesics.

Problems

261

Exercise 4.12. Give the map SU(2) → SO(3) of Remark 4.29 explicitly. Exercise 4.13. Give a formula for the Hopf map ℎ ∶ 𝑆 3 → 𝑆 2 , and show that the fibers ℎ−1 (𝑝) are circles. Considering 𝑆3 = ℝ3 ∪ {∞}, via the stereographic projection, show that the preimage of the equator of 𝑆2 by ℎ is a torus of revolution 𝑇 ⊂ ℝ3 and that the fibers in this torus are circles of type (1, 1) in 𝐻1 (𝑇) ≅ ℤ2 , where we have taken the basis formed by the meridian and the parallel of 𝑇. Exercise 4.14. Which connected surfaces with 𝜅 ≡ 0 are symmetric and which are homogeneous? Exercise 4.15. Let Λ ⊂ ℝ2 be a lattice. Decide if the set of distances {||𝑣|| | 𝑣 ∈ Λ} determines the lattice Λ. Exercise 4.16. Let Λ ⊂ ℝ2 be a lattice. Does the set of angles {∠(𝑣, 𝑤) | 𝑣, 𝑤 ∈ Λ−{0}} determine Λ, up to multiplication by a scalar? Exercise 4.17. Let 𝑀 be a flat Riemannian manifold. Prove that the parallel transport between two points of 𝑀 does not depend on the choice of path joining them, for two homotopic piecewise 𝐶 1 paths. Exercise 4.18. Prove that there are 17 different isomorphism classes of groups of plane tessellations by parallelograms. Exercise 4.19. Let 𝑀 be a connected smooth manifold, and let g be a non-degenerate symmetric (0, 2)-tensor. Prove that the signature of g𝑝 is constant for 𝑝 ∈ 𝑀. Exercise 4.20. Prove that 𝑆2 does not admit semi-Riemannian (non-Riemannian) metrics. Exercise 4.21. Describe the Beltrami-Klein model, the Poincaré ball model, and the Poincaré half space model of the hyperbolic 𝑛-space ℍ𝑛 , for 𝑛 ≥ 3. Exercise 4.22. In the hyperbolic plane, we call a curve perpendicular to a family of limiting parallel geodesics a horocycle. Show that a curve 𝑐 is a horocycle if and only if its geodesic curvature function is 𝑘𝑔 ≡ 1. Show that the horocycles are preserved by isometries and that given two horocycles (parametrized by arc length) there is always an isometry sending one to the other. Exercise 4.23. Prove that there is a unique triangle in ℍ2 (up to isometry) with edges of given length 𝑙1 , 𝑙2 , 𝑙3 . Prove that there is a unique triangle in ℍ2 (up to isometry) with angles 𝛼, 𝛽, 𝛾, for 𝛼 + 𝛽 + 𝛾 < 𝜋. Exercise 4.24. Determine which manifolds are SU(1, 1), U(1, 1) and PU(1, 1). +

Exercise 4.25. Give explicit formulas for the isomorphisms PO (2, 1) ≅ PSU(1, 1) ≅ PSL(2, ℝ). Exercise 4.26. Prove that the map 𝜑 of (4.28) satisfies that 𝜑(𝐵 2 ) = 𝐵 2 . Prove directly that 𝜑∗ g𝑃 = g𝑃 using the expression (4.30). −

Exercise 4.27. Classify the isometries Isom (ℍ2 ). They can be of two types: • Reflection with respect to a geodesic (with local model 𝜑(𝑧) = 𝑧 in ℍ2𝑃𝐷 ). • Composition of a reflection with respect to a geodesic with a translation along the same geodesic called a glide reflection. 𝑧+𝑎

Exercise 4.28. Determine when 𝜑(𝑧) = 𝑒i𝜃 1+𝑎𝑧 is a rotation, an ideal rotation or a translation, of ℍ2𝑃𝐷 in terms of 𝑎 and 𝜃. Exercise 4.29. Prove that a group Γ < PSL(2, ℝ) acts properly on 𝐻 if and only if it is discrete.

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262

5 1 Exercise 4.30. Let Γ = ⟨𝐴⟩ < O(2, 1) be the group generated by 𝐴 = 4( 3 0 it is discrete, but it does not act properly on ℝ3 − {(0, 0, 0)}.

3 5 0

0 0 ). Prove that 4

Exercise 4.31. Let 𝐴𝐵𝐶 be a triangle in a 2-dimensional simply connected space form, whose sides 𝐵𝐶, 𝐶𝐴, 𝐴𝐵 have lengths 𝑎, 𝑏, 𝑐, respectively, and whose angles at vertices 𝐴, 𝐵, 𝐶 are 𝛼, 𝛽, 𝛾, respectively. Show the following formulas, according to the type of geometry: • For 𝕊2 , we have

sin 𝑎 sin 𝛼

=

sin 𝑏 sin 𝛽

=

sin 𝑐 . sin 𝛾

• For 𝔼2 , we have

𝑎 sin 𝛼

=

𝑏 sin 𝛽

=

𝑐 . sin 𝛾

• For ℍ2 , we have

sinh 𝑎 sin 𝛼

=

sinh 𝑏 sin 𝛽

=

sinh 𝑐 . sin 𝛾

This is known as the law of sines in triangle geometry. What is the law of cosines? Exercise 4.32. Prove the isomorphism of groups PSL(2, ℤ) ≅ ℤ2 ∗ ℤ3 , taking generators 𝜓(𝑧) = −1/𝑧 and 𝜙(𝑧) = (𝑧 − 1)/𝑧. Exercise 4.33. Determine the dimension of the Teichmüller space of metrics of curvature −1 for the non-orientable surface 𝑋𝑘 , 𝑘 ≥ 3. Conclude that the map from the Teichmüller space for 𝑋𝑘 to the Teichmüller space for Σ𝑘−1 , 𝑘 ≥ 3, is not surjective. Exercise 4.34. Prove, using the Preissman theorem (Remark 3.71), that if a compact connected 𝑛-manifold admits a metric of constant curvature, then it can be of only one sign.

References and extra material Basic reading. The book [Cox] has a classical approach to geometry. For hyperbolic geometry, we refer the student to [Iv1] and [Rat]. [Cox] H.S.M. Coxeter, Introduction to Geometry, Wiley Classics Library, Wiley, 1969. [Iv1] B. Iversen, Hyperbolic Geometry, London Mathematical Society Student Texts, Vol. 25, Cambridge University Press, 2009. [Rat] J. Ratcliffe, Foundations of Hyperbolic Manifolds, Graduate Texts in Mathematics, Vol. 149, 2nd Edition, Springer, 2006. Further reading. The following topics can be studied to enlarge the content analysed in this chapter. • Quaternions. The definition and properties of the quaternions, in particular in terms of matrices. As an application, it gives the relation between SO(4) and SU(2) × SU(2). [Rod] L. Rodman, Topics in Linear Quaternion Algebra, Princeton Series in Applied Mathematics, Princeton University Press, 2014. • Poincaré space. The description of the Poincaré dodecahedral space or the lens spaces of Example 4.34 are given in: [S-T] H. Seifert, W. Threlfall, A Textbook of Topology, Academic Press, 1980. • Spherical geometry. The classical viewpoint of the geometry of 𝕊2 , with applications to astronomy. [Bru] G.V. Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton University Press, 2013. • Hopf fibration. It is worthwhile to give an explicit description of the Hopf fibration in Remark 4.30, and prove that 𝜋3 (𝑆2 ) = ℤ, using the long exact sequence of homotopy for a fibration (see page 134).

References and extra material

263

[Lyo] D.W. Lyons, An elementary introduction to the Hopf fibration, Mathematics Magazine, 76(2), April 2003. • Bieberbach groups. These are the fundamental groups of flat manifolds (Remark 4.54). [Cha] L. Charlap, Bieberbach Groups and Flat Manifolds, Universitext, Springer, 1986. • Orbifolds and chrystallographic groups. Determine the orbifolds obtained by actions of finite groups of isometries on the plane and the sphere, recovering the 17 chrystallographic groups of the plane (cf. Exercise 4.18). [Co1] J. Conway, The orbifold notation for surface groups, In Groups, Combinatorics and Geometry. London Mathematical Society Lecture Note Series 165, 438-447, 1992, Cambridge University Press. +

• Lorentz group. The study of the group O> (3, 1) and the existence of a double cover SL(2, ℂ) → + O> (3, 1). [Nab] G. Naber, The Geometry of Minkowski Spacetime, Applied Mathematical Sciences, Vol. 92, Springer, 1992. • Hyperbolic geometry. A detailed study of the isometries of the hyperbolic plane, its classification, and the discrete subgroups of PSL(2, ℝ) can be found in [Iv1]. • Teichmüller space. Look at chapter 4 of [Jo2] or [I-T]. [Jo2] J. Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics, 3rd Edition, Universitext, Springer, 2006. [I-T] Y. Imayoshi, M. Taniguchi, An Introduction to Teichmüller Spaces, Springer, 1992. References. [Co2] J. Conway, Functions of One Complex Variable I, Vol. 1, Graduate Texts in Mathematics, Springer, 1995. [C-S] J. Conway, D. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A. K. Peters, 2003. [Euc] Euclid, Euclid’s Elements, Green Lion Press, 1st Edition (translated by T.L. Heath), 2002. [Gol] W.M. Goldman, Locally homogeneous geometric manifolds, Proceedings of the International Congress of Mathematicians Hyderabad, India, 717-744, 2010. [Gro] A. Grothendieck, Techniques de construction en géométrie analytique. I. Description axiomatique de l’espace de Teichmüller et de ses variantes, Séminaire Henri Cartan 13, Exposés No. 7 and 8. Paris, 1960-1961. [Hen] D.W. Henderson, D. Taimina, Experiencing Geometry, Pearson, 3rd Edition, 2004. [Kat] S. Katok, Fuchsian Groups, University of Chicago Press, 1992. [K-W] J. Kazdan, F. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Diff. Geom. 10, 113-134, 1975. [Kee] L. Keen, Canonical polygons for finitely generated Fuchsian groups, Acta Math. 115, 1-16, 1966. [Mi6] J.W. Milnor, Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. 6, 9-24, 1982. [MSW] D. Mumford, C. Series, D. Wright, Indra’s Pearls: The Vision of Felix Klein, Cambridge University Press, 2002. [O’N] B. O’Neill, Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, Academic Press, 1983.

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[Poi] H. Poincaré, Cinquième complément a l’analysis situs, Rend. Circ. Mat. Palermo, 18, 45-110, 1904. [War] F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, Vol. 94, 1983. [Wee] J. Weeks, The Shape of Space, 2nd Edition, CRC Press, 2002.

Chapter 5

Complex geometry

We enter now into the study of complex manifolds, i.e., manifolds modelled on open subsets of complex vector spaces for which we can define the notion of holomorphic functions. As we have encountered in Chapter 3, extra geometric structure produces refined topological invariants. For complex manifolds, we get Dolbeault cohomology, which enhances de Rham cohomology suitably. It is natural to introduce Riemannian structures on complex manifolds, leading to an important subclass of complex manifolds, called Kähler manifolds. For them, Dolbeault cohomology gives a bigrading on the de Rham cohomology. Finally, smooth projective varieties are complex manifolds defined by polynomial equations in the complex projective space. These are the principal objects studied in algebraic geometry, and they are Kähler manifolds. As in previous chapters, we shall focus on dimension 2, i.e., compact connected complex curves, which are oriented surfaces with a complex structure. We will see that they always have a plane model (can be described as the zero locus of a polynomial in the complex projective plane). The degree-genus formula relates the genus of the surface (its topology) with the degree of the polynomial (which is algebraic geometric information). We shall also look at the classification of compact complex curves, which is akin to the classification of surfaces of constant curvature studied in Chapter 4. We end with a detailed analysis of the case of elliptic curves, that is when the genus is 𝑔 = 1.

5.1. Complex manifolds We start by reviewing the fundamental facts of complex analysis that we will need within this chapter. These can be found in [Ahl]. Later we shall give the basic results of complex manifolds, without entering into the technical proofs. These will be used to study complex curves. The theory of complex manifolds can be found in [Huy], [Wel].

5.1.1. Review of complex analysis. Let us take open sets 𝑈 ⊂ ℂ𝑛 and 𝑉 ⊂ ℂ𝑚 and a function 𝑓 ∶ 𝑈 → 𝑉. Recall that 𝑓 is called holomorphic if, for all 𝑧0 ∈ 𝑈, there exists 265

266

5. Complex geometry

a ℂ-linear map 𝐿𝑧0 ∶ ℂ𝑛 → ℂ𝑚 such that 𝑓(𝑧0 + ℎ) − 𝑓(𝑧0 ) − 𝐿𝑧0 ℎ = 0. |ℎ| ,ℎ→0

lim 𝑛

ℎ∈ℂ

In that case, the linear map 𝐿𝑧0 is called the (complex) differential of 𝑓 at 𝑧0 and denoted 𝑑𝑧ℂ0 𝑓. Note that it coincides with the (real) differential, so 𝑓 is differentiable as a map from 𝑈 ⊂ ℝ2𝑛 to 𝑉 ⊂ ℝ2𝑚 . The set of holomorphic maps from 𝑈 to 𝑉 is denoted Hol(𝑈, 𝑉). If 𝑓 is invertible and the inverse is also holomorphic in 𝑉, then 𝑓 is called a biholomorphism. Let us start by reviewing the case of one variable. The set of holomorphic functions 𝑓 ∶ 𝑈 ⊂ ℂ → ℂ is denoted Hol(𝑈). For 𝑓 ∈ Hol(𝑈), we write 𝑓(𝑧) = 𝑢(𝑧) + i𝑣(𝑧) with 𝑢, 𝑣 ∶ 𝑈 → ℝ real functions (the real and imaginary parts of 𝑓). We also write 𝑧 = 𝑥 + i𝑦. As a function 𝑓 ∶ 𝑈 ⊂ ℝ2 → ℝ2 , we have 𝑓(𝑥, 𝑦) = (𝑢(𝑥, 𝑦), 𝑣(𝑥, 𝑦)). (1) If 𝑓 is holomorphic, the complex differential at 𝑧0 , 𝑑𝑧ℂ0 𝑓 ∶ ℂ → ℂ is given by a complex number 𝑓(𝑧0 + ℎ) − 𝑓(𝑧0 ) . ℎ In particular, such a limit should be independent of the way ℎ approaches 0. Writing 𝑧0 = 𝑥 + i𝑦 and taking ℎ → 0 along the horizontal and vertical axis, we obtain the equalities, 𝑓′ (𝑧0 ) = lim

ℎ→0

𝑓(𝑥 + ℎ + i𝑦) − 𝑓(𝑥 + i𝑦) 𝜕𝑓 = (𝑧 ) ℎ 𝜕𝑥 0 ℎ∈ℝ,ℎ→0 𝜕𝑢 𝜕𝑣 = (𝑧 ) + i (𝑧0 ), 𝜕𝑥 0 𝜕𝑥 𝑓(𝑥 + i(𝑦 + ℎ)) − 𝑓(𝑥 + i𝑦) 𝜕𝑓 𝑓′ (𝑧0 ) = lim = −i (𝑧0 ) iℎ 𝜕𝑦 ℎ∈ℝ,ℎ→0 𝜕𝑢 𝜕𝑣 = −i (𝑧0 ) + (𝑧0 ). 𝜕𝑦 𝜕𝑦 Since such equations hold for all 𝑧0 ∈ 𝑈, 𝑓 must satisfy the so-called CauchyRiemann equations 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕𝑣 = , =− . 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 These equations are, in fact, a sufficient condition, so 𝑓 = 𝑢 + i𝑣 is a holomorphic function if and only if it satisfies (5.1). 𝑓′ (𝑧0 ) =

(5.1)

lim

(2) We introduce the Wirtinger operators on complex valued smooth functions, 𝜕 1 𝜕 𝜕 𝜕 1 𝜕 𝜕 = ( − i ), = ( + i ). 𝜕𝑧 2 𝜕𝑥 𝜕𝑦 2 𝜕𝑥 𝜕𝑦 𝜕𝑧 If 𝑓 ∶ 𝑈 ⊂ ℂ → ℂ is holomorphic, then by (5.1), 𝑓′ (𝑧) = 𝜕 𝑓 𝜕𝑧

𝜕𝑓 𝜕𝑥

𝜕𝑓

= −i 𝜕𝑦 , so

= 𝑓′ (𝑧), which justifies the definition of the first Wirtinger operator. The second one can be understood as a conjugate of the first one, using 𝑧 = 𝑥 − i𝑦. 𝜕 But more relevant, a function 𝑓 is holomorphic if and only if 𝜕𝑧 𝑓 = 0, since the Cauchy-Riemann equations (5.1) are

𝜕𝑓 𝜕𝑥

𝜕𝑓

= −i 𝜕𝑦 .

5.1. Complex manifolds

267

(3) A function 𝑓 ∶ 𝑈 ⊂ ℂ → ℂ is holomorphic if and only if it satisfies the Cauchy theorem, that is, for every piecewise 𝐶 1 contractible loop 𝛾 ∶ 𝑆 1 → 𝑈 it holds ∫𝛾 𝑓 𝑑𝑧 = 0 (Exercise 5.1). (4) A holomorphic function 𝑓 = 𝑢 + i𝑣 is automatically a 𝐶 ∞ function, i.e., 𝑢, 𝑣 ∶ 𝑈 ⊂ ℝ2 → ℝ are both 𝐶 ∞ . Furthermore, a holomorphic function is analytic, that is, for any 𝑧0 ∈ 𝑈 there exists a ball 𝐵𝜖 (𝑧0 ) around 𝑧0 in which 𝑓 has an expression as uniformly convergent power series ∞

𝑓(𝑧) = ∑ 𝑎𝑘 (𝑧 − 𝑧0 )𝑘 . 𝑘=0

The coefficients 𝑎𝑘 can be easily computed from 𝑓 as the Taylor coefficients 1 𝑎𝑘 = 𝑘! 𝑓(𝑘) (𝑧0 ), where 𝑓(𝑘) denotes the 𝑘th complex derivative. In particular, if 𝑓(𝑧0 ) = 0, it is said that 𝑧0 is a zero of 𝑓. In that case, we have that 𝑓(𝑧) = ∞ ∑𝑘=𝑛 𝑎𝑘 (𝑧 − 𝑧0 )𝑘 for some 𝑛 ≥ 1, 𝑎𝑛 ≠ 0. Such 𝑛 is called the order of the zero 𝑧0 , and write 𝑛 = ord𝑧0 𝑓. If 𝑓(𝑧0 ) ≠ 0, we say ord𝑧0 𝑓 = 0. Therefore ̃ ̃ 0 ) ≠ 0, and 𝑛 = ord𝑧 𝑓. 𝑓(𝑧) = (𝑧 − 𝑧0 )𝑛 𝑓(𝑧), where 𝑓 ̃ is holomorphic, 𝑓(𝑧 0 (5) Let 𝑓 ∶ 𝑈 ⊂ ℂ → ℂ be a holomorphic map. Interpreted as a map 𝑓 ∶ 𝑈 ⊂ ℝ2 → ℝ2 , the real differential 𝑑𝑧0 𝑓 ∶ ℝ2 → ℝ2 is a linear map whose matrix is 𝑑𝑧0 𝑓 = ( 𝜕ᵆ

𝜕ᵆ 𝜕𝑥 𝜕𝑣 𝜕𝑥

𝜕𝑣

𝜕ᵆ 𝜕𝑦 𝜕𝑣 𝜕𝑦

)=(

𝛼 −𝛽 ), 𝛽 𝛼 𝜕𝑣

𝜕ᵆ

where 𝛼 = 𝜕𝑥 (𝑧0 ) = 𝜕𝑦 (𝑧0 ) and 𝛽 = 𝜕𝑥 (𝑧0 ) = − 𝜕𝑦 (𝑧0 ), and 𝑓′ (𝑧0 ) = 𝛼 + i𝛽. This corresponds with the description of the complex numbers as 𝑎 −𝑏 | 2 × 2-matrices, ℂ = {( ) |𝑎, 𝑏 ∈ ℝ} ⊂ 𝑀2×2 (ℝ). Note that the multi𝑏 𝑎 plicative structure of ℂ is that inherited from 𝑀2×2 (ℝ) (this is usually called the doubling process for going from ℝ to ℂ = ℝ2 ). 𝑎 −𝑏 ), with (𝑎, 𝑏) ≠ (0, 0), is a dilation of ℝ2 which 𝑏 𝑎 preserves orientation (Remark 4.12). Recall that a dilation is a linear map 𝜑𝐴 ∶ ℝ2 → ℝ2 , 𝜑𝐴 (𝑣) = 𝐴 𝑣, such that ||𝜑𝐴 (𝑣)|| = 𝜆||𝑣||, for all 𝑣, and some fixed 𝜆 > 0. This is equivalent to 𝐴𝑡 𝐴 = 𝜆2 Id. By Lemma 4.13, this is equivalent to 𝜑𝐴 preserving angles. Therefore if 𝜑𝐴 is a dilation, then either 𝜆 cos 𝜃 −𝜆 sin 𝜃 𝜆 cos 𝜃 𝜆 sin 𝜃 𝐴=( ) or 𝐴 = ( ). The first type is 𝜆 sin 𝜃 𝜆 cos 𝜃 −𝜆 sin 𝜃 −𝜆 cos 𝜃 a rotation composed with a homothety and preserves orientation, the second type is a reflection composed with a homothety and reverses orientation.

(6) A matrix 𝐴 = (

(7) A consequence of the previous items is that 𝑓 ∶ 𝑈 → ℂ is holomorphic and 𝑓′ (𝑧) ≠ 0 for all 𝑧 ∈ 𝑈 if and only if 𝑓 ∶ (𝑈, g𝑠𝑡𝑑 ) → (ℝ2 , g𝑠𝑡𝑑 ) is a 𝐶 1 map (as a real map), it is locally conformal (Definition 4.11), and it preserves the orientation of ℝ2 . This proves Lemma 4.16.

268

5. Complex geometry

(8) A map 𝑓 ∶ 𝑈 ⊂ ℂ → ℂ is called antiholomorphic if the map 𝑓(𝑧) is holomor𝜕 𝜕 phic. Conjugating 𝜕𝑧 𝑓(𝑧) = 0, this is equivalent to 𝜕𝑧 𝑓(𝑧) = 0. As conjugation is an orientation reversing isometry of ℝ2 , we have that 𝑓 ∶ 𝑈 → ℂ 𝜕𝑓 is antiholomorphic and 𝜕𝑧 ≠ 0 for all 𝑧 ∈ 𝑈 if and only if 𝑓 ∶ (𝑈, g𝑠𝑡𝑑 ) → (ℝ2 , g𝑠𝑡𝑑 ) is a 𝐶 1 and locally conformal map that reverses the orientation of ℝ2 . (9) Another important concept in complex analysis is that of meromorphic functions. Roughly speaking, these are holomorphic functions with some (nice) singularities. Let 𝑈 ⊂ ℂ be an open set, and let 𝑆 ⊂ 𝑈 be a discrete subset. A function 𝑓 ∶ 𝑈 − 𝑆 → ℂ is a meromorphic function on 𝑈 if 𝑓 is holomorphic on 𝑈 − 𝑆 and, for every 𝑧0 ∈ 𝑆 there exists 𝑛 > 0 such that 𝑓(𝑧)(𝑧 − 𝑧0 )𝑛 is bounded around 𝑧0 . In this case, 𝑓(𝑧)(𝑧 − 𝑧0 )𝑛 can be extended to a unique holomorphic function on a ball 𝐵𝜖 (𝑧0 ) ⊂ 𝑈. The point 𝑧0 is called a pole of 𝑓 and the minimum 𝑛 such that 𝑓(𝑧)(𝑧 − 𝑧0 )𝑛 is bounded near 𝑧0 is called the order of the pole 𝑧0 . We write ord𝑧0 𝑓 = −𝑛. Observe that for a pole 𝑧0 of order 𝑛, 𝑛 ≥ 1, we have that 𝑓(𝑧) → ∞, when 𝑧 → 𝑧0 . Meromorphic functions admit a power series expansion around a pole 𝑧0 of order 𝑛 of the form ∞

𝑓(𝑧) = ∑ 𝑎𝑖 (𝑧 − 𝑧0 )𝑖 = 𝑖=−𝑛

𝑎−(𝑛−1) 𝑎−𝑛 𝑎 + + ⋯ + −1 𝑧 − 𝑧0 (𝑧 − 𝑧0 )𝑛 (𝑧 − 𝑧0 )𝑛−1 + 𝑎0 + 𝑎1 (𝑧 − 𝑧0 ) + 𝑎2 (𝑧 − 𝑧0 )2 + ⋯ ,

with 𝑎−𝑛 ≠ 0. This kind of series is known as Laurent series, and the coefficient 𝑎−1 is called the residue of 𝑓 at 𝑧0 . Note that we can write 𝑓(𝑧) = ̃ ̃ 0 ) ≠ 0. (𝑧 − 𝑧0 )−𝑛 𝑓(𝑧), where 𝑓 ̃ is holomorphic and 𝑓(𝑧 (10) Meromorphic functions can be described locally as quotients of holomorphic functions. If 𝑓 ∶ 𝑈 − 𝑆 → ℂ is meromorphic with a set of poles 𝑆 and 𝑧0 ∈ 𝑆, then on a ball 𝐵𝜖 (𝑧0 ) ⊂ 𝑈, we have 𝑓 = 𝑔/ℎ, for holomorphic functions 𝑔(𝑧), and ℎ(𝑧) = (𝑧 − 𝑧0 )𝑛 , −𝑛 = ord𝑧0 𝑓. Conversely, if 𝑓 = 𝑔/ℎ on an open subset 𝑉 ⊂ 𝑈 and 𝑔, ℎ are holomorphic on 𝑉, then let 𝑛 = ord𝑧0 𝑔, 𝑚 = ord𝑧0 ℎ ≥ 0. ̃ So 𝑔(𝑧) = (𝑧 − 𝑧0 )𝑛 𝑔(𝑧), ̃ ℎ(𝑧) = (𝑧 − 𝑧0 )𝑚 ℎ(𝑧), where 𝑔,̃ ℎ ̃ are holomorphic ̃ and 𝑔(𝑧 ̃ 0 ) ≠ 0, ℎ(𝑧0 ) ≠ 0. Then 𝑓 = 𝑔/ℎ = (𝑧 − 𝑧0 )𝑛−𝑚 𝑔/̃ ℎ,̃ where 𝑓 ̃ = 𝑔/̃ ℎ ̃ ̃ 0 ) ≠ 0. So 𝑓 is meromorphic and has a zero of is holomorphic on 𝑉, and 𝑓(𝑧 order 𝑛 − 𝑚 if 𝑛 > 𝑚, or a pole of order 𝑚 − 𝑛 if 𝑚 > 𝑛. (11) An easy calculation with the Wirtinger operators shows that for a smooth function 𝑓, its differential is 𝜕𝑓 𝜕𝑓 𝑑𝑥 + 𝑑𝑦 𝜕𝑥 𝜕𝑥 𝜕𝑓 𝜕𝑓 1 𝜕𝑓 1 𝜕𝑓 = ( − i ) (𝑑𝑥 + i𝑑𝑦) + ( + i ) (𝑑𝑥 − i𝑑𝑦) 2 𝜕𝑥 𝜕𝑦 2 𝜕𝑥 𝜕𝑦 𝜕𝑓 𝜕𝑓 = 𝑑𝑧 + 𝑑𝑧 . 𝜕𝑧 𝜕𝑧

𝑑𝑓 =

Let us move on to the case of several variables. Given an open set 𝑈 ⊂ ℂ𝑛 , we denote Hol(𝑈) the set of holomorphic functions 𝑓 ∶ 𝑈 ⊂ ℂ𝑛 → ℂ. We denote 𝑧𝑗 = 𝑥𝑗 + i𝑦𝑗 , 1 ≤ 𝑗 ≤ 𝑛, and 𝑓(𝑧1 , . . . , 𝑧𝑛 ) = 𝑢(𝑥1 , 𝑦1 , . . . , 𝑥𝑛 , 𝑦𝑛 ) + i𝑣(𝑥1 , 𝑦1 , . . . , 𝑥𝑛 , 𝑦𝑛 ).

5.1. Complex manifolds

269

If 𝑓 is holomorphic, taking the limit along only one axis, we have that 𝑓 must be holomorphic in each variable separately, that is 𝑧𝑗 ↦ 𝑓(𝑧1 , . . . , 𝑧𝑗 , . . . , 𝑧𝑛 ) is holomorphic as a function of 𝑧𝑗 , for each 1 ≤ 𝑗 ≤ 𝑛. Therefore 𝑓 satisfies the Cauchy-Riemann equations 𝜕𝑢 𝜕𝑣 = , 𝜕𝑥𝑗 𝜕𝑦𝑗

𝜕𝑢 𝜕𝑣 =− , 𝜕𝑦𝑗 𝜕𝑥𝑗

for all 1 ≤ 𝑗 ≤ 𝑛. Again, these Cauchy-Riemann equations are also a sufficient condition for holomorphicity (this is called the Osgood lemma [Nar]). In particular, this proves that 𝑓 is holomorphic if and only if it is holomorphic in each variable separately. For this reason, most of the notions of one complex variable analysis translate to the case of several complex variables. In particular, if we take the Wirtinger operators (5.2)

𝜕 1 𝜕 𝜕 = ( −i ), 𝜕𝑧𝑗 2 𝜕𝑥𝑗 𝜕𝑦𝑗

𝜕 1 𝜕 𝜕 = ( +i ), 2 𝜕𝑥𝑗 𝜕𝑦𝑗 𝜕𝑧𝑗

then a function 𝑓 ∶ 𝑈 ⊂ ℂ𝑛 → ℂ is holomorphic if and only if 1 ≤ 𝑗 ≤ 𝑛.

𝜕 𝑓 𝜕𝑧𝑗

= 0 for all

As in the case of one variable, a holomorphic function 𝑓 = 𝑢 + i𝑣 is automatically an analytic function, that is, for any 𝑎 = (𝑎1 , . . . , 𝑎𝑛 ) ∈ 𝑈, there exists a ball 𝐵𝜖 (𝑎) around 𝑎 on which 𝑓 has an expression as a uniformly convergent power series ∞

(5.3)

𝑓(𝑧) =

∑

𝑐 𝑘1 ⋯𝑘𝑛 (𝑧1 − 𝑎1 )𝑘1 ⋯ (𝑧𝑛 − 𝑎𝑛 )𝑘𝑛 .

𝑘1 ,. . .,𝑘𝑛 =0

The coefficients 𝑐 𝑘1 ⋯𝑘𝑛 can be computed from 𝑓 as the Taylor coefficients 𝑐 𝑘1 ⋯𝑘𝑛 = 1 𝜕𝑘1 +⋯+𝑘𝑛 𝑓 𝑘1 ! ⋯ 𝑘𝑛 ! 𝜕𝑧𝑘1 ⋯𝜕𝑧𝑘𝑛𝑛 1

(𝑎).

We can also define meromorphic functions on several complex variables as local quotients of holomorphic functions. Let 𝑈 ⊂ ℂ𝑛 be an open subset. A meromorphic function on 𝑈 is a function 𝑓 ∶ 𝑈 − 𝑆 → ℂ, where 𝑆 ⊂ 𝑈 is a closed subset with no interior, such that 𝑓 is holomorphic on 𝑈 − 𝑆, and for any 𝑎 ∈ 𝑆, there is an open neighbourhood 𝑉 ⊂ 𝑈 of 𝑎 and holomorphic functions 𝑔, ℎ on 𝑉 such that 𝑓 = 𝑔/ℎ on 𝑉 − 𝑆. There is no Laurent series expression for 𝑓, and 𝑆 is not discrete (actually, 𝑆 is the zero set of ℎ on the subset 𝑉). Finally, given a function 𝑓 ∶ 𝑈 ⊂ ℂ𝑛 → 𝑉 ⊂ ℂ𝑚 , write as (𝑤 1 , . . . , 𝑤 𝑚 ) = 𝑓(𝑧1 , . . . , 𝑧𝑛 ) with 𝑤 1 , . . . , 𝑤 𝑚 ∶ 𝑈 → ℂ. Then 𝑓 is a holomorphic map if and only if 𝑤 𝑘 are holomorphic functions for each 1 ≤ 𝑘 ≤ 𝑚. Remark 5.1. There is a natural map 𝚤 ∶ 𝑀𝑚×𝑛 (ℂ) ↪ 𝑀2𝑚×2𝑛 (ℝ) that extends the map ℂ ↪ 𝑀2×2 (ℝ). It consists of assigning to a complex linear map 𝜑𝐴 ∶ ℂ𝑛 → ℂ𝑚 the same map interpreted as a real linear map 𝜑𝐴 ∶ ℝ2𝑛 → ℝ2𝑚 . Take a complex matrix 𝐴 = (𝑎𝑗𝑘 ), and write 𝑎𝑗𝑘 = 𝛼𝑗𝑘 + i𝛽𝑗𝑘 ∈ ℂ, where 𝛼𝑗𝑘 , 𝛽𝑗𝑘 ∈ ℝ. So 𝜑𝐴 (𝑒 𝑘 ) = ∑ 𝑎𝑗𝑘 𝑒𝑗′ , where (𝑒 𝑘 ) and (𝑒𝑗′ ) are the standard bases of ℂ𝑛 and ℂ𝑚 , respectively. The standard

270

5. Complex geometry

bases of ℝ2𝑛 and ℝ2𝑚 are (𝑒 1 , i𝑒 1 , . . . , 𝑒 𝑛 , i𝑒 𝑛 ) and (𝑒′1 , i𝑒′1 , . . . , 𝑒′𝑚 , i𝑒′𝑚 ). Then 𝜑𝐴 (𝑒 𝑘 ) = ∑(𝛼𝑗𝑘 𝑒𝑗′ + 𝛽𝑗𝑘 i𝑒𝑗′ ) and 𝜑𝐴 (i𝑒 𝑘 ) = ∑(−𝛽𝑗𝑘 𝑒𝑗′ + 𝛼𝑗𝑘 i𝑒𝑗′ ). Thus 𝛼 ⎛ 11 𝛽 ⎜ 11 ⋮ 𝚤(𝐴) = ⎜ ⎜ ⋮ ⎜ 𝛼𝑚1 ⎝ 𝛽𝑚1

(5.4)

−𝛽11 𝛼11

−𝛽𝑚1 𝛼𝑚1

... ... ... ... ⋱ ⋱ ... ... ... ...

𝛼1𝑛 𝛽1𝑛

𝛼𝑚𝑛 𝛽𝑚𝑛

−𝛽1𝑛 𝛼1𝑛 ⋮ ⋮ −𝛽𝑚𝑛 𝛼𝑚𝑛

⎞ ⎟ ⎟. ⎟ ⎟ ⎠

For a holomorphic map 𝑓 ∶ 𝑈 ⊂ ℂ𝑛 → 𝑉 ⊂ ℂ𝑚 , (𝑤 1 , . . . , 𝑤 𝑚 ) = 𝑓(𝑧1 , . . . , 𝑧𝑛 ), the 𝜕𝑤 complex differential at 𝑧 ∈ 𝑈 is given by 𝑑𝑧ℂ 𝑓 = ( 𝜕𝑧 𝑗 ) . We write 𝑧𝑘 = 𝑥𝑘 + i𝑦 𝑘 and 𝑘

𝑗,𝑘

𝑤𝑗 = 𝑢𝑗 + i𝑣𝑗 , and 𝜕𝑤𝑗 = 𝛼𝑗𝑘 + i𝛽𝑗𝑘 , 𝜕𝑧𝑘

where 𝛼𝑗𝑘 =

𝜕𝑢𝑗 𝜕𝑣𝑗 = , 𝜕𝑥𝑘 𝜕𝑦 𝑘

𝛽𝑗𝑘 =

𝜕𝑢𝑗 𝜕𝑣𝑗 =− . 𝜕𝑦 𝑘 𝜕𝑥𝑘

Then the differential as a real function 𝑓 ∶ 𝑈 ⊂ ℝ2𝑛 → ℝ2𝑚 is ⎛ ⎜ ⎜ 𝑑𝑧 𝑓 = 𝚤(𝑑𝑧ℂ 𝑓) = ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎝

𝜕ᵆ1 𝜕𝑥1 𝜕𝑣1 𝜕𝑥1

⋮ ⋮

𝜕ᵆ1 𝜕𝑦1 𝜕𝑣1 𝜕𝑦1

... ⋱

𝜕ᵆ1 𝜕𝑥𝑛 𝜕𝑣1 𝜕𝑥𝑛

𝜕ᵆ1 𝜕𝑦𝑛 𝜕𝑣1 𝜕𝑦𝑛

𝜕ᵆ𝑚 𝜕𝑥𝑛 𝜕𝑣𝑚 𝜕𝑥𝑛

− 𝜕𝑦𝑚

𝜕ᵆ1 𝜕𝑥𝑛 𝜕𝑣1 𝜕𝑥𝑛

− 𝜕𝑥1 ⎞ 𝑛 𝜕ᵆ1 ⎟ 𝜕𝑥𝑛 ⋮ ⎟ ⎟. ⋮ ⎟ 𝜕𝑣 − 𝜕𝑥𝑚 ⎟ 𝑛 𝜕ᵆ𝑚 ⎟ ⎠ 𝜕𝑥𝑛

⋱

𝜕ᵆ𝑚 𝜕𝑥1 𝜕𝑣𝑚 𝜕𝑥1

− 𝜕𝑦𝑚

𝜕ᵆ1 𝜕𝑥1 𝜕𝑣1 𝜕𝑥1

− 𝜕𝑥1

𝜕ᵆ𝑚 𝜕𝑥1 𝜕𝑣𝑚 𝜕𝑥1

− 𝜕𝑥𝑚

⋮ ⋮

...

𝜕ᵆ

1 𝜕𝑣𝑚

𝜕𝑦1 𝜕𝑣

𝜕ᵆ1 𝜕𝑥1

1

...

...

...

...

⋱ ⋱

𝜕𝑣

1 𝜕ᵆ𝑚

𝜕𝑥1

...

...

𝜕ᵆ𝑚 𝜕𝑥𝑛 𝜕𝑣𝑚 𝜕𝑥𝑛

⋮ ⋮

𝜕ᵆ

𝑛

𝜕𝑣𝑚 𝜕𝑦𝑛

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

𝜕𝑣

𝑎 −𝑏 ). 𝑏 𝑎 Equivalently, 𝑑𝑧 𝑓 is in the image of 𝑀𝑚×𝑛 (ℂ) ⊂ 𝑀2𝑚×2𝑛 (ℝ), or said otherwise, 𝑑𝑧 𝑓 is a complex linear map for every 𝑧 ∈ 𝑈. Hence 𝑓 is holomorphic if and only if every 2×2 block of 𝑑𝑧 𝑓 is of the form (

Remark 5.2. For 𝑚 = 𝑛, the inclusion of Remark 5.1 induces a group monomorphism 𝚤 ∶ GL(𝑛, ℂ) → GL(2𝑛, ℝ). It is interesting to note that det(𝚤(𝐴)) = | det(𝐴)|2 , for 𝐴 ∈ 𝑀𝑛×𝑛 (ℂ). To prove this, write 𝐴 = 𝑃 −1 𝐽𝑃, where 𝐽 is a Jordan matrix. So det(𝚤(𝐴)) = det(𝚤(𝐽)), and det(𝐴) = det(𝐽), hence it is enough to check the formula for Jordan matrices, in particular for lower triangular matrices. But for such matrices, the determinant is the product of the elements in the diagonal, so it is enough to prove for (1 × 1)-matrices, i.e., complex numbers. It follows from det(𝚤(𝑎 + i𝑏)) = 𝑎 −𝑏 det( ) = 𝑎2 + 𝑏2 = |𝑎 + i𝑏 |2 . 𝑏 𝑎

5.1. Complex manifolds

271

5.1.2. Complex manifolds. In the same vein as differential geometry is a generalization of real analysis to manifolds, complex geometry is a generalization of complex analysis to manifolds. As we will see, the rigid properties of holomorphic functions endow complex geometry with more powerful algebraic tools, reaching the realm of algebraic geometry. The notion of a complex manifold is analogous to the differentiable case given in section 1.1.2. Definition 5.3. A complex manifold is a topological manifold 𝑀 of dimension 2𝑛 together with an atlas 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )}, 𝜑𝛼 ∶ 𝑈𝛼 ⊂ 𝑀 → 𝑉𝛼 ⊂ ℝ2𝑛 = ℂ𝑛 , such that the changes of charts 𝑛 𝑛 𝜑𝛽 ∘ 𝜑−1 𝛼 ∶ 𝜑 𝛼 (𝑈𝛼 ∩ 𝑈 𝛽 ) ⊂ ℂ → 𝜑 𝛽 (𝑈𝛼 ∩ 𝑈 𝛽 ) ⊂ ℂ

are biholomorphic functions for all 𝛼, 𝛽. The atlas 𝒜 is called a complex atlas. Remark 5.4. • Since a biholomorphism is in particular a diffeomorphism, a complex manifold 𝑀 is also a differentiable manifold of dimension 2𝑛. It is customary to say that 𝑀 has complex dimension 𝑛, and to denote it as dimℂ 𝑀 = 𝑛, and real dimension 2𝑛, which will be denoted as usual as dim 𝑀 = 2𝑛. • For 𝑛 = 1, it is customary to call these manifolds complex curves despite the fact that, topologically, they are surfaces. For 𝑛 = 2 they are called complex surfaces, but they are topological 4-manifolds. • In analogy with the case of smooth manifolds, the atlas 𝒜 can be completed to a maximal atlas in a unique way. It is usual to consider that the complex atlas is maximal. • Any complex manifold is orientable. By Remark 5.1, the differential 𝑑𝑧 𝑓 of a biholomorphism lies in the image of 𝚤 ∶ GL(𝑛, ℂ) ↪ GL(2𝑛, ℝ). Then by Remark 5.2, det(𝑑𝑧 𝑓) = | det 𝚤(𝑑𝑧ℂ 𝑓)|2 > 0, so 𝑓 preserves orientation. Thus, if we orient arbitrarily one chart, then all other charts are oriented in a unique way (assuming that 𝑀 is connected), defining a global orientation for the complex manifold. More is true, as there is a natural orientation for 𝑀. Observe that 𝑇𝑝 𝑀 ≅ ℂ𝑛 = ℝ2𝑛 has a canonical orientation given by writing the canonical complex coordinates as 𝑧𝑗 = 𝑥𝑗 + i𝑦𝑗 , and take (𝜕𝑥1 , 𝜕𝑦1 , 𝜕𝑥2 , 𝜕𝑦2 , . . . , 𝜕𝑥𝑛 , 𝜕𝑦𝑛 ) as an oriented real basis. So 𝑀 has a natural orientation induced by the canonical orientation of the charts. Example 5.5. (1) An open set 𝑈 ⊂ ℂ, is itself a complex manifold with the inclusion map as chart. (2) The sphere 𝑆 2 can be endowed with a complex structure. Let 𝑒 0 = (1, 0, 0), −𝑒 0 = (−1, 0, 0) be the north and south poles of 𝑆 2 . The stereographic projections (4.6) and (4.12) from 𝑒 0 and −𝑒 0 can be seen, respectively, as maps 𝜑 ∶ 𝑆 2 − {𝑒 0 } → ℂ and 𝜑′ ∶ 𝑆 2 − {−𝑒 0 } → ℂ (recall that the map 𝜑′ in (4.12) is the stereographic projection from the south pole, composed with a reflection, so that it is orientation preserving). The change of coordinates

272

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𝜑′ ∘ 𝜑−1 ∶ ℂ − {0} → ℂ − {0} is given by 𝑧 ↦ 1/𝑧, which is a biholomorphism (see page 208). Therefore, this atlas defines a complex structure on 𝑆 2 . (3) We already saw that 𝑆 2 is diffeomorphic to ℂ𝑃 1 , via (4.10). Under this diffeomorphism, the charts 𝜑, 𝜑′ become 𝜙0 ∶ 𝑈0 → ℂ, 𝜙0 ([𝑧0 , 𝑧1 ]) = 𝑧1 /𝑧0 and 𝜙1 ∶ 𝑈1 → ℂ, 𝜙1 ([𝑧0 , 𝑧1 ]) = 𝑧0 /𝑧1 , where 𝑈0 = {[𝑧0 , 𝑧1 ] ∈ ℂ𝑃 1 | 𝑧0 ≠ 0} and 𝑈1 = {[𝑧0 , 𝑧1 ] ∈ ℂ𝑃 1 | 𝑧1 ≠ 0}. The change of charts is 𝜙1 ∘ 𝜙−1 0 (𝑧) = 1/𝑧. (4) In general, the complex projective space ℂ𝑃 𝑛 has a natural complex structure. This will be studied in section 5.2.3. On a complex manifold we have a notion of holomorphic functions. Let 𝑀 be a complex manifold with complex atlas 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )}, and let 𝑈 ⊂ 𝑀 be an open set. A function 𝑓 ∶ 𝑈 → ℂ is said to be holomorphic if, for all 𝑈𝛼 , the function 𝑓 ∘ 𝜑−1 𝛼 ∶ 𝜑𝛼 (𝑈 ∩ 𝑈𝛼 ) ⊂ ℂ𝑛 → ℂ is a holomorphic function in the usual sense. This defines a sheaf (Definition 1.13) 𝒪𝑀 , called the sheaf of holomorphic functions or structure sheaf, such that for an open subset 𝑈 ⊂ 𝑀, 𝒪𝑀 (𝑈) = {𝑓 ∶ 𝑈 → ℂ | 𝑓 is holomorphic on 𝑈} . As in the case of differentiable manifolds (see Theorem 1.16), for a topological manifold 𝑀 it is equivalent to have a complex atlas and a structure sheaf of holomorphic functions. More precisely, suppose that 𝑀 is a topological manifold of dimension 2𝑛, and 𝒪𝑀 is a subsheaf of the sheaf of continuous functions such that, for all 𝑝 ∈ 𝑀 there is an open neighbourhood 𝑈 𝑝 and a chart 𝜑 ∶ 𝑈 → 𝑉 ⊂ ℝ2𝑛 = ℂ𝑛 that induces an isomorphism 𝜑∗ ∶ 𝒪𝑀 (𝑈) → Hol(𝑉), 𝑓 ↦ 𝑓 ∘ 𝜑−1 . Then 𝑀 admits a unique structure of a complex manifold such that 𝒪𝑀 is the associated sheaf of holomorphic functions on 𝑀. The notion of holomorphic map between complex manifolds is again a straightforward generalization of a smooth map between differentiable manifolds. A map between complex manifolds 𝑓 ∶ 𝑀 → 𝑁 is called holomorphic if 𝑓∗ 𝒪𝑁 (𝑉) ⊂ 𝒪𝑀 (𝑓−1 (𝑉)) for all open sets 𝑉 ⊂ 𝑁, or equivalently, if 𝜓𝛽 ∘ 𝑓 ∘ 𝜑−1 𝛼 ∶ 𝜑 𝛼 (𝑈𝛼 ) → 𝜓 𝛽 (𝑉 𝛽 ) is holomorphic for all 𝛼, 𝛽 with 𝑓(𝑈𝛼 ) ⊂ 𝑉 𝛽 , where {(𝑈𝛼 , 𝜑𝛼 )} is a complex atlas for 𝑀 and {(𝑉 𝛽 , 𝜓𝛽 )} is a complex atlas for 𝑁. A biholomorphism is an invertible holomorphic map with holomorphic inverse. The set of biholomorphisms 𝑓 ∶ 𝑀 → 𝑀 is denoted Bihol(𝑀). Remark 5.6. (1) Let 𝑀 be a connected complex manifold, and let 𝑓 ∶ 𝑀 → ℂ be a holomorphic function. If there is 𝑝 ∈ 𝑀 such that 𝑓(𝑝) = 0 and all derivatives of 𝑓 at 𝑝 vanish, then 𝑓 = 0. We see this as follows: first note that by (5.3), the condition that 𝑓(𝑝) = 0 and all derivatives of 𝑓 at 𝑝 vanish is equivalent to the existence of some 𝑈 𝑝 ⊂ 𝑀 such that 𝑓|𝑈 ≡ 0. Then the set 𝐴 = 𝜕𝑟 𝑓 {𝑝 ∈ 𝑀|𝑓(𝑝) = 0, 𝜕𝑧 ⋯𝜕𝑧 (𝑝) = 0, for all 𝑟 ≥ 1, 𝑖𝑘 } = {𝑝 ∈ 𝑀| exists 𝑈 𝑝 𝑖1

𝑖𝑟

such that 𝑓|𝑈 ≡ 0} is closed (by the first description) and open (by the second description). By connectedness, 𝐴 = 𝑀, and hence 𝑓 = 0 everywhere. (2) If 𝑓, 𝑔 ∈ 𝒪𝑀 (𝑀) satisfy for some 𝑝 ∈ 𝑀, that 𝑓(𝑝) = 𝑔(𝑝) and all derivatives of 𝑓 and 𝑔 at 𝑝 coincide, then 𝑓 = 𝑔.

5.1. Complex manifolds

273

(3) A holomorphic function on a compact connected complex manifold 𝑀 is constant. Let 𝑓 ∶ 𝑀 → ℂ be a holomorphic function. By compactness, there is a point 𝑝0 ∈ 𝑀 where |𝑓| achieves its maximum. Take a local complex chart (𝑈, 𝜑) around 𝑝0 with 𝜑(𝑝0 ) = (0, . . . , 0) and 𝜑(𝑈) = 𝐵𝜖 (0) ⊂ ℂ𝑛 . Then ̃ 1 , . . . , 𝑧𝑛 ) = 𝑓 ∘ 𝜑−1 is holomorphic and achieves a local maximum at 𝑓(𝑧 (0, . . . , 0). Considering 𝑓 ̃ as a function of 𝑧1 and using the maximum princĩ 1 , 0, . . . , 0) is constant. Using the argument now with ple (Exercise 5.1), 𝑓(𝑧 ̃ 1 , 𝑧2 , . . . , 0) is constant. Recursively, 𝑧2 ↦ 𝑓(𝑧1 , 𝑧2 , 0, . . . , 0), we get that 𝑓(𝑧 ̃ we get that 𝑓 is constant and hence 𝑓 is constant on 𝑈. By (2) above, 𝑓 is constant. (4) These results also hold for (real) analytic manifolds and analytic functions (Remark 1.20). Note that a complex manifold is in particular a (real) analytic manifold. Remark 5.7. • Let 𝑈 ⊂ ℂ be an open set, seen as a complex manifold. Then a meromorphic function 𝑓 on 𝑈 is the same as a holomorphic function 𝑓 ̃ ∶ 𝑈 → ℂ𝑃 1 . To see that, recall that, locally on open sets 𝑉 ⊂ 𝑈, we have 𝑓 = 𝑔/ℎ with 𝑔, ℎ ∶ 𝑉 → ℂ holomorphic functions. We can do this in such a way as to have either 𝑔(𝑧) ≠ 0 or ℎ(𝑧) ≠ 0 for all 𝑧 ∈ 𝑉. Then we define 𝑓|̃ 𝑉 ∶ 𝑉 → ℂ𝑃 1 ̃ = [ℎ(𝑧), 𝑔(𝑧)]. It is easy to check that these local definitions agree on by 𝑓(𝑧) the overlaps so they define a map 𝑓 ̃ ∶ 𝑈 → ℂ𝑃 1 . Let us see that this map is holomorphic. If 𝑆 is the set of poles of 𝑓, then 𝜑0 ∘ 𝑓|̃ 𝑉 −𝑆 = 𝑔/ℎ = 𝑓, where 𝜑0 is the chart in Example 5.5(3), so 𝑓 ̃ is holomorphic on 𝑈 − 𝑆. If 𝑧0 ∈ 𝑆, then ℎ(𝑧0 ) = 0 and 𝑔(𝑧) ≠ 0 for all 𝑧 ∈ 𝑉. So 𝜑1 ∘ 𝑓|̃ 𝑉 = ℎ(𝑧)/𝑔(𝑧) is holomorphic on 𝑉. Under the identification (4.11), ℂ𝑃1 ≅ ℂ = ℂ∪{∞}, by [𝑧0 , 𝑧1 ] ↦ 𝑧1 /𝑧0 , a meromorphic function 𝑓 on 𝑈 is understood as a holomorphic function 𝑓 ∶ 𝑈 → ℂ. A point 𝑧0 ∈ 𝑈 is a pole of 𝑓 when 𝑓(𝑧0 ) = ∞. • The above generalizes straightaway to complex curves. Given a complex manifold 𝑀 of complex dimension 1, a meromorphic function on 𝑀 is defined as a holomorphic function 𝑓 ∶ 𝑀 → ℂ𝑃 1 . This means that in a complex chart 𝜑 ∶ 𝑈 ⊂ 𝑀 → ℂ, 𝑓 ∘ 𝜑−1 is a meromorphic function on 𝜑(𝑈) ⊂ ℂ, i.e., locally the quotient of two holomorphic functions. • For a complex manifold 𝑀 of arbitrary dimension, we define a meromorphic function as a function 𝑓 ∶ 𝑀 − 𝑆 → ℂ, where 𝑆 ⊂ 𝑀 is a closed subset of empty interior, which is locally the quotient of two holomorphic functions. 𝑛 This is equivalent to 𝑓 ∘ 𝜑−1 𝛼 being a meromorphic function on 𝜑 𝛼 (𝑈𝛼 ) ⊂ ℂ for every complex chart (𝑈𝛼 , 𝜑𝛼 ). Therefore on (small) open subsets, 𝑓 = 𝑔/ℎ, ̃ with 𝑔, ℎ holomorphic, hence we have a map 𝑓 ̃ ∶ 𝑉 − 𝑆 ′ → ℂ𝑃 1 , 𝑓(𝑧) = ′ [ℎ(𝑧), 𝑔(𝑧)], but this is not defined on the set 𝑆 where 𝑔, ℎ vanish simultaneously. Globalizing, we have a holomorphic map 𝑓 ̃ ∶ 𝑀 − 𝑆 ′ → ℂ𝑃 1 , but in general it cannot be extended to the whole of 𝑀. • The importance of meromorphic functions stems from the fact that there are many of them on a large class of compact complex manifolds (i.e., smooth

274

5. Complex geometry

projective varieties, see Remark 5.47(9)), whereas there are only constant complex valued holomorphic functions (Remark 5.6(3)). 𝑛

We have a category 𝐂𝐌𝐚𝐧 consisting of complex manifolds of complex dimen𝑛 sion 𝑛, and whose morphisms are holomorphic maps, and the subcategory 𝐂𝐌𝐚𝐧𝑐 of compact complex manifolds of complex dimension 𝑛. Since a complex manifold is in 𝑛 particular a differentiable oriented manifold, we have a forgetful functor 𝐂𝐌𝐚𝐧𝑐 → 2𝑛 2𝑛 𝐃𝐌𝐚𝐧𝑜𝑟,𝑐 , to the category 𝐃𝐌𝐚𝐧𝑜𝑟,𝑐 of compact, oriented smooth 2𝑛-manifolds. and a map on the classifying lists (5.5)

𝑐𝑜

𝑐𝑜

𝕃𝐂𝐌𝐚𝐧𝑛 → 𝕃𝐃𝐌𝐚𝐧2𝑛 . 𝑐

𝑜𝑟,𝑐

For the case of complex curves (𝑛 = 1), we will see later that every smooth (compact, connected) oriented surface admits a compatible complex structure (Remark 5.49(5)), so (5.5) is surjective. However, in general this complex structure is not unique (it is only unique in the case of genus 𝑔 = 0), so (5.5) is not injective. In the general case 𝑛 > 1, the map is neither surjective nor injective. For example, the spheres 𝑆 2𝑛 for 𝑛 = 2 and 𝑛 ≥ 4 cannot admit a complex structure. It is a long standing problem whether 𝑆 6 admits or not a complex structure. For another example, we introduce new notation: given an oriented manifold 𝑀, we denote by 𝑀 the same manifold 𝑀 with the opposite orientation. Then ℂ𝑃 2 does not admit a complex structure, i.e., any complex structure on ℂ𝑃 2 must induce the usual orientation of ℂ𝑃 2 . The complex tangent space. Let 𝑀 be a complex manifold of complex dimension 𝑛, and let 𝑧 ∈ 𝑀. Analogously to the smooth case, we can define the holomorphic tangent space at 𝑧, 𝒯𝑧 𝑀. Let 𝒪𝑀,𝑧 be the set of germs of holomorphic functions at 𝑝, i.e., holomorphic functions 𝑓 ∶ 𝑈 → ℂ defined on some open neighbourhood 𝑈 of 𝑧. Then, in analogy with section 3.1.1, we define 𝒯𝑧 𝑀 as the space of derivations on 𝒪𝑀,𝑧 , i.e., ℂ-linear maps 𝑋𝑧 ∶ 𝒪𝑀,𝑧 → ℂ satisfying the Leibniz rule 𝑋𝑧 (𝑓 𝑔) = 𝑋𝑧 (𝑓)𝑔(𝑧) + 𝑓(𝑧)𝑋𝑧 (𝑔), for all 𝑓, 𝑔 ∈ 𝒪𝑀,𝑧 . As in the smooth case, 𝒯𝑧 𝑀 is a ℂ-vector space of dimension 𝑛. To give a basis, we fix a complex chart 𝜑 = (𝑧1 , . . . , 𝑧𝑛 ) ∶ 𝑈 → ℂ𝑛 around 𝑧. Then 𝒯𝑧 𝑀 is generated by the 𝜕 𝜕 holomorphic partial derivations 𝜕𝑧 , . . . , 𝜕𝑧 given by 1

𝑛

𝜕(𝑓 ∘ 𝜑−1 ) 𝜕 𝑓= (𝑧), 𝜕𝑧𝑖 𝜕𝑧𝑖 for 𝑓 ∈ 𝒪𝑀,𝑧 , where the derivatives are the usual holomorphic derivatives as described 𝜕 in section 5.1.1. Leaving the point 𝑧 free, we have the derivations 𝜕𝑧 on the sheaf 𝒪𝑀 . 𝑖 This definition is consistent with the changes of charts. If 𝜓 = (𝑤 1 , . . . , 𝑤 𝑛 ) is another complex chart around 𝑧, then we have the change rule 𝑛

𝜕𝑤𝑗 𝜕 𝜕 =∑ . 𝜕𝑧𝑖 𝑗=1 𝜕𝑧𝑖 𝜕𝑤𝑗 The complex manifold 𝑀 also has a differentiable structure giving us the usual tangent space at 𝑧, 𝑇𝑧 𝑀, which is a 2𝑛-dimensional real vector space. In order to relate 𝑇𝑧 𝑀 with 𝒯𝑧 𝑀, let us consider the complexification 𝑇𝑧 𝑀ℂ = 𝑇𝑧 𝑀 ⊗ℝ ℂ (see section

5.1. Complex manifolds

275

2.4.2 for the tensor product). This is a 2𝑛-dimensional complex vector space whose 𝜕 𝜕 𝜕 𝜕 basis is given by 𝜕𝑥 , 𝜕𝑦 , . . . , 𝜕𝑥 , 𝜕𝑦 , where 𝑧𝑖 = 𝑥𝑖 +i𝑦 𝑖 , 1 ≤ 𝑖 ≤ 𝑛, are the coordinate 1 1 𝑛 𝑛 functions of 𝜑. As done in (5.2), we introduce the derivations 𝜕 1 𝜕 𝜕 = ( −i ), 𝜕𝑧𝑖 2 𝜕𝑥𝑖 𝜕𝑦 𝑖

𝜕 1 𝜕 𝜕 = ( +i ). 2 𝜕𝑥𝑖 𝜕𝑦 𝑖 𝜕𝑧𝑖

These are derivations on smooth functions 𝐶 ∞ (−, ℂ), but they are characterized by the 𝜕 𝜕 fact that restricted to 𝒪𝑀 ⊂ 𝐶 ∞ (−, ℂ), 𝜕𝑧 vanishes identically and 𝜕𝑧 coincides with 𝑖

𝑖

the derivation of 𝒯𝑧 𝑀 denoted in the same way. Clearly, another basis for 𝑇𝑧 𝑀ℂ , since we can recover (5.6)

𝜕 𝜕 𝜕 𝜕 , . . . , 𝜕𝑧 , 𝜕𝑧 , . . . , 𝜕𝑧 𝜕𝑧1 𝑛 1 𝑛

is

𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 = + = 2 Re( = i( − ), ) = −2 Im( ). 𝜕𝑥𝑖 𝜕𝑧𝑖 𝜕𝑧𝑖 𝜕𝑧𝑖 𝜕𝑦 𝑖 𝜕𝑧𝑖 𝜕𝑧𝑖 𝜕𝑧𝑖

Therefore, we have a decomposition 𝑇𝑧 𝑀ℂ = ⟨

𝜕 𝜕 𝜕 𝜕 , ... , , ... , ⟩⊕⟨ ⟩. 𝜕𝑧1 𝜕𝑧𝑛 𝜕𝑧1 𝜕𝑧𝑛

This decomposition is preserved by the holomorphic changes of charts, so it is intrinsic. 𝜕 𝜕 𝜕 𝜕 We will denote 𝑇𝑧1,0 𝑀 = 𝒯𝑧 𝑀 = ⟨ 𝜕𝑧 , . . . , 𝜕𝑧 ⟩ and 𝑇𝑧0,1 𝑀 = 𝒯𝑧 𝑀 = ⟨ 𝜕𝑧 , . . . , 𝜕𝑧 ⟩, so 1 𝑛 1 𝑛 we can write 𝑇𝑧 𝑀ℂ = 𝑇𝑧1,0 𝑀 ⊕ 𝑇𝑧0,1 𝑀.

(5.7)

Passing to differential forms, let us consider the complexified cotangent space 𝜕 𝜕 𝜕 𝜕 𝑇𝑧∗ 𝑀ℂ = 𝑇𝑧∗ 𝑀⊗ℝ ℂ. Then, taking the dual elements of the basis 𝜕𝑧 , . . . , 𝜕𝑧 , 𝜕𝑧 , . . . , 𝜕𝑧 1 𝑛 1 𝑛 for 𝑇𝑧 𝑀ℂ , we have the 1-forms 𝑑𝑧1 , . . . , 𝑑𝑧𝑛 , 𝑑𝑧1 , . . . , 𝑑𝑧𝑛 ∈ 𝑇𝑧∗ 𝑀ℂ characterized by 𝑑𝑧𝑖 (

𝜕 ) = 𝛿 𝑖𝑗 , 𝜕𝑧𝑗

𝑑𝑧𝑖 (

𝜕 ) = 0, 𝜕𝑧𝑗

𝑑𝑧𝑖 (

𝜕 ) = 0, 𝜕𝑧𝑗

𝑑𝑧𝑖 (

𝜕 ) = 𝛿 𝑖𝑗 , 𝜕𝑧𝑗

where 𝛿 𝑖𝑗 is the Kronecker delta. It is a straightforward computation to show that if 𝑑𝑥1 , 𝑑𝑦1 , . . . , 𝑑𝑥𝑛 , 𝑑𝑦𝑛 is the standard dual basis for 𝑇𝑧∗ 𝑀ℂ associated to the real coordinates, then 𝑑𝑧𝑗 = 𝑑𝑥𝑗 + i𝑑𝑦𝑗 , 𝑑𝑧𝑗 = 𝑑𝑥𝑗 − i𝑑𝑦𝑗 . Therefore, 𝑑𝑧1 , . . . , 𝑑𝑧𝑛 , 𝑑𝑧1 , . . . , 𝑑𝑧𝑛 is a basis for 𝑇𝑧∗ 𝑀ℂ . Denoting Ω1,0 𝑧 (𝑀) = ⟨𝑑𝑧1 , . . . , 𝑑𝑧𝑛 ⟩ and Ω0,1 𝑧 (𝑀) = ⟨𝑑𝑧1 , . . . , 𝑑𝑧𝑛 ⟩, we have a decomposition 0,1 Ω1𝑧 (𝑀)ℂ = 𝑇𝑧∗ 𝑀ℂ = Ω1,0 𝑧 (𝑀) ⊕ Ω𝑧 (𝑀).

This is the dual to the decomposition (5.7). Working globally, we consider 1-forms on 𝑀 with complex coefficients Ω1 (𝑀)ℂ = Ω1 (𝑀) ⊗ℝ ℂ , for which we have a decomposition Ω1 (𝑀)ℂ = Ω1,0 (𝑀) ⊕ Ω0,1 (𝑀), where 𝜔 ∈ Ω1,0 (𝑀) if 𝜔𝑧 ∈ Ω1,0 𝑧 (𝑀) for all 𝑧 ∈ 𝑀 (and analogously for Ω0,1 (𝑀)). For higher degree forms, we take exterior powers to define the spaces of (𝑝, 𝑞)forms as (5.8)

Ω𝑝,𝑞 (𝑀) = (

𝑝

⋀

Ω1,0 (𝑀)) ∧ (

𝑞

⋀

Ω0,1 (𝑀)) .

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5. Complex geometry

Then we have a bigraduation on the spaces of complex valued forms. For 𝑘 ≥ 0, Ω𝑘 (𝑀)ℂ =

⨁

Ω𝑝,𝑞 (𝑀).

𝑝+𝑞=𝑘

Equivalently, a form 𝜔 ∈ Ω𝑘 (𝑀)ℂ lives in Ω𝑝,𝑞 (𝑀) if and only if, locally, it has the form ∑

𝜔=

𝑓𝐼𝐽 𝑑𝑧𝑖1 ∧ ⋯ ∧ 𝑑𝑧𝑖𝑝 ∧ 𝑑𝑧𝑗1 ∧ ⋯ ∧ 𝑑𝑧𝑗𝑞 ,

|𝐼|=𝑝,|𝐽|=𝑞

where 𝐼 = {𝑖1 , . . . , 𝑖𝑝 }, 𝐽 = {𝑗1 , . . . , 𝑗𝑞 } are multi-indices as in section 2.5. We use the notation 𝑑𝑧𝐼 = 𝑑𝑧𝑖1 ∧ ⋯ ∧ 𝑑𝑧𝑖𝑝 and 𝑑𝑧𝐽 = 𝑑𝑧𝑗1 ∧ ⋯ ∧ 𝑑𝑧𝑗𝑞 , so we can write 𝜔 = ∑ 𝑓𝐼𝐽 𝑑𝑧𝐼 ∧ 𝑑𝑧𝐽 . Regarding the exterior differential 𝑑 ∶ Ω𝑘 (𝑀)ℂ → Ω𝑘+1 (𝑀)ℂ , let us define the operators 𝜕 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝+1,𝑞 (𝑀),

𝜕 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝,𝑞+1 (𝑀),

called the holomorphic and antiholomorphic exterior differentials, respectively. Take 𝜔 = ∑ 𝑓𝐼𝐽 𝑑𝑧𝐼 ∧ 𝑑𝑧𝐽 ∈ Ω𝑝,𝑞 (𝑀) and define 𝜕𝜔 = ∑ ∑ 𝐼,𝐽 𝛼

Since 𝑑𝑓 = ∑𝛼

𝜕𝑓𝐼𝐽 𝑑𝑧𝛼 ∧ 𝑑𝑧𝐼 ∧ 𝑑𝑧𝐽 , 𝜕𝑧𝛼

𝜕𝑓 𝑑𝑧𝛼 𝜕𝑧𝛼

+ ∑𝛼

𝜕𝑓 𝜕𝑧𝛼

𝜕𝜔 = ∑ ∑ 𝐼,𝐽 𝛼

𝜕𝑓𝐼𝐽 𝑑𝑧𝛼 ∧ 𝑑𝑧𝐼 ∧ 𝑑𝑧𝐽 . 𝜕𝑧𝛼

𝑑𝑧𝛼 for a smooth function 𝑓 (see section 5.1.1(11)),

we have that 𝑑𝜔 = 𝜕𝜔 + 𝜕𝜔, for any 𝜔 ∈ Ω𝑘 (𝑀)ℂ . Therefore there is a decomposition 𝑑 = 𝜕 +𝜕. It is remarkable that this property characterizes complex structures (Exercise 5.8). 5.1.3. Almost complex structures. As we have seen, a complex structure on a manifold 𝑀 makes the tangent spaces 𝑇𝑧 𝑀 complex vector spaces. This is a natural structure that can be described by a tensor, but it turns out to be weaker than having a complex atlas. Let us introduce this notion, but first we need some bit of linear algebra. Let 𝑉 be an even dimensional real vector space. If we give 𝑉 the structure of a complex vector space such that the multiplication 𝑚 ∶ ℂ × 𝑉 → 𝑉, (𝜆, 𝑣) ↦ 𝑚(𝜆, 𝑣) restricts to the usual multiplication for 𝜆 ∈ ℝ, then we have a (real) endomorphism 𝐽 ∶ 𝑉 → 𝑉 given as 𝐽(𝑣) = 𝑚(i, 𝑣), satisfying 𝐽 2 = − Id. Conversely, if 𝑉 is a real vector space and 𝐽 ∶ 𝑉 → 𝑉 is an endomorphism with 𝐽 2 = − Id, then we can give 𝑉 the structure of a complex vector space by defining the multiplication 𝑚 ∶ ℂ × 𝑉 → 𝑉, 𝑚(𝜆, 𝑣) = 𝑚(𝛼 + i𝛽, 𝑣) = 𝛼 𝑣 + 𝛽𝐽(𝑣), for 𝜆 = 𝛼 + i𝛽 ∈ ℂ. Therefore, to give a complex structure (𝑉, 𝑚) is the same as to give (𝑉, 𝐽) as above. Given a complex vector space (𝑉, 𝐽), we can complexify the real vector space 𝑉 as 𝑉ℂ = 𝑉 ⊗ ℂ, and extend 𝐽 to 𝑉ℂ . As 𝐽 2 + Id = 0, the map 𝐽 diagonalizes on 𝑉ℂ with eigenvalues ±i. We denote by 𝑉 1,0 the i-eigenspace of 𝐽 on 𝑉ℂ , and by 𝑉 0,1 the (−i)eigenspace of 𝐽 on 𝑉ℂ , so 𝑉ℂ = 𝑉 1,0 ⊕ 𝑉 0,1 . Clearly 𝑉 0,1 = 𝑉 1,0 , where 𝑉 0,1 denotes the

5.1. Complex manifolds

277

conjugate vector space to 𝑉 0,1 (Exercise 5.3). If 𝑛 = dimℂ 𝑉, then dimℝ 𝑉 = dimℂ 𝑉ℂ = 2𝑛, and dimℂ 𝑉 1,0 = dimℂ 𝑉 0,1 = 𝑛. The projections onto 𝑉 1,0 and 𝑉 0,1 are given by 1 𝜋1,0 ∶ 𝑉ℂ → 𝑉 1,0 , 𝜋1,0 (𝑣) = (𝑣 − i𝐽𝑣), 2 1 0,1 𝜋0,1 ∶ 𝑉ℂ → 𝑉 , 𝜋0,1 (𝑣) = (𝑣 + i𝐽𝑣). 2 Clearly, there is an isomorphism 𝜋1,0 |𝑉 ∶ 𝑉 → 𝑉 1,0 , which is an isomorphism of complex vector spaces (𝑉, 𝐽) ≅ (𝑉 1,0 , i).

V 0;1

iV

V 1;0 π1;0

V

Now we globalize these notions. Definition 5.8. Let 𝑀 be an 2𝑛-dimensional differentiable manifold. A (1, 1)-tensor 𝐽, seen as 𝐽 ∶ 𝔛(𝑀) → 𝔛(𝑀), with 𝐽 2 = − Id, is called an almost complex structure on 𝑀. Given two almost complex manifolds (𝑀, 𝐽) and (𝑀 ′ , 𝐽 ′ ), we say that a smooth map 𝑓 ∶ 𝑀 → 𝑀 ′ is pseudoholomorphic if 𝑓∗ ∘ 𝐽 = 𝐽 ′ ∘ 𝑓∗ . This means that an almost complex manifold is a smooth manifold such that the tangent spaces are (given the structure of) complex vector spaces. An almost complex manifold is always orientable. Indeed, the almost complex structure induces a natural orientation on the complex vector spaces (𝑇𝑝 𝑀, 𝐽) that varies continuously, giving a global orientation, as has happened for complex manifolds. Example 5.9. • Consider ℂ𝑛 with coordinates (𝑧1 , . . . , 𝑧𝑛 ). The standard almost complex structure is the one given as 𝐽𝑠𝑡𝑑 (𝑣) = i𝑣, for 𝑧 ∈ ℂ𝑛 and 𝑣 ∈ 𝑇𝑧 ℂ𝑛 = ℂ𝑛 . If we write 𝑧1 = 𝑥1 + i𝑦1 , . . . , 𝑧𝑛 = 𝑥𝑛 + i𝑦𝑛 , then the canonical basis of 𝑇𝑧 ℂ𝑛 𝜕 𝜕 𝜕 𝜕 is ( 𝜕𝑥 , 𝜕𝑦 , . . . , 𝜕𝑥 , 𝜕𝑦 ) and 1

1

𝑛

𝑛

𝜕 𝜕 𝜕 𝜕 𝐽𝑠𝑡𝑑 ( , 𝐽𝑠𝑡𝑑 ( . )= )=− 𝜕𝑥𝑘 𝜕𝑦 𝑘 𝜕𝑦 𝑘 𝜕𝑥𝑘 • Any complex manifold 𝑀 is, in a natural way, an almost complex manifold. Let 𝜑 ∶ 𝑈 ⊂ 𝑀 → ℂ𝑛 be complex coordinates. We define 𝐽 on 𝑈 as 𝐽 = (𝜑∗ )−1 ∘ 𝐽𝑠𝑡𝑑 ∘ 𝜑∗ , that is, in such a way that 𝜑 ∶ (𝑈, 𝐽) → (ℂ𝑛 , 𝐽𝑠𝑡𝑑 ) is a pseudobiholomorphism. As the changes of charts are biholomorphisms of (ℂ𝑛 , 𝐽𝑠𝑡𝑑 ), this 𝐽 is globally defined, independently of charts.

278

5. Complex geometry

Note that, on a complex chart (𝑧1 = 𝑥1 +i𝑦1 , . . . , 𝑧𝑛 = 𝑥𝑛 +i𝑦𝑛 ), 𝐽 is given 𝜕 𝜕 𝜕 𝜕 as 𝐽 ( 𝜕𝑥 ) = 𝜕𝑦 , 𝐽 ( 𝜕𝑦 ) = − 𝜕𝑥 . Observe that the two definitions of 𝑇𝑧1,0 𝑀, 𝑘

𝑘

𝑘

as the space spanned by 𝐽(

𝑘

𝜕 𝜕 , . . . , 𝜕𝑧 𝜕𝑧1 𝑛

and as the 𝑖-eigenspace of 𝐽, agree, since

𝜕 1 𝜕 𝜕 1 𝜕 𝜕 𝜕 −i +i . )=𝐽( ( )) = ( )=i 𝜕𝑧𝑗 2 𝜕𝑥𝑗 𝜕𝑦𝑗 2 𝜕𝑦𝑗 𝜕𝑥𝑗 𝜕𝑧𝑗

This gives natural isomorphisms (𝑇𝑧 𝑀, 𝐽) ≅ (𝑇𝑧1,0 𝑀, i) ≅ 𝒯𝑧 𝑀. The same computation shows that 𝑇 0,1 𝑀 is the (−i)-eigenspace of 𝐽. On the contrary, the existence of an almost complex structure does not guarantee the existence of a complex structure. Given an almost complex manifold (𝑀, 𝐽), to give 𝑀 a complex structure such that 𝐽 agrees with the one induced by the complex charts means that we can find charts 𝜑 ∶ 𝑈 ⊂ 𝑀 → ℂ𝑛 which satisfy 𝜑∗ ∘ 𝐽 = 𝐽𝑠𝑡𝑑 ∘ 𝜑∗ . Certainly, if there is an atlas 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )} such that (𝜑𝛼 )∗ ∘ 𝐽 = 𝐽𝑠𝑡𝑑 ∘ (𝜑𝛼 )∗ for all 𝛼, then the changes of charts 𝑓 = 𝜑𝛽 ∘ (𝜑𝛼 )−1 satisfy that 𝑓∗ ∘ 𝐽𝑠𝑡𝑑 = 𝐽𝑠𝑡𝑑 ∘ 𝑓∗ . This means that 𝑓 is a biholomorphism (Remark 5.1). Hence 𝒜 is a complex atlas compatible with 𝐽. When an almost complex structure 𝐽 admits a compatible complex atlas, we say that 𝐽 is integrable. The integrability condition is given by a deep theorem. Theorem 5.10 (Newlander-Nirenberg). Let (𝑀, 𝐽) be an almost complex manifold. Let us define the Nijenhuis tensor, 𝑁𝐽 ∶ 𝔛(𝑀) × 𝔛(𝑀) → 𝔛(𝑀) by 𝑁𝐽 (𝑋, 𝑌 ) = [𝑋, 𝑌 ] − [𝐽𝑋, 𝐽𝑌 ] + 𝐽[𝐽𝑋, 𝑌 ] + 𝐽[𝑋, 𝐽𝑌 ], for 𝑋, 𝑌 ∈ 𝔛(𝑀). Then 𝐽 is integrable if and only if 𝑁𝐽 vanishes everywhere. Proof. The proof of this result is quite difficult, including hard analysis techniques, and will not be given here (see [N-N]). However, we will provide a proof for the case of compact complex curves (i.e., surfaces, which are the main focus of this book) in Corollary 6.53 by an indirect route. Here we show how the Nijenhuis tensor appears and why its vanishing is a necessary condition, proving the easy direction of the theorem. Suppose that 𝑀 is a complex manifold, and let 𝔛(𝑇 1,0 𝑀) be its space of holomorphic vector fields, i.e., a vector field 𝑋 ∈ 𝔛(𝑇 1,0 𝑀) if 𝑋𝑧 ∈ 𝑇𝑧1,0 𝑀 for all 𝑧 ∈ 𝑀. 𝜕 Observe that 𝑇 1,0 𝑀 is closed under the Lie bracket since, if we have 𝑋 = ∑ 𝑋 𝑖 𝜕𝑧 and 𝑖

𝜕

𝑌 = ∑ 𝑌 𝑖 𝜕𝑧 in 𝔛(𝑇 1,0 𝑀), then 𝑖

[𝑋, 𝑌 ] = ∑ 𝑋 𝑗 𝑖,𝑗

𝜕𝑌 𝑖 𝜕 𝜕𝑋 𝑖 𝜕 − 𝑌𝑖 ∈ 𝔛(𝑇 1,0 𝑀). 𝜕𝑧𝑗 𝜕𝑧𝑖 𝜕𝑧𝑗 𝜕𝑧𝑖

Using the isomorphism 𝜋1,0 ∶ (𝑇𝑀, 𝐽) → (𝑇 1,0 𝑀, 𝑖), any 𝑋, 𝑌 ∈ 𝔛(𝑇 1,0 𝑀) are written 1 ̃ 𝑌 = 𝜋1,0 (𝑌 ̃ ) = 1 (𝑌 ̃ − i𝐽 𝑌 ̃ ), for vector fields 𝑋,̃ 𝑌 ̃ ∈ as 𝑋 = 𝜋1,0 (𝑋)̃ = 2 (𝑋̃ − i𝐽 𝑋), 2 𝔛(𝑀). Then 4[𝑋, 𝑌 ] = [𝑋̃ − i𝐽 𝑋,̃ 𝑌 ̃ − i𝐽 𝑌 ̃ ] = [𝑋,̃ 𝑌 ̃ ] − [𝐽 𝑋,̃ 𝐽 𝑌 ̃ ] − i ([𝐽 𝑋,̃ 𝑌 ̃ ] + [𝑋,̃ 𝐽 𝑌 ̃ ]) .

5.1. Complex manifolds

279

Since [𝑋, 𝑌 ] ∈ 𝔛(𝑇 1,0 𝑀), we must have 4[𝑋, 𝑌 ] = 𝑍̃ − i𝐽 𝑍̃ for a unique 𝑍̃ ∈ 𝔛(𝑀). So it must be 𝑍̃ = [𝑋,̃ 𝑌 ̃ ] − [𝐽 𝑋,̃ 𝐽 𝑌 ̃ ] and 𝐽 𝑍̃ = [𝐽 𝑋,̃ 𝑌 ̃ ] + [𝑋,̃ 𝐽 𝑌 ̃ ]. Therefore 0 = 𝑍̃ + 𝐽 (𝐽 𝑍)̃ = [𝑋,̃ 𝑌 ̃ ] − [𝐽 𝑋,̃ 𝐽 𝑌 ̃ ] + 𝐽[𝐽 𝑋,̃ 𝑌 ̃ ] + 𝐽[𝑋,̃ 𝐽 𝑌 ̃ ] = 𝑁𝐽 (𝑋,̃ 𝑌 ̃ ). It is easy to see that 𝑁𝐽 is a tensor (Exercise 5.8). Thus, for complex manifolds, the Nijenhuis tensor vanishes. □ The Newlander-Nirenberg theorem is a sort of Frobenius integrability theorem. Let us briefly explain this. Given a differentiable manifold 𝑀, a distribution 𝒟 consists of giving vector subspaces 𝒟𝑝 ⊂ 𝑇𝑝 𝑀, for every 𝑝 ∈ 𝑀, of some fixed dimension 𝑠. These subspaces vary smoothly in the sense that, for all 𝑝 ∈ 𝑀, there exists an open neighbourhood 𝑈 of 𝑝 and a set of vector fields 𝑋1 , . . . , 𝑋𝑠 ∈ 𝔛(𝑈) such that 𝒟𝑞 = ⟨𝑋1,𝑞 , . . . , 𝑋𝑠,𝑞 ⟩, for all 𝑞 ∈ 𝑈. A distribution is called integrable if through each point there exists a submanifold 𝑁 ⊂ 𝑀 such that 𝒟𝑝 = 𝑇𝑝 𝑁 for all 𝑝 ∈ 𝑁. The space of vector fields 𝑋 of 𝑀 such that 𝑋𝑝 ∈ 𝒟𝑝 for all 𝑝 ∈ 𝑀 is denoted by 𝔛(𝒟). The conditions for the integrability of 𝒟 are given by the Frobenius theorem, whose proof is not very difficult, and can be found in [War]. Theorem 5.11 (Frobenius). A distribution 𝒟 on a differentiable manifold 𝑀 is integrable if and only if [𝑋, 𝑌 ] ∈ 𝔛(𝒟) for all 𝑋, 𝑌 ∈ 𝔛(𝒟). In the case of the Newlander-Nirenberg theorem, we consider the distribution 𝒟 = 𝑇 1,0 𝑀 ⊂ 𝑇𝑀ℂ . By the proof of Theorem 5.10, the vanishing of the Nijenhuis tensor is equivalent to 𝑋, 𝑌 ∈ 𝔛(𝑇 1,0 𝑀) ⟹ [𝑋, 𝑌 ] ∈ 𝔛(𝑇 1,0 𝑀). Thus the Newlander-Nirenberg theorem is a complex analogue of the Frobenius theorem, saying that 𝑇 1,0 𝑀 ⊂ 𝑇𝑀ℂ is integrable if and only if there are complex coordinates for 𝑀 compatible with 𝐽. Remark 5.12. (1) The Nijenhuis tensor is a sort of “curvature tensor” for the almost complex structure 𝐽. By the Newlander-Nirenberg theorem, its vanishing allows us to find charts 𝜑 ∶ (𝑈, 𝐽) → (ℂ𝑛 , 𝐽𝑠𝑡𝑑 ) for which 𝐽 becomes standard. In a similar vein, for a Riemannian manifold (𝑀, g), the vanishing of the curvature allows us to find charts 𝜑 ∶ (𝑈, g) → (ℝ𝑛 , g𝑠𝑡𝑑 ) for which g becomes standard (flat), via Theorem 3.93. (2) Many geometric structures are controlled by tensors, such as the Riemannian or almost complex structures, or the symplectic structures that we will encounter later (Definition 5.21). Soft geometric structures are those for which it is always possible to find charts on which they become standard. For them, the only geometric information is global, that is topological. Riemannian metrics and almost complex structure are non-soft, and have local invariants. (3) An almost complex manifold may have very few (or none) non-constant pseudoholomorphic functions 𝑓 ∶ (𝑈, 𝐽) → (ℂ, 𝐽𝑠𝑡𝑑 ), with 𝑈 ⊂ 𝑀 an open subset. This is because the Cauchy-Riemann equations for almost complex manifolds, that is, the equations 𝑓∗ ∘ 𝐽 = 𝐽𝑠𝑡𝑑 ∘ 𝑓∗ , are overdetermined (this means

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that there are many more equations than unknowns). For this reason, an almost complex manifold cannot be defined by a sheaf of functions as we did for a complex manifold. (4) Actually, an almost complex structure is integrable exactly when (locally) there are many pseudoholomorphic functions. If we have an open subset 𝑈 ⊂ 𝑀 and pseudoholomorphic functions 𝑧𝑖 ∶ (𝑈, 𝐽) → (ℂ, 𝐽𝑠𝑡𝑑 ), 1 ≤ 𝑖 ≤ 𝑛, such that 𝑑𝑧1 , . . . , 𝑑𝑧𝑛 are ℂ-linearly independent, then 𝜑 = (𝑧1 , . . . , 𝑧𝑛 ) ∶ (𝑈, 𝐽) → (ℂ𝑛 , 𝐽𝑠𝑡𝑑 ) is a local biholomorphism and hence it gives a complex chart (maybe in a smaller neighbourhood of a point). If such open sets 𝑈 cover 𝑀, then 𝑀 is complex and 𝐽 is integrable. (5) If (𝑀, 𝐽) is a complex manifold and 𝑁 ⊂ 𝑀 is a smooth submanifold such that 𝐽(𝑇𝑝 𝑁) = 𝑇𝑝 𝑁 ⊂ 𝑇𝑝 𝑀, for all 𝑝 ∈ 𝑁, then (𝑁, 𝐽|𝑁 ) is a complex manifold. Certainly, 𝐽|𝑁 is an almost complex structure, and the Nijenhuis tensor satisfies 𝑁𝐽|𝑁 = 0 since 𝑁𝐽 = 0. Theorem 5.10 gives the result. (6) Let (𝑀, 𝐽) be an almost complex manifold. We can define Ω1,0 (𝑀) as the annihilator of 𝑇 0,1 𝑀 and Ω0,1 (𝑀) as the annihilator of 𝑇 1,0 𝑀. Taking exterior products as in (5.8), we obtain a bigrading of the forms, and Ω𝑘 (𝑀)ℂ =

⨁

Ω𝑝,𝑞 (𝑀).

𝑝+𝑞=𝑘

Now we can restrict the differential 𝑑 ∶ Ω𝑘 (𝑀)ℂ → Ω𝑘+1 (𝑀)ℂ to Ω𝑝,𝑞 (𝑀) ⊂ Ω𝑘 (𝑀)ℂ , with 𝑝 + 𝑞 = 𝑘, to obtain a map 𝑑 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑘+1 (𝑀)ℂ =

⨁

Ω𝑟,𝑠 (𝑀).

𝑟+𝑠=𝑘+1

Let 𝜋𝑟,𝑠 ∶ Ω𝑘+1 (𝑀)ℂ → Ω𝑟,𝑠 (𝑀) be the projections, and define 𝜕 = 𝜋𝑝+1,𝑞 ∘ 𝑑 and 𝜕 = 𝜋𝑝,𝑞+1 ∘ 𝑑, as in the integrable case. Then 𝑁𝐽 = 0 if and only if 𝑑 = 𝜕 + 𝜕 if and only if 𝜕2 = 0 (Exercise 5.8). Example 5.13. The spheres 𝑆 2 and 𝑆 6 are the unique spheres that admit an almost complex structure. We know that 𝑆 2 has an integrable complex structure because 𝑆 2 ≅ ℂ𝑃 1 . However, let us construct the almost complex structure of 𝑆 2 starting with the quaternions ℍ. This is a normed division algebra, that is an algebra in which every nonzero element has an inverse and there is a norm with |𝑞 ⋅ 𝑞′ | = |𝑞| |𝑞′ |, for 𝑞, 𝑞′ ∈ ℍ. In concrete terms, ℍ = ℝ⟨1, i, j, k⟩ ≅ ℝ4 , with i2 = j2 = k2 = −1, ij = −ji = k, jk = −kj = i, ki = −ik = j. Note that ℍ is non-commutative but it is associative. The imaginary quaternions are Im ℍ = ℝ⟨i, j, k⟩, and we take 𝑆 2 = 𝑆(Im ℍ), the 2-sphere in Im ℍ = ℝ3 . The almost complex structure is defined by 𝐽𝑝 (𝑥) = Im(𝑝𝑥), for 𝑝 ∈ 𝑆 2 and 𝑥 ∈ 𝑇𝑝 𝑆 2 ⊂ Im ℍ. We mention that this 𝐽 is actually the usual (integrable) almost complex structure of 𝑆 2 . The almost complex structure on 𝑆 6 is defined in a similar vein, but starting with the octonions 𝕆. This is an 8-dimensional normed division algebra which is neither commutative nor associative. As vector space, 𝕆 = ℝ⟨1, 𝑒 1 , . . . , 𝑒 7 ⟩ ≅ ℝ8 , and Im 𝕆 = ℝ⟨𝑒 1 , . . . , 𝑒 7 ⟩ ≅ ℝ7 . We will not write the product explicitly but mention that 𝑒2𝑖 = −1. We consider the sphere 𝑆 6 = 𝑆(Im 𝕆) and define the almost complex structure by 𝐽𝑝 ∶ 𝑇𝑝 𝑆 → 𝑇𝑝 𝑆, 𝐽𝑝 (𝑥) = Im(𝑝𝑥), for 𝑝 ∈ 𝑆 6 and 𝑥 ∈ 𝑇𝑝 𝑆 6 ⊂ Im 𝕆. The properties of

5.1. Complex manifolds

281

the multiplication of the octonions [C-S] show that 𝐽𝑝2 = − Id. However, this almost complex structure is non-integrable, so whether there exists another almost complex structure that would be integrable remains an open problem. Recall that ℝ, ℂ, ℍ, 𝕆 are the unique normed division algebras extending ℝ, so this method can only be used to provide almost complex structures for 𝑆 2 and 𝑆 6 . Example 5.14. The question of whether a smooth manifold 𝑀 can be endowed with a complex structure (surjectivity of (5.5)) can be split into two questions: • Does 𝑀 have an almost complex structure? This is an easy question that can be studied via homotopy theory arguments. • Can we solve the Newlander-Nirenberg problem 𝑁𝐽 = 0? This is a hard analytical question. In the case of the spheres 𝑆 2𝑛 , 𝑛 ≠ 1, 3, the first step fails. In real dimension 4, we have that ℂ𝑃2 admits a complex structure, whereas ℂ𝑃2 #ℂ𝑃2 does not admit an almost complex structure, and ℂ𝑃 2 #ℂ𝑃 2 #ℂ𝑃2 does admit almost complex structures, but not an integrable complex structure. 5.1.4. Dolbeault cohomology. The existence of a complex structure has consequences in the topology of a manifold, which we want to explore. Within this section, let us take a complex manifold 𝑀. As we mentioned in section 5.1.2, the complex structure of 𝑀 induces a decomposition of the exterior differential 𝑑 = 𝜕+𝜕. Using the bigrading Ω𝑝,𝑞 (𝑀), we obtain the following double cochain complex. ⋮O

⋮O

𝜕

⋯

𝜕

𝜕

/ Ω𝑝+1,𝑞−1 (𝑀) O

𝜕

𝜕

⋯

𝜕

⋯

𝜕

/ Ω𝑝+1,𝑞 (𝑀) O

𝜕

/ Ω𝑝+1,𝑞+1 (𝑀) O

𝜕

/ Ω𝑝,𝑞−1 (𝑀) O

𝜕

𝜕 𝜕

⋮O

/ Ω𝑝−1,𝑞−1 (𝑀) O 𝜕

/ Ω𝑝,𝑞 (𝑀) O

𝜕

/ Ω𝑝,𝑞+1 (𝑀) O

𝜕

/⋯

𝜕

/⋯

𝜕

/ Ω𝑝−1,𝑞 (𝑀) O

𝜕

/ Ω𝑝−1,𝑞+1 (𝑀) O

𝜕

⋮

/⋯

𝜕

𝜕 𝜕

𝜕

𝜕

⋮

⋮

Moreover, since 𝑑 2 = 0 we have 0 = 𝑑 2 = (𝜕 + 𝜕)2 = 𝜕2 + (𝜕𝜕 + 𝜕𝜕) + 𝜕2 , where 𝜕2 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝+2,𝑞 (𝑀), 𝜕𝜕 + 𝜕𝜕 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝+1,𝑞+1 (𝑀) and 𝜕2 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝,𝑞+2 (𝑀). As the components have all different bigrade, we must have (5.9)

𝜕2 = 0,

𝜕𝜕 + 𝜕𝜕 = 0,

2

𝜕 = 0.

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5. Complex geometry

In particular, the rows of the double complex form a cochain complex 𝜕

𝜕

𝜕

𝜕

⋯ ⟶ Ω𝑝,𝑞−1 (𝑀) ⟶ Ω𝑝,𝑞 (𝑀) ⟶ Ω𝑝,𝑞+1 (𝑀) ⟶ ⋯ whose cohomology is an invariant of the complex structure of 𝑀. This complex is called Dolbeault complex, and the antiholomorphic differential 𝜕 is usually referred to as the Dolbeault operator. Definition 5.15. Let 𝑀 be a complex manifold with Dolbeault operator 𝜕 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝,𝑞+1 (𝑀). We define the (𝑝, 𝑞)-Dolbeault cohomology as the complex vector space 𝐻 𝑝,𝑞 (𝑀) = 𝐻 𝑞 (Ω𝑝,• , 𝜕) =

ker(𝜕 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝,𝑞+1 (𝑀)) im(𝜕 ∶ Ω𝑝,𝑞−1 (𝑀) → Ω𝑝,𝑞 (𝑀))

.

The dimensions ℎ𝑝,𝑞 = dim 𝐻 𝑝,𝑞 (𝑀) are called the Hodge numbers of 𝑀. If 𝑀 is nonconnected with connected components 𝑀 = ⨆𝜆∈Λ 𝑀𝜆 , then 𝐻 𝑝,𝑞 (𝑀) = ∏𝜆∈Λ 𝐻 𝑝,𝑞 (𝑀𝜆 ), as in Example 2.121(2). Remark 5.16. For a non-integrable almost complex manifold, 𝜕2̄ ≠ 0 (Exercise 5.8). Therefore Dolbeault cohomology cannot be defined in this setting. Sheaf cohomology. Cohomology of sheaves is probably the right framework to understand cohomology theories. We are not going to give all the details, for which we refer the reader to [Iv2], but at least we shall explain briefly how to interpret Dolbeault cohomology as a sheaf cohomology. Recall that sheaves were introduced in Definition 1.13. Let us fix a topological space 𝑋, and let 𝒜 be an Abelian category. A sheaf on 𝒜 is a contravariant functor 𝐎𝐩𝐞𝐧(𝑋) → 𝒜 satisfying the properties of Definition 1.13. Given two sheaves ℱ, 𝒢, a morphism of sheaves is a natural transformation 𝑓 ∶ ℱ → 𝒢 (Definition 2.137). A sheaf ℐ on 𝑋 is called injective if for any sheaf monomorphism 𝑖 ∶ ℱ → 𝒢 and any morphism 𝑓 ∶ ℱ → ℐ there exists a morphism 𝑓 ̃ ∶ 𝒢 → ℐ such that the following diagram commutes. ℐO _? ? 𝑓̃ ? 𝑓 ? /ℱ /𝒢 0 𝑖

It can be proven (see [Iv2]) that for any sheaf ℱ on 𝑋 there exists an exact sequence (5.10)

𝑑0

𝑑1

𝑑2

𝑑3

0 ⟶ ℱ ⟶ ℐ0 ⟶ ℐ1 ⟶ ℐ2 ⟶ ⋯

with ℐ 𝑖 injective sheaves. Here, a sequence of sheaves ℱ1 → ℱ2 → ℱ3 is exact if, when evaluated in small open sets 𝑈 ⊂ 𝑋 (in the usual topology), the sequence ℱ1 (𝑈) → ℱ2 (𝑈) → ℱ3 (𝑈) is exact. The sequence (5.10) is called an injective resolution of ℱ and it is denoted by ℱ → ℐ • . For this resolution, we can take its global sections obtaining a cochain complex 𝑑1

𝑑2

𝑑3

ℐ 0 (𝑋) ⟶ ℐ 1 (𝑋) ⟶ ℐ 2 (𝑋) ⟶ ⋯

5.1. Complex manifolds

283

The sheaf cohomology of ℱ, denoted 𝐻 𝑘 (𝑋, ℱ), is defined as the chain cohomology 𝐻 𝑘 (𝑋, ℱ) = 𝐻 𝑘 (ℐ • (𝑋), 𝑑 • ). Conditions (1) and (2) of Definition 1.13 imply that ℱ(𝑋) = 𝐻 0 (𝑋, ℱ). The deep theorems come here: first, it can be proven that the cohomology does not depend on the chosen resolution. Given a sheaf, it is difficult to find out an injective resolution for that sheaf (injective sheaves tend to be “too big”), so it can be hard to compute the cohomology via the definition. However, alternative sheaves can be used. An acyclic sheaf is a sheaf 𝒢 such that 𝐻 𝑘 (𝑋, 𝒢) = 0 for all 𝑘 > 0. Then an acyclic resolution is a resolution ℱ → 𝒢 • such that all 𝒢 𝑙 , 𝑙 ≥ 0, are acyclic sheaves. We have that cohomology can be computed by using 𝒢 • instead of ℐ • , so 𝐻 𝑘 (𝑋, ℱ) = 𝐻 𝑘 (𝒢 • (𝑋), 𝑑 • ). The task now consists of finding out natural classes of acyclic sheaves that can be used depending on the situation, and fortunately there are several (flasque, flabby, fine,. . . ). Here we shall use the case of fine sheaves, which are sheaves that have partitions of unity. In particular, if 𝑀 is a differentiable manifold and 𝒢 is a sheaf on 𝑀 such that 𝒢(𝑈) is a 𝐶 ∞ (𝑈)-module for every open set 𝑈 ⊂ 𝑀, then 𝒢 is a fine sheaf, so in particular 𝒢 is acyclic, and sheaf cohomology can be computed using these sheaves. Example 5.17. (1) Let 𝑀 be a differentiable manifold, and let ℝ be the (locally) constant ℝ sheaf on 𝑀 (that is, ℝ(𝑈) = ℝ, for each connected open set 𝑈 ⊂ 𝑋). Let Ω𝑘 be the sheaf of 𝑘-forms, that is Ω𝑘 (𝑈) is the space of 𝑘-forms on the open set 𝑈 ⊂ 𝑀. Observe that, in this abstract context, the Poincaré lemma says that if 𝑈 is a small ball, then 𝑑

𝑑

𝑑

0 ⟶ ℝ ⟶ Ω0 (𝑈) ⟶ Ω1 (𝑈) ⟶ Ω2 (𝑈) ⟶ ⋯ 𝑑

is an exact sequence. This means that ℝ → Ω• is an exact sequence of sheaves, so it is a resolution of ℝ. Moreover, the sheaves Ω𝑘 are fine sheaves, so cohomology can be computed using this resolution. Hence, 𝐻 𝑘 (𝑀, ℝ) = 𝐻 𝑘 (Ω• (𝑀), 𝑑) ker(𝑑 ∶ Ω𝑘 (𝑀) → Ω𝑘+1 (𝑀)) 𝑘 = 𝐻𝑑𝑅 (𝑀, ℝ). im(𝑑 ∶ Ω𝑘−1 (𝑀) → Ω𝑘 (𝑀)) That is, sheaf cohomology for ℝ is the de Rham cohomology. =

(2) Let 𝑀 be a complex manifold, and let Ω 𝑝 be the sheaf of holomorphic 𝑝forms on 𝑀. The elements of Ω 𝑝 (𝑈) are forms of type ∑ 𝑓𝐼 (𝑧)𝑑𝑧𝐼 , where 𝑓𝐼 ∈ 𝒪𝑀 (𝑈). Therefore Ω 𝑝 (𝑈) = {𝜔 ∈ Ω𝑝,0 (𝑈) | 𝜕 𝜔 = 0} = ker(𝜕 ∶ Ω𝑝,0 (𝑈) → Ω𝑝,1 (𝑈)) . Then a 𝜕-Poincaré lemma (proposition 1.3.8 in [Huy]) says that for a ball 𝑈 in 𝑀, 𝜕

𝜕

𝜕

0 ⟶ Ω 𝑝 (𝑈) ⟶ Ω𝑝,0 (𝑈) ⟶ Ω𝑝,1 (𝑈) ⟶ Ω𝑝,2 (𝑈) ⟶ ⋯

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5. Complex geometry

𝜕

is an exact sequence. Therefore Ω 𝑝 → Ω𝑝,• is a fine resolution, and the cohomology of Ω 𝑝 can be computed by means of Ω𝑝,• , obtaining 𝐻 𝑞 (𝑀, Ω 𝑝 ) = 𝐻 𝑞 (Ω𝑝,• , 𝜕) =

ker(𝜕 ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑝,𝑞+1 (𝑀)) im(𝜕 ∶

Ω𝑝,𝑞−1 (𝑀)

→

Ω𝑝,𝑞 (𝑀))

= 𝐻 𝑝,𝑞 (𝑀).

Therefore Dolbeault cohomology can be viewed as the sheaf cohomology of the sheaf Ω 𝑝 , which captures subtle information of the complex structure of 𝑀. (3) Note that 𝐻 𝑝,0 (𝑀) = 𝐻 0 (𝑀, Ω 𝑝 ) = Ω 𝑝 (𝑀), which is the space of global holomorphic 𝑝-forms. In particular, 𝐻 0,0 (𝑀) = 𝒪𝑀 (𝑀). If 𝑀 is compact connected, then 𝐻 0,0 (𝑀) = ℂ by Remark 5.6(3), so ℎ0,0 = 1.

5.2. Kähler manifolds In this section, we focus on a certain type of complex manifold of high importance, called Kähler manifolds. For defining them, we need to introduce metric concepts on complex manifolds which allow an interplay of Riemannian geometry and complex geometry. An important subclass of Kähler manifolds is given by the so-called smooth projective varieties, which is the set of zeroes of polynomials in the complex projective space. This gives way to a very rich interplay between complex algebraic geometry and the differential geometry of Kähler manifolds. 5.2.1. Hermitian metrics. Hermitian metrics are the complex analogue to scalar products on real vector spaces. Definition 5.18. Let 𝑉 be a complex vector space. A Hermitian metric on 𝑉 is a map h ∶ 𝑉 × 𝑉 → ℂ such that: (1) It is ℂ-linear in the first entry, i.e., h(𝛼𝑢 + 𝛽𝑣, 𝑤) = 𝛼h(𝑢, 𝑤) + 𝛽h(𝑣, 𝑤), for all 𝑢, 𝑣, 𝑤 ∈ 𝑉 and 𝛼, 𝛽 ∈ ℂ. (2) It is ℂ-antilinear in the second entry, i.e., h(𝑢, 𝛼𝑣 + 𝛽𝑤) = 𝛼h(𝑢, 𝑣) + 𝛽h(𝑢, 𝑤). (3) It is skew symmetric in the sense that h(𝑢, 𝑣) = h(𝑣, 𝑢), for all 𝑢, 𝑣 ∈ 𝑉. (4) It is positive definite, i.e., h(𝑢, 𝑢) > 0 for all 𝑢 ≠ 0. Remark 5.19. Condition (2) is redundant. It follows from (1) and (3). The expression of a Hermitian metric in coordinates is as follows. Take a (complex) basis (𝑢1 , . . . , 𝑢𝑛 ) of 𝑉. For vectors 𝑥 = ∑ 𝑥𝑖 𝑢𝑖 and 𝑦 = ∑ 𝑦𝑗 𝑢𝑗 , we have that h(𝑥, 𝑦) = ∑ 𝑥𝑖 𝑦𝑗 h(𝑢𝑖 , 𝑢𝑗 ) = 𝑥𝑡 𝐻𝑦, where 𝐻 = (ℎ𝑖𝑗 ) = (h(𝑢𝑖 , 𝑢𝑗 ))𝑛𝑖,𝑗=1 is a Hermitian matrix, i.e., 𝐻 ∗ = 𝐻 with 𝐻 ∗ = 𝐻 𝑡 the conjugate transpose. Hermitian matrices behave similarly to symmetric matrices over ℝ. In particular, they can be diagonalized, with some real eigenvalues 𝜆1 , . . . , 𝜆𝑛 , and the number of positive, negative, and zero eigenvalues characterize the canonical form. A Hermitian matrix corresponds to a Hermitian metric if all its eigenvalues are positive. In that case there are bases (𝑒 1 , . . . , 𝑒 𝑛 ) of 𝑉, called orthonormal bases, such that h(𝑒 𝑖 , 𝑒𝑗 ) = 𝛿 𝑖𝑗 .

5.2. Kähler manifolds

285

Example 5.20. The standard Hermitian metric on ℂ𝑛 is given by h(𝑧, 𝑤) = ∑𝑖 𝑧𝑖 𝑤𝑖 . The sphere 𝑆 2𝑛−1 ⊂ ℂ𝑛 is given by the complex vectors of length 1 with respect to this metric. We can split a Hermitian metric on a complex vector space (𝑉, 𝐽) into its real and imaginary parts, by putting h = g − i𝜔, where g(𝑢, 𝑣) = Re h(𝑢, 𝑣) and 𝜔(𝑢, 𝑣) = − Im h(𝑢, 𝑣) are two (real) bilinear maps on 𝑉. The choice of the sign minus for 𝜔 will be justified later. Let us see some properties of g and 𝜔. (1) The bilinear map g is a scalar product on 𝑉, i.e., it is symmetric and positive definite. Since h(𝑣, 𝑢) = h(𝑢, 𝑣), taking real parts, we have g(𝑣, 𝑢) = g(𝑢, 𝑣), for all 𝑢, 𝑣 ∈ 𝑉. Moreover, g(𝑢, 𝑢) = h(𝑢, 𝑢) > 0 for 𝑢 ≠ 0. (2) The bilinear map 𝜔 is antisymmetric. Taking imaginary parts on h(𝑣, 𝑢) = h(𝑢, 𝑣), we have 𝜔(𝑣, 𝑢) = −𝜔(𝑢, 𝑣), for all 𝑢, 𝑣 ∈ 𝑉. (3) The map 𝐽 defines an isometry on the Euclidean space (𝑉, g). It also preserves 𝜔. From h(𝐽𝑢, 𝐽𝑣) = h(i𝑢, i𝑣) = h(𝑢, 𝑣), it follows that g(𝐽𝑢, 𝐽𝑣) = g(𝑢, 𝑣) and 𝜔(𝐽𝑢, 𝐽𝑣) = 𝜔(𝑢, 𝑣). (4) We have the identity g(𝑢, 𝑣) = 𝜔(𝑢, 𝐽𝑣). From h(𝐽𝑢, 𝑣) = ih(𝑢, 𝑣) we get g(𝐽𝑢, 𝑣) − i𝜔(𝐽𝑢, 𝑣) = i(g(𝑢, 𝑣) − i𝜔(𝑢, 𝑣)). Hence g(𝐽𝑢, 𝑣) = 𝜔(𝑢, 𝑣) and g(𝑢, 𝑣) = −𝜔(𝐽𝑢, 𝑣). Also we have g(𝑢, 𝑣) = g(𝐽𝑢, 𝐽𝑣) = −𝜔(𝐽 2 𝑢, 𝐽𝑣) = 𝜔(𝑢, 𝐽𝑣). (5) Interpret g as a map ĝ ∶ 𝑉 → 𝑉 ∗ , 𝑢 ↦ g(𝑢, −), and 𝜔 as a map 𝜔̂ ∶ 𝑉 → 𝑉 ∗ , 𝑢 ↦ 𝜔(𝑢, −). As 𝜔(𝑢, 𝑣) = g(𝐽𝑢, 𝑣), we have 𝜔̂ = ĝ ∘ 𝐽. This means that knowing two of the three elements g, 𝐽, 𝜔, the third is uniquely defined. The Hermitian metric is recovered from them. (6) If (𝑉, 𝐽) is a complex vector space, and g is a scalar product on 𝑉 so that g(𝐽𝑢, 𝐽𝑣) = g(𝑢, 𝑣), then the map 𝜔 defined as 𝜔(𝑢, 𝑣) = g(𝐽𝑢, 𝑣) is antisymmetric, since 𝜔(𝑣, 𝑢) = g(𝐽𝑣, 𝑢) = g(𝑢, 𝐽𝑣) = g(𝐽𝑢, 𝐽 2 𝑣) = −g(𝐽𝑢, 𝑣) = −𝜔(𝑢, 𝑣). Therefore h = g − i𝜔 is a Hermitian metric on 𝑉. Certainly, h(𝑣, 𝑢) = h(𝑢, 𝑣), h(i𝑢, 𝑣) = g(𝐽𝑢, 𝑣) − i𝜔(𝐽𝑢, 𝑣) = 𝜔(𝑢, 𝑣) + ig(𝑢, 𝑣) = ih(𝑢, 𝑣), and h(𝑢, 𝑢) = g(𝑢, 𝑢) > 0 for 𝑢 ≠ 0. (7) Analogously, if 𝜔 is a non-degenerate antisymmetric bilinear form invariant by 𝐽 and g(𝑢, 𝑣) = 𝜔(𝑢, 𝐽𝑣) is positive definite, then h(𝑢, 𝑣) = g(𝑢, 𝑣) − i𝜔(𝑢, 𝑣) is a Hermitian metric. (8) The bilinear map 𝜔 is non-degenerate, that is, if 𝜔(𝑢, 𝑣) = 0 for all 𝑣 ∈ 𝑉, then 𝑢 = 0. This is equivalent to the injectivity of 𝜔.̂ But 𝜔̂ = ĝ ∘ 𝐽 ∶ 𝑉 → 𝑉 → 𝑉 ∗ is an isomorphism because g is non-degenerate and 𝐽 is an isomorphism. The appearance of an antisymmetric bilinear map 𝜔 as before will play an important role in the study of Kähler manifolds. This motivates the following. Definition 5.21. Let 𝑉 be a real vector space, and let 𝜔 ∶ 𝑉 × 𝑉 → ℝ be a bilinear map. We say that 𝜔 is symplectic if it is antisymmetric and non-degenerate. Let us write 𝜔 and g in coordinates, for a Hermitian vector space (𝑉, 𝐽, h). Take a ℂbasis 𝐵ℂ = (𝑒 1 , . . . , 𝑒 𝑛 ) for (𝑉, 𝐽) and the corresponding ℝ-basis 𝐵ℝ = (𝑒 1 , 𝐽𝑒 1 , . . . , 𝑒 𝑛 , 𝐽𝑒 𝑛 )

286

5. Complex geometry

as an ℝ-vector space. Write ℎ𝑖𝑗 = h(𝑒 𝑖 , 𝑒𝑗 ) = 𝛼𝑖𝑗 − i𝛽 𝑖𝑗 . Then g(𝑒 𝑖 , 𝑒𝑗 ) = g(𝐽𝑒 𝑖 , 𝐽𝑒𝑗 ) = 𝛼𝑖𝑗 , g(𝐽𝑒 𝑖 , 𝑒𝑗 ) = −g(𝑒 𝑖 , 𝐽𝑒𝑗 ) = 𝛽 𝑖𝑗 , and 𝜔(𝑒 𝑖 , 𝑒𝑗 ) = 𝜔(𝐽𝑒 𝑖 , 𝐽𝑒𝑗 ) = 𝛽 𝑖𝑗 , 𝜔(𝑒 𝑖 , 𝐽𝑒𝑗 ) = −𝜔(𝐽𝑒 𝑖 , 𝑒𝑗 ) = 𝛼𝑖𝑗 . Note that 𝛼𝑖𝑗 = 𝛼𝑗𝑖 and 𝛽 𝑖𝑗 = −𝛽𝑗𝑖 (in particular 𝛽 𝑖𝑖 = 0). The matrices of g and 𝜔 are thus ⎛ ⎜ g=⎜ ⎜ ⎝

(5.11)

𝛼11 𝛽11 𝛼21 𝛽22 ⋮

𝛽 ⎛ 11 −𝛼11 ⎜ 𝜔 = ⎜ 𝛽21 ⎜ −𝛼21 ⎝ ⋮

−𝛽11 𝛼11 −𝛽21 𝛼21 ⋮ 𝛼11 𝛽11 𝛼21 𝛽21 ⋮

𝛼12 𝛽12 𝛼22 𝛽22 𝛽12 −𝛼12 𝛽22 −𝛼22

−𝛽12 𝛼12 −𝛽22 𝛼22

... ⎞ ... ⎟ ⎟, ⎟ ⋱ ⎠

𝛼12 𝛽12 𝛼22 𝛽22

... ⎞ ... ⎟ ⎟. ⎟ ⋱ ⎠

Hermitian manifolds. Let us now study the concept of a Hermitian metric on a complex manifold. In analogy with the Riemannian case, this is just a smooth assignment of Hermitian metrics on each tangent space. Definition 5.22. Let 𝑀 be a complex manifold of dimension 𝑛. A Hermitian metric on 𝑀 consists of a map which assigns, to each 𝑝 ∈ 𝑀, a Hermitian metric h𝑝 ∶ 𝒯𝑝 𝑀 × 𝒯𝑝 𝑀 → ℂ. These ℎ𝑝 vary smoothly on 𝑝 in the sense that, for every 𝑝 ∈ 𝑀, there exists 𝜕

𝜕

holomorphic coordinates (𝑧1 , . . . , 𝑧𝑛 ) ∶ 𝑈 → ℂ𝑛 around 𝑝 so that ℎ𝑖𝑗 (𝑧) = h𝑧 ( 𝜕𝑧 , 𝜕𝑧 ) 𝑖 𝑗 are smooth functions on 𝑈. Let 𝐽 be the (integrable) almost complex structure of 𝑀. Recall from section 5.1.3 that, for any 𝑝 ∈ 𝑀, there is an isomorphism of complex vector spaces 𝐴 = 𝜋1,0 ∶ 1 (𝑇𝑝 𝑀, 𝐽) → (𝑇𝑝1,0 𝑀, i) = (𝒯𝑝 𝑀, i) given, for 𝑣 ∈ 𝑇𝑝 𝑀, by 𝐴(𝑣) = 2 (𝑣 − i𝐽(𝑣)). With this 𝐴, we can translate the Hermitian metric h on 𝒯𝑝 𝑀 to another Hermitian metric on (𝑇𝑝 𝑀, 𝐽), denoted by h again, by means of (5.12)

h(𝑢, 𝑣) = h(𝐴(𝑢), 𝐴(𝑣)),

for 𝑢, 𝑣 ∈ 𝑇𝑝 𝑀. Remark 5.23. • Definition 5.22 can be done in the same way for almost complex manifolds. We call almost Hermitian to an almost complex manifolds with a Hermitian metric. • We can extend the Riemannian metric g = Re(h) on 𝑇𝑝 𝑀 to a metric g̃ on the complexification 𝑇𝑝 𝑀ℂ = 𝑇𝑝 𝑀 ⊕ i𝑇𝑝 𝑀 by declaring the two summands orthogonal and setting ||𝑋1 + i𝑋2 ||2g̃ = ||𝑋1 ||2g + ||𝑋2 ||2g , for 𝑋1 , 𝑋2 ∈ 𝑇𝑝 𝑀. However, this metric g̃ does not coincide with the initial metric on 𝑇𝑝1,0 𝑀 since 2 2 2 𝜕 ‖ 1‖ 𝜕 ‖ ‖1 𝜕 ‖ 𝜕 ‖ ‖‖ 𝜕𝑧 ‖‖ = ‖‖ 2 ( 𝜕𝑥 − i 𝜕𝑦 )‖‖ = 2 ‖‖ 𝜕𝑥 ‖‖ . 𝑖 g̃ 𝑖 𝑖 𝑖 g g̃

5.2. Kähler manifolds

287

1

Actually, g|̃ 𝑇𝑝1,0 𝑀 = 2 g. The conjugation, given by

∑ 𝑤𝑖

𝜕 𝜕 = ∑ 𝑤𝑖 , 𝜕𝑧 𝜕𝑧𝑖 𝑖

is a ℂ-antilinear isomorphism between 𝑇𝑝0,1 𝑀 and 𝑇𝑝1,0 𝑀. Using it, we can transform the Hermitian metric h on 𝒯𝑝 𝑀 into a tensor on 𝑇𝑝1,0 𝑀 ⊗ 𝑇𝑝0,1 𝑀, also denoted h, by h(𝑢, 𝑣) = h(𝑢, 𝑣) for 𝑢 ∈ 𝑇𝑝1,0 𝑀 and 𝑣 ∈ 𝑇𝑝0,1 𝑀. Since both h(𝑢, −) and the conjugation are antilinear, this form is ℂ-bilinear and, thus, a tensor. For this reason, it is customary to identify the Hermitian metric h with this tensor and to write it on coordinates as h = ∑ ℎ𝑖𝑗 𝑑𝑧𝑖 ⊗ 𝑑𝑧𝑗 .

(5.13)

𝑖,𝑗

By the results above, taking g = Re(h) and 𝜔 = − Im(h), we decompose h = g−i𝜔, with g a Riemannian metric and 𝜔 ∈ Ω2 (𝑀) non-degenerate. Let us write ℎ𝑖𝑗 = h (

𝜕 𝜕 , ) = 𝛼𝑖𝑗 − i𝛽 𝑖𝑗 . 𝜕𝑧𝑖 𝜕𝑧𝑗

The map 𝐴 sends the real basis 𝜕 𝜕 𝜕 𝜕 , , ... , , ) 𝜕𝑥1 𝜕𝑦1 𝜕𝑥𝑛 𝜕𝑦𝑛

of 𝑇𝑝 𝑀

𝜕 𝜕 𝜕 𝜕 ,i , ... , ,i ) 𝜕𝑧1 𝜕𝑧1 𝜕𝑧𝑛 𝜕𝑧𝑛

of 𝒯𝑝 𝑀.

( to the real basis (

The matrix of h defined by (5.12) on (𝑇𝑝 𝑀, 𝐽) is (ℎ𝑖𝑗 ), with respect to the complex ba𝜕

𝜕

sis ( 𝜕𝑥 , . . . , 𝜕𝑥 ). This implies that the matrices of g and 𝜔, in the standard basis of 1 𝑛 𝑇𝑝 𝑀 ≅ ℝ2𝑛 , are given by (5.11). Therefore, if ⋅ denotes the symmetric product of tensors (Example 3.3(3)), 𝑛

g = ∑ 𝛼𝑖𝑗 𝑑𝑥𝑖 ⋅ 𝑑𝑥𝑗 − 2𝛽 𝑖𝑗 𝑑𝑥𝑖 ⋅ 𝑑𝑦𝑗 + 𝛼𝑖𝑗 𝑑𝑦 𝑖 ⋅ 𝑑𝑦𝑗 , 𝑖,𝑗=1

𝜔 = ∑(𝛽 𝑖𝑗 𝑑𝑥𝑖 ∧ 𝑑𝑥𝑗 + 𝛽 𝑖𝑗 𝑑𝑦 𝑖 ∧ 𝑑𝑦𝑗 ) + ∑ 𝛼𝑖𝑗 𝑑𝑥𝑖 ∧ 𝑑𝑦𝑗 . 𝑖 0, for every 𝑘 = 0, . . . , 𝑛.

290

5. Complex geometry

(3) The manifold 𝑀 = 𝑆 1 × 𝑆 3 admits a complex structure, but it cannot be given any Kähler structure since 𝑏2 = 0. To endow 𝑀 with a complex structure, let us consider the (non-compact and simply connected) complex manifold 𝑁̃ = ℂ2 −{0} and take the group Γ = ⟨𝜑⟩ ≅ ℤ, generated by 𝜑(𝑧, 𝑤) = (2𝑧, 2𝑤), which acts freely and properly on 𝑁.̃ By Exercise 5.6, the quotient 𝑁 = (ℂ2 −{0})/⟨𝜑⟩ has a complex structure. On the other hand, 𝑁 is diffeomorphic to 𝑀 = 𝑆 1 ×𝑆 3 . To see it, let us take 𝜓 ∶ ℂ2 −{0} → ℝ×𝑆 3 , 𝜓(𝑧, 𝑤) = (log 𝑟, 𝑞), where 𝑟 = ||(𝑧, 𝑤)|| and 𝑞 = (𝑧, 𝑤)/𝑟. The action of 𝜑 in these coordinates is given by 𝜑̂ = 𝜓 ∘ 𝜑 ∘ 𝜓−1 (𝑠, 𝑞) = (log(2𝑟), 𝑞) = (log 𝑟 + log 2, 𝑞) = (𝑠 + log 2, 𝑞), with 𝑠 = log 𝑟. This implies that 𝑁 = (ℂ2 − {0})/⟨𝜑⟩ ≅ (ℝ/(log 2)ℤ) × 𝑆 3 ≅ 𝑆1 × 𝑆3 . (4) As 𝜔 ∈ Ω1,1 (𝑀), 𝑑𝜔 = (𝜕 + 𝜕)𝜔 = 𝜕𝜔 + 𝜕𝜔 = 0 implies that 𝜕𝜔 = 0 and 𝜕𝜔 = 0, since each summand lies in Ω2,1 (𝑀) and Ω1,2 (𝑀), respectively. Hence [𝜔] ∈ 𝐻 1,1 (𝑀). (5) A manifold (𝑀, 𝐽, h) is Kähler if and only if, for each 𝑝 ∈ 𝑀, there exists a complex chart 𝜑 = (𝑧1 , . . . , 𝑧𝑛 ) with 𝜑(𝑝) = 0 so that the Hermitian metric 𝜕ℎ 𝜕ℎ satisfies h(𝑧) = Id +𝑂(|𝑧|2 ), i.e., 𝜕𝑧𝑗𝑘 (0) = 𝜕𝑧𝑗𝑘 (0) = 0, for all 𝑖, 𝑗, 𝑘 (Exercise 𝑖 𝑖 5.15). Note that for a Riemannian manifold (𝑀, g), there are always coordinates so that g = Id +𝑂(|𝑥|2 ) (this is the content of Exercise 3.11). The Kähler condition is thus a specific feature of complex geometry. (6) A complex submanifold of a Kähler manifold is Kähler. Certainly, if (𝑀, 𝐽, h = g − i𝜔) is Kähler and 𝑁 ⊂ 𝑀 is a complex submanifold, then h𝑁 = 𝑖∗ h is a Hermitian metric on 𝑁, where 𝑖 ∶ 𝑁 ↪ 𝑀 denotes the inclusion. Then h𝑁 = g𝑁 − i𝜔𝑁 where g𝑁 = 𝑖∗ g is the induced Riemannian metric and 𝜔𝑁 = 𝑖∗ 𝜔 is the fundamental form of 𝑁. So 𝑑𝜔𝑁 = 0. The Kähler form 𝜔 plays a key role in the topology and geometry of Kähler manifolds. In fact, there is an important branch of geometry, named symplectic geometry, devoted to studying manifolds (not necessarily complex) which admit a 2-form analogous to the fundamental form 𝜔 of a Kähler manifold. Definition 5.29. A symplectic manifold is a pair (𝑀, 𝜔) where 𝑀 is a smooth manifold and 𝜔 ∈ Ω2 (𝑀) is a non-degenerate form with 𝑑𝜔 = 0 Remark 5.30. • By Exercise 5.12, a symplectic manifold must be of even (real) dimension, say dim 𝑀 = 2𝑛, where 𝜔𝑛 ≠ 0. • A Kähler manifold (𝑀, 𝐽, h) is the same as a complex and symplectic manifold (𝑀, 𝐽, 𝜔) such that g(𝑢, 𝑣) = 𝜔(𝑢, 𝐽𝑣) is a Riemannian metric. In particular, Kähler manifolds are symplectic. For this reason, sometimes the Kähler manifold is referred to as (𝑀, 𝐽, 𝜔). • However, there are symplectic manifolds which are not Kähler (even manifolds which are complex and symplectic at the same time but do not admit Kähler structures). For instance, there are compact symplectic manifolds with 𝑏1 odd, in contrast with compact Kähler manifolds, which have

5.2. Kähler manifolds

291

𝑏1 even (Remark 5.32(3)). The first example of such manifold is the KodairaThurston manifold [O-T]. • A symplectic manifold is oriented with volume form 𝜈 = 𝜔𝑛 . A compact symplectic manifold satisfies that 𝑏2𝑘 > 0, for 0 ≤ 𝑘 ≤ 𝑛 (as in Remark 5.28(2)), so the spheres 𝑆 2𝑛 , 𝑛 ≥ 2, do not admit symplectic structures. • The Darboux theorem says that, on a symplectic manifold (𝑀, 𝜔), for any point 𝑝 ∈ 𝑀 there is a chart (𝑈, (𝑥1 , . . . , 𝑥2𝑛 )) such that 𝜔 = 𝑑𝑥1 ∧ 𝑑𝑥2 + ⋯ + 𝑑𝑥2𝑛−1 ∧𝑑𝑥2𝑛 . This is a standard model. Therefore symplectic structures have no local invariants (such as the curvature or the Nijenhuis tensor). These are soft geometric structures, and hence the main questions in symplectic geometry are global (topological). Compact Kähler manifolds exhibit a strong interplay between their complex and differentiable structure. One of the most important instances of this is the Hodge decomposition theorem, that relates de Rham cohomolgy with Dolbeault cohomology. Theorem 5.31 (Hodge). If (𝑀, 𝐽, h) is a compact Kähler manifold, then de Rham cohomology admits a direct sum decomposition for all 𝑘 ≥ 0, 𝑘 𝐻𝑑𝑅 (𝑀, ℂ) ≅

⨁

𝐻 𝑝,𝑞 (𝑀),

𝑝+𝑞=𝑘

where 𝐻 𝑝,𝑞 (𝑀) are Dolbeault cohomology spaces, and 𝐻 𝑝,𝑞 (𝑀) ≅ 𝐻 𝑞,𝑝 (𝑀). The proof of this theorem uses hard analytic techniques (cf. Remark 6.40(4)). Remark 5.32. (1) The decomposition of Theorem 5.31 depends on Hermitian structure, however the terms are only dependent on topology and complex structure, respectively. (2) If 𝑀 is a compact Kähler manifold, then Theorem 5.31 says that 𝑏𝑘 = ∑𝑝+𝑞=𝑘 ℎ𝑝,𝑞 and ℎ𝑝,𝑞 = ℎ𝑞,𝑝 , where ℎ𝑝,𝑞 are Hodge numbers (Definition 5.15). (3) In particular, if 𝑘 = 2𝑟 + 1 is an odd number, we have that 𝑏2𝑟+1 = ℎ2𝑟+1,0 + ℎ2𝑟,1 + ⋯ + ℎ𝑟+1,𝑟 + ℎ𝑟,𝑟+1 + ⋯ + ℎ1,2𝑟 + ℎ0,2𝑟+1 = 2(ℎ2𝑟+1,0 + ⋯ + ℎ𝑟+1,𝑟 ) is an even number. This is a strong topological restriction for a manifold to be Kähler. (4) By Remark 5.28(4), [𝜔] ∈ 𝐻 1,1 (𝑀), so [𝜔𝑘 ] ∈ 𝐻 𝑘,𝑘 (𝑀) by Exercise 5.10. Clearly [𝜔]𝑛 ≠ 0 in 𝐻 𝑛,𝑛 (𝑀), so [𝜔]𝑘 ≠ 0 in 𝐻 𝑘,𝑘 (𝑀) for 𝑘 = 1, . . . , 𝑛. Therefore ℎ𝑘,𝑘 > 0 for 0 ≤ 𝑘 ≤ 𝑛. This shows again that 𝑏2𝑘 > 0 (Remark 5.28(2)). (5) The 𝑛 × 𝑛 matrix (ℎ𝑝,𝑞 (𝑀))𝑝,𝑞 is called the Hodge diamond of 𝑀. It presents many properties. It is symmetric with respect to the principal diagonal since ℎ𝑝,𝑞 = ℎ𝑞,𝑝 . It is also symmetric with respect to the antidiagonal, since ℎ𝑛−𝑝,𝑛−𝑞 = ℎ𝑝,𝑞 . This follows from Serre duality (Remark 6.40(8)), which

292

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says that 𝐻 𝑛−𝑝,𝑛−𝑞 (𝑀) ≅ 𝐻 𝑝,𝑞 (𝑀)∗ . We draw the Hodge diamond like this, where the rows add up to the Betti numbers. ℎ𝑛,𝑛 ℎ

𝑛−1,𝑛

ℎ

𝑏2𝑛 𝑛,𝑛−1

⋱ ℎ0,𝑛

𝑏2𝑛−1 ⋱

ℎ1,𝑛−1 . . . ℎ𝑛−1,1

⋱

ℎ𝑛,0 ⋱

ℎ

0,1

ℎ

ℎ

1,0

0,0

⋮ 𝑏𝑛 ⋮ 𝑏1 𝑏0

(6) Consider a compact connected complex curve (𝐶, 𝐽). So 𝐶 is an oriented compact connected surface of some genus 𝑔 ≥ 0. Then we know that 𝑏1 = 2𝑔, 1 i.e., 𝐻𝑑𝑅 (𝐶, ℂ) ≅ ℂ2𝑔 , and hence ℎ1,0 = ℎ0,1 = 𝑔, that is 𝐻 0,1 (𝐶) ≅ ℂ𝑔 and 1,0 𝐻 (𝐶) ≅ ℂ𝑔 . By Example 5.17(3), 𝐻 1,0 (𝐶) = 𝐻 0 (𝐶, Ω 1 ) = Ω 1 (𝐶) is the vector space of global holomorphic differentials on 𝐶. The sheaf Ω 1 is usually referred to, in algebraic geometry, as the canonical sheaf of the curve. The equality (5.15)

𝑔 = dim Ω 1 (𝐶) serves as a definition of the genus from the point of view of algebraic geometry. Formula (5.15) gives a very strong relation between the topology and the complex geometry of a complex curve [Ful].

5.2.3. Complex projective space. The complex projective space is the natural ambient for doing algebraic geometry. It turns out to be a Kähler manifold, and this property is inherited with all its complex submanifolds (the smooth projective varieties) by Remark 5.28(6). This produces a bridge between complex differential geometry and complex algebraic geometry. Definition 5.33. The complex projective space is defined as the quotient ℂ𝑃𝑛 = (ℂ𝑛+1 − {0})/ ∼ , where 𝑧 ∼ 𝑤 if and only if 𝑧 = 𝜆𝑤 for some non-zero 𝜆 ∈ ℂ. The equivalence classes are denoted as [𝑧] = [𝑧0 , . . . , 𝑧𝑛 ]. Note that ℂ𝑃 𝑛 has a natural topology as a quotient space. It also comes with a standard holomorphic atlas as follows. Let 𝑈 𝑖 = {[𝑧0 , . . . , 𝑧𝑛 ] | 𝑧𝑖 ≠ 0} ⊂ ℂ𝑃 𝑛 be the usual 𝑧 𝑧 𝑧 affine open sets, and define 𝜑𝑖 ∶ 𝑈 𝑖 → ℂ𝑛 by 𝜑𝑖 ([𝑧0 , . . . , 𝑧𝑛 ]) = ( 𝑧0 , . . . , 𝑧ˆ𝑖 , . . . , 𝑧𝑛 ). 𝑖

𝑖

𝑖

Its inverse is 𝜑−1 ˆ𝑖 , . . . , 𝑤 𝑛 ) = [𝑤 0 , . . . , 1, . . . , 𝑤 𝑛 ]. The changes of charts 𝜑𝑖 ∘ 𝑖 (𝑤 0 , . . . , 𝑤 𝜑𝑗−1 ∶ 𝜑𝑗 (𝑈 𝑖 ∩ 𝑈 𝑗 ) → 𝜑𝑖 (𝑈 𝑖 ∩ 𝑈 𝑗 ) are biholomorphisms, as they are given by (we take 𝑖 < 𝑗 in this formula) 𝑤𝑗−1 1 𝑤𝑗+1 𝑤 𝑤 𝑤 ˆ (5.16) 𝜑𝑖 ∘ 𝜑𝑗−1 (𝑤 0 , . . . , 𝑤 ˆ𝑗 , . . . , 𝑤 𝑛 ) = ( 0 , . . . , 𝑖 , . . . , , , , ... , 𝑛 ) . 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑤𝑖 𝑛 From this we conclude that ℂ𝑃 has a complex atlas. Actually the changes of coordinates are given by rational functions, i.e., quotients of polynomials. This gives ℂ𝑃 𝑛 the structure of an algebraic variety (Remark 1.23).

5.2. Kähler manifolds

293

5.2.3.1. CW-complex structure. Note that ℂ𝑃 𝑛 = 𝑈0 ⊔ 𝐻, where 𝑈0 = {[1, 𝑧1 , . . . , 𝑧𝑛 ]} ≅ ℂ𝑛 is the standard affine space and 𝐻 = {[0, 𝑧1 , . . . , 𝑧𝑛 ]} ≡ ℂ𝑃 𝑛−1 is the hyperplane at infinity. Note that 𝑈0 = ℂ𝑛 is homeomorphic to an open disc via 𝑓 ∶ 𝐵 2𝑛 → 𝑈0 , 𝑧1 𝑧𝑛 𝑓(𝑧1 , . . . , 𝑧𝑛 ) = [1, 1−|𝑧| , . . . , 1−|𝑧| ] = [1 − |𝑧|, 𝑧1 , . . . , 𝑧𝑛 ]. From this formula, we see that 𝑓 extends continuously to a function 𝑓 ̃ ∶ 𝐷2𝑛 → ℂ𝑃 𝑛 . The restriction is given by (5.17)

𝑞𝑛−1 = 𝑓|̃ 𝜕𝐷2𝑛 ∶ 𝜕𝐷2𝑛 = {|𝑧| = 1} → 𝐻 = ℂ𝑃 𝑛−1 , 𝑞𝑛−1 (𝑧1 , . . . , 𝑧𝑛 ) = [0, 𝑧1 , . . . , 𝑧𝑛 ].

With this description, we can see ℂ𝑃 𝑛 as the space obtained by attaching a 2𝑛-cell to ℂ𝑃 𝑛−1 with gluing map 𝑞𝑛−1 (Example 1.33(2)). Hence ℂ𝑃 𝑛 = ℂ𝑃 𝑛−1 ∪𝑞𝑛−1 𝐷2𝑛 . Remark 5.34. Note that the map (5.17) is the map 𝑞𝑛−1 ∶ 𝑆 2𝑛−1 → ℂ𝑃 𝑛−1 , where 𝜋

𝑞𝑛 ∶ 𝑆 2𝑛+1 ⊂ ℂ𝑛+1 − {0} ⟶ ℂ𝑃 𝑛 ,

𝑞𝑛 (𝑧0 , . . . , 𝑧𝑛 ) = [𝑧0 , . . . , 𝑧𝑛 ].

The map 𝑞𝑛 is called the Hopf map. Note that 𝑞1 ∶ 𝑆 3 → ℂ𝑃 1 = 𝑆 2 has already appeared in Remark 4.30. The map 𝑞𝑛 is a quotient map, where (𝑧0 , . . . , 𝑧𝑛 ) ∼ (𝑤 0 , . . . , 𝑤 𝑛 ) if and only if (𝑧0 , . . . , 𝑧𝑛 ) = 𝜆(𝑤 0 , . . . , 𝑤 𝑛 ), for some 𝜆 ∈ ℂ∗ . As (𝑧0 , . . . , 𝑧𝑛 ), (𝑤 0 , . . . , 𝑤 𝑛 ) ∈ 𝑆 2𝑛+1 , we must have |𝜆| = 1. Hence the fibers of 𝑞𝑛 are the orbits of the 𝑆 1 action 𝑒i𝜃 ⋅ (𝑧0 , . . . , 𝑧𝑛 ) = (𝑒i𝜃 𝑧0 , . . . , 𝑒i𝜃 𝑧𝑛 ). We can write ℂ𝑃 𝑛 = 𝑆 2𝑛+1 /𝑆 1 . Note, in particular, that ℂ𝑃𝑛 is compact and connected being the quotient of a compact connected space. This gives ℂ𝑃𝑛 the structure of a CW-complex (Definition 2.104), whose skeleta are as follows. Take 𝑋0 = {𝑝} a single point and 𝑋2𝑘 = ℂ𝑃 𝑘 for 𝑘 = 1, . . . , 𝑛. By the discussion above, 𝑋2𝑘 = 𝑋2𝑘−2 ∪𝑞𝑘−1 𝐷2𝑘 and also 𝑋2𝑘+1 = 𝑋2𝑘 , for all 𝑘 ≥ 0. Therefore there are only cells of even dimension, one for each dimension 2𝑘, 0 ≤ 𝑘 ≤ 𝑛. The cellular homology follows (see page 99) from this description. The chain complex is 𝐶𝑊 𝐶𝑊 𝐶2𝑘 (ℂ𝑃𝑛 ) = ℤ⟨𝐷2𝑘 ⟩, and 𝐶2𝑘+1 (ℂ𝑃 𝑛 ) = 0, for 0 ≤ 𝑘 ≤ 𝑛. Hence, the chain complex is ℤ⟨𝐷2𝑛 ⟩ → 0 → ℤ⟨𝐷2𝑛−2 ⟩ → 0 → ⋯ → 0 → ℤ⟨𝐷2 ⟩ → 0 → ℤ⟨𝐷0 ⟩. 𝐶𝑊 The boundary maps 𝜕𝑘 ∶ 𝐶𝑘𝐶𝑊 (ℂ𝑃 𝑛 ) → 𝐶𝑘+1 (ℂ𝑃𝑛 ) are identically zero since either the target or the domain is zero. Therefore 𝐻𝑖 (ℂ𝑃 𝑛 ) = {

ℤ⟨[𝐷𝑖 ]⟩, 0,

𝑖 even, 0 ≤ 𝑖 ≤ 2𝑛, otherwise.

The generator 𝐿𝑘 = [𝐷2𝑘 ] is defined as the image of the 2𝑘-cell 𝐷2𝑘 → ℂ𝑃 𝑛 , which is a linear projective subspace of complex dimension 𝑘, 𝐿𝑘 = ℂ𝑃 𝑘 ⊂ ℂ𝑃 𝑛 . Summing up, (5.18)

𝐻2𝑘 (ℂ𝑃 𝑛 ) = ℤ⟨𝐿𝑘 ⟩,

0 ≤ 𝑘 ≤ 𝑛.

Therefore 𝑏2𝑘 = 1 for 0 ≤ 𝑘 ≤ 𝑛, and the other Betti numbers are zero. Let us finally check that ℂ𝑃 𝑛 is simply connected. This is just a consequence that it has no 1-cells. We do it by induction on 𝑛. First ℂ𝑃 1 = 𝑆 2 is simply connected. Now suppose that ℂ𝑃𝑛−1 is simply connected. On ℂ𝑃𝑛 = 𝑈0 ⊔ ℂ𝑃 𝑛−1 , we take open sets 𝑈 = 𝑈0 ≅ ℂ𝑛 and 𝑉 = ℂ𝑃 𝑛 − {𝑝0 }, where 𝑝0 is the origin in 𝑈0 ≅ 𝐵 2𝑛 . We retract radially to see that 𝑉 ∼ ℂ𝑃 𝑛−1 . The intersection 𝑈 ∩ 𝑉 ≅ 𝐵 2𝑛 − {𝑝0 } ∼ 𝑆 2𝑛−1 is connected. Hence we can apply the Seifert-van Kampen theorem to conclude that ℂ𝑃 𝑛 is simply connected (Example 2.25(2)).

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5. Complex geometry

5.2.3.2. The Fubini-Study metric. In this section we are going to construct a Hermitian metric on ℂ𝑃 𝑛 that makes it a Kähler manifold, in a natural way. In the case of ℂ𝑃 1 (treated in section 4.1.3), we saw that the metric of constant curvature is invariant under all projectivities induced by elements in U(2) < GL(2, ℂ) (Proposition 4.27). As these projectivities act transitively on the projective space, this makes it homogeneous (Definition 3.85). For ℂ𝑃 𝑛 , 𝑛 ≥ 2, we cannot aim for a metric of constant curvature, because it is compact and simply connected but not homeomorphic to 𝑆 2𝑛 (which is the only compact simply connected space form). Therefore we shall look for a homogeneous non-isotropic metric that satisfies that the Hermitian matrices 𝐴 ∈ U(𝑛 + 1) < GL(𝑛 + 1, ℂ) induce projectivities 𝜑𝐴 ∶ ℂ𝑃 𝑛 → ℂ𝑃 𝑛 which are isometries. Remark 5.35. By Definition 4.25, the unitary group is given as U(𝑛) = {𝐴 ∈ GL(𝑛, ℂ) | 𝐴∗ 𝐴 = Id}, where 𝐴∗ = 𝐴𝑡 . Giving ℂ𝑛 the standard Hermitian metric h(𝑣, 𝑤) = 𝑣𝑡 𝑤 (Example 5.20), the matrices 𝐴 ∈ U(𝑛) correspond to complex linear maps 𝜑𝐴 ∶ ℂ𝑛 → ℂ𝑛 , 𝜑𝐴 (𝑣) = 𝐴𝑣, which are Hermitian isometries. Clearly h(𝜑𝐴 (𝑣), 𝜑𝐴 (𝑤)) = (𝐴𝑣)𝑡 𝐴𝑤 = 𝑣𝑡 𝐴𝑡 𝐴 𝑤 = 𝑣𝑡 𝑤 = h(𝑣, 𝑤), since 𝐴𝑡 𝐴 = Id implies also 𝐴𝑡 𝐴 = Id. We will not discuss further, but just note that a matrix is in U(𝑛) when its columns form a Hermitian orthonormal basis of ℂ𝑛 , and that U(𝑛) acts transitively on unitary vectors of ℂ𝑛 (Exercise 5.4). The projectivities induced by elements of U(𝑛 + 1) are denoted PU(𝑛 + 1) < PGL(𝑛 + 1, ℂ) and act transitively on ℂ𝑃 𝑛 . Therefore if we want a metric invariant by U(𝑛 + 1), then it is determined by its value at one point. Note that it is not guaranteed that taking a metric at one point and moving it around, we can construct a homogeneous metric, since it may not be well defined (this happens if two different maps take the initial point to the same point, but the transformed metrics do not coincide). So, we decide to define the sought Hermitian metric for ℂ𝑃𝑛 in an indirect way, by giving its fundamental form. Definition 5.36. We define the fundamental form of ℂ𝑃𝑛 as the 2-form 𝜔 given, in homogeneous coordinates 𝑧 = [𝑧0 , . . . , 𝑧𝑛 ], by 𝜔=

i i 𝜕𝜕 log |𝑧|2 = 𝜕𝜕 log(|𝑧0 |2 + ⋯ + |𝑧𝑛 |2 ). 2 2

The form 𝜔 satisfies the following properties: (1) The form 𝜔 is a well defined 2-form on ℂ𝑃 𝑛 . Take the coordinates on 𝑈0 , (𝑤 1 , . . . , 𝑤 𝑛 ) ↦ [1, 𝑤 1 , . . . , 𝑤 𝑛 ]. We introduce 𝑤 0 = 1 and write 𝑤 = (𝑤 0 , 𝑤 1 , . . . , 𝑤 𝑛 ). So in these coordinates we have (5.19)

𝜔=

i i 𝜕𝜕 log(1 + |𝑤 1 |2 + ⋯ + |𝑤 𝑛 |2 ) = 𝜕𝜕 log |𝑤|2 . 2 2

Now consider another chart 𝑈 𝑗 with coordinates 𝑤′ = (𝑤′0 , . . . , 𝑤𝑗′ = 1, . . . , 𝑤′𝑛 ), i

i

in which 𝜔 has the form 𝜔 = 2 𝜕𝜕 log(1+∑𝑖≠𝑗 |𝑤′𝑖 |2 ) = 2 𝜕𝜕 log |𝑤|2 . By (5.16), 1

the change of coordinates is given by 𝑤′ = 𝑤 𝑤, that is 𝑤′ = 𝑓(𝑤)𝑤 for some 𝑗 holomorphic function 𝑓. Let us check that the form 𝜔 is well defined, i.e.,

5.2. Kähler manifolds

295

that is independent of the choice of coordinates. Certainly, i i i 𝜕𝜕 log |𝑤′ |2 = 𝜕𝜕(log |𝑓(𝑤)|2 + log |𝑤|2 ) = 𝜕𝜕 log |𝑤|2 , 2 2 2 using that for a holomorphic function, 𝜕𝜕 log |𝑓|2 = 𝜕𝜕 log(𝑓𝑓) = 𝜕𝜕 log 𝑓 + 𝜕𝜕 log 𝑓 = 0. This holds since 𝜕 log 𝑓 = 0 because 𝑓 is holomorphic, and 𝜕(log 𝑓) = 0 because log 𝑓 is antiholomorphic (section 5.1.1, item (8)), and using that 𝜕𝜕 = −𝜕𝜕 (see (5.9)). (2) Clearly 𝜔 ∈ Ω1,1 (ℂ𝑃 𝑛 ), since it is of the form 𝜕𝜕 of a function. ̄ ∶ (3) Given a matrix 𝐴 ∈ U(𝑛 + 1), let us denote 𝜑𝐴 ∶ ℂ𝑛+1 → ℂ𝑛+1 and 𝜑𝐴 ∗ 𝑛 𝑛 ̄ leaves 𝜔 invariant since 𝜑𝐴 𝜔 = ℂ𝑃 → ℂ𝑃 the maps induced by 𝐴. Then 𝜑𝐴 i i 2 2 𝜕𝜕 log(|𝐴𝑧| ) = 𝜕𝜕 log(|𝑧| ) = 𝜔. We have used that |𝐴𝑧|2 = |𝑧|2 because 2 2 𝐴 preserves the standard Hermitian metric on ℂ𝑛+1 . (4) Let us compute 𝜔 in the local coordinates 𝑤 = (𝑤 1 , . . . , 𝑤 𝑛 ) of 𝑈0 , using (5.19). 𝑛 This time we do not include the coordinate 𝑤 0 = 1 and so |𝑤|2 = ∑𝑖=1 |𝑤 𝑖 |2 . The formula for 𝜔 is now 𝜔=

∑ 𝑤 𝑖 𝑑𝑤𝑖 𝜕|𝑤|2 i i i 𝜕𝜕 log(1 + |𝑤|2 ) = 𝜕 ( ) = 𝜕( ) 2 2 2 1 + |𝑤| 2 1 + |𝑤|2

2 i (∑ 𝑑𝑤 𝑖 ∧ 𝑑𝑤𝑖 )(1 + |𝑤| ) − (∑𝑖 𝑤𝑖 𝑑𝑤 𝑖 ) ∧ (∑𝑗 𝑤𝑗 𝑑𝑤𝑗 ) 2 (1 + |𝑤|2 )2 2 i (∑𝑖 𝑑𝑤 𝑖 ∧ 𝑑𝑤𝑖 )(1 + |𝑤| ) − ∑𝑖,𝑗 𝑤𝑖 𝑤𝑗 𝑑𝑤 𝑖 ∧ 𝑑𝑤𝑗 = . 2 (1 + |𝑤|2 )2

=

i

(5) At the origin (𝑤 1 , . . . , 𝑤 𝑛 ) = (0, . . . , 0), we have 𝜔 = 2 ∑ 𝑑𝑤𝑗 ∧ 𝑑𝑤𝑗 . This corresponds to a Hermitian metric at this point, by item (8) in page 285. By (4) above, the form 𝜔 is a homogeneous and well defined (1, 1)-form. So it corresponds to a Hermitian metric at all points of ℂ𝑃 𝑛 . Thus 𝜔 is the fundamental form of a Hermitian metric h, which on the chart 𝑈0 is given as ℎ𝑗𝑘 (𝑤) =

𝑤𝑗 𝑤 𝑘 1 𝛿 − . 1 + |𝑤|2 𝑗𝑘 (1 + |𝑤|2 )2

(6) The group U(𝑛 + 1) leaves invariant the complex structure on ℂ𝑃𝑛 and the form 𝜔. Thus it leaves also invariant the associated Riemannian metric g, and the Hermitian metric h. Hence the group PU(𝑛 + 1) is a subgroup of the group of isometries of (ℂ𝑃𝑛 , g), and it acts transitively. The projective space is a homogeneous Riemannian manifold. (7) The metric h is called the Fubini-Study metric, and it is denoted h𝐹𝑆 . The fundamental form is denoted 𝜔𝐹𝑆 and the Riemannian metric g𝐹𝑆 . This Hermitian metric is uniquely determined by the properties of being invariant under U(𝑛 + 1) and being the standard Hermitian form at the origin (0, . . . , 0) ∈ 𝑈0 (in the affine coordinates).

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(8) For ℂ𝑃 1 , the Fubini-Study metric is given by h𝐹𝑆 =

1 𝑤𝑤 1 − = . 1 + |𝑤|2 (1 + |𝑤|2 )2 (1 + |𝑤|2 )2 |𝑑𝑤|2

1

The corresponding Riemannian metric is g𝐹𝑆 = (1+|𝑤|2 )2 . This is g𝐹𝑆 = 4 g𝑆2 , that is a quarter of the standard round metric on the sphere. Therefore in this case the Fubini-Study metric is of constant curvature 𝜅(g𝐹𝑆 ) ≡ 4. Also 1 1 area(ℂ𝑃 1 , g𝐹𝑆 ) = 4 area(𝑆 2 , g𝑆2 ) = 4 4𝜋 = 𝜋. (9) The fact that 𝜔 is closed follows easily since 𝜕2 = 0 and 𝜕2 = 0. Indeed, i i i 𝜕𝜔 = 2 𝜕𝜕𝜕 log |𝑧|2 = 0 and 𝜕𝜔 = 2 𝜕𝜕𝜕 log |𝑧|2 = − 2 𝜕𝜕𝜕 log |𝑧|2 = 0, using again that 𝜕𝜕 + 𝜕𝜕 = 0. Then 𝑑𝜔 = (𝜕 + 𝜕)𝜔 = 0. (10) The last item also follows from homogeneity (Exercise 5.14). (11) If ℂ𝑃𝑘 ⊂ ℂ𝑃 𝑛 is a projective linear complex subspace and h𝐹𝑆 is the FubiniStudy Hermitian metric for ℂ𝑃𝑛 , then h𝐹𝑆 |ℂ𝑃 𝑘 is the Fubini-Study Hermitian metric for ℂ𝑃 𝑘 . We have proved the following result. Proposition 5.37. The projective space with the Fubini-Study metric (ℂ𝑃 𝑛 , h𝐹𝑆 ) is a Kähler manifold. By item (9) above, 𝜔 ∈ Ω1,1 (ℂ𝑃𝑛 ) is closed and [𝜔] ∈ 𝐻 1,1 (ℂ𝑃𝑛 ). By Remark 5.32(4), we have that [𝜔]𝑘 ≠ 0 for 1 ≤ 𝑘 ≤ 𝑛. We already know that 𝑏2𝑘 = 1 and ℎ𝑘,𝑘 > 0, therefore ℎ𝑘,𝑘 = 1, 0 ≤ 𝑘 ≤ 𝑛, and the other Hodge numbers are zero. Then 𝐻 𝑘,𝑘 (ℂ𝑃 𝑛 ) = 𝐻 2𝑘 (ℂ𝑃𝑛 , ℂ) = ℂ⟨[𝜔𝑘 ]⟩. As 𝜔 is a real 2-form, we also have 𝐻 2𝑘 (ℂ𝑃𝑛 , ℝ) = ℝ⟨[𝜔𝑘 ]⟩. By (5.18), 𝐻2 (ℂ𝑃𝑛 ) = ℤ⟨𝐿1 ⟩, where 𝐿1 = ℂ𝑃 1 ⊂ ℂ𝑃 𝑛 is a projective line. By item (11) above, 𝜔|ℂ𝑃 1 is the Fubini-Study metric of ℂ𝑃1 . Hence ∫ 𝜔 = ∫ 𝜔 = 𝜋, ℂ𝑃 1 𝜔 Ω𝐹𝑆 = 𝜋 is

𝐿1

by item (8) above. This means that

2 the generator of 𝐻𝑑𝑅 (ℂ𝑃 𝑛 , ℤ).

Remark 5.38. We have not defined singular cohomology (see Remark 2.142), but we can give a working definition of integral de Rham cohomology. Let 𝑀 be a smooth 𝑘 𝑘 manifold. We define 𝐻𝑑𝑅 (𝑀, ℤ) ⊂ 𝐻𝑑𝑅 (𝑀) as the set of cohomology classes [𝛼] ∈ 𝑘 𝐻𝑑𝑅 (𝑀) such that ⟨[𝛼], [𝑇]⟩ ∈ ℤ for any [𝑇] ∈ 𝐻𝑘 (𝑀, ℤ), with the pairing (2.20). 𝑘 Remark 5.39. The group 𝐻𝑑𝑅 (𝑀, ℤ) is the image of the integral singular cohomology 𝑘 𝑘 𝑘 𝐻 (𝑀, ℤ) → 𝐻 (𝑀, ℝ) ≅ 𝐻𝑑𝑅 (𝑀) (Remark 2.142). As the product can be defined on singular cohomology (Remark 2.165), we have that the elements of 𝐻 • (𝑀, ℤ) can be 2 multiplied. In the case of ℂ𝑃𝑛 , we have that [Ω𝐹𝑆 ] ∈ 𝐻𝑑𝑅 (ℂ𝑃 𝑛 , ℤ) is the Poincaré dual of [𝐿𝑛−1 ] (section 2.6.4). First, ⟨[𝐿1 ], [𝐿𝑛−1 ]⟩ = 𝐿1 ⋅ 𝐿𝑛−1 = 1 by Proposition 2.163, noting that a (generic) line and hyperplane intersect transversally in one point. Next, ⟨[𝐿1 ], [𝐿𝑛−1 ]⟩ = 1 = ⟨[𝐿1 ], [Ω𝐹𝑆 ]⟩, so Ψ([𝐿𝑛−1 ]) = [Ω𝐹𝑆 ] in the notation of section 2.6.4. Now taking transverse hyperplanes 𝐻1 , . . . , 𝐻𝑛−𝑘 , we have that 𝐿𝑘 = 𝐻1 ∩ ⋯ ∩ 𝐻𝑛−𝑘 is a projective subspace of complex dimension 𝑘. Hence Ψ([𝐿𝑘 ]) = [Ω𝑛−𝑘 𝐹𝑆 ], by section 2𝑘 2.6.5, and 𝐻𝑑𝑅 (ℂ𝑃𝑛 , ℤ) = [Ω𝑘𝐹𝑆 ] by (5.18).

5.2. Kähler manifolds

297

5.2.3.3. Projective varieties. Now we study complex submanifolds of the complex projective space. These are the main object of study in (complex) algebraic geometry. Definition 5.40. Let 𝐹1 , . . . , 𝐹𝑟 ∈ ℂ[𝑧0 , . . . , 𝑧𝑛 ] be a collection of homogeneous polynomials. The zero set 𝑋 = 𝑉(𝐹1 , . . . , 𝐹𝑟 ) = {[𝑧0 , . . . , 𝑧𝑛 ] ∈ ℂ𝑃 𝑛 | 𝐹𝑗 (𝑧0 , . . . , 𝑧𝑛 ) = 0, 𝑗 = 1, . . . , 𝑟} ⊂ ℂ𝑃 𝑛 is called a projective variety. Remark 5.41. Note that if 𝐹 is a homogeneous polynomial of degree 𝑑, then 𝐹(𝜆𝑧0 , . . . , 𝜆𝑧𝑛 ) = 𝜆𝑑 𝐹(𝑧0 , . . . , 𝑧𝑛 ), for 𝜆 ∈ ℂ∗ . Therefore the value 𝐹(𝑧0 , . . . , 𝑧𝑛 ) is not a well defined number for a point [𝑧0 , . . . , 𝑧𝑛 ] ∈ ℂ𝑃 𝑛 , but the vanishing condition 𝐹(𝑧0 , . . . , 𝑧𝑛 ) = 0 is well characterized. In algebraic geometry language, the polynomials 𝐹1 , . . . , 𝐹𝑟 generate an ideal 𝐼(𝐹1 , . . . , 𝐹𝑟 ) = {∑ 𝑃𝑗 𝐹𝑗 |𝑃𝑗 ∈ ℂ[𝑧0 , . . . , 𝑧𝑛 ], 1 ≤ 𝑗 ≤ 𝑟} ⊂ ℂ[𝑧0 , . . . , 𝑧𝑛 ]. For any homogeneous ideal 𝐼 ⊂ ℂ[𝑧0 , . . . , 𝑧𝑛 ] (that is, an ideal generated by homogeneous elements), we can define 𝑋 = 𝑉(𝐼) = {[𝑧] ∈ ℂ𝑃 𝑛 | 𝐹(𝑧) = 0, for all 𝐹 ∈ 𝐼}. By Noetherianity, any ideal is finitely generated, so 𝐼 = 𝐼(𝐹1 , . . . , 𝐹𝑟 ), for some homogeneous 𝐹1 , . . . , 𝐹𝑟 , and 𝑋 is a projective variety. Analogously, given any projective variety 𝑋 ⊂ ℂ𝑃 𝑛 , there is an ideal 𝐼(𝑋) = {𝐹 homogeneous | 𝐹|𝑋 ≡ 0} ⊂ ℂ[𝑧0 , . . . , 𝑧𝑛 ]. Such ideal is radical, that is an ideal 𝐼 such that if 𝑓𝑚 ∈ 𝐼, for some 𝑚 > 1, then 𝑓 ∈ 𝐼. There is a correspondence between radical ideals and projective varieties (this is the Hilbert Nullstellensatz), given by 𝐼 ↦ 𝑉(𝐼) and 𝑋 ↦ 𝐼(𝑋). Therefore we shall assume that the ideal 𝐼 = 𝐼(𝐹1 , . . . , 𝐹𝑟 ) is always radical. Let 𝑋 = 𝑉(𝐼) with 𝐼 = 𝐼(𝐹1 , . . . , 𝐹𝑟 ) radical. Given 𝑧 ∈ 𝑋, we denote Jac(𝐹1 , . . . , 𝐹𝑟 ) = 𝜕𝐹 ( 𝜕𝑧 𝑖 ) 𝑗

1≤𝑖≤𝑟 0≤𝑗≤𝑛

. If rk Jac(𝐹𝑖 )(𝑧) = 𝑠, for all 𝑧 ∈ 𝑋, and 𝑠 ≥ 0 a fixed integer, then we say that

𝑋 is a smooth projective variety of dimension 𝑑 = 𝑛 − 𝑠. As always, a smooth projective variety of dimension 1 is called a smooth projective curve. Proposition 5.42. If 𝑋 ⊂ ℂ𝑃 𝑛 is a smooth projective variety of dimension 𝑑, then 𝑋 is a complex manifold of dimℂ 𝑋 = 𝑑. Proof. We start by denoting 𝐹 = (𝐹1 , . . . , 𝐹𝑟 ) ∶ ℂ𝑛+1 → ℂ𝑟 , and note that 𝑑𝑧ℂ 𝐹 = Jac(𝐹𝑖 )(𝑧). Therefore its rank is well defined independently of the choice of coordinates. With this said, we can choose coordinates so that 𝑝 = [1, 0, . . . , 0] ∈ 𝑈0 . We take affine coordinates (𝑤 1 , . . . , 𝑤 𝑛 ) ↦ [1, 𝑤 1 , . . . , 𝑤 𝑛 ] and consider the dehomogenized polynomials 𝐹𝑖̂ (𝑤 1 , . . . , 𝑤 𝑛 ) = 𝐹𝑖 (1, 𝑤 1 , . . . , 𝑤 𝑛 ). Clearly

ˆ𝑖 𝜕𝐹 (0) 𝜕𝑤𝑗

=

𝜕𝐹𝑖 (𝑝) 𝜕𝑧𝑗

for 𝑘 ≥ 1, 𝑝 = (1, 0, . . . , 0). Now observe that for a homoge-

neous polynomial 𝐹(𝑧0 , . . . , 𝑧𝑛 ) of some degree 𝑑 > 0, we have the Euler identity saying that 𝑛 𝜕𝐹 ∑ 𝑧𝑖 =𝑑⋅𝐹. 𝜕𝑧 𝑖 𝑖=0

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5. Complex geometry

𝜕𝐹

As 𝐹(𝑝) = 0, this implies that 𝜕𝑧 (𝑝) = 0. Applying this to our polynomials, we get that 0 rk Jac(𝐹𝑖̂ )(0) = rk Jac(𝐹𝑖 )(𝑝). This means that the rank can be computed in the affine chart. Now as rk Jac(𝐹𝑖̂ )(0) = 𝑛 − 𝑑, we have some minor, which we can assume by 𝜕𝐹

reordering that it is ( 𝜕𝑤𝑖 ) 𝑗

1≤𝑖≤𝑛−𝑑 𝑑+1≤𝑗≤𝑛

, which is invertible. By the implicit function theorem

for holomorphic functions [Sha], there are neighbourhoods 𝑈 ⊂ ℂ𝑛 of (0, . . . , 0) and a holomorphic function 𝜓 ∶ 𝐵𝜖 (0) ⊂ ℂ𝑑 → ℂ𝑛−𝑑 , with 𝜓(0) = 0, satisfying that 𝑌 = {(𝑤 1 , . . . , 𝑤 𝑛 ) ∈ 𝑈 | 𝐹1 (𝑤 1 , . . . , 𝑤 𝑛 ) = ⋯ = 𝐹𝑛−𝑑 (𝑤 1 , . . . , 𝑤 𝑛 ) = 0} = {(𝑤′ , 𝜓(𝑤′ )) ∈ 𝑈 | 𝑤′ = (𝑤 1 , . . . , 𝑤 𝑑 ) ∈ 𝐵𝜖 (0)}.

(wd+1 ; : : : ; wn) Y

wd

w1 Write 𝑤 = (𝑤′ , 𝑤″ ) with 𝑤′ = (𝑤 1 , . . . , 𝑤 𝑑 ), 𝑤″ = (𝑤 𝑑+1 , . . . , 𝑤 𝑛 ). Take the biholomorphism 𝜑 ∶ 𝑈 ⊂ ℂ𝑛 → 𝑈 ′ ⊂ ℂ𝑛 defined by 𝜑(𝑤′ , 𝑤″ ) = (𝑤′ , 𝑤″ − 𝜓(𝑤′ )), then 𝜑(𝑌 ) = (ℂ𝑛−𝑘 × {0}) ∩ 𝑈 ′ . Denote 𝐺 𝑖 = 𝐹𝑖̂ ∘ 𝜑−1 , the transformed of the 𝐹𝑖̂ . 𝜕𝐺 Then 𝜕𝑤𝑖 ≡ 0, for 1 ≤ 𝑖 ≤ 𝑛 − 𝑑 and 1 ≤ 𝑗 ≤ 𝑑. So for 𝑖 > 𝑛 − 𝑑, we have that 𝑗

𝜕𝐺𝑖 (𝑤′ ) 𝜕𝑤𝑗

= 0, for all 1 ≤ 𝑗 ≤ 𝑑, since otherwise rk Jac(𝐺 𝑖 )(𝑤′ ) > 𝑛 − 𝑑. Using Ex-

ercise 5.25, we see that this implies that 𝐺 𝑖 (𝑤′ , 0) ≡ 0 for all 𝑖 > 𝑛 − 𝑑. Therefore, 𝑋 ∩ 𝑈 = 𝑉(𝐹1 , . . . , 𝐹𝑟 ) ∩ 𝑈 ≅ 𝑉(𝐺 1 , . . . , 𝐺𝑛−𝑑 ) ∩ 𝑈 ′ ≅ 𝐵𝜖 (0) ⊂ ℂ𝑑 . Thus 𝑋 is a smooth complex manifold of dimℂ 𝑋 = 𝑑. □ Let 𝑀 ⊂ ℂ𝑃 𝑛 be a complex submanifold. Then 𝑀 inherits a Kähler structure by taking h = h𝐹𝑆 |𝑀 (Remark 5.28(5)). In this way we obtain many examples of Kähler manifolds, since in particular all smooth projective varieties fit here. Further𝜔 more, the fundamental form of 𝑀 is 𝜔 = 𝜔𝐹𝑆 |𝑀 , hence taking Ω = 𝜋 , we have that 𝜔 2 2 [Ω] ∈ 𝐻𝑑𝑅 (𝑀, ℤ), since it is the image of [Ω𝐹𝑆 ] = [ 𝜋𝐹𝑆 ] ∈ 𝐻𝑑𝑅 (ℂ𝑃 𝑛 , ℤ) under the 2 2 restriction map 𝐻𝑑𝑅 (ℂ𝑃𝑛 , ℤ) → 𝐻𝑑𝑅 (𝑀, ℤ). Therefore we have that the Kähler form satisfies 2 [Ω] ∈ 𝐻 1,1 (𝑀) ∩ 𝐻𝑑𝑅 (𝑀, ℤ).

5.2. Kähler manifolds

299

That is, for any smooth projective variety 𝑀 ⊂ ℂ𝑃 𝑛 , they are integral Kähler classes. There is a surprising converse that (another time more) links topology and (complex) geometry. Theorem 5.43 (Kodaira embedding). Let (𝑀, 𝐽, 𝜔) be a compact Kähler manifold such 𝜔 2 that [ 𝜋 ] ∈ 𝐻 1,1 (𝑀)∩𝐻𝑑𝑅 (𝑀, ℤ). Then there is a holomorphic embedding 𝜙 ∶ 𝑀 → ℂ𝑃 𝑁 for some large 𝑁 > 0. This is a strong result in some sense similar to the Whitney embedding theorem for smooth manifolds. It serves to go from abstract complex geometry to (projective) algebraic geometry, since it says that complex manifolds can be understood as complex submanifolds of projective space. The embedding given by Theorem 5.43 is holomorphic (it respects 𝐽) but it is not isometric (it does not respect the Hermitian metric, or equivalently, 𝜙∗ [𝜔𝐹𝑆 ] = [𝜔] but 𝜙∗ 𝜔𝐹𝑆 ≠ 𝜔 in general). The proof of Theorem 5.43 is hard and can be found in [Wel]. It consists basically of finding enough global meromorphic functions 𝑓1 , . . . , 𝑓𝑁 on 𝑀 that define maps to ℂ, and setting 𝜙(𝑝) = [1, 𝑓1 (𝑝), . . . , 𝑓𝑁 (𝑝)]. When cleaning denominators, we have locally that 𝜙(𝑝) = [𝜙0 (𝑝), 𝜙1 (𝑝), . . . , 𝜙𝑁 (𝑝)], as 𝑓𝑗 = 𝜙𝑗 /𝜙0 , 𝑗 = 1, . . . , 𝑁, are quotients of holomorphic functions. Definition 5.44. A Hodge manifold is a compact Kähler manifold (𝑀, 𝐽, 𝜔) such that 𝜔 2 (𝑀, ℤ). [ 𝜋 ] ∈ 𝐻 1,1 (𝑀) ∩ 𝐻𝑑𝑅 Note that the factor accompanying 𝜔 is irrelevant, since Kähler forms can be multiplied by positive real numbers, and clearly they remain Kähler forms. Finally, we combine Theorem 5.43 with the following result of Chow. Theorem 5.45 (Chow). If 𝑋 ⊂ ℂ𝑃 𝑁 is a compact locally holomorphic set (that is, for every point 𝑝 ∈ 𝑋, there is a neighbourhood 𝑈 𝑝 such that 𝑈 ∩ 𝑋 is the locus of vanishing of some holomorphic functions), then 𝑋 is a projective variety (i.e., defined globally by the vanishing of some homogeneous polynomials). Remark 5.46. By Theorem 5.43, a Hodge manifold can be embedded as a complex submanifold of some ℂ𝑃 𝑁 . By Theorem 5.45, this is defined by homogeneous polynomials, that is it is a smooth projective variety. Therefore Hodge manifolds are exactly the smooth projective varieties. This gives a clean correspondence between complex differential geometry and (complex) algebraic geometry. Remark 5.47. We do not want to enter into the systematic treatment of algebraic varieties, for which we refer to [Har]. For instance, we will not talk about schemes or Spec, which are standard notions in algebraic geometry. But we briefly recall some facts that we will use later: (1) Any projective variety 𝑋 ⊂ ℂ𝑃 𝑛 admits the structure of an algebraic variety (Remark 1.23). (2) The Zariski topology in ℂ𝑃𝑛 is the one whose closed subsets are the (algebraic) projective varieties (hence the open subsets are the complement of projective varieties). This induces a Zariski topology on every 𝑋 ⊂ ℂ𝑃 𝑛 . Zariski

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closed subsets are closed in standard topology. A (Zariski) open subset of a projective variety of dimension 𝑑 is called a quasiprojective variety (or just a variety) of dimension 𝑑. (3) For any connected projective variety 𝑋 ⊂ ℂ𝑃 𝑛 , there is a well defined notion of dimension 𝑑 = dimℂ 𝑋. The points satisfying the condition of Proposition 5.42 are called regular or smooth points. The set of regular points 𝑋 reg ⊂ 𝑋 is Zariski open in 𝑋 and it is a smooth complex submanifold of complex dimension 𝑑. The complement 𝑋 sing is a (Zariski) closed subset consisting of the non-smooth or singular points. Those are the ones for which rk Jac(𝐹1 , . . . , 𝐹𝑟 )(𝑧) < 𝑛 − 𝑑, for 𝐹1 , . . . , 𝐹𝑟 defining equations of 𝑋. Moreover, 𝑋 sing is also a projective variety and dimℂ 𝑋 sing < 𝑑. (4) 𝑋 is called a hypersurface if dimℂ 𝑋 = 𝑛 − 1. In this case, 𝑋 = 𝑉(𝐹) for a single homogeneous polynomial 𝐹 ∈ ℂ[𝑧0 , . . . , 𝑧𝑛 ] (Exercise 5.27). (5) 𝑋 ⊂ ℂ𝑃 𝑛 is irreducible if for any decomposition 𝑋 = 𝑋1 ∪ 𝑋2 , with 𝑋1 , 𝑋2 ⊂ 𝑋 both Zariski closed, then either 𝑋 = 𝑋1 or 𝑋 = 𝑋2 . This is equivalent to 𝑋 reg being connected and dense in 𝑋. In particular, a smooth connected projective variety is always irreducible (Exercise 5.26), and an irreducible projective variety is connected. (6) 𝑋 = 𝑉(𝐹) is an irreducible hypersurface if and only if 𝐹 is an irreducible 𝑛 𝑛 polynomial. Certainly, let 𝐹 = 𝐹1 1 ⋯ 𝐹𝑘 𝑘 be the (unique) factorization of 𝐹 ∈ ℂ[𝑧0 , . . . , 𝑧𝑛 ] (recall that ℂ[𝑧0 , . . . , 𝑧𝑛 ] is a unique factorization domain), where 𝐹1 , . . . , 𝐹 𝑘 are distinct irreducible polynomials, and 𝑛𝑗 > 0. Then 𝑋 = 𝑉(𝐹) = 𝑉(𝐹1 ) ∪ ⋯ ∪ 𝑉(𝐹 𝑘 ), where 𝑉(𝐹𝑖 ) are Zariski closed, 𝑖 = 1, . . . , 𝑘. If 𝑋 is irreducible, then 𝑋 = 𝑉(𝐹𝑖 ) for some 𝑖. (7) If 𝜋 ∶ 𝑋 → 𝑌 is an algebraic map between projective varieties (such map is in particular holomorphic at the regular points) and the fibers 𝐹𝑦 = 𝜋−1 (𝑦), 𝑦 ∈ 𝑌 , have dimension dimℂ 𝐹𝑦 ≤ 𝑘, then we have that dimℂ 𝑋 ≤ 𝑘 + dimℂ 𝑌 . Moreover, if 𝑋 is irreducible and, for a Zariski open subset 𝑈 ⊂ 𝑌 , all the fibers have the same dimension 𝑘 (this is called the generic dimension of the fiber), then we have dimℂ 𝑋 = 𝑘 + dimℂ 𝑌 . (8) Let 𝑥 ∈ 𝑋 reg . We define the tangent space at 𝑥 as the 𝑑-dimensional complex projective linear space 𝐭𝑥 𝑋 ⊂ ℂ𝑃 𝑛 such that 𝑥 ∈ 𝐭𝑥 𝑋 and 𝑇𝑥 (𝐭𝑥 𝑋) = 𝑇𝑥 𝑋 reg .

5.3. Complex curves Definition 5.48. A complex curve, also called a Riemann surface, is a complex manifold of complex dimension 1. Remark 5.49. (1) The underlying differentiable manifold of a complex curve is a smooth surface. As complex manifolds are always orientable, this surface is orientable. Therefore a compact connected complex curve is an orientable surface of some genus 𝑔 ≥ 0.

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301

(2) Let us take any oriented Riemannian surface (𝑆, g). For any 𝑝 ∈ 𝑆, we define 𝐽𝑝 ∶ 𝑇𝑝 𝑆 → 𝑇𝑝 𝑆 by decreeing that, for any 𝑣 ∈ 𝑇𝑝 𝑆, 𝑣 ≠ 0, 𝐽𝑝 𝑣 is the unique vector with ||𝐽𝑝 𝑣|| = ||𝑣||, g𝑝 (𝐽𝑝 𝑣, 𝑣) = 0, and (𝑣, 𝐽𝑝 𝑣) is an oriented basis. Otherwise said, 𝐽𝑝 is the rotation on 𝑇𝑝 𝑆 of angle 𝜋/2 in the positive direction. Then we have that 𝐽𝑝2 = − Id, hence 𝐽 is an almost complex structure on 𝑆. Moreover, observe that 𝐽 is an isometry. (3) On an (oriented) surface 𝑆, any almost complex structure 𝐽 is integrable. Take any vector field 𝑋. Then the Nijenhuis tensor satisfies (5.20)

𝑁𝐽 (𝑋, 𝑋) = [𝑋, 𝑋] − [𝐽𝑋, 𝐽𝑋] + 𝐽[𝐽𝑋, 𝑋] + 𝐽[𝑋, 𝐽𝑋] = 0, 𝑁𝐽 (𝑋, 𝐽𝑋) = [𝑋, 𝐽𝑋] − [𝐽𝑋, −𝑋] + 𝐽[𝐽𝑋, 𝐽𝑋] + 𝐽[𝑋, −𝑋] = 0. For a point 𝑝 ∈ 𝑆, if 𝑋 ≠ 0, then (𝑋(𝑝), 𝐽𝑋(𝑝)) is a basis for 𝑇𝑝 𝑆. Hence 𝑁𝐽 = 0 at 𝑝. This can be done at all points, so 𝑁𝐽 ≡ 0. By Newlander-Nirenberg Theorem 5.10, 𝐽 is integrable. This will be reproved in Corollary 6.53 without using the NewlanderNirenberg theorem.

(4) Note that the calculation in (5.20) works for an almost complex manifold (𝑀, 𝐽) of any dimension 2𝑛. Therefore for 𝐽 to be non-integrable, it has to happen that some 𝑁𝐽 (𝑒 1 , 𝑒 3 ) ≠ 0, with vectors 𝑒 1 , 𝑒 2 = 𝐽𝑒 1 , 𝑒 3 ∈ 𝑇𝑝 𝑀, 𝑝 ∈ 𝑀, and 𝑒 3 linearly independent with 𝑒 1 , 𝑒 2 . (5) Putting together (2) and (3), we obtain that every oriented Riemannian surface (𝑆, g) can be endowed with a natural structure of a complex curve. By item (7) on page 285, we can build a Hermitian metric by taking 𝜔(𝑋, 𝑌 ) = g(𝐽𝑋, 𝑌 ) and h = g−i 𝜔. By Remark 5.28(1), 𝑑𝜔 = 0, so automatically (𝑆, 𝐽, 𝜔) is a Kähler manifold. Therefore, every oriented Riemannian surface can be endowed with a unique Kähler structure extending the Riemannian structure. (6) Conversely, if (𝐶, 𝐽) is a complex curve, then any Hermitian metric h = g−i 𝜔 endows 𝑆 with a Riemannian metric g such that 𝐽 is the rotation of angle 𝜋/2 in the positive direction, by Example 5.26. Note that Hermitian metrics always exist (Exercise 5.5). Thus there is a bijection between Hermitian complex curves and Riemannian surfaces. This is the historic reason for calling Riemann surfaces to the complex curves. Note thus the distinction between a Riemann surface (when 𝐽 has been chosen) and a Riemannian surface (when g and hence also 𝐽 have been chosen). (7) In particular, any orientable surface Σ𝑔 , 𝑔 ≥ 0, admits complex structures. Take a compact connected oriented Riemannian surface (𝑆, g) and endow it with its natural Kähler structure. Recall that, since 𝑆 is compact and oriented, we have 2 𝐻𝑑𝑅 (𝑆, ℤ) = ⟨[𝛼]⟩ with ∫𝑆 𝛼 = 1. Therefore, if we take 𝜆 = ∫𝑆 𝜔 = Vol(𝑆) > 0, then 𝜔 2 [𝜔] = 𝜆[𝛼]. So [ 𝜆 ] ∈ 𝐻𝑑𝑅 (𝑆, ℤ) and 𝑆 is a Hodge manifold, taking the Kähler form 𝜋 𝜔̃ = 𝜆 𝜔. Hence, by Remark 5.46, 𝑆 is a smooth connected algebraic curve embedded in ℂ𝑃 𝑁 , for some 𝑁 > 0 large enough.

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5.3.1. Embedding a curve in ℂ𝑃 3 . In the next sections, we will develop some tools of algebraic geometry in order to study algebraic curves in detail and get some deep topological implications. We start with a smooth complex irreducible algebraic curve 𝐶 ⊂ ℂ𝑃 𝑁 , for some 𝑁 ≥ 3. Observe that in particular, 𝐶 is connected. In this section, we will prove that there exists a projection 𝜑 ∶ ℂ𝑃 𝑁 − 𝐿 → ℂ𝑃 3 such that 𝜑|𝐶 ∶ 𝐶 → ℂ𝑃 3 is an embedding (here 𝐿 is the center of the projection, which is a linear subspace not intersecting 𝐶). In this way, we end up having an embedding of our complex curve as 𝐶 ⊂ ℂ𝑃 3 . This is similar to the Whitney embedding theorem, which asserts that if 𝑀 is a smooth 𝑛-manifold, then there is an embedding 𝑀 ⊂ ℝ2𝑛+1 . In order to achieve this, we look for a projection 𝜑 ∶ ℂ𝑃 𝑁 −{𝑝0 } → ℂ𝑃 𝑁−1 such that 𝜑|𝐶 ∶ 𝐶 → ℂ𝑃 𝑁−1 is an embedding, 𝑝0 ∉ 𝐶 being the center of the projection, for any 𝑁 ≥ 4. Iterating this until 𝑁 = 4, we obtain the desired result. To get that the restriction 𝜑|𝐶 is an embedding, we have to choose the point 𝑝0 suitably. For that, we introduce the so-called secant and tangent varieties of 𝐶, denoted Sec (𝐶) and Tan (𝐶), respectively. To be precise, Sec (𝐶) ⊂ ℂ𝑃 𝑁 is the (Zariski) closure of the set of all projective lines joining two different points of 𝐶 and Tan (𝐶) ⊂ ℂ𝑃 𝑁 is the (Zariski) closure of the space of all the (complex) tangent lines of 𝐶. Let us see that both are algebraic subsets of ℂ𝑃 𝑁 and estimate their dimensions. Let 𝑇 ⊂ 𝐶 2 × ℂ𝑃 𝑁 be the subset of triples (𝑝, 𝑞, 𝑟) ∈ 𝐶 × 𝐶 × ℂ𝑃 𝑁 such that 𝑝, 𝑞, 𝑟 are collinear. Algebraically, it means that 𝑝0 𝑇 = {(𝑝, 𝑞, 𝑟) ∈ 𝐶 2 × ℂ𝑃 𝑁 || rk( ⋮ 𝑝𝑛

𝑞0 ⋮ 𝑞𝑛

𝑟0 ⋮ ) ≤ 2} . 𝑟𝑛

This set 𝑇 is a projective variety, since it is a zero set of all the order 3 minors of the matrix above. Let us consider the open set 𝑈 = {(𝑥, 𝑦) ∈ 𝐶 2 | 𝑥 ≠ 𝑦} ⊂ 𝐶 2 . Projecting onto the first two factors 𝜋1 ∶ 𝑇 → 𝐶 2 , we have that, for any (𝑥, 𝑦) ∈ 𝑈 ⊂ 𝐶 2 , 𝜋−1 1 (𝑥, 𝑦) is the projective line joining 𝑥 and 𝑦, so the fibers of 𝜋1 have generic dimension 1, hence 2 dimℂ 𝜋−1 1 (𝑈) = dimℂ 𝐶 + 1 = 3, by Remark 5.47(7). The variety Sec (𝐶) is the closure −1 of the open set 𝜋2 (𝜋1 (𝑈)) ⊂ ℂ𝑃 𝑁 , where 𝜋2 ∶ 𝐶 2 × ℂ𝑃 𝑁 → ℂ𝑃 𝑁 is the projection onto the last factor, thus dimℂ Sec (𝐶) ≤ 3. For the tangent variety, we take 𝑇 ′ ⊂ 𝐶 × ℂ𝑃 𝑛 as the set of pairs (𝑥, 𝑝) ∈ 𝐶 × ℂ𝑃 𝑁 such that 𝑝 ∈ 𝐭𝑥 𝐶, the tangent line to 𝐶 at 𝑥 inside ℂ𝑃 𝑛 . To see that 𝑇 ′ is an algebraic variety, define 𝐶𝑖 = 𝐶 ∩ 𝑈 𝑖 and 𝑇𝑖′ = 𝑇 ′ ∩ (𝐶𝑖 × 𝑈 𝑖 ), where 𝑈 𝑖 are the usual affine charts (see page 292). If 𝐶 = 𝑉(𝐹1 , . . . , 𝐹𝑟 ), let 𝐹1̂𝑖 , . . . , 𝐹𝑟̂𝑖 be the dehomogenized polynomials in the variable 𝑧𝑖 . We have that 𝑇𝑖′ = {(𝑧, 𝑝 = 𝑧 + 𝑣) ∈ 𝐶𝑖 × 𝑈 𝑖 | 𝑑𝑧 𝐹1̂𝑖 (𝑣) = 0, . . . , 𝑑𝑧 𝐹𝑟̂𝑖 (𝑣) = 0} , which is a quasi-projective variety of 𝐶𝑖 × 𝑈 𝑖 . The set 𝑇 ′ is the union of the Zariski closure of all 𝑇𝑖′ , so it is a projective variety of ℂ𝑃𝑁 . Take 𝜋1 ∶ 𝐶 × ℂ𝑃 𝑁 → 𝐶 and 𝜋2 ∶ 𝐶 × ℂ𝑃 𝑁 → ℂ𝑃 𝑁 , the projections onto the first and second factors, respectively. Observe that, for any 𝑥 ∈ 𝐶, 𝜋−1 1 (𝑥) = {𝑥} × 𝐭𝑥 𝐶, which is a complex line. Thus dimℂ 𝑇 ′ = dimℂ 𝐶 + 1 = 2 and, since Tan (𝐶) = 𝜋2 (𝑇 ′ ) we have dimℂ Tan (𝐶) ≤ 2. Therefore, as dimℂ (Sec (𝐶) ∪ Tan (𝐶)) < 4, if 𝑁 ≥ 4, then we can choose a point 𝑝0 ∈ ℂ𝑃 𝑁 − (Sec (𝐶) ∪ Tan (𝐶)). Fix any of such 𝑝0 , and let 𝐻 ⊂ ℂ𝑃 𝑁 be a hyperplane

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not containing 𝑝0 . Let 𝜑 ∶ ℂ𝑃 𝑁 − {𝑝0 } → 𝐻 ≅ ℂ𝑃 𝑁−1 be the projection from 𝑝0 to 𝐻, that is, for 𝑥 ∈ ℂ𝑃 𝑁 − {𝑝0 }, 𝜑(𝑥) is the intersection of the line joining 𝑥 and 𝑝0 with 𝐻. After a change of coordinates in ℂ𝑃𝑁 , we can take 𝑝0 = [1, 0, . . . , 0] and 𝐻 = {𝑧0 = 0}, so we have the explicit formula 𝜑([𝑧0 , 𝑧1 , . . . , 𝑧𝑁 ]) = [𝑧1 , . . . , 𝑧𝑁 ]. Observe that 𝜑 is injective on 𝐶 since, if 𝑥, 𝑦 ∈ 𝐶 are different points satisfying 𝜑(𝑥) = 𝜑(𝑦), then 𝑥𝑖 = 𝑦 𝑖 for 𝑖 > 0, and thus the complex line passing through 𝑥 and 𝑦 contains 𝑝0 . In that case, we would have 𝑝0 ∈ Sec (𝐶), contrary to assumption. To check that 𝜑|𝐶 is an embedding, it remains to see that the complex differential 𝑑𝑥 𝜑|𝐶 ∶ 𝑇𝑥 𝐶 → 𝑇𝜑(𝑥) ℂ𝑃 𝑁−1 is a monomorphism for 𝑥 ∈ 𝐶. Suppose for instance that 𝑥 lies in the chart 𝑈1 . This chart has coordinates (𝑧0 , 𝑧2 , . . . , 𝑧𝑁 ), where the projection is given by 𝜑(𝑧0 , 𝑧2 , . . . , 𝑧𝑁 ) = (𝑧2 , . . . , 𝑧𝑁 ). In these coordinates, let 𝑣 = (𝑣 0 , 𝑣 2 , . . . , 𝑣 𝑛 ) be a vector spanning 𝑇𝑥 𝐶. Then the tangent line 𝐭𝑥 𝐶 is the closure of 𝑥 + ℂ𝑣. As 𝑝0 ∉ 𝐭𝑥 𝐶, we have that 𝑣 is not vertical, i.e., there is some 𝑣𝑗 ≠ 0 with 𝑗 ≥ 2. Then 𝑑𝑥 𝜑|𝐶 (𝑣 0 , 𝑣 2 , . . . , 𝑣 𝑁 ) = (𝑣 2 , . . . , 𝑣 𝑁 ) ≠ 0, showing that 𝑑𝑥 𝜑|𝐶 is a monomorphism, as we wanted to prove. 5.3.2. Plane curves. Let 𝐶 = 𝑉(𝐹) ⊂ ℂ𝑃 2 be an irreducible projective plane curve. This is the zero set in ℂ𝑃 2 of an irreducible homogeneous polynomial 𝐹(𝑧0 , 𝑧1 , 𝑧2 ) (Remark 5.47(4)), of some degree 𝑑 > 0. By Remark 5.47(3), 𝐶 reg is a smooth connected complex curve, and 𝐶 sing is a 0-dimensional projective variety, that is a finite collection of points. Take 𝑝 ∈ 𝐶. Choosing suitable coordinates, we can assume that 𝑝 = [1, 0, 0]. We consider the affine chart 𝑈0 = ℂ2 , and the dehomogenized polynomial 𝑓(𝑧1 , 𝑧2 ) = 𝐹(1, 𝑧1 , 𝑧2 ) ∈ ℂ[𝑧1 , 𝑧2 ], which is a (non-homogeneous) polynomial of degree 𝑑 > 0 with 𝑓(0, 0) = 0. We can write 𝑓(𝑧1 , 𝑧2 ) = 𝑓1 (𝑧1 , 𝑧2 ) + 𝑓2 (𝑧1 , 𝑧2 ) + ⋯ + 𝑓𝑑 (𝑧1 , 𝑧2 ), where 𝑓𝑘 are homogeneous polynomials of degree 𝑘. In this context, observe that 𝐶 𝜕𝑓 𝜕𝑓 is smooth at 𝑝 ∈ 𝐶 if 𝑓1 (𝑧1 , 𝑧2 ) = 𝑎𝑧1 + 𝑏𝑧2 ≠ 0, since ( 𝜕𝑧 (0), 𝜕𝑧 (0)) = (𝑎, 𝑏). 1 2 Moreover, in that case the tangent line at 𝑝 is the projective line 𝐭𝑝 𝐶 given by the closure of 𝑝 + ℂ(𝑏, −𝑎) ⊂ ℂ2 , that is the line with equation 𝑓1 = 0. If 𝑓1 = 0, then 𝑝 is a singular point of 𝐶. In that case, we say that 𝑟 = min{𝑘 | 𝑓𝑘 ≠ 0} is the order of the singularity at 𝑝, and the set (5.21)

𝐭𝐜𝑝 𝐶 = {(𝑧1 , 𝑧2 ) ∈ ℂ2 | 𝑓𝑟 (𝑧1 , 𝑧2 ) = 0}

is called the tangent cone at 𝑝, where the overline means Zariski closure. Observe that for 𝑟 = 1 the tangent cone is precisely the tangent line. When 𝑟 = 2, we say that 𝑝 is a double point. In general, the tangent cone (5.21) is a collection of lines through 𝑟 𝑝 (possibly with multiplicity). This means that 𝑓𝑟 (𝑧1 , 𝑧2 ) = ∏𝑗=1 ℓ𝑗 (𝑧1 , 𝑧2 ), where ℓ𝑗 (𝑧1 , 𝑧2 ) = 𝑎𝑗 𝑧1 + 𝑏𝑗 𝑧2 are linear factors (which may be repeated). To check that, first take the factors 𝑧1 so that 𝑓𝑟 (𝑧1 , 𝑧2 ) = 𝑧𝑎1 𝑔(𝑧1 , 𝑧2 ), where 𝑔 is homogeneous of ∗ degree 𝑟 − 𝑎 and contains a non-zero monomial 𝛼𝑧𝑟−𝑎 ̂ 2 ) = 𝑔(1, 𝑧2 ) 2 , 𝛼 ∈ ℂ . Let 𝑔(𝑧 be the dehomogenization, which has degree 𝑟 − 𝑎. Let 𝜆1 , . . . , 𝜆𝑟−𝑎 ∈ ℂ be the roots

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of 𝑔(𝑧 ̂ 2 ). So 𝑔(𝑧 ̂ 2 ) = 𝛼 ∏(𝑧2 − 𝜆𝑗 ), which yields that 𝑔(𝑧1 , 𝑧2 ) = 𝛼 ∏(𝑧2 − 𝜆𝑗 𝑧1 ) after rehomogenization. Then 𝑓𝑟 (𝑧1 , 𝑧2 ) = 𝛼𝑧𝑎1 ∏(𝑧2 − 𝜆𝑗 𝑧1 ), as asserted. It is known [Ful] that when all linear factors of the tangent cone are different (nonproportional), then 𝐶 consists of 𝑟 smooth branches around 𝑝. This means that there 𝑟 is an 𝜖 > 0 so that 𝐶 ∩ 𝐵𝜖 (0) = ⋃𝑗=1 𝐶𝑗 , where 𝐶𝑗 is a smooth complex curve inside 𝑟

𝐵𝜖 (0) ⊂ ℂ2 , and 𝐭𝐜𝑝 𝐶 = ⋃𝑗=1 𝐭𝑝 𝐶𝑗 . Note that although 𝐶 is globally irreducible, the intersection 𝐶 ∩ 𝐵𝜖 (0) is reducible consisting of 𝑟 different complex submanifolds (which 𝑟 are only defined locally). If 𝑓𝑟 = ∏𝑗=1 ℓ𝑗 , then 𝐶𝑗 and 𝑉(ℓ𝑗 ) are biholomorphic, for 𝑗 = 1, . . . , 𝑟. Such a point is called a node of order 𝑟 or a nodal point. A node of order 2 is called simply a node. Around a node of order 𝑟, a projective curve looks like the wedge of 𝑟 discs (topologically). Remark 5.50. When there are repeated factors in the decomposition of 𝑓𝑟 , there appear other types of singularities with more complexity (see Figure 5.1). We mention two of them. • In the curve 𝐶 = {𝑧22 − 𝑧31 = 0}, the singular point 𝑝 = 0 is called a cusp. In this case, the map 𝑡 ↦ (𝑡2 , 𝑡3 ) is a local parametrization around 0, so 𝐶 has only one branch, but 𝑝 is a double point. The tangent cone is given by 𝑧22 = 0. Consider a closed small ball 𝐵𝜖̄ (0) around 0 with boundary the 3sphere 𝑆𝜖3 = 𝜕𝐵𝜖̄ (0). Then 𝐶 ∩ 𝑆𝜖3 ⊂ 𝑆𝜖3 is a closed 1-dimensional loop in 𝑆 3 , that is, a knot. In the particular case of 𝐶 defined by 𝑧22 − 𝑧31 = 0, the knot is the so-called (2, 3)-torus knot, that is, the closed curve generated (in the torus viewed as a square with word 𝑎𝑏𝑎−1 𝑏−1 ) by the line with slope 2/3. In general, algebraic singularities are related to knots, providing a bridge between singularity theory and topology. • The curve 𝐶 = {𝑧22 − 𝑧41 = (𝑧2 + 𝑧21 )(𝑧2 − 𝑧21 ) = 0}. Then 𝐶 has two branches at 0, given by 𝑧2 − 𝑧21 = 0 and 𝑧2 − 𝑧21 = 0. These two branches share the same tangent, and the tangent cone is 𝑧22 = 0. Such singularity is called a tacnode. The set 𝐶 ∩ 𝑆𝜖3 ⊂ 𝑆𝜖3 is a link (a collection of several knots tangled among them).

Figure 5.1. Singularities of plane curves: node (left), cusp (center), tacnode (right).

Projecting the curves to ℂ𝑃 2 with nodal singularities. Our objective now is to prove that, given a smooth projective curve 𝐶 ⊂ ℂ𝑃 3 , there exists a projection 𝜑 ∶ ℂ𝑃3 − {𝑝0 } → ℂ𝑃 2 such that 𝜑|𝐶 ∶ 𝐶 → ℂ𝑃 2 is an immersion onto a plane projective curve 𝐶 ′ = 𝜑(𝐶) with only nodes as singularities. This means that 𝜑 is generically

5.3. Complex curves

305

injective, except for the nodes 𝑞 ∈ 𝐶 ′ for which (𝜑|𝐶 )−1 (𝑞) = {𝑞1 , 𝑞2 }, and the two branches at 𝑞 are 𝐶1 = 𝜑(𝐵𝜖 (𝑞1 )) ∩ 𝐶 and 𝐶2 = 𝜑(𝐵𝜖 (𝑞2 )) ∩ 𝐶. We have to choose 𝑝0 conveniently. First note that if 𝐶 is contained in a plane, then it is already a plane curve, and we are finished. So assume that 𝐶 is not contained in a plane, and in particular 𝐶 is not a line. Even more, 𝐶 cannot contain a line since 𝐶 is irreducible. Let 𝑇 be the set of pairs (𝑥, 𝑦) ∈ 𝐶 2 such that the tangent lines to 𝐶 at 𝑥 and 𝑦 are coplanar. Since two lines are coplanar if and only if they intersect, 𝑇 is also the set of pairs of points whose tangent lines intersect. Hence 𝑇 is a projective variety. Now, let CopSec (𝐶) = {(𝑥, 𝑦, 𝑝) ∈ 𝑇 × ℂ𝑃 3 | 𝑝 ∈ 𝐿𝑥,𝑦 } be the set of secants at points with coplanar tangent lines, where 𝐿𝑥,𝑦 denotes the line through 𝑥 and 𝑦. As argued before, dimℂ CopSec (𝐶) = dim 𝑇 + 1. In order to prove that dimℂ CopSec (𝐶) ≤ 2, we need to prove that dimℂ 𝑇 ≤ 1. To see that, we need to see that 𝑇 is a proper subset of 𝐶 2 . Suppose, to the contrary, that 𝑇 = 𝐶 2 . We claim that in such a case ⋂𝑥∈𝐶 𝐭𝑥 𝐶 ≠ ∅. If all the tangents are equal, the result is trivial, so we can suppose that 𝑥, 𝑦 ∈ 𝐶 have different, but intersecting, tangents. Let {𝑧0 } = 𝐭𝑥 𝐶 ∩ 𝐭𝑦 𝐶, 𝜋 the plane containing 𝐭𝑥 𝐶 and 𝐭𝑦 𝐶, and 𝑍 = {𝑧 ∈ 𝐶 | 𝑧0 ∈ 𝐭𝑧 𝐶}. Since 𝐶 is not planar, 𝜋 ∩ 𝐶 is a finite set of points and thus, 𝑈 = 𝐶 − 𝜋 is an open dense set in 𝐶. Given 𝑧 ∈ 𝑈, by hypothesis 𝐭𝑧 𝐶 intersects 𝐭𝑥 𝐶 and 𝐭𝑦 𝐶 but, since 𝑧 ∉ 𝜋, it must intersect them at 𝑧0 . Therefore 𝑈 ⊂ 𝑍 and, since 𝑍 is closed, it must be 𝑍 = 𝐶, finishing the proof of the claim.

z

tz C

π

y ty C

z0

C x

tx C

Hence, if 𝑇 = 𝐶 2 , then all the tangent lines of 𝐶 intersect at some point 𝑝 ∈ ℂ𝑃 3 . Consider an affine chart 𝑈 𝑖 = ℂ3 with 𝑝 ∈ 𝑈 𝑖 , and let 𝐶𝑖 = 𝐶 ∩ 𝑈 𝑖 . Fix a smooth point 𝑥0 ∈ 𝐶𝑖 and take a parametrization 𝛾 ∶ 𝐵𝜖 (0) ⊂ ℂ → 𝐶𝑖 ⊂ ℂ3 of 𝐶 around 𝑥0 . As all the tangents meet, we have 𝛾(𝑧) = 𝑝 + 𝜆(𝑧)𝛾′ (𝑧) for some holomorphic function 𝜆 ∶ 𝐵𝜖 (0) → ℂ. In that case, differentiating, we obtain 𝛾′ (𝑧) = 𝜆′ (𝑧)𝛾′ (𝑧) + 1−𝜆′ 𝜆(𝑧)𝛾″ (𝑧). This means that 𝛾″ (𝑧) = 𝜆 𝛾′ (𝑧). As in the smooth case (where this would mean that there is no normal acceleration), this yields that 𝛾(𝑧) is locally a complex line. Just observe that the solution to this complex ordinary differential equation is 1−𝜆′ 𝛾′ (𝑧) = 𝑒𝐹(𝑧) 𝑣 0 , where 𝑣 0 ∈ ℂ3 is a fixed vector and 𝐹 ′ = 𝜆 . This implies that 𝛾 parametrizes locally a line. By algebraicity, this holds globally, so 𝐶 is a line, contrary to our assumption. Thus 𝑇 ≠ 𝐶 2 . In order to avoid singularities with 𝑟 > 2 branches, we need to eliminate triples of points projecting onto the same point. We define TriSec (𝐶) as the set of points of ℂ𝑃 3 aligned with three different points of 𝐶. We consider the projective variety 𝑇 ′ ⊂ 𝐶 3 of

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triples (𝑥, 𝑦, 𝑧) ∈ 𝐶 3 with 𝑥, 𝑦, 𝑧 collinear. Let 𝜋1 ∶ 𝑇 ′ → 𝐶 2 be the projection onto the first two factors, and let us check that 𝜋1 (𝑇 ′ ) is a proper subset of 𝐶 2 . Suppose on the contrary that 𝜋1 (𝑇 ′ ) = 𝐶 2 , and take two points 𝑥, 𝑦 ∈ 𝐶 such that the tangents 𝐭𝑥 𝐶, 𝐭𝑦 𝐶 do not intersect (possible by the conclusions above). The line 𝐿𝑥,𝑦 through 𝑥, 𝑦 intersects 𝐶 in (at least) another point 𝑧 ∈ 𝐶. Moving 𝑥 in a neighbourhood 𝑥′ ∈ 𝑈 𝑥 ⊂ 𝐶, and 𝑦 in a neighbourhood 𝑦′ ∈ 𝑉 𝑦 ⊂ 𝐶, we can see that the image of 𝑧′ = 𝑧(𝑥′ , 𝑦′ ) ∈ 𝐿𝑥′ ,𝑦′ ∩ 𝐶 moves in two independent directions. Therefore the points 𝑧′ fill a 4-dimensional ball 𝐵𝜖4 (𝑧) ⊂ ℂ𝑃 3 . This is a contradiction since in that case 𝐶 would contain a 4-ball, proving that dimℂ 𝜋1 (𝑇 ′ ) ≤ 1. As the fibers of 𝜋1 are the intersections 𝐿𝑥,𝑦 ∩ 𝐶 − {𝑥, 𝑦}, they are finite. Thus dimℂ 𝑇 ′ ≤ 1. Finally take the projection 𝜋 ∶ TriSec (𝐶) → 𝑇 ′ , whose fibers are lines, concluding that dimℂ TriSec (𝐶) ≤ 1 + 1 = 2. As a conclusion of the previous discussion, 𝑈 = ℂ𝑃 3 − (Tan (𝐶) ∪ TriSec (𝐶) ∪ CopSec (𝐶)) is a non-empty open set. Take a point 𝑝0 ∈ 𝑈 and consider the projection 𝜑 ∶ ℂ𝑃 3 − {𝑝0 } → 𝐻 onto a plane 𝐻 ⊂ ℂ𝑃 3 not containing 𝑝0 . As in the proof in section 5.3.1, 𝜑|𝐶 ∶ 𝐶 → ℂ𝑃 2 is an immersion. We also have to assure that the point 𝑝0 lies in finitely many secants, so that 𝜑|𝐶 is a generically one-to-one map. To do so, consider the secant variety Sec (𝐶) which has dimension dimℂ Sec (𝐶) ≤ 3. If dimℂ Sec (𝐶) < 3, then we take 𝑝0 ∈ 𝑈 ∩ (ℂ𝑃 3 − Sec (𝐶)) not lying in any secant. Otherwise, 𝜋2 ∶ Sec (𝐶) → ℂ𝑃 3 must have a finite fiber over some dense open set 𝑈 ′ ⊂ ℂ𝑃 3 , and we can take 𝑝0 ∈ 𝑈 ∩ 𝑈 ′ so that 𝑝0 only lies in finitely many secants. Therefore 𝜑|𝐶 is an embedding except at finitely many points. As 𝑝0 does not lie in trisecants, when this happens there are exactly two different points 𝑞1 , 𝑞2 ∈ 𝐶 with 𝑞 = 𝜑(𝑞1 ) = 𝜑(𝑞2 ). As 𝑝0 ∉ CopSec (𝐶), the tangent lines 𝐭𝑞1 𝐶, 𝐭𝑞2 𝐶 are not coplanar, so their images under 𝜑, which are the tangents to the two branches of 𝐶 ′ = 𝜑(𝐶) at 𝑞, are distinct. This implies that 𝑞 ∈ 𝐶 ′ is a node. Theorem 5.51. Let 𝐶 be a compact connected complex curve. Then there is an immersion 𝜑 ∶ 𝐶 → ℂ𝑃 2 , whose image 𝐶 ′ = 𝜑(𝐶) = 𝑉(𝐹) is an irreducible plane projective curve with only nodes. The curve 𝐶 ′ is called a planar model of 𝐶. We call degree of the curve 𝐶 ′ = 𝑉(𝐹) to the integer deg 𝐶 ′ = deg 𝐹 > 0. We recall that 𝐶 ′ is a line if deg 𝐶 ′ = 1, a conic if deg 𝐶 ′ = 2, and we call 𝐶 ′ a cubic if deg 𝐶 ′ = 3. Bézout theorem. Let 𝐶 = 𝑉(𝐹) be an irreducible plane projective curve of some degree 𝑑 > 0. Let 𝐿 ⊂ ℂ𝑃 2 be a projective line such that 𝐿 intersects 𝐶 only at regular points and transversally, i.e., 𝐿 is not tangent to 𝐶 at the intersection points. This can be arranged easily. Take a point 𝑞 ∉ 𝐶, fix coordinates such that 𝑞 = [0, 0, 1] and look at vertical lines {𝑧1 = 𝑐}, 𝑐 ∈ ℂ, in the affine coordinates (𝑧1 , 𝑧2 ) (that is, lines through 𝑞). The set of points of 𝐶 whose tangent is vertical gives an algebraic condition, hence it is a finite set. So we can choose 𝑐 ∈ ℂ so that the line {𝑧1 = 𝑐} is not tangent to 𝐶 and does not pass through 𝐶 sing . With this choice, all the intersection points are transverse and 𝐿 ∩ 𝐶 has exactly 𝑑 points. To see this, parametrize 𝐿 by (𝑐, 𝑧2 ) so that 𝐿 ∩ 𝐶 are the solutions of the equation (5.22)

𝑓(𝑧2 ) = 𝐹(1, 𝑐, 𝑧2 ) = 0.

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307

The polynomial 𝐹(𝑧0 , 𝑧1 , 𝑧2 ) contains a non-zero monomial 𝑧𝑑2 , as 𝐹(0, 0, 1) ≠ 0. So 𝑓(𝑧) has degree 𝑑, hence it has 𝑑 roots. Let us see that all of them are simple: if 𝜆 is a 𝜕𝐹 double root of 𝑓(𝑧2 ), then 𝑓(𝜆) = 0 and 𝑓′ (𝜆) = 0. Then 𝐹(𝑝) = 0 and 𝜕𝑧 (𝑝) = 0 for 2 𝑝 = (1, 𝑐, 𝜆) = 0. So 𝑝 ∈ 𝐶 is a point with a vertical tangent, contrary to assumption. Moreover, since both 𝐿 and 𝐶 reg are complex manifolds, the intersection is oriented so all the intersection numbers are 1. This computation can be put in the topological context of section 2.6.5 (especially (2.25)), recalling that the homology 𝐻2 (ℂ𝑃2 ) = ℤ⟨[𝐿]⟩ is generated by the fundamental class of any line (see (5.18)). We have that ⟨[𝐿], [𝐿]⟩ = 𝐿1 ⋅ 𝐿2 = 1 ∈ 𝐻0 (ℂ𝑃2 ) ≅ ℤ, by taking two distinct lines 𝐿1 , 𝐿2 , which clearly intersect transversally, using Proposition 2.163. By the computation above, we have ⟨[𝐶], [𝐿]⟩ = 𝐶 ⋅ 𝐿 = 𝑑, so it must be [𝐶] = 𝑑 [𝐿] ∈ 𝐻2 (ℂ𝑃2 ). This is a topological definition of the degree of a plane curve. Theorem 5.52 (Bézout). Let 𝐶1 , 𝐶2 be irreducible plane projective curves of degrees 𝑑1 and 𝑑2 . Then ⟨[𝐶1 ], [𝐶2 ]⟩ = 𝑑1 𝑑2 . In particular, if 𝐶1 and 𝐶2 intersect transversally with all the intersection points regular, then |𝐶1 ∩ 𝐶2 | = 𝑑1 𝑑2 . Proof. As [𝐶1 ] = 𝑑1 [𝐿] and [𝐶2 ] = 𝑑2 [𝐿], we have ⟨[𝐶1 ], [𝐶2 ]⟩ = 𝑑1 𝑑2 [𝐿]2 = 𝑑1 𝑑2 . The second part follows from Proposition 2.163. □ Theorem 5.52 also holds if 𝐶1 and 𝐶2 are reducible and even if they have multiple components (that is, the defining polynomials have repeated factors). This can be deduced by the bilinearity of the intersection product. From the Bézout formula, we get that any two plane curves intersect. In particular this implies that if 𝐶 is a smooth plane curve, it is connected since any two components must intersect. Thus by Remark 5.47(5), any smooth plane curve is irreducible. Remark 5.53. If 𝐶1 and 𝐶2 do not intersect transversally, but they do not share any irreducible component, we have a formula [Ful] ∑

𝐼𝑝 (𝐶1 , 𝐶2 ) = 𝑑1 𝑑2 ,

𝑝∈𝐶1 ∩𝐶2

where 𝐼𝑝 (𝐶1 , 𝐶2 ) is a positive integer called the intersection index of 𝐶1 and 𝐶2 at 𝑝. It can be defined as the number of intersection points when we perturb slightly 𝐶1 and 𝐶2 around 𝑝 until they intersect transversally. In this sense, 𝐼𝑝 (𝐶1 , 𝐶2 ) is the expected number of intersection points if 𝐶1 and 𝐶2 intersected transversally. This index can be computed in a purely algebraic way as follows. Suppose that 𝐶1 is smooth at 𝑝 and let us work in an affine neighbourhood of 𝑝. Let 𝜓 ∶ 𝐵𝜖 (0) ⊂ ℂ → 𝐶1 be a holomorphic parametrization of a neighbourhood of 𝑝 with 𝜓(0) = 𝑝. Let 𝐶2 = 𝑉(𝐹2 ), for some homogeneous polynomial 𝐹2 ∈ ℂ[𝑧0 , 𝑧1 , 𝑧2 ], and let 𝐹2̂ be a suitable dehomogenization. Then the holomorphic function 𝑓 = 𝐹2̂ ∘ 𝜓 ∶ 𝐵𝜖 (0) → ℂ has a zero of some order 𝑘 ≥ 1 at 0, and we declare 𝐼𝑝 (𝐶1 , 𝐶2 ) = 𝑘. Observe that, if 𝑘 = 1, both curves intersect transversally at 𝑝. In general, we can modify 𝐶2 slightly around 𝑝, e.g., by taking 𝑉(𝐹2̂ − 𝑎), for 𝑎 ∈ ℂ with |𝑎| small enough. Then the zero of

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multiplicity 𝑘 becomes 𝑘 different zeros for 𝑓 − 𝑎 = (𝐹2̂ − 𝑎) ∘ 𝜓. This gives a link with our topological definition. In the case that 𝑝 is a nodal point of 𝐶1 , we take the local branches 𝑋1 , . . . , 𝑋𝑟 of 𝐶1 at 𝑝. The theory of Newton-Puiseux series [New] says that every branch can always be holomorphically parametrized, so 𝐼𝑝 (𝑋𝑖 , 𝐶2 ) can be defined as above, and 𝐼𝑝 (𝐶1 , 𝐶2 ) = ∑𝑖 𝐼𝑝 (𝑋𝑖 , 𝐶2 ). 5.3.3. Degree-genus formula. This section is devoted to proving the following result, which links the topological invariant of a complex curve, its genus, with its algebraic invariants, its degree and number of nodes of a planar model. Theorem 5.54. Let 𝐶 be a compact connected complex curve of genus 𝑔 ≥ 0, and suppose that its plane model 𝜑 ∶ 𝐶 → 𝐶 ′ ⊂ ℂ𝑃 2 has degree 𝑑 > 0 and 𝛿 ≥ 0 nodes. Then, it holds that (𝑑 − 1)(𝑑 − 2) 𝑔= − 𝛿. 2 Proof. In a similar vein as in sections 5.3.1 and 5.3.2, we will find a suitable point 𝑃0 ∈ ℂ𝑃 2 to project from it onto a complex line 𝐿 ⊂ ℂ𝑃 2 , in such a way that we get a map 𝐶 → ℂ𝑃 1 which will be a ramified cover (Definition 2.60). More explicitly, let 𝜑 ∶ 𝐶 → ℂ𝑃 2 be an immersion onto a plane model 𝐶 ′ = 𝜑(𝐶) ⊂ ℂ𝑃 with nodal singularities, and let 𝑁 ⊂ 𝐶 ′ be the set of nodes. We will pick 𝑃0 outside 𝐶 ′ and outside all the tangent lines through the points of 𝑁. Observe that, since 𝑁 is a finite set, there are finitely many of such lines. Let 𝐿 ⊂ ℂ𝑃 2 be a line with 𝑃0 ∉ 𝐿, and let 𝜋 ∶ ℂ𝑃 2 − {𝑃0 } → 𝐿 = ℂ𝑃 1 be the projection from 𝑃0 to 𝐿. Take 𝑞 = 𝜋 ∘ 𝜑 ∶ 𝐶 → ℂ𝑃 1 . We claim that 𝑞 is a ramified cover of degree 𝑑. Moreover, for the ramification points 𝑅 ⊂ 𝐶, we will prove that 2

(5.23)

∑ (𝑚𝑝 − 1) = 𝑑(𝑑 − 1) − 2𝛿, 𝑝∈𝑅

where 𝑚𝑝 is the ramification index at 𝑝. The formula (2.4) for computing the EulerPoincaré characteristic of a ramified cover (which in the context of complex curves, is also known as Hurwitz formula) yields 2 − 2𝑔 = 𝜒(𝐶) = (deg 𝑞) 𝜒(ℂ𝑃 1 ) − ∑ (𝑚𝑝 − 1) = 2𝑑 − 𝑑(𝑑 − 1) + 2𝛿, 𝑝∈𝑅 (𝑑−1)(𝑑−2)

from where we infer 𝑔 = −𝛿, as required. In order to prove that 𝑞 is a ramified 2 cover and to compute the ramification indices 𝑚𝑝 for 𝑝 ∈ 𝑅, we take coordinates such that 𝑃0 = [0, 0, 1] ∉ 𝐶 ′ and 𝐿 = {𝑧2 = 0}. In these coordinates, 𝜋 ∶ ℂ𝑃 2 − {𝑃0 } → ℂ𝑃 1 is 𝜋([𝑧0 , 𝑧1 , 𝑧2 ]) = [𝑧0 , 𝑧1 ]. Moreover, we work in the coordinate chart 𝑈0 = {𝑧0 ≠ 0} in which 𝜋 ∶ 𝑈0 → ℂ ⊂ ℂ𝑃 1 is given by 𝜋(𝑧1 , 𝑧2 ) = 𝑧1 . The computations for the other coordinate chart 𝑈1 = {𝑧1 ≠ 0} are completely analogous. Let 𝐶 ′ = 𝑉(𝐹), for 𝐹 ∈ ℂ[𝑧0 , 𝑧1 , 𝑧2 ] irreducible and homogeneous of degree 𝑑. The ̂ 1 , 𝑧2 ) = 0, where 𝐹(𝑧 ̂ 1 , 𝑧2 ) = 𝐹(1, 𝑧1 , 𝑧2 ) ∈ ℂ[𝑧1 , 𝑧2 ] is the equation for 𝐶 ′ ∩ 𝑈0 is 𝐹(𝑧 ̂ 1 , 𝑧2 ) = 𝑓0 (𝑧1 , 𝑧2 ) + 𝑓1 (𝑧1 , 𝑧2 ) + ⋯ + 𝑓𝑑 (𝑧1 , 𝑧2 ), dehomogenized polynomial. Note 𝐹(𝑧 where 𝑓𝑘 (𝑧1 , 𝑧2 ) is the homogeneous piece of degree 𝑘 and 𝐹(𝑧0 , 𝑧1 , 𝑧2 ) = 𝑓0 (𝑧1 , 𝑧2 )𝑧𝑑0 + 𝑓1 (𝑧1 , 𝑧2 )𝑧𝑑−1 + ⋯ + 𝑓𝑑 (𝑧1 , 𝑧2 ). As 𝐹(0, 0, 1) = 𝑓𝑑 (0, 1) ≠ 0, this means that 𝑓𝑑 ≠ 0, so 0

5.3. Complex curves

309

𝐹 ̂ is of degree 𝑑. Maybe after dividing by the coefficient of 𝑧𝑑2 to make this polynomial 𝑑 𝑑−𝑗 𝑗 monic, we have 𝑓𝑑 (𝑧1 , 𝑧2 ) = 𝑧𝑑2 + ∑𝑗=1 𝑎𝑗 𝑧2 𝑧1 , for some 𝑎𝑗 ∈ ℂ. (1) Let 𝑝′ = 𝜑(𝑝) ∈ 𝐶 ′ be a smooth point with non-vertical tangent (see Figure 𝜕𝐹̂ 5.2). In that case, we have that 𝜕𝑧 (𝑝′ ) ≠ 0. Using a translation, we can sup2 pose that 𝑝′ = (0, 0) (such translation does not spoil the previous properties of our choice of coordinates). By the implicit function theorem, there exists a holomorphic map 𝜓 ∶ 𝑉1 ⊂ ℂ → 𝑉2 ⊂ ℂ, where 𝑉1 and 𝑉2 are neighbourhoods of 0 and 𝜓(0) = 0, such that 𝐶 ′ ∩ 𝐵𝜖 (𝑝′ ) = {(𝑧1 , 𝜓(𝑧1 )) | 𝑧1 ∈ 𝑉1 } , for 𝜖 > 0 small enough. Now we consider the neighbourhood around 𝑝 ∈ 𝐶 given by 𝑈 = 𝜑−1 (𝐶 ′ ∩ 𝐵𝜖 (𝑝′ )). Then the map 𝑞|𝑈 can be written as the composition 𝜑

𝜋

𝑞 ∶ 𝑈 ⟶ 𝐶 ′ ∩ 𝐵𝜖 (𝑝′ ) ⟶ 𝑉1 . For 𝜖 > 0 small enough, 𝜑|𝑈 is a biholomorphism. Also 𝜋 ∶ 𝐶 ′ ∩ 𝐵𝜖 (𝑝′ ) → 𝑉1 , 𝜋(𝑧1 , 𝑧2 ) = 𝑧1 , is a biholomorphism whose inverse is the holomorphic map 𝑧1 ↦ (𝑧1 , 𝜓(𝑧1 )). Therefore, 𝑞|𝑈 is a biholomorphism. (2) Let 𝑝′ ∈ 𝑁 ⊂ 𝐶 ′ be a nodal point (see Figure 5.2). In that case, there exists 𝜖 > 0 such that 𝐶 ′ ∩ 𝐵𝜖 (𝑝′ ) is the union of two smooth branches 𝐶1 , 𝐶2 ⊂ 𝐵𝜖 (𝑝) with 𝐶1 ∩ 𝐶2 = {𝑝′ }. There exists different points 𝑝1 , 𝑝2 ∈ 𝐶 with 𝜑(𝑝1 ) = 𝜑(𝑝2 ) = 𝑝′ , such that 𝜑 is a biholomorphism between a neighbourhood around 𝑝1 and 𝐶1 , and also between a neighbourhood of 𝑝2 and 𝐶2 . Since the tangents at 𝑝′ along 𝐶𝑖 , 𝑖 = 1, 2 are not vertical, the proof above shows that 𝑞 ∶ 𝐶 → ℂ𝑃 1 is a local biholomorphism around 𝑝 𝑖 and 𝜋(𝑝′ ), for 𝑖 = 1, 2. (3) Let 𝑝′ = 𝜑(𝑝) ∈ 𝐶 ′ be a smooth point with vertical tangent (see Figure 5.2). 𝜕𝐹̂ 𝜕𝐹 In this case 𝜕𝑧 (𝑝′ ) = 0. Since 𝑝′ is a smooth point, it must be 𝜕𝑧 (𝑝′ ) ≠ 0. 2 1 Again after a translation, we can suppose that 𝑝′ = (0, 0). By the implicit function theorem, there exists a holomorphic map 𝜓 ∶ 𝑉2 → 𝑉1 , 𝜓(0) = 0, such that (5.24)

𝐶 ′ ∩ 𝐵𝜖 (𝑝′ ) = {(𝜓(𝑧2 ), 𝑧2 ) | 𝑧2 ∈ 𝑉2 } , for 𝜖 > 0 small enough. Take 𝑈 = 𝜑−1 (𝐶 ′ ∩ 𝐵𝜖 (𝑝′ )), so that we have the following diagram. 𝑈

𝜑

/ 𝐶 ′ ∩ 𝐵𝜖 (𝑝′ ) O 𝜓×Id

𝑉2

/𝑉 u: 1 u u uu uu𝜓 u u uu 𝜋

Hence 𝑞|𝑈 = 𝜓∘(𝜓×Id)−1 ∘𝜑. Since 𝜑 and 𝜓×Id are local biholomorphisms at 𝑝 and 𝑝′ , respectively, we have that the ramification index of 𝑞 at 𝑝 coincides with the ramification index of 𝜓 ∶ 𝑉2 → 𝑉1 at 0.

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5. Complex geometry

P0 = [0; 0; 1]

p03 p01

p02

Figure 5.2. The point 𝑝1′ is in the situation of (1), 𝑝2′ is as in (2), and 𝑝3′ as in (3).

(4) Let 𝑓 ∶ 𝑈 ⊂ ℂ → 𝑈 ′ ⊂ ℂ be a holomorphic map with a zero at 0 of order 𝑘 ≥ 1. This means that 𝑓(𝑧) = 𝑧𝑘 𝑔(𝑧) with 𝑔(𝑧) a holomorphic function with 𝑔(0) ≠ 0. There exists a holomorphic function ℎ with ℎ(𝑧)𝑘 = 𝑔(𝑧) (maybe after reducing 𝑈), so 𝑓(𝑧) = (𝑧 ℎ(𝑧))𝑘 . The function 𝜙(𝑧) = 𝑧 ℎ(𝑧) satisfies that 𝜙′ (0) = ℎ(0) ≠ 0 so it is a biholomorphism around 0. Taking the coordinate 𝑤 = 𝜙(𝑧), we have that 𝑓 ̃ = 𝑓 ∘ 𝜙−1 ∶ 𝐵𝜖 (0) → 𝐵𝛿 (0) is given by ̃ 𝑓(𝑤) = 𝑤𝑘 , for 𝜖, 𝛿 > 0 small enough. Therefore 𝑓 is a ramified cover around 0, and the ramification index is 𝑘. (5) Thus the ramification index 𝑚𝑝 for a point 𝑝 as in step (3) coincides with the order 𝑘 of the zero of the map 𝜓 defining (5.24). This equals the multiplicity of the tangency of 𝐶 ′ at 𝑝′ , that is, the intersection index 𝐼𝑝′ (𝐭𝑝′ 𝐶 ′ , 𝐶 ′ ) = 𝑘, of 𝐶 ′ and its tangent line 𝐭𝑝′ 𝐶 ′ . Certainly, using Remark 5.53, we parametrize the curve 𝐶 ′ by (𝑧1 , 𝑧2 ) = (𝜓(𝑧2 ), 𝑧2 ), and substitute the parametrization into the equation of the tangent 𝐭𝑝′ 𝐶 ′ , which is 𝑧1 = 0, to get the function 𝜓(𝑧2 ), which has a zero of order 𝑘. We say that a tangent is an ordinary tangent when 𝑘 = 2, and an inflection point when 𝑘 ≥ 3; see Figure 5.3.

Figure 5.3. Ordinary tangent (left) and inflection point (right).

5.3. Complex curves

311

(6) At this point, it is easy to see that 𝑞 is a ramified cover. First, observe that all the fibers of 𝑞 are finite since, for 𝑥 ∈ 𝐿 ≅ ℂ𝑃 1 , we already know that 𝑞−1 (𝑥) is a discrete set of a compact set. Let 𝑥 ∈ ℂ𝑃 1 and 𝑞−1 (𝑥) = {𝑝1 , . . . , 𝑝𝑠 }. By (1), (2) and (3) above, there exists open neighbourhoods 𝑈1 , . . . , 𝑈𝑠 of 𝑝1 , . . . , 𝑝𝑠 , respectively, and 𝑉1 , . . . , 𝑉𝑠 of 𝑥 such that 𝑞|𝑈𝑖 ∶ 𝑈 𝑖 → 𝑉 𝑖 is a ramified cover. Take 𝑉 = ⋂𝑖 𝑉 𝑖 and 𝑈 = 𝑞−1 (𝑉), so 𝑞|𝑈 ∶ 𝑈 → 𝑉 is a ramified cover. Thus this holds globally, and 𝑞 is a ramified cover whose ramification points are the points 𝑅 = 𝜑−1 (𝑅𝑣 ) where 𝑅𝑣 ⊂ 𝐶 ′ is the set of points with vertical tangent. (7) Let 𝑥 ∈ 𝐿 ≅ ℂ𝑃 1 . The preimage 𝑞−1 (𝑥) is the set 𝜑−1 (𝐿𝑃0 ,𝑥 ∩ 𝐶 ′ ), where 𝐿𝑃0 ,𝑥 is the line through 𝑃0 and 𝑥. If 𝑥 ∉ 𝜋(𝑁) and it is a regular value of 𝑞, then all points in 𝑞−1 (𝑥) are of the type (1). That means that the line 𝐿𝑃0 ,𝑥 is not tangent to 𝐶 ′ at the points of 𝐿𝑃0 ,𝑥 ∩ 𝐶 ′ , so the intersection is transverse. By Bézout Theorem 5.52, |𝐿𝑃0 ,𝑥 ∩ 𝐶 ′ | = 𝑑, so the degree of the ramified cover is 𝑑. 𝜕𝐹

(8) Consider the projective variety 𝑌 ′ = 𝑉 ( 𝜕𝑧 ), which is a projective curve of 2 degree 𝑑−1. As 𝐶 ′ is irreducible, the intersection 𝐶 ′ ∩𝑌 ′ is a finite collection of 𝜕𝐹̂ points. The points 𝐶 ′ ∩ 𝑌 ′ are the points with vertical tangent (where 𝜕𝑧 = 0 2

in the affine coordinates), and the singular points of 𝐶 ′ (where both 𝜕𝐹̂ 𝜕𝑧2

𝜕𝐹̂ 𝜕𝑧1 ′

=

0, = 0 in the affine coordinates). Thus 𝐶 ′ ∩ 𝑌 ′ = 𝑅𝑣 ⊔ 𝑁. Since 𝐶 has degree 𝑑 and 𝑌 ′ has degree 𝑑 − 1, Theorem 5.52 yields 𝑑(𝑑 − 1) = ∑ 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) = ∑ 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) + ∑ 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ).

(5.25)

𝑝′ ∈𝐶 ′ ∩𝑌 ′

𝑝′ ∈𝑅𝑣

𝑝′ ∈𝑁

(9) 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) = 2 for 𝑝′ ∈ 𝑁. Around a node, 𝐶 ′ has two branches 𝐶1 , 𝐶2 , none of which have a vertical tangent. Then 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) = 𝐼𝑝′ (𝐶1 , 𝑌 ′ ) + 𝐼𝑝′ (𝐶2 , 𝑌 ′ ). Let us see that 𝐼𝑝′ (𝐶𝑖 , 𝑌 ′ ) = 1 for both branches. Locally 𝐶𝑖 ∩ 𝐵𝜖 (𝑝′ ) = 𝑉(𝑓𝑗 ), for a holomorphic function 𝑓𝑖 defined on the ball, where 𝐹 ̂ = 𝑓1 ⋅ 𝑓2 . Then 𝜕𝐹̂ 𝜕𝑓 𝜕𝑓 = 𝑓1 𝜕𝑧 2 + 𝜕𝑧 1 𝑓2 . As 𝐶1 is smooth with non-vertical tangent, we have that 𝜕𝑧 2

𝜕𝑓1 (𝑝′ ) 𝜕𝑧2

2

2

≠ 0. Therefore 𝜕𝑓 𝜕𝐹 ̂ )) = 𝐼𝑝′ (𝐶1 , 𝑉 ( 1 𝑓2 )) 𝜕𝑧2 𝜕𝑧2 = 𝐼𝑝′ (𝐶1 , 𝑉(𝑓2 )) = 𝐼𝑝′ (𝐶1 , 𝐶2 ) = 1,

𝐼𝑝′ (𝐶1 , 𝑌 ′ ) = 𝐼𝑝′ (𝐶1 , 𝑉 (

because they have different tangents. The second equality is seen by parametrizing 𝐶1 and inserting the parametrization in both equations, to see that the 𝜕𝑓 summand 𝑓1 𝜕𝑧 2 is irrelevant for the intersection index. 2

(10) Let 𝑝′ = 𝜑(𝑝) ∈ 𝑅𝑣 ⊂ 𝐶 ′ be a smooth point with vertical tangent. Then 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) = 𝑚𝑝 − 1, with 𝑚𝑝 the ramification index. To do the computation, suppose that 𝑝′ = (0, 0) as before. Then the local parametrization of 𝐶 ′ is of the form (𝜓(𝑧), 𝑧), with the notations of (3). Then 𝑓(𝑧1 , 𝑧2 ) = 𝑧1 − 𝜓(𝑧2 ) defines the curve 𝐶 ′ ∩ 𝐵𝜖 (𝑝′ ). Therefore 𝐹 ̂ = ℎ 𝑓, for some holomorphic 𝜕𝐹̂ 𝜕ℎ 𝜕𝑓 function ℎ with ℎ ≠ 0 on 𝐵𝜖 (𝑝′ ). Then 𝜕𝑧 = 𝜕𝑧 𝑓 + ℎ 𝜕𝑧 . We substitute the 2

2

2

312

5. Complex geometry

parametrization (𝜓(𝑧), 𝑧) into this function to obtain 𝜕𝑓 𝜕𝐹 ̂ (𝜓(𝑧), 𝑧) = ℎ(𝜓(𝑧), 𝑧) (𝜓(𝑧), 𝑧) = ℎ(𝜓(𝑧), 𝑧) 𝜓′ (𝑧), 𝜕𝑧2 𝜕𝑧2 since

𝜕𝑓 𝜕𝑧2

= 𝜓′ (𝑧). Therefore 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) is the order of the zero of 𝜓′ (𝑧) at

𝑧 = 0. But if 𝜓(𝑧) = 𝑧𝑘 𝑔(𝑧) with 𝑔(0) ≠ 0, then 𝜓′ (𝑧) = 𝑧𝑘−1 (𝑘 𝑔(𝑧) + 𝑧𝑔′ (𝑧)). So the order of the zero of 𝜓′ (𝑧) is 𝑘 − 1, whence 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) = 𝑚𝑝 − 1. (11) Plugging (8) and (9) into (5.25) and recalling that 𝛿 = |𝑁| is the number of nodes, we get 𝑑(𝑑 − 1) = ∑ 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) + ∑ 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) = ∑ (𝑚𝑝 − 1) + 2𝛿. 𝑝′ ∈𝑅𝑣

𝑝′ ∈𝑁

𝑝∈𝑅

□

This completes the proof of (5.23).

Example 5.55. Let 𝐶 be a compact connected complex curve of genus 𝑔 with a model 𝐶 ′ of degree 𝑑. (1) If 𝑑 = 1, then, since 𝑔 ≥ 0, we must have 𝛿 = 0 and 𝑔 = 0. Therefore 𝐶 ′ is smooth and diffeomorphic to 𝑆 2 . Actually, for a homogeneous polynomial of degree 1, 𝐹 = 𝑎𝑧0 + 𝑏𝑧1 + 𝑐𝑧2 ∈ ℂ[𝑧0 , 𝑧1 , 𝑧2 ], 𝐶 ′ = 𝑉(𝐹) = {𝑎𝑧0 + 𝑏𝑧1 + 𝑐𝑧2 = 0} is a projective line, biholomorphic to ℂ𝑃 1 . (2) If 𝑑 = 2, then we also have 𝛿 = 0 so 𝐶 ′ is smooth and 𝑔 = 0. In this case 𝐶 ′ is a conic. We can construct a biholomorphism 𝐶 ′ ≅ ℂ𝑃 1 explicitly as follows. Pick any 𝑝0 ∈ 𝐶 ′ and a line 𝐿 with 𝑝0 ∉ 𝐿. Then for any 𝑝 ∈ 𝐶 ′ draw the line 𝐿𝑝0 ,𝑝 through 𝑝0 and 𝑝 (in the case of 𝑝 = 𝑝0 , 𝐿𝑝0 ,𝑝0 is the tangent to 𝐶 ′ at 𝑝0 ). Then we define 𝜑 ∶ 𝐶 ′ → 𝐿,

𝜑(𝑝) = 𝐿 ∩ 𝐿𝑝0 ,𝑝 .

Geometrically this is a projection from a point 𝑝0 of the curve to a line. It is similar to the projection used in the proof of Theorem 5.54, with the difference that the center of the projection lies in the curve. Note that a stereographic projection is of this type.

p '(p)

p0

C0 L

5.3. Complex curves

313

In coordinates, we arrange that the equation of the conic 𝐶 ′ is 𝑧21 + 𝑧22 = 1 (all smooth conics are projectively equivalent). Take 𝑝0 = (1, 0) and the line 𝐿 = {𝑧1 = 0}. Take (0, 𝑡) ∈ 𝐿, parametrized by 𝑡 ∈ ℂ. The line through 𝑝0 and (0, 𝑡) intersects the conic at (5.26)

(𝑧1 , 𝑧2 ) = 𝜑−1 (𝑡) = (

1 − 𝑡2 2𝑡 , ), 1 + 𝑡2 1 + 𝑡2

giving the isomorphism 𝜑−1 ∶ 𝐿 → 𝐶 ′ . The point 𝑡 = ∞ corresponds to the tangent at 𝑝0 , and hence 𝜑−1 (∞) = 𝑝0 . In projective coordinates, we can write 𝜑−1 ∶ ℂ𝑃 1 → 𝐶 ′ , 𝜑−1 ([𝑡0 , 𝑡1 ]) = [1 + 𝑡2 , 1 − 𝑡2 , 2𝑡] = [𝑡02 + 𝑡12 , 𝑡02 − 𝑡12 , 2𝑡0 𝑡1 ]. This is a global parametrization of the curve 𝐶 ′ . (3) The projective curves 𝐶 ′ with 𝑔 = 0 are called rational curves. In this case, by (5.27) below, the desingularized curve 𝐶 is biholomorphic to ℂ𝑃1 , and thus there exists a parametrization 𝜑 ∶ ℂ𝑃 1 → 𝐶 ′ . Observe that, unlike in the smooth situation, only few curves can be globally parametrized, namely only those with genus 𝑔 = 0. (4) In general, the number of nodes is bounded. As 𝑔 ≥ 0, we have that 𝛿 ≤ (𝑑−1)(𝑑−2) (𝑑−1)(𝑑−2) . Also smooth projective curves can only have genus 𝑔 = , 2 2 with 𝑑 > 0 an integer. This does not cover all possible values, so for studying complex curves via plane models, we are forced to deal with curves with (at least nodal) singularities. (5) If 𝐶 ′ is a plane model with nodes of order 𝑟 ≥ 2, then the formula in Theorem 5.54 gets modified as follows. Let 𝑁 be the set of nodes, let 𝑟𝑝′ ≥ 2 be the order of 𝑝′ ∈ 𝑁, and let 𝛿𝑝′ = 𝑟𝑝′ (𝑟𝑝′ −1)/2. Then 𝑔 = (𝑑−1)(𝑑−2)/2−∑𝑝′ ∈𝑁 𝛿𝑝′ (Exercise 5.30). (6) If 𝑑 = 3, then 𝐶 ′ is a cubic plane curve. Clearly 𝛿 ≤ 1. If 𝛿 = 1, then 𝑔 = 0 and so 𝐶 ′ is a rational curve with one node. As an example, let 𝐶 ′ with equation 𝑧22 = 𝑧31 + 𝑧21 . This has a node at the origin with tangents 𝑧2 = ±𝑧1 .

L

p

p0

'(p)

To parametrize it, we work as before but taking the node 𝑝0 = (0, 0) as the center of the projection. Let 𝐿 be the line 𝑧1 = 1, and identify 𝑡 ∈ ℂ with

314

5. Complex geometry

1

C0 C0 1

1 Figure 5.4. Elliptic curve 𝑧22 = 𝑧31 − 𝑧1 . It has genus 𝑔 = 1.

(1, 𝑡) ∈ 𝐿. The line through 𝑝0 and (1, 𝑡) is 𝑧2 = 𝑡 𝑧1 , which intersects 𝐶 ′ at (𝑧1 , 𝑧2 ) = 𝜑−1 (𝑡) = (𝑡2 − 1, 𝑡3 − 𝑡), giving the parametrization. (7) If 𝑑 = 3 and 𝛿 = 0, then 𝐶 ′ is smooth of genus 𝑔 = 1. So 𝐶 ′ is diffeomorphic to a torus. These curves are called elliptic curves. We will do a thorough study of them in section 5.5. They cannot be parametrized by rational functions, but we will see that they admit a sort of holomorphic parametrization. As an example of a smooth cubic, take the equation 𝑧22 = 𝑧31 − 𝑧1 depicted in Figure 5.4. What we draw are the real points, which are actually two circles inside the surface 𝐶 ′ ≅ 𝑇 2 (including the point at infinity). (8) The above can be used to get changes of variables to convert integrals with radicals into primitives of rational functions. This allows us to solve integration by quadratures (that is, explicitly compute primitives). For instance, to 𝑑𝑥 compute ∫ , we set 𝑦 = √1 − 𝑥2 , and use the parametrization (5.26). 2 Then 𝑑𝑥 =

√1−𝑥 4𝑡 𝑑𝑡, (1+𝑡2 )2

∫

hence

𝑑𝑥 √1 − 𝑥 2

=∫

𝑑𝑥 1 + 𝑡2 4𝑡 =∫ 𝑑𝑡, 𝑦 2𝑡 (1 + 𝑡2 )2

which is the integral of a rational function, and hence explicitly computable. 𝑃(𝑥,𝑦) In general, suppose that we want to compute ∫ 𝑄(𝑥,𝑦) 𝑑𝑥, where 𝑃, 𝑄 are polynomials, and 𝑦 is an algebraic function of 𝑥, that is, it satisfies a polynomial equation 𝐹(𝑥, 𝑦) = 0. If the curve 𝐶 = 𝑉(𝐹) is of genus 0, then it can be parametrized as 𝜑 ∶ ℂ𝑃 1 → 𝐶, 𝜑(𝑡) = (𝑥(𝑡), 𝑦(𝑡)), where 𝑥(𝑡), 𝑦(𝑡) are ratio𝑃(𝑥,𝑦) 𝑃(𝑥(𝑡),𝑦(𝑡)) nal functions. Then we can substitute ∫ 𝑄(𝑥,𝑦) 𝑑𝑥 = ∫ 𝑄(𝑥(𝑡),𝑦(𝑡)) 𝑥′ (𝑡)𝑑𝑡 and reduce the integral to a rational integral. (9) A surprising connection with arithmetic is provided by Belyi theorem [GGD]. It asserts that a smooth plane algebraic curve 𝐶 ′ = 𝑉(𝐹) admits an equation whose coefficients are algebraic numbers, i.e., 𝐹 ∈ ℚ[𝑧0 , 𝑧1 , 𝑧2 ], if and only if 𝐶 ′ admits a ramified cover over ℂ𝑃 1 with only three ramification values in

5.4. Classification of complex curves

315

ℂ𝑃 1 . Here ℚ is the algebraic closure of ℚ, that is 𝜆 ∈ ℚ ⊂ ℂ if and only if 𝜆 is the root of a polynomial with rational coefficients.

5.4. Classification of complex curves In this section, we deal with the problem of the classification of compact connected complex curves. As we will see, this classification has a moduli space, being close to the classification of metrics of constant curvature in Chapter 4. Therefore a smooth (compact, connected) oriented (real) surface will admit infinitely many non-isomorphic complex structures when 𝑔 ≥ 1. Let 𝐶 be a compact connected complex curve and consider the universal cover 𝜋 ∶ 𝐶 ̃ → 𝐶. By Exercise 5.6, 𝐶 ̃ has a natural complex structure such that 𝜋 is a holomorphic map, and the deck transformations Γ = Deck(𝜋) are biholomorphisms of 𝐶.̃ In this way, the complex structure of 𝐶 is recovered from the complex structure of 𝐶 ̃ as the ̃ for a subgroup Γ < Bihol(𝐶)̃ acting freely and properly. quotient 𝐶 ≅ 𝐶/Γ, By the classification of surfaces we know that 𝐶 is homeomorphic to some Σ𝑔 and therefore the universal cover 𝐶 ̃ is homeomorphic either to 𝑆 2 or ℝ2 . Therefore, in order to classify compact complex curves, we must focus first in the simply connected case. The result is given by a deep theorem of Poincaré and Koebe, which is a classical result in the theory of one complex variable. It can be found in [F-K]. We shall give a proof in the relevant case that we use, later in Corollary 6.55 by using analytical arguments. Theorem 5.56 (Uniformization theorem). Let 𝐶 ̃ be a simply connected complex curve, i.e., dimℂ 𝐶 ̃ = 1. Then 𝐶 ̃ is biholomorphic to one (and only one) of the following: (I) ℂ = ℂ𝑃 1 , (II) ℂ, (III) 𝔻 = {𝑧 ∈ ℂ | |𝑧| < 1}. Remark 5.57. • The uniqueness in Theorem 5.56 is easy. First ℂ is the only space compact, so it is not biholomorphic to ℂ or 𝔻. Also, if 𝑓 ∶ ℂ → 𝔻 is a holomorphic map, then it is a holomorphic and bounded map on the whole of ℂ. By the Liouville theorem (Exercise 5.1(3)), 𝑓 is constant, and hence it cannot be a biholomorphism. So ℂ ≇ 𝔻. • In the case that 𝐶 ̃ = 𝑈, where 𝑈 ⊂ ℂ is an open set of the complex plane, the uniformization theorem is basically the Riemann mapping theorem. This says that if 𝑈 ⊂ ℂ is simply connected and 𝑈 ≠ ℂ, then 𝑈 is biholomorphic to the disc 𝔻. • In the uniformization theorem, the hypothesis of second countable for the definition of manifold in Definition 1.1 is not necessary [C-R]. Now we move to the classification of complex curves case by case according to its universal cover.

316

5. Complex geometry

5.4.1. Case (I). The universal cover is ℂ. Let 𝐶 be a connected complex curve and suppose that the universal cover 𝐶 ̃ → 𝐶 satisfies that 𝐶 ̃ ≅ ℂ. Let Γ < Bihol(ℂ) be the group of deck transformations. By Proposition 4.22, Bihol(ℂ) = Mob(ℂ). A group Γ of biholomorphisms of ℂ acting freely and properly on ℂ must be Γ = {Id}. This follows 𝑐+𝑑𝑧 because any Möbius map 𝜑 always has a fixed point, since 𝜑(𝑧) = 𝑎+𝑏𝑧 = 𝑧 has at least one solution 𝑧 ∈ ℂ. The conclusion is that 𝐶 ≅ ℂ, so the list of complex curves in Case (I) is (5.27)

𝕃𝐂𝐌𝐚𝐧(I) = {ℂ}.

Note that in this case the genus of 𝐶 is 𝑔 = 0. Also the map 𝕃𝐂𝐌𝐚𝐧(I) → 𝕃𝐃𝐌𝐚𝐧𝑔=0 to the 𝑜𝑟 list of smooth compact connected surfaces of genus 0 is a bijection. 5.4.2. Case (II). The universal cover is ℂ. Let 𝐶 be now a connected complex curve such that its universal cover 𝜋 ∶ 𝐶 ̃ → 𝐶 satisfies that 𝐶 ̃ ≅ ℂ. So far, we are not going to assume that 𝐶 is compact, so that we also deal with the non-compact case. Lemma 5.58. Let 𝜑 ∶ ℂ → ℂ be a biholomorphism of ℂ. Then 𝜑(𝑧) = 𝑎𝑧 + 𝑏, for some 𝑎, 𝑏 ∈ ℂ, 𝑎 ≠ 0. Proof. Let 𝜑 ∶ ℂ → ℂ be a biholomorphism. Then as a map 𝜑 ∶ ℝ2 → ℝ2 it is a homeomorphism, so it extends to the Alexandroff compactification continuously 𝜑̂ ∶ 𝑆 2 → 𝑆 2 as a homeomorphism. Otherwise said, 𝜑 maps compact sets to compact sets. The neighbourhoods of ∞ in the Alexandroff compactification 𝑆 2 = ℝ2 ∪ {∞} are the ̂ sets 𝑈 = (ℝ2 − 𝐾) ∪ {∞}, where 𝐾 ⊂ ℝ2 is a compact set. So setting 𝜑(∞) = ∞, we have that 𝜑̂ maps the neighbourhoods of ∞ to neighbourhoods of ∞, proving that 𝜑̂ is continuous at ∞. The same argument applies to the inverse, giving that 𝜑̂ is a homeomorphism. Now we check that 𝜑̂ is a biholomorphism of ℂ = ℂ ∪ {∞} = 𝑆 2 . It remains to see ̂ that it is holomorphic at 𝑧 = ∞. To prove this, we look at the map 𝑤 = 𝜑(𝑧), using the charts 𝑧′ = 1/𝑧 and 𝑤′ = 1/𝑤 around ∞. The map 𝜑̂ is given by 𝑤′ = 𝑔(𝑧′ ) where 𝑤′ =

1 1 1 = = = 𝑔(𝑧′ ). ′) 𝑤 ̂ ̂ 𝜑(𝑧) 𝜑(1/𝑧

We know that 𝑔 is continuous and 𝑔(0) = 0. Take a ball 𝐵𝜖 (0) such that 𝑔(𝐵𝜖 (0)) ⊂ 𝐵1 (0). Then 𝑔 ∶ 𝐵𝜖 (0) − {0} → 𝐵1 (0) − {0} is holomorphic and bounded. Therefore it cannot have a singularity at 𝑧′ = 0, and hence it extends holomorphically to the origin. This means that 𝑔 is holomorphic at 𝑧′ = 0, meaning that 𝜑̂ is holomorphic at ∞ ∈ ℂ. ̂ , we get that 𝜑̂ is a biholomorphism of ℂ. Doing the same with the inverse 𝜑−1 ̂ Applying Proposition 4.22, 𝜑̂ ∈ Mob(ℂ). As 𝜑(∞) = ∞, it must be of the form ̂ = 𝑎𝑧 + 𝑏, for some 𝑎, 𝑏 ∈ ℂ, 𝑎 ≠ 0. 𝜑(𝑧) □ Let Γ be a subgroup of biholomorphisms of ℂ acting freely and properly on ℂ. By Lemma 5.58, any 𝜑 ∈ Γ is of the form 𝜑(𝑧) = 𝑎𝑧 + 𝑏, for some 𝑎, 𝑏 ∈ ℂ, 𝑎 ≠ 0. If 𝜑(𝑧) = 𝑎𝑧 + 𝑏 has no fixed points on ℂ, it must be 𝑎 = 1. So 𝜑(𝑧) = 𝑧 + 𝑏 = 𝜏𝑏 (𝑧) is a translation. This means that Γ is a subgroup of the group of translations. By Proposition 4.41, it is a free Abelian group of translations defined by the lattice Λ = {𝑏 ∈ ℂ | 𝜏𝑏 ∈ Γ}.

5.4. Classification of complex curves

317

Let Λ be a lattice of ℂ, and let 𝐶 = ℂ/Λ. We have the following cases: • If rank Λ = 0, then 𝐶 = ℂ is the complex plane. • If rank Λ = 1, then Λ = ℤ⟨𝑢0 ⟩, for some non-zero complex number 𝑢0 ≠ 0. Then 𝐶 = ℂ/Λ is topologically a cylinder. Note that the biholomorphism 𝐹 ∶ ℂ → ℂ, 𝐹(𝑧) = 𝑢0 𝑧, sends Λ0 = ℤ to Λ = ℤ⟨𝑢0 ⟩, inducing a biholomorphism 𝐹 ̄ ∶ ℂ/ℤ → ℂ/Λ. Therefore 𝐶 ≅ ℂ/ℤ. This means that there is only one complex structure of this type for the cylinder. A different description of this manifold can be given as follows. The map 𝐺 ∶ ℂ → ℂ∗ , 𝐺(𝑧) = 𝑒2𝜋i𝑧 , is surjective and a local biholomorphism, since 𝐺 ′ (𝑧) ≠ 0, for all 𝑧 ∈ ℂ. Also 𝐺(𝑧 + 𝑚) = 𝑒2𝜋i(𝑧+𝑚) = 𝑒2𝜋i𝑧 = 𝐺(𝑧), for 𝑚 ∈ ℤ. Hence it induces a biholomorphism 𝐺̄ ∶ ℂ/ℤ → ℂ∗ . So the complex cylinder is ℂ∗ . • If rank Λ = 2, then Λ = ⟨𝑢1 , 𝑢2 ⟩ is given by a lattice generated by two ℝlinearly independent complex numbers. The quotient 𝐶 = ℂ/Λ = ℝ2 /Λ is topologically a 2-torus, and hence it is a compact connected surface of genus 𝑔 = 1. A complex curve of the type 𝐶 = ℂ/Λ is called a complex torus. Actually, when the complex curve 𝐶 is topologically a torus, then it must lie in case (II), hence it is a complex torus (Corollary 5.63). Now we move to the classification of complex tori. In analogy with Proposition 4.42, we have the following result. Lemma 5.59. Let Λ and Λ′ be lattices of ℂ, and let 𝐶 = ℂ/Λ and 𝐶 ′ = ℂ/Λ′ the corresponding complex tori. Then 𝐶 and 𝐶 ′ are biholomorphic if and only if there exists 𝑎 ∈ ℂ∗ such that 𝑎Λ = Λ′ . Proof. Suppose that 𝑓 ∶ 𝐶 → 𝐶 ′ is a biholomorphism. Denote 𝜋 ∶ ℂ → 𝐶 = ℂ/Λ and 𝜋′ ∶ ℂ → 𝐶 ′ = ℂ/Λ′ the universal covers of 𝐶 and 𝐶 ′ . We lift the map 𝑓 to the universal covers to get a map 𝑓 ̃ ∶ ℂ → ℂ such that 𝜋′ ∘ 𝑓 ̃ = 𝑓 ∘ 𝜋. The map 𝑓 ̃ is clearly a local biholomorphism, and a cover, hence it is a biholomorphism. By Lemma 5.58, ̃ = 𝑎𝑧 + 𝑏, for some 𝑎, 𝑏 ∈ ℂ, 𝑎 ≠ 0. 𝑓(𝑧) Now for every 𝑢 ∈ Λ, we have that 𝜋′ ∘ 𝑓 ̃ ∘ 𝜏ᵆ = 𝑓 ∘ 𝜋 ∘ 𝜏ᵆ = 𝑓 ∘ 𝜋 = 𝜋′ ∘ 𝑓,̃ which ̃ ) = 𝜋′ , and hence 𝑓 ̃ ∘ 𝜏ᵆ ∘ 𝑓−1 ̃ = 𝜏𝑣 for some 𝑣 ∈ Λ′ . implies that 𝜋′ ∘ (𝑓 ̃ ∘ 𝜏ᵆ ∘ 𝑓−1 ̃ ̃ Moreover, since 𝑓(𝜏ᵆ (𝑧)) = 𝑎(𝑧 + 𝑢) + 𝑏 = 𝜏𝑣 (𝑓(𝑧)) = 𝑎𝑧 + 𝑏 + 𝑣, it must be 𝑣 = 𝑎𝑢. ̃ (𝑤) = 𝑎−1 𝑤 − 𝑏𝑎−1 is the lifting of This means that 𝑎Λ ⊂ Λ′ . On the other hand, 𝑓−1 −1 ′ the map 𝑓 ∶ 𝐶 → 𝐶 to the universal covers. Hence the same argument gives that 𝑎−1 Λ′ ⊂ Λ. It follows that 𝑎Λ = Λ′ . Conversely, if two lattices Λ and Λ′ satisfy that 𝑎Λ = Λ′ for some 𝑎 ∈ ℂ∗ , then ̃ the map 𝑓 ̃ ∶ ℂ → ℂ, 𝑓(𝑧) = 𝑎𝑧 induces a biholomorphism from 𝐶 = ℂ/Λ to 𝐶 ′ = ℂ/Λ′ . □ Therefore a complex torus 𝐶 = ℂ/Λ is characterized by the lattice Λ modulo the action of ℂ∗ by multiplication. By (4.19), the set of lattices is described as +

ℛ = GL(2, ℝ)/ GL(2, ℤ) ≅ GL (2, ℝ)/ SL(2, ℤ),

318

5. Complex geometry

where GL(2, ℝ) accounts for the choice of a basis of a lattice, and we have to quotient by the different possible changes of basis for the lattice, which is given by the action of GL(2, ℤ). The second description is given when we choose only the oriented basis of a lattice. Note that for a complex torus 𝐶 = ℂ/Λ, there is a natural orientation. Therefore the second description of ℛ is more suitable for our purposes. Let ℳ𝑇ℂ2 be the moduli space of complex tori. By Lemma 5.59, we have +

ℳ𝑇ℂ2 = ℂ∗ \ℛ = ℂ∗ \ GL (2, ℝ)/ SL(2, ℤ). The action of 𝑎 ∈ ℂ∗ is given by taking the inclusion 𝚤 ∶ ℂ∗ = GL(1, ℂ) ↪ GL(2, ℝ), and for each matrix 𝑀 ∈ GL(2, ℝ) corresponding to a basis of Λ, we let it act by matrix multiplication on the left, as (𝑎, 𝑀) ↦ 𝚤(𝑎) ⋅ 𝑀. Recall that we have a classification of oriented Riemannian flat tori given as (see (4.23)) + ℳ𝑇𝑜𝑟2 = SO(2)\ GL (2, ℝ)/ SL(2, ℤ). Every complex torus 𝐶 = ℂ/Λ admits a Riemannian metric of zero curvature, by writing it as 𝐶 = ℝ2 /Λ and giving it the induced metric from (ℝ2 , g𝑠𝑡𝑑 ). Moreover, this metric is compatible with the complex structure, that is it comes from a Hermitian metric as in Example 5.26. The map that associates to each Riemannian metric g its natural complex structure is +

(5.28)

ℳ𝑇𝑜𝑟2 = SO(2)\ GL (2, ℝ)/ SL(2, ℤ) +

⟶ ℳ𝑇ℂ2 = ℂ∗ \ℛ = ℂ∗ \ GL (2, ℝ)/ SL(2, ℤ).

Recall the polar decomposition of a non-zero complex number as 𝑧 = 𝜌 𝑒i𝜃 , where 𝜌 = |𝑧| ∈ ℝ>0 and 𝑒i𝜃 ∈ 𝑆 1 . This gives an identification ℂ∗ = ℝ>0 × 𝑆 1 . Mapping with 𝚤 ∶ ℂ∗ → GL(2, ℝ), 𝚤(𝑆 1 ) = SO(2) and 𝚤(ℝ>0 ) = {𝜆 Id |𝜆 ∈ ℝ>0 }. Therefore 𝚤(ℂ∗ ) = ℝ>0 ⋅ SO(2) < GL(2, ℝ), where ℝ>0 are understood as the homotheties of positive ratio. So +

ℳ𝑇ℂ2 = ℂ∗ \ GL (2, ℝ)/ SL(2, ℤ) (5.29)

+

= (ℝ>0 ⋅ SO(2))\ GL (2, ℝ)/ SL(2, ℤ) +

= ℝ>0 \(SO(2)\ GL (2, ℝ)/ SL(2, ℤ)) = ℳ𝑇𝑜𝑟2 /ℝ>0 , where the action of ℝ>0 on ℳ𝑇𝑜𝑟2 corresponds, at the level of lattices, to (𝜆, Λ) ↦ 𝜆Λ. This action changes the area of the lattice by multiplying it by a factor 𝜆2 . The map (5.28) is surjective, which means that every complex torus 𝐶 = ℂ/Λ admits compatible flat Riemannian structures and the different such structures are determined by a factor which is given by fixing the total area (or alternatively, by fixing the length of the shortest closed geodesic). Equivalently, ℳ𝑇ℂ2 ≅ ℳ𝑇𝑜𝑟,1 2 , described in Remark 4.78. So ℳ𝑇𝑜𝑟2 ≅ ℳ𝑇ℂ2 × ℝ>0 . In particular, ℳ𝑇ℂ2 is a manifold of dimension 2. The description in (4.32) goes over to the current situation. Theorem 5.60. There is a natural homeomorphism ℳ𝑇ℂ2 ≅ 𝐻/ PSL(2, ℤ), where 𝐻 = {𝜏 ∈ ℂ| Im 𝜏 > 0} is the upper half-plane, and PSL(2, ℤ) acts by Möbius transformations. Proof. Let Λ ⊂ ℂ be a rank 2 oriented lattice, so that Λ = ⟨𝑢1 , 𝑢2 ⟩ for 𝑢1 , 𝑢2 ∈ ℂ∗ a −1 pair of ℝ-linearly independent vectors. We can multiply Λ by 𝑢−1 1 and we get 𝑢1 Λ = ᵆ2 ᵆ2 ⟨1, ᵆ ⟩ = ⟨1, 𝜏⟩, for 𝜏 = ᵆ . As (𝑢1 , 𝑢2 ) is a positive basis, the basis (1, 𝜏) is also oriented, 1

1

5.4. Classification of complex curves

319

which is equivalent to Im(𝜏) > 0, or 𝜏 ∈ 𝐻. In this way, we obtain a normalised representative of every equivalence class of lattices whose first vector is 1 and whose second + vector is 𝜏 ∈ 𝐻. In terms of the quotient (5.29), we have proved that ℂ∗ \ GL (2, ℝ) ≅ 𝐻. ℂ Hence ℳ𝑇 2 ≅ 𝐻/ PSL(2, ℤ). Let us see how SL(2, ℤ) acts on 𝐻. For 𝜏 ∈ 𝐻, denote Λ𝜏 = ⟨1, 𝜏⟩. Consider a 𝑎 𝑐 matrix 𝐴 = ( ) ∈ SL(2, ℤ). It acts on the basis (1, 𝜏) of Λ𝜏 by changing it to 𝑏 𝑑 (𝑎 + 𝑏𝜏, 𝑐 + 𝑑𝜏). Then Λ𝜏 = ⟨1, 𝜏⟩ = ⟨𝑎 + 𝑏𝜏, 𝑐 + 𝑑𝜏⟩ ∼ ⟨1,

𝑐 + 𝑑𝜏 ⟩ = ⟨1, 𝜏′ ⟩ = Λ𝜏′ , 𝑎 + 𝑏𝜏

𝑐+𝑑𝜏

𝑐+𝑑𝜏

with 𝜏′ = 𝑎+𝑏𝜏 . So the action of 𝐴 ∈ SL(2, ℤ) is given as 𝜏′ = 𝑎+𝑏𝜏 , that is, it acts by Möbius transformations on 𝐻. Finally, recall that the action of SL(2, ℤ) factors through an action of PSL(2, ℤ). □ Summing up, the classification of complex curves in Case (II) is given by the list 𝕃𝐂𝐌𝐚𝐧(II) = {ℂ, ℂ∗ } ⊔ (𝐻/ PSL(2, ℤ)). Let us describe geometrically the moduli space 𝐻/ PSL(2, ℤ). By Remark 4.78, a fundamental domain for the action of PSL(2, ℤ) on 𝐻 is given by Figure 4.5 with 𝑑 = 1, that is 1 1 𝒟̂ = {(𝛼, 𝛽) | 𝛼2 + 𝛽 2 ≥ 1, − ≤ 𝛼 ≤ , 𝛽 > 0} 2 2 1 = {𝜏 ∈ 𝐻 | |𝜏| ≥ 1, | Re(𝜏)| ≤ } . 2 1

1

As described in Remark 4.78, we have to glue the boundary sides by ( 2 , 𝛽) ∼ (− 2 , 𝛽), for 𝛽 ≥

√3 , 2

and (𝛼, 𝛽) ∼ (−𝛼, 𝛽), for 𝛼2 + 𝛽 2 = 1. In terms of the complex variable 𝜏, 1

1

the boundary Re(𝜏) = ± 2 is glued via 𝜏 ↦ 𝜏 + 1, and the boundary |𝜏| = 1, | Re(𝜏)| ≤ 2 is glued by 𝜏 = 𝑒i𝜃 ↦ −1/𝜏 = 𝑒i(𝜋−𝜃) . Consider the maps 𝜑(𝜏) = 𝜏 + 1 and 𝜓(𝜏) = −1/𝜏, that generate the action of PSL(2, ℤ) on 𝐻 (Remark 4.78). The tessellation on 𝐻 produced by this action is as follows.

a

^ D

b

-1=2

a

b

1=2

Figure 5.5. Tessellation of the action of PSL(2, ℤ) on 𝐻. The region 𝒟̂ is the fundamental domain.

320

5. Complex geometry

The action of PSL(2, ℤ) on 𝐻 is not free. There are two fixed points on 𝒟.̂ First, the point 𝜏1 = i is a fixed point of 𝜓, with isotropy Γ𝜏1 = ⟨𝜓⟩ ≅ ℤ2 . The point 𝜏2 = 𝜋i

1

√3

𝑒 3 = 2 + 2 i ∈ 𝐻 is a fixed point of 𝜙 = 𝜑 ∘ 𝜓, 𝜙(𝑧) = (𝑧 − 1)/𝑧. As 𝜙3 = Id, the isotropy of 𝜏2 is Γ𝜏2 = ⟨𝜙⟩ ≅ ℤ3 . Following the philosophy mentioned on page 229, the moduli ℳ𝑇ℂ2 , that parametrizes complex structures on a torus, can itself be given a complex structure in a natural way. First, observe that we can compactify ℳ𝑇ℂ2 as we did in (4.24), by adding the complex cylinder, ℳ ℂ𝑇 2 = ℳ𝑇ℂ2 ⊔ {ℂ∗ }. In terms of the lattice Λ𝜏 = ℤ⟨1, 𝜏⟩, the cylinder corresponds to the “point” 𝜏∞ = i∞. When 𝜏 → i∞, the lattice Λ𝜏 → Λ𝜏∞ = ℤ, and 𝐶𝜏 = ℂ/Λ𝜏 → ℂ∗ = ℂ/ℤ, in some sense. Therefore, ℳ ℂ𝑇 2 ≅ 𝐻/ PSL(2, ℤ) ⊔ {𝜏∞ }, and we give the topology where the neighbourhoods of 𝜏∞ are the images of {𝜏 ∈ 𝐻 | Im(𝜏) > 𝑅}. Proposition 5.61. The moduli space ℳ ℂ𝑇 2 has a complex structure, and it is biholomorphic to ℂ. Proof. First note that 𝑋 = ℳ ℂ𝑇 2 is homeomorphic to 𝑆 2 by (4.24). Let 𝜋 ∶ 𝐻⊔{𝜏∞ } → 𝑋 be the quotient map, and denote 𝑣 1 = 𝜋(𝜏1 ), 𝑣 2 = 𝜋(𝜏2 ), 𝑣∞ = 𝜋(𝜏∞ ). If 𝑇1 , 𝑇2 ⊂ 𝐻 𝜋i are the orbits of 𝜏1 = i, 𝜏2 = 𝑒 3 , respectively, then the action of Γ = PSL(2, ℤ) on 𝐻 − (𝑇1 ∪ 𝑇2 ) is free and proper, and it acts by biholomorphisms. Then by Exercise 5.6, (𝐻 − (𝑇1 ∪ 𝑇2 ))/Γ = 𝑋 − {𝑣 1 , 𝑣 2 , 𝑣∞ } has an induced complex structure. Let us produce complex charts for the remaining three points, such that the changes of charts are biholomorphic. For 𝑣 1 = 𝜋(𝜏1 ), with 𝜏1 = i, the isotropy is given by Γ𝜏1 = ⟨𝜓⟩ ≅ ℤ2 . Hence a neighbourhood of 𝑣 1 is of the form 𝑈1 = 𝜋(𝑊 𝜏1 ) ≅ 𝑊 𝜏1 /⟨𝜓⟩, where 𝑊 𝜏1 is a neighbourhood of 𝜏1 . By Exercise 5.21, there is a holomorphic chart 𝑓 ∶ 𝑊 𝜏1 → 𝐵𝜖 (0) such that 𝜓′ = 𝑓 ∘ 𝜓 ∘ 𝑓−1 is given by 𝜓′ (𝑤) = −𝑤. Then 𝑊 𝜏1 /⟨𝜓⟩ ≅ 𝐵𝜖 (0)/⟨𝜓′ ⟩. Finally, we take the homeomorphism 𝐵𝜖 (0)/⟨𝜓′ ⟩ ≅ 𝐵𝜖2 (0), 𝑤 ↦ 𝑤2 , and composing, we have the required chart 𝑈1 ≅ 𝐵𝜖2 (0). This is compatible with the complex atlas on 𝑋 − {𝑣 1 , 𝑣 2 , 𝑣∞ }, since 𝑈1 − {𝑣 1 } → 𝐵𝜖2 (0) − {0} is holomorphic, as it is induced from 𝑊 𝜏1 − {𝜏1 } → 𝐵𝜖2 (0) − {0}, 𝑧 ↦ 𝑓(𝑧)2 . The argument for 𝑣 2 is analogous. For 𝑣∞ , take 𝐷𝑅 = {𝜏 ∈ 𝐻 | Im(𝜏) > 𝑅} and 𝑓 ∶ 𝐷𝑅 → 𝐵𝜖 (0), 𝑓(𝜏) = 𝑒2𝜋i𝜏 . Then 𝑓 descends to the quotient to give a biholomorphism 𝑓 ̄ ∶ 𝐷𝑅 /⟨𝜑⟩ → 𝐵𝜖 (0) − {0}. We declare the complex chart 𝑈∞ = 𝜋(𝐷𝑅 ) ∪ {𝑣∞ } and 𝑓 ̃ ∶ 𝑈∞ → 𝐵𝜖 (0), where 𝑓 ̃ = 𝑓 ̄ ̃ ∞ ) = 0. on 𝑈∞ − {𝑣∞ } and 𝑓(𝑣 Now 𝑋 is a complex curve and it is homeomorphic to 𝑆 2 . By (5.27), 𝑋 is biholomorphic to ℂ. □ A direct corollary is that ℳ𝑇ℂ2 = 𝐻/ PSL(2, ℤ) ≅ ℂ (i.e., they are biholomorphic). Remark 5.62. By Remark 4.78, ℳ ℂ𝑇 2 has a Riemannian metric of constant curvature 𝜅 ≡ −1, with two orbifold points and one cusp (see Figure 4.12). This metric is compatible with the complex structure of Proposition 5.61. This agrees with Exercise 6.23.

5.4. Classification of complex curves

321

5.4.3. Case (III). The universal cover is 𝔻. Let 𝐶 be a compact connected complex curve such that the universal cover 𝐶 ̃ → 𝐶 satisfies that 𝐶 ̃ = 𝔻. Then 𝐶 = 𝔻/Γ, where Γ is a subgroup of the group of biholomorphisms of 𝔻 acting freely and properly. As mentioned in Theorem 4.66, the biholomorphisms of 𝔻 are given as 𝑧+𝑎 Bihol(𝔻) = {𝜑 ∶ 𝔻 → 𝔻 || 𝜑(𝑧) = 𝑒i𝜃 , |𝑎| < 1, 𝜃 ∈ [0, 2𝜋)} 1 + 𝑎𝑧 = Isom+ (ℍ2𝑃𝐷 ). It coincides with the group of orientation preserving isometries of ℍ2𝑃𝐷 = 𝐵 2 = 𝔻 endowed with the Poincaré metric g𝑃 of constant curvature −1 (section 4.3.3). Thus we can endow 𝔻 with the metric g𝑃 and Γ acts freely and properly by isometries. So there is a metric g on 𝐶 = 𝔻/Γ induced by 𝑔𝑃 of constant curvature −1. In particular, this implies that 𝐶 has genus 𝑔 ≥ 2. Therefore, the classification of complex curves in Case (III) coincides with the classification of oriented connected surfaces with metrics of constant curvature −1. This means that 𝑐𝑜 𝕃𝐂𝐌𝐚𝐧(III) = 𝕃𝐑𝐢𝐞𝐦2 . 𝐾≡−1,𝑜𝑟

We said in section 4.3.5 that, for fixed genus 𝑔 ≥ 2, this moduli space ℳ𝑔 is an orbifold of dimension 6𝑔 − 6. It can be proved that it has a complex structure. Thus it is a complex orbifold of complex dimension 3𝑔 − 3. The associated Teichmüller space is a smooth complex manifold of complex dimension 3𝑔 − 3. We collect the results that we have achieved on the classification of compact complex curves. Corollary 5.63. Let 𝐶 be a compact connected complex curve, and let 𝐶 ̃ be the universal cover of 𝐶. (1) 𝐶 ̃ = ℂ if and only if 𝐶 has genus 𝑔 = 0. (2) 𝐶 ̃ = ℂ if and only if 𝐶 has genus 𝑔 = 1. (3) 𝐶 ̃ = 𝔻 if and only if 𝐶 has genus 𝑔 ≥ 2. Proof. We have seen that if the curve 𝐶 is in Case (I), then it must be 𝑔 = 0. If 𝐶 is in Case (II), then it must be 𝑔 = 1. And if 𝐶 is in Case (III), then it must be 𝑔 ≥ 2. Therefore, as these cases are excluding, the converse holds. □ There is a natural map that associates to every oriented Riemannian surface (𝑆, g) a complex curve (𝑆, 𝐽), that is 𝕃𝐑𝐢𝐞𝐦2 → 𝕃𝐂𝐌𝐚𝐧1 (Remark 5.49). Restricting to metrics 𝑜𝑟 of constant curvature on compact connected surfaces, we have a map (5.30)

𝑐𝑜

𝕃𝐑𝐢𝐞𝐦2

𝐾≡𝑘0 ,𝑜𝑟,𝑐

𝑐𝑜

⟶ 𝕃𝐂𝐌𝐚𝐧1 , 𝑐

where 𝑘0 = 0, ±1. Depending on the genus, we have for a compact connected complex curve 𝐶 the following cases: • If 𝑔 = 0, then 𝐶 = ℂ𝑃 1 admits a unique metric of constant curvature 𝜅 ≡ 1, 𝑐𝑜 up to isometry (Corollary 4.33). Thus both 𝕃𝐑𝐢𝐞𝐦2 and 𝕃𝐂𝐌𝐚𝐧(I) are just 𝐾≡1,𝑜𝑟,𝑐

one point, and (5.30) is bijective.

𝑐

322

5. Complex geometry

• If 𝑔 = 1, then 𝐶 is a complex torus, so it admits metrics of constant curvature 𝜅 ≡ 0. By (5.29), the map ℳ𝑇𝑜𝑟2 → ℳ𝑇ℂ2 = 𝐻/ PSL(2, ℤ) is surjective and the fibers are parametrized by a positive number, which corresponds to fixing the total area of the flat metric. Therefore (5.30) is surjective and the fibers are parametrized by ℝ>0 . • If 𝑔 ≥ 2, then 𝐶 admits a unique metric of constant curvature 𝜅 ≡ −1 (section 6.3.1). The map (5.30) is bijective in this case. Therefore all compact connected complex curves admit compatible metrics of constant curvature. Compare this with Theorem 4.82. Remark 5.64. Another question is how many compatible metrics of constant curvature are there for a given (compact, connected) complex curve (𝐶, 𝐽) of genus 𝑔. For 𝑔 ≥ 2, 𝐶 admits a unique metric of constant curvature 𝜅 ≡ −1. For 𝑔 = 1, 𝐶 admits a unique flat metric g for each value of area(𝐶, g). The moduli of flat metrics compatible with 𝐽 is thus ℝ>0 . However for 𝑔 = 0, 𝐶 admits many distinct metrics of constant curvature 𝜅 ≡ 1. The moduli of such metrics is PGL(2, ℂ)/ PU(2), which has (real) dimension 3. Certainly, the curve 𝐶 is biholomorphic to ℂ𝑃1 ≅ 𝑆 2 . In particular, it has the round metric g𝑆2 . Now suppose that g is another metric compatible with the complex structure and of constant curvature 𝜅 ≡ 1. Then (𝑆 2 , g) is a simply connected Riemannian surface with constant curvature 1, and by the uniqueness of the simply connected space forms (Theorem 3.108), it is isometric to 𝕊2 = (𝑆 2 , g𝑆2 ). Let 𝜓 ∶ (𝑆 2 , g) → (𝑆 2 , g𝑆2 ) be an isometry (which we may assume is orientation preserving). Let h, h𝑆2 be the Hermitian metrics associated to g, g𝑆2 , respectively (cf. Example 5.26). Then there is a smooth function 𝜇 > 0 such that h = 𝜇 h𝑆2 , and hence g = 𝜇 g𝑆2 , meaning that g and g𝑆2 are conformal metrics (Definition 4.11). This in turn implies that 𝜓 ∶ (𝑆 2 , g𝑆2 ) → (𝑆 2 , g𝑆2 ) is a conformal map, and hence a biholomorphism by Remark 4.17. By Proposition 4.22, 𝜓 ∈ Mob(ℂ) = PGL(2, ℂ). Thus there is a surjective map PGL(2, ℂ) → ℳ = {g ∈ ℳet(𝑆 2 )|g compatible with 𝐽, 𝜅g ≡ 1}, given by 𝜓 ↦ 𝜓∗ g𝑆2 . Here 𝜅g denotes the curvature of g. This gives a transitive action of PGL(2, ℂ) on ℳ. The isotropy of g𝑆2 is the subgroup {𝜓|𝜓∗ g𝑆2 = g𝑆2 } = Isom+ (𝑆 2 , g𝑆2 ) = PU(2), by Proposition 4.27. Hence ℳ ≅ PGL(2, ℂ)/ PU(2). Remark 5.65. The discrepancy between the discussion on (5.30) and Remark 5.64 only happens for 𝑔 = 0. This is due to the fact that Bihol(ℂ𝑃 1 ) ≠ Isom(𝕊2 ). On the other hand, for a compact connected surface 𝑆 of 𝑔 ≥ 1 with a metric of constant curvature, we have that Bihol(𝑆, 𝐽) = Isom(𝑆, g). For the case of 𝜅 ≡ 0, one has to use that the total area is fixed. Remark 5.66. In higher dimensions, a natural question is whether a compact complex manifold 𝑀 of complex dimension 𝑛 > 1 admits a Hermitian metric satisfying some nice geometric property. This is similar to Remark 3.86 in the smooth setting. Very relevant cases are metrics with holonomy inside SU(𝑛) (see Exercise 5.9), called CalabiYau metrics, and Kähler-Einstein metrics (see Remark 6.58(6)).

5.5. Elliptic curves

323

5.5. Elliptic curves Elliptic curves are a very important class of Riemann surfaces. Due to their algebraic and analytic properties, they are cornerstone in different areas like number theory and cryptography. This centrality arises from the different points of view that can be used to treat them, some of which we will show in this section. Definition 5.67. An elliptic curve is a compact connected complex curve of genus 1. Using the degree-genus formula (Theorem 5.54), we see that any smooth cubic plane curve 𝐶 (i.e., 𝑑 = 3 and 𝛿 = 0) has genus 𝑔 = 1, so 𝐶 is an elliptic curve. A key fact, that we aim to prove, is that all elliptic curves are isomorphic to smooth cubic plane curves. Before entering into this, we want to do a detailed analysis of plane cubic curves. 5.5.1. Plane cubics. Let 𝐶 = 𝑉(𝐹) ⊂ ℂ𝑃 2 be a smooth plane curve with 𝐹 ∈ ℂ[𝑧0 , 𝑧1 , 𝑧2 ] homogeneous of degree 3. For any line 𝐿 ⊂ ℂ𝑃 2 , the intersection 𝐿 ∩ 𝐶 consists of three points, counted with multiplicity. That is, 𝐿 ∩ 𝐶 can be three different points with transversal intersection, or one point with transversal intersection and another point at which 𝐿 is an ordinary tangent to 𝐶, or 𝐿 ∩ 𝐶 is a single point 𝑝 ∈ 𝐶 with intersection index 3 (see Figure 5.6). In the later case, the point 𝑝 is an inflection point of 𝐶 (see item (5) in the proof of Theorem 5.54).

Figure 5.6. Three intersection points (left), intersection point and tangent (center), and inflection point (right).

Lemma 5.68. Every smooth plane cubic curve has, at least, one inflection point. Proof. Let 𝐶 = 𝑉(𝐹) be a smooth plane cubic curve with 𝐹 ∈ ℂ[𝑧0 , 𝑧1 , 𝑧2 ] homogeneous of degree 𝑑 = 3. The Hessian of 𝐹 is the matrix of the second derivatives 𝜕2 𝐹

Hess 𝐹 = ( 𝜕𝑧 𝜕𝑧 ), whose entries are polynomials of degree 𝑑 − 2 = 1. Then the 𝑖

𝑗

polynomial 𝐺 = det(Hess 𝐹) has degree 3(𝑑 − 2) = 3. We are going to show that the inflections are the points of 𝑉(𝐹) ∩ 𝑉(𝐺). First note how 𝐺 behaves under change of coordinates 𝑧 = 𝐴𝑤, for 𝐴 ∈ GL(3, ℂ). ̃ ̃ ̃ Write in the new coordinates 𝐹(𝑤) = 𝐹(𝐴𝑤) = 𝐹(𝑧) and 𝐺(𝑤) = det(Hess 𝐹(𝑤)). ̃ ̃ We have Hess 𝐹(𝑤) = 𝐴𝑡 (Hess 𝐹(𝐴𝑤)) 𝐴, so 𝐺(𝑤) = (det 𝐴)2 det(Hess 𝐹(𝐴𝑤)) = (det 𝐴)2 𝐺(𝐴𝑤) = (det 𝐴)2 𝐺(𝑧). Therefore the curve 𝑉(𝐺) is intrinsically defined.

324

5. Complex geometry

Now let 𝑝0 ∈ 𝐶 be any point. After a change of coordinates, we can suppose that 𝑝0 = [1, 0, 0] and that the tangent line to 𝐶 at 𝑝0 is 𝑧2 = 0. Then, maybe after dividing by the (non-zero) coefficient of 𝑧20 𝑧2 , 𝐹 has the form 𝐹(𝑧0 , 𝑧1 , 𝑧2 ) = 𝑧20 𝑧2 + 𝑎𝑧0 𝑧21 + 𝑏𝑧0 𝑧1 𝑧2 + 𝑐𝑧0 𝑧22 + 𝑑𝑧31 + 𝑒𝑧21 𝑧2 + 𝑓𝑧1 𝑧22 + 𝑔𝑧32 , for some 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔 ∈ ℂ. Hence the Hessian of 𝐹 at 𝑝0 is 0 0 2 (Hess 𝐹)(𝑝0 ) = ( 0 2𝑎 𝑏 ) . 2 𝑏 2𝑐 So 𝐺(𝑝0 ) = det(Hess 𝐹)(𝑝0 ) = −8𝑎. Therefore 𝑝0 ∈ 𝑉(𝐹) ∩ 𝑉(𝐺) if and only if 𝑎 = 0. On the other hand, in the coordinate chart 𝑈0 the curve 𝐶 is given by the dehomô 𝑦) = 𝐹(1, 𝑥, 𝑦) = 𝑦+𝑎𝑥2 +𝑏𝑥𝑦+𝑐𝑦2 +𝑑𝑥3 +𝑒𝑥2 𝑦+𝑓𝑥𝑦2 +𝑔𝑦3 . genized polynomial 𝐹(𝑥, The tangent line at 𝑝0 = (0, 0) is parametrized by 𝜓(𝑡) = (𝑡, 0), so the index 𝐼𝑝0 (𝐶, 𝐭𝑝0 𝐶) ̃ is the order of 𝑡 = 0 in 𝐹(𝜓(𝑡)) = 𝑎𝑡2 + 𝑑𝑡3 , which is greater than 2 if and only if 𝑎 = 0. So 𝑝0 is an inflection if and only if 𝑝0 ∈ 𝑉(𝐹) ∩ 𝑉(𝐺). Finally, by Bézout Theorem 5.52 we have that ∑𝑝∈𝑆 𝐼𝑝 (𝑉(𝐹), 𝑉(𝐺)) = 3𝑑(𝑑 − 2) = 9. Thus, 𝑉(𝐹) ∩ 𝑉(𝐺) has at least one point, and at most nine points. Actually it has exactly nine points (Exercise 5.31). □ Remark 5.69. In general, the same proof as above shows that a degree 𝑑 plane projective curve has at most 3𝑑(𝑑 − 2) inflection points. Therefore, in the proof of Theorem 5.54, we can choose the projection point 𝑝0 not lying in any tangent through an inflection point. This implies in particular that all ramification indices are 𝑚𝑝 = 2, and hence the intersection indices in the proof are 𝐼𝑝′ (𝐶 ′ , 𝑌 ′ ) = 1, thus assuring that all intersections 𝐶 ′ ∩ 𝑌 ′ are transversal. Now we want to manipulate the equation of the cubic to get a simplified equation. Take a smooth plane cubic curve 𝐶, and take 𝑝0 ∈ 𝐶 an inflection point. After a change of coordinates, we can suppose that 𝑝0 = [0, 0, 1] and that the line at infinity 𝑧0 = 0 is the tangent line to 𝐶 at 𝑝0 . Observe that this is a different choice than that used in Lemma 5.68. Then 𝐹 has the form 𝐹(𝑧0 , 𝑧1 , 𝑧2 ) = 𝑧0 𝑧22 + 𝑏𝑧0 𝑧1 𝑧2 + 𝑐𝑧20 𝑧2 + 𝑑𝑧31 + 𝑒𝑧0 𝑧21 + 𝑓𝑧20 𝑧1 + 𝑔𝑧30 . As 𝐹 is not reducible, it must be 𝑑 ≠ 0. Now we dehomogenize in the affine chart 𝑈0 = {𝑧0 ≠ 0}, and we get the affine equation of 𝐶, ̂ 𝑦) = 𝐹(1, 𝑥, 𝑦) = 𝑦2 + 𝑏𝑥𝑦 + 𝑐𝑦 + 𝑑𝑥3 + 𝑒𝑥2 + 𝑓𝑥 + 𝑔 = 0. 𝐹(𝑥, 𝑏

𝑐

With the affine change of variables 𝑦 ↦ 𝑦 − 2 𝑥 − 2 , we rewrite the equation as 𝑦2 − 𝑄(𝑥) = 0, for some cubic polynomial 𝑄(𝑥) ∈ ℂ[𝑥]. Let 𝜆1 , 𝜆2 , 𝜆3 ∈ ℂ be the roots of 𝑄(𝑥). Then the cubic is defined by ̂ 𝑦) = 𝑦2 − 𝑎(𝑥 − 𝜆1 )(𝑥 − 𝜆2 )(𝑥 − 𝜆3 ) = 0, 𝐹(𝑥, for some 𝑎 ≠ 0. Observe that the curve is smooth if and only if the three roots 𝜆1 , 𝜆2 , 𝜆3 𝜕𝐹̂ 𝜕𝐹̂ are different (just compute 𝜕𝑥 = 0, 𝜕𝑦 = 0). See Figure 5.7.

5.5. Elliptic curves

λ1

325

λ2

λ1

λ3

λ2 = λ 3

Figure 5.7. The curve 𝑦2 = (𝑥 − 𝜆1 )(𝑥 − 𝜆2 )(𝑥 − 𝜆3 ), for three different roots (left), and when one root is repeated (right).

Moreover we can take the change of variables 𝑥̃ = (𝜆1 , 𝜆2 , 𝜆3 ) to (0, 1, 𝜆) where 𝜆 = 𝑦̃ =

𝑎−1/2 𝑦, (𝜆2 −𝜆1 )−3/2

𝜆3 −𝜆1 𝜆2 −𝜆1

𝑥−𝜆1 𝜆2 −𝜆1

which sends the roots

is the single ratio of the three roots. Taking

the equation of the cubic becomes 𝑦2̃ − 𝑥(̃ 𝑥̃ − 1)(𝑥̃ − 𝜆) = 0.

Definition 5.70. A smooth plane cubic is in standard form if we have chosen affine ̂ 𝑦) = 𝑦2 − 𝑥(𝑥 − 1)(𝑥 − 𝜆), for some coordinates so that its equation becomes 𝐹(𝑥, 𝜆 ∈ ℂ − {0, 1}. Remark 5.71. • In the standard form, the projection 𝜋 ∶ ℂ𝑃 2 − {[0, 0, 1]} → ℂ𝑃 1 given by (𝑥, 𝑦) ↦ 𝑥, defines a ramified cover 𝜋 ∶ 𝐶 → ℂ𝑃 1 of degree two ramifying at 0, 1, 𝜆 and ∞ with multiplicity 2 (Exercise 5.35). Thus, by the Hurwitz formula for ramified covers (2.4), 2 − 2𝑔 = 𝜒(𝐶) = (deg 𝜋) 𝜒(ℂ𝑃 1 ) − ∑ (𝑚𝑝 − 1) = 2 ⋅ 2 − 4 ⋅ 1 = 0, 𝑝∈{0,1,𝜆,∞}

so 𝑔 = 1 agrees with the fact that 𝐶 is an elliptic curve. • The value 𝜆 ∈ ℂ − {0, 1} parametrizes the set of smooth plane cubic curves. However 𝜆 is defined after choosing an ordering of the roots 𝜆1 , 𝜆2 , 𝜆3 of 𝑄, which is given by an element of the symmetric group 𝑆 3 of permutations of three elements. Therefore the smooth plane cubic curves in standard form are parametrized by the moduli space 𝒩𝑠𝑡𝑑 = (ℂ − {0, 1})/𝑆 3 . The orbit of 𝜆 under the group 𝑆 3 is the set of all possible simple ratios of three 1 1 𝜆−1 𝜆 points {𝜆, 1 − 𝜆, 𝜆 , 1−𝜆 , 𝜆 , 𝜆−1 }. Let us come back to the expression of a smooth cubic curve 𝐶 given by an equâ 𝑦) = 𝑦2 − 𝑄(𝑥), for some cubic polynomial 𝑄(𝑥) ∈ ℂ[𝑥]. Write 𝑄(𝑥) = tion 𝐹(𝑥, 𝑎(𝑥 −𝜆1 )(𝑥 −𝜆2 )(𝑥 −𝜆3 ) with 𝜆1 , 𝜆2 , 𝜆3 the three roots of 𝑄 and 𝑎 ≠ 0. The discriminant

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5. Complex geometry

of 𝐹 is defined as the discriminant of 𝑄(𝑥), which is Δ = 𝑎4 (𝜆1 −𝜆2 )2 (𝜆2 −𝜆3 )2 (𝜆3 −𝜆1 )2 . This is an important object because it serves to determine whether 𝐶 is smooth. Clearly Δ = 0 when 𝑄(𝑥) has multiple roots, so 𝐶 is smooth if and only if Δ ≠ 0. Moreover, Δ is invariant by permutation of the roots, hence it has an expression on the coefficients ̂ 𝑦) = of 𝑄(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 = 0. For instance, for the standard form 𝐹(𝑥, 𝑦2 − 𝑥(𝑥 − 1)(𝑥 − 𝜆), we have Δ = 𝜆2 (𝜆 − 1)2 . ̂ 𝑦) = 𝑦2 − (𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑) = 0, for some Consider the cubic given by 𝐹(𝑥, 𝑏

𝑎, 𝑏, 𝑐, 𝑑 ∈ ℂ with 𝑎 ≠ 0. Performing the change of variables 𝑥̃ = (𝑎/4)1/3 (𝑥 + 3𝑎 ), we get an equation of the form 𝐹(𝑥,̃ 𝑦) = 𝑦2 − (4𝑥3̃ − 𝐴𝑥̃ − 𝐵) = 0, for some 𝐴, 𝐵 ∈ ℂ. Definition 5.72. A smooth plane cubic is on Weierstrass form if we have chosen affine ̂ 𝑦) = 𝑦2 − (4𝑥3 − 𝐴𝑥 − 𝐵) = 0. coordinates so that its equation becomes 𝐹(𝑥, Suppose that 𝐶 is given in Weierstrass form 𝑦2 = 4𝑥3 − 𝐴𝑥 − 𝐵. Writing 4𝑥3 − 𝐴𝑥 − 𝐵 = 𝑎(𝑥 − 𝜆1 )(𝑥 − 𝜆2 )(𝑥 − 𝜆3 ), we get that 𝑎 = 4, 𝜆1 + 𝜆2 + 𝜆3 = 0, 𝐴 = −4(𝜆1 𝜆2 + 𝜆1 𝜆3 + 𝜆2 𝜆3 ), 𝐵 = 4𝜆1 𝜆2 𝜆3 . The discriminant is a polynomial of degree 6 on the roots 𝜆𝑖 , so Δ = 𝑎4 (𝛼 𝐴3 + 𝛽 𝐵 2 ), for some 𝛼, 𝛽 ∈ ℂ. These numbers can be computed looking at two particular cases, like (𝜆1 , 𝜆2 , 𝜆3 ) = (0, 1, −1), (1, 2, −3), for which (𝐴, 𝐵) = (4, 0), (28, −24), and Δ = 44 ⋅ 4, 44 ⋅ 400, respectively. This yields the expression Δ = 16 (𝐴3 − 27𝐵 2 ) . Let us understand the freedom in the Weierstrass form of a cubic curve. When 𝐶 is given in Weierstrass form, the coefficients 𝐴, 𝐵 determine the roots of 𝑄(𝑥) up to ordering. As the center of mass of the roots is fixed at the origin, 𝜆1 + 𝜆2 + 𝜆3 = 0, we are not allowed to translate in the horizontal direction. Still we may do a homothety 𝑥 ↦ 𝜇𝑥, where 𝜇 ∈ ℂ∗ . To keep the coefficient 4𝑥3 , we have to accompany it with a corresponding homothety 𝑦 ↦ 𝜇3/2 𝑦. This changes 𝜆𝑖 ↦ 𝜇𝜆𝑖 and (𝐴, 𝐵) ↦ (𝜇2 𝐴, 𝜇3 𝐵). Therefore the moduli space of cubic forms in Weierstrass form is given as 𝒩 𝑊 = {(𝐴, 𝐵) ∈ ℂ2 | Δ = 16 (𝐴3 − 27𝐵 2 ) ≠ 0}/ℂ∗ , where 𝜇 ∈ ℂ∗ acts by (𝐴, 𝐵) ↦ (𝜇2 𝐴, 𝜇3 𝐵). Remark 5.73. For integers 𝑑0 , 𝑑1 > 0 with gcd(𝑑0 , 𝑑1 ) = 1, we call the weighted projective line to ℂ𝑃[𝑑1 0 ,𝑑1 ] = (ℂ2 − {0})/ℂ∗ , with ℂ∗ acting as (𝑧0 , 𝑧1 ) ↦ (𝜇𝑑0 𝑧0 , 𝜇𝑑1 𝑧1 ). This space ℂ𝑃[𝑑1 0 ,𝑑1 ] has actually an orbifold structure (Exercise 5.36). It also has a natural complex structure. Moreover, as a complex manifold, ℂ𝑃[𝑑1 0 ,𝑑1 ] is biholomorphic 𝑑

𝑑

to ℂ𝑃 1 , via [𝑤 0 , 𝑤 1 ] ↦ [𝑤 01 , 𝑤 1 0 ]. Using the weighted projective line, we can write 1 𝒩 𝑊 = ℂ𝑃[2,3] − {[3, 1]} .

There is a natural isomorphism 𝒩𝑠𝑡𝑑 ≅ 𝒩 𝑊 , which comes from putting a cubic on standard form into the Weierstrass form. To obtain it, let 𝑄(𝑥) = 𝑥(𝑥 − 1)(𝑥 − 𝜆) = 𝑥3 − (𝜆 + 1)𝑥2 + 𝜆𝑥.

5.5. Elliptic curves

3 Observe that 𝑄(√ 4𝑥 + 3

√4

327

𝜆+1 ) 3

1

3

√4 2 1 (𝜆 − 𝜆 + 1)𝑥 − 27 3 3 2

= 4𝑥3 −

(2𝜆3 − 3𝜆2 − 3𝜆 + 2). Hence

𝐴 = 3 (𝜆2 − 𝜆 + 1) and 𝐵 = 27 (2𝜆 − 3𝜆 − 3𝜆 + 2). The map ℎ ∶ ℂ − {0, 1} → ℂ2 , ℎ(𝜆) = (𝐴, 𝐵), is algebraic and ℎ(1 − 𝜆) = ℎ(𝜆), ℎ(1/𝜆) = (𝜆−2 𝐴, 𝜆−3 𝐵). Hence it descends to a map 1 ℎ ̄ ∶ 𝒩𝑠𝑡𝑑 = (ℂ − {0, 1})/𝑆 3 ⟶ 𝒩 𝑊 = ℂ𝑃[2,3] − {[3, 1]},

(5.31)

3 √ 4 2 1 ̄ ℎ([𝜆]) =[ (𝜆 − 𝜆 + 1), (2𝜆3 − 3𝜆2 − 3𝜆 + 2)] , 3 27

which is an isomorphism (i.e., a biholomorphism). 1 Take the isomorphism 𝐽𝑊 ∶ ℂ𝑃[2,3] → ℂ𝑃 1 , given as the composition of isomor1 phisms ℂ𝑃[2,3] → ℂ𝑃 1 → ℂ𝑃 1 , [𝐴, 𝐵] ↦ [𝐴3 , 𝐵 2 ] ↦ [𝐴3 − 27𝐵2 , 𝐴3 ]. It is arranged in such a way that 𝐽𝑊 ([3, 1]) = [0, 1]. Hence it gives an isomorphism

𝐴3 . 𝐴3 − 27𝐵 2 This is called the 𝐽-invariant of the elliptic curve on Weierstrass form.

(5.32)

1 𝐽𝑊 ∶ 𝒩 𝑊 = ℂ𝑃[2,3] − {[3, 1]} → ℂ, 𝐽𝑊 ([𝐴, 𝐵]) =

The 𝐽-invariant of the elliptic curve in standard form is the composition 𝐽𝑠𝑡𝑑 = 𝐽𝑊 ∘ ℎ ̄ ∶ 𝒩𝑠𝑡𝑑 → ℂ, which is given by 𝐽𝑠𝑡𝑑 (𝜆) =

4 (𝜆2 − 𝜆 + 1)3 . 27 𝜆2 (𝜆 − 1)2

This formula yields an isomorphism 𝐽𝑠𝑡𝑑 ∶ (ℂ − {0, 1})/𝑆 3 → ℂ. 5.5.2. The Weierstrass ℘ function. Let us now start with an abstract elliptic curve, that is a compact connected complex curve 𝐶 of genus 1. By Corollary 5.63, 𝐶 = ℂ/Λ, with Λ ⊂ ℂ a lattice. In this section we are going to prove that 𝐶 is isomorphic to a smooth plane cubic curve. This requires finding an explicit holomorphic embedding 𝜑 ∶ 𝐶 → ℂ𝑃 2 , such that 𝜑(𝐶) = 𝑉(𝐹), for some homogeneous polynomial 𝐹(𝑧0 , 𝑧1 , 𝑧2 ) of degree 3. Looking at the affine chart 𝑈0 ⊂ ℂ𝑃 2 , we have an embedding 𝜑 ∶ 𝐶0 → 𝑈0 , ̂ 𝜑(𝑧) = (𝑥(𝑧), 𝑦(𝑧)), such that 𝐹(𝑥(𝑧), 𝑦(𝑧)) = 0, where 𝐶0 = 𝜑−1 (𝑈0 ) and 𝐹 ̂ ∈ ℂ[𝑥, 𝑦] is the dehomogenized polynomial. The functions 𝑥, 𝑦 ∶ 𝐶0 → ℂ are holomorphic and can be extended to 𝐶 as meromorphic functions 𝑥, 𝑦 ∶ 𝐶 → ℂ. Therefore, the key point for finding an embedding 𝐶 → ℂ𝑃 2 is to construct meromorphic functions on 𝐶 and to see which algebraic relations they satisfy. For 𝐶 = ℂ/Λ, if 𝑓 ∶ 𝐶 → ℂ𝑃 1 is a meromorphic function, we lift it to the universal cover 𝜋 ∶ ℂ → 𝐶 = ℂ/Λ, to get a meromorphic function 𝑓 ̃ ∶ ℂ → ℂ𝑃 1 . ℂ JJ JJ 𝑓 ̃ JJ JJ 𝜋 JJ $  / ℂ𝑃 1 𝐶 = ℂ/Λ 𝑓

̃ + 𝑤) = 𝑓(𝑧) ̃ for any 𝑤 ∈ Λ. It is usually said Observe that 𝑓 ̃ is Λ-periodic, that is 𝑓(𝑧 that 𝑓 ̃ is doubly periodic. We want to construct Λ-periodic meromorphic functions. First some easy comments.

328

5. Complex geometry

(1) If 𝑓 is holomorphic, then 𝑓 ̃ ∶ ℂ → ℂ is a periodic holomorphic function. So ̃ | 𝑧 ∈ ℂ} = sup{𝑓(𝑧) ̃ | 𝑧 ∈ 𝐷} < ∞ where 𝐷 ⊂ ℂ is a fundamental sup{𝑓(𝑧) domain of Λ. By the Liouville theorem (Exercise 5.1(3)), as 𝑓 is an entire function (i.e., a holomorphic function defined on the whole plane ℂ) which is bounded, then it is constant. (2) Suppose that 𝑓 ∶ 𝐶 → ℂ𝑃 1 is a meromorphic function with a single pole of order 1 at 𝑝0 ∈ 𝐶. Then 𝑔(𝑧) = 1/𝑓(𝑧), 𝑔 ∶ 𝐶 → ℂ𝑃 1 is also a meromorphic function with a single zero of order 1 at 𝑝0 . In particular, it is a local biholomorphism near 𝑝0 . Since 𝑔 is a ramified cover (Exercise 5.23), then 𝑔 is of degree 1, hence 𝑔 is a homeomorphism (actually a biholomorphism by item (4) in the proof of Theorem 5.54). This is impossible since 𝐶 has genus 1 and ℂ𝑃 1 has genus 0. (3) Suppose that 𝑓 ∶ 𝐶 → ℂ𝑃 1 has a single pole. Then its order must be at least 2, and 𝑓 ̃ ∶ ℂ → ℂ has poles in a translate of the lattice 𝑧0 + Λ, for some 𝑧0 ∈ ℂ. ̃ + 𝑧0 ) is also meromorphic and Λ-periodic and has the Note that 𝑔(𝑧) ̃ = 𝑓(𝑧 poles at Λ. So we can assume that the poles are at the points of Λ. In order to build a meromorphic Λ-periodic function 𝑓 ̃ ∶ ℂ → ℂ with poles at Λ of order 𝑟 ≥ 2, the first guess is to take 1 . (𝑧 − 𝑤)𝑟 𝑤∈Λ

̃ = ∑ 𝑓(𝑧)

(5.33)

Unfortunately, this is not convergent for 𝑟 = 2. 1 𝑤𝑟

Lemma 5.74. The sum ∑𝑤∈Λ−{0}

converges absolutely if and only if 𝑟 > 2.

Proof. Take 𝜔1 , 𝜔2 ∈ Λ generators of Λ. Then, there exists 𝑐 1 , 𝑐 2 > 0 such that, if 𝑤 = 𝑛𝜔1 + 𝑚𝜔2 ∈ Λ for 𝑛, 𝑚 ∈ ℤ, it holds 𝑐 1 (𝑛2 + 𝑚2 )𝑟/2 ≤ |𝑤|𝑟 ≤ 𝑐 2 (𝑛2 + 𝑚2 )𝑟/2 . Then 1 2 we have the ratio of convergence 1 ≤ 𝐶 𝑅2−𝑟 , 𝑟 |𝑤| 𝑤∈Λ−{0} ∑

|𝑤|≥𝑅

for 𝑅 > 0, where 𝐶 is some constant. For that, observe that up to a constant factor, the ∞ 1 ratio of convergence is controlled by the integral ∫𝑅 𝜌𝑟−1 𝑑𝜌 = (𝑟 − 2)𝑅2−𝑟 .

5.5. Elliptic curves

329

The divergence of (5.33) for 𝑟 = 2 comes from the constant term at 𝑧 = 0. That is, 1 1 writing 𝑓(𝑧) = 𝑧2 +∑𝑤∈Λ−{0} (𝑧−𝑤)2 , the second part, at 𝑧 = 0, behaves as the divergent 1

expression ∑𝑤∈Λ−{0} 𝑤2 . The idea behind making this sum into a convergent sum is to “renormalize” by subtracting this constant term. Definition 5.76. Let Λ ⊂ ℂ be a rank 2 lattice. We define the Weierstrass elliptic function, ℘ ∶ ℂ → ℂ as the series 1 1 1 + ∑ ( − ). 𝑧2 𝑤∈Λ−{0} (𝑧 − 𝑤)2 𝑤2

℘(𝑧) =

Proposition 5.77. The Weierstrass elliptic function converges absolutely and uniformly on 𝐵𝑅̄ (0) − Λ, for any 𝑅 > 0. It defines a meromorphic function with poles of order 2 at the points of Λ. ̄ (0). Observe that, since Λ ∩ 𝐵𝑅+1 (0) Proof. Let us fix 𝑅 > 0, and let Λ′ = Λ − 𝐵𝑅+1 is a finite set, it is enough to study the convergence of the series taking the sum over 𝑤 ∈ Λ′ . To do so, fix 𝑤 ∈ Λ′ . For |𝑧| ≤ 𝑅, we have that |𝑧/𝑤| < 1. Then we have the power series expansion ∞

1 𝑑 1 𝑑 1/𝑤 𝑑 1 𝑛 ∑ =− ( 𝑧 )= ( )= 𝑑𝑧 𝑧 − 𝑤 𝑑𝑧 1 − 𝑧/𝑤 𝑑𝑧 𝑛=0 𝑤𝑛+1 (𝑧 − 𝑤)2 ∞

= ∑ 𝑛=1

∞

𝑛

𝑤

𝑧𝑛−1 = 𝑛+1

1 𝑛+1 𝑛 +∑ 𝑧 . 𝑤2 𝑛=1 𝑤𝑛+2

So for 𝑧 ∈ 𝐵𝑅̄ (0), the sum can be written as ∞

∑ (

(5.34)

𝑤∈Λ′

1 1 𝑛+1 − ) = ∑ ∑ 𝑛+2 𝑧𝑛 . 𝑤 (𝑧 − 𝑤)2 𝑤2 𝑤∈Λ′ 𝑛=1

We want to interchange the order of summation in (5.34), for which we need to prove 1 ′ that this sum converges absolutely. Let us define 𝑆𝑚 = ∑𝑤∈Λ′ 𝑤𝑚 . By Lemma 5.74 ′ ′ and Remark 5.75, 𝑆𝑚 is a convergent series for 𝑚 > 2 and |𝑆𝑚 | ≤ 𝐶 (𝑅 + 1)2−𝑚 , for some constant 𝐶 > 0. Then for |𝑧| ≤ 𝑅, ∞ 𝑛 |∞ | |∞ | | ∑ ∑ 𝑛 + 1 𝑧𝑛 | = | ∑ (𝑛 + 1)𝑆 ′ 𝑧𝑛 | ≤ 𝐶 ∑ (𝑛 + 1) ( 𝑅 ) < ∞. 𝑛+2 | | | | 𝑅+1 | |𝑛=1 𝑤∈Λ′ 𝑤𝑛+2 | |𝑛=1 𝑛=1 ∞

𝑛+1

Hence ∑𝑛=1 ∑𝑤∈Λ′ 𝑤𝑛+2 𝑧𝑛 converges absolutely and uniformly. Therefore (5.34) also converges absolutely and uniformly on 𝐵𝑅̄ (0), and we can interchange the order of summation, obtaining that ∞

∞

∞

𝑛+1 𝑛 1 ′ 𝑧 = ∑ (𝑛 + 1) ( ∑ 𝑛+2 ) 𝑧𝑛 = ∑ (𝑛 + 1)𝑆𝑛+2 𝑧𝑛 . 𝑛+2 𝑤 𝑤 𝑤∈Λ′ 𝑛=1 𝑛=1 𝑤∈Λ′ 𝑛=1 ∑ ∑

Thus we have the expression for ℘ in 𝐵𝑅̄ (0), ℘(𝑧) =

1 ∑ + 𝑧2 𝑤∈Λ∩𝐵

𝑅+1 (0)

∞

(

1 1 ′ − 2 ) + ∑ (𝑛 + 1)𝑆𝑛+2 𝑧𝑛 , 2 𝑤 (𝑧 − 𝑤) 𝑛=1

330

5. Complex geometry

where the first two summands are a finite sum of meromorphic functions, and the last one is holomorphic on 𝐵𝑅̄ (0). Hence ℘ is a meromorphic function for |𝑧| ≤ 𝑅 with double poles at Λ ∩ 𝐵𝑅̄ (0). Since 𝑅 > 0 is arbitrary, the result follows. □ Remark 5.78. The computations above actually show that the power series expansion of ℘ in a small disc around 0 is ∞

℘(𝑧) = where 𝑆𝑚 = ∑𝑤∈Λ−{0}

1 + ∑ (𝑛 + 1)𝑆𝑛+2 𝑧𝑛 , 𝑧2 𝑛=1

1 . 𝑤𝑚

Corollary 5.79. The derivative of ℘ is given by −2 , (𝑧 − 𝑤)3 𝑤∈Λ

℘′ (𝑧) = ∑

where the sum converges absolutely and uniformly on sets 𝐵𝑅̄ (0) − Λ, 𝑅 > 0. Moreover, it is a Λ-periodic meromorphic odd function with triple poles at every point of Λ. Proof. Taking a term-by-term differentiation of the formula in Definition 5.76, we get the given formula. As it is the derivative of a meromorphic function with poles at Λ, it has poles at the same points and the order is one unit more. Let us see that it is odd and Λ-periodic. First −2 −2 −2 =− ∑ =− ∑ = −℘(𝑧), 3 3 (−𝑧 − 𝑤) (𝑧 + 𝑤) (𝑧 − 𝑤′ )3 𝑤∈Λ 𝑤∈Λ 𝑤′ ∈Λ

℘′ (−𝑧) = ∑

doing the change 𝑤′ = −𝑤. Second, let 𝑤 0 ∈ Λ, then −2 −2 = ∑ = ℘(𝑧), 3 (𝑧 + 𝑤 − 𝑤) (𝑧 − 𝑤 ′ )3 0 ′ 𝑤∈Λ 𝑤 ∈Λ

℘(𝑧 + 𝑤 0 ) = ∑

doing the change 𝑤′ = 𝑤 − 𝑤 0 .

□

Corollary 5.80. The Weierstrass elliptic function ℘ is Λ-periodic and even. Proof. To prove that ℘ is an even function, consider 𝑓(𝑧) = ℘(𝑧) − ℘(−𝑧), which is a meromorphic function such that 𝑓′ (𝑧) = ℘′ (𝑧) + ℘′ (−𝑧) = 0, by Corollary 5.79. Then 𝑓(𝑧) is constant, say 𝑓(𝑧) = 𝑐 0 , for some 𝑐 0 ∈ ℂ, and all 𝑧 ∈ ℂ − Λ. Now the development as power series around 𝑧 = 0 of ℘ in Remark 5.78 says that ℘(𝑧) = 1 + 𝑎1 𝑧 + 𝑎2 𝑧2 + ⋯, so 𝑓(𝑧) = ℘(𝑧) − ℘(−𝑧) has no constant term. Thus 𝑐 0 = 0 and 𝑧2 ℘(𝑧) = ℘(−𝑧). To prove Λ-periodicity, take 𝑤 0 ∈ Λ. Without loss of generality, we can suppose 1 1 that 2 𝑤 0 ∉ Λ since, otherwise, we can argue with 2 𝑤 0 . Consider 𝑔(𝑧) = ℘(𝑧 + 𝑤 0 ) − ℘(𝑧), so 𝑔′ (𝑧) = ℘′ (𝑧+𝑤 0 )−℘′ (𝑧) = 0, by Corollary 5.79, and then 𝑔(𝑧) = 𝑐 0 , for some 1 1 1 𝑐 0 ∈ ℂ, and all 𝑧 ∈ ℂ − Λ. Taking 𝑧 = − 2 𝑤 0 , we have 𝑐 0 = ℘( 2 𝑤 0 ) − ℘(− 2 𝑤 0 ) = 0. Therefore ℘(𝑧 + 𝑤 0 ) = ℘(𝑧) for all 𝑧 ∈ ℂ − Λ. □ Remark 5.81. Since ℘ is even, 𝑆 2𝑙+1 = 0 for 𝑙 ≥ 0. This can also be seen as opposite elements of the lattice cancel pairwise in the sum 𝑆 2𝑙+1 = ∑𝑤∈Λ−{0} 𝑤−(2𝑙+1) .

5.5. Elliptic curves

331

Given a complex curve 𝐶 and a point 𝑝0 ∈ 𝐶, we denote by 𝐿(𝑛𝑝0 ) the vector space of meromorphic functions on 𝐶 with a single pole at 𝑝0 of order at most 𝑛, and by 𝑙(𝑛𝑝0 ) its complex dimension. Proposition 5.82. For any complex curve, 𝑙(𝑛𝑝0 ) ≤ 𝑙((𝑛 − 1)𝑝0 ) + 1, for 𝑛 ≥ 1. Proof. We define the 𝑛th residue map at 𝑝0 , res 𝑛𝑝0 ∶ 𝐿(𝑛𝑝0 ) → ℂ as follows. Given a meromorphic function 𝑓 ∶ 𝐶 → ℂ with a pole in 𝑝0 or order at most 𝑛, let us take a coordinate chart around 𝑝0 in which 𝑝0 corresponds to 𝑧 = 0. In these coordinates, the Laurent expansion of 𝑓 around 0 has the form 𝑎−(𝑛−1) 𝑎 𝑎 𝑓(𝑧) = −𝑛 + 𝑛−1 + ⋯ + 1 + 𝑎0 + 𝑎1 𝑧 + 𝑎2 𝑧2 + ⋯ . 𝑛 𝑧 𝑧 𝑧 Then, we define res 𝑛𝑝0 (𝑓) = 𝑎−𝑛 . This depends on the choice of coordinate, but we will not care about it. Observe that ker(res 𝑛𝑝0 ) = 𝐿((𝑛 − 1)𝑝0 ) so we have 𝐿(𝑛𝑝0 )/𝐿((𝑛 − 1)𝑝0 ) ≅ im(res 𝑛𝑝0 ) ⊂ ℂ. Therefore either 𝑙(𝑛𝑝0 ) = 𝑙((𝑛 − 1)𝑝0 ) if res 𝑛𝑝0 = 0, or 𝑙(𝑛𝑝0 ) = 𝑙((𝑛 − 1)𝑝0 ) + 1 if res 𝑛𝑝0 is surjective. □ Fix an elliptic curve 𝐶 = ℂ/Λ, with projection 𝜋 ∶ ℂ → ℂ/Λ, and let 𝑝0 = 𝜋(0). We know that meromorphic functions in 𝐿(𝑛𝑝0 ) correspond to Λ-periodic meromorphic functions on ℂ such that they have poles of order at most 𝑛 at the points of Λ. By item (2) on page 328, 𝐿(𝑝0 ) = ℂ consists only of constant functions. So 𝑙(𝑝0 ) = 1, and hence 𝑙(𝑛𝑝0 ) ≤ 𝑛 for 𝑛 ≥ 1, by Proposition 5.82. An opposite estimate is as follows. On 𝐶 we have the meromorphic function ℘ with a double pole at 𝑝0 . The derivative ℘′ is a meromorphic function with a triple pole at 𝑝0 . Analogously the 𝑘th derivative ℘(𝑘) is a meromorphic function with a pole of order 𝑘 + 2, for all 𝑘 ≥ 0. This means that res 𝑛𝑝0 is surjective for 𝑛 ≥ 2. Therefore 𝑙(𝑛𝑝0 ) = 𝑙((𝑛 − 1)𝑝0 ) + 1 for 𝑛 ≥ 2, and hence (5.35)

𝑙(𝑛𝑝0 ) = 𝑛, for all 𝑛 ≥ 1.

Remark 5.83. Behind the formula (5.35) there is a very deep theorem of complex curves, known as the Riemann-Roch theorem (which has generalizations to higher dimensional varieties). Given a compact complex curve 𝐶, a divisor on 𝐶 is a finite sum 𝐷 = ∑ 𝑛𝑖 𝑝 𝑖 with 𝑛𝑖 ∈ ℤ and 𝑝 𝑖 ∈ 𝐶. The set of divisors forms an Abelian group with pointwise addition, and we define deg 𝐷 = ∑ 𝑛𝑖 . Given a divisor 𝐷 = ∑ 𝑛𝑖 𝑝 𝑖 , we define 𝐿(𝐷) as the vector space of meromorphic functions 𝑓 ∶ 𝐶 → ℂ𝑃 1 such that ord𝑝𝑖 (𝑓) + 𝑛𝑖 ≥ 0, that is if 𝑛𝑖 < 0, then 𝑓 must have a zero of order at least −𝑛𝑖 at 𝑝 𝑖 and, if 𝑛𝑖 ≥ 0 then 𝑓 can have at 𝑝 𝑖 a pole of order at most 𝑛𝑖 . The (complex) dimension of this vector space is denoted by 𝑙(𝐷). This agrees with the previous notation for 𝐷 = 𝑛𝑝0 . In this setting, the Riemann-Roch theorem asserts that for a smooth compact connected complex curve 𝐶, there exists a divisor, called the canonical divisor and that is denoted by 𝐾𝐶 , such that (5.36)

𝑙(𝐷) − 𝑙(𝐾𝐶 − 𝐷) = deg 𝐷 − 𝑔 + 1.

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5. Complex geometry

The canonical divisor 𝐾𝐶 is associated to the canonical sheaf Ω 1 (Example 5.17(2)). More concretely, 𝐾𝐶 is the divisor of zeros and poles of a global meromorphic 1-form of Ω 1 (𝐶). For an elliptic curve, Ω 1 (𝐶) has a section, namely 𝑑𝑧, without zeros and poles, and so 𝐾𝐶 = 0. If we take 𝐷 = 𝑛𝑝0 for 𝑛 ≥ 1, the space 𝐿(−𝑛𝑝0 ) is the set of holomorphic functions on 𝐶 with a zero of order 𝑛 ≥ 0 which, as we saw, is trivial. Hence the Riemann-Roch theorem (5.36) says that 𝑙(𝑛𝑝0 ) = deg(𝑛𝑝0 ) − 𝑔 + 1 + 𝑙(−𝑛𝑝0 ) = 𝑛 − 1 + 1 + 0 = 𝑛. Coming back to the elliptic curve, observe that ℘2 is a meromorphic function with a single pole of order 4. Thus 𝐿(4𝑝0 ) = ⟨1, ℘, ℘′ , ℘2 ⟩. Analogously 𝐿(5𝑝0 ) = ⟨1, ℘, ℘′ , ℘2 , ℘℘′ ⟩. However, the next step 𝐿(6𝑝0 ) = ⟨1, ℘, ℘′ , ℘2 , ℘℘′ , ℘3 , (℘′ )2 ⟩ contains seven generators for a 6-dimensional vector space. So there must be a linear relation between them. This gives an algebraic equation relating ℘, ℘′ . Proposition 5.84. Given an elliptic curve 𝐶 = ℂ/Λ, we have (℘′ )2 = 4 ℘3 − 𝑔2 ℘ − 𝑔3 , where 𝑔2 = 60 𝑆 4 and 𝑔3 = 140 𝑆 6 . Proof. Let us define the function 𝐹 ∶ ℂ → ℂ𝑃 1 , 𝐹(𝑧) = (℘′ )2 − 4 ℘3 + 𝑔2 ℘ + 𝑔3 . We want to check that 𝐹 is holomorphic and 𝐹(0) = 0. Then, as 𝐹 is Λ-periodic, it must be constant and, since 𝐹(0) = 0, it must be 𝐹 = 0, finishing the proof. Since 𝐹 is Λ-periodic and the unique poles of 𝐹 may happen at the points of Λ, we only need to check that 𝐹 has a zero at 𝑧 = 0. For this, let us compute the Laurent series of 𝐹 around 𝑧 = 0. By Remark 5.78, 1 + 3𝑆 4 𝑧2 + 5𝑆 6 𝑧4 + 𝑂(𝑧6 ), 𝑧2 −2 ℘′ (𝑧) = 3 + 6𝑆 4 𝑧 + 20𝑆 6 𝑧3 + 𝑂(𝑧5 ). 𝑧 ℘(𝑧) =

So we have 3 9𝑆 1 1 2 4 6 + 3𝑆 𝑧 + 5𝑆 𝑧 + 𝑂(𝑧 )) = 6 + 24 + 15𝑆 6 + 𝑂(𝑧2 ), 4 6 𝑧2 𝑧 𝑧 2 24𝑆 −2 4 (℘′ )2 (𝑧) = ( 3 + 6𝑆 4 𝑧 + 20𝑆 6 𝑧3 + 𝑂(𝑧5 )) = 6 − 2 4 − 80𝑆 6 + 𝑂(𝑧2 ). 𝑧 𝑧 𝑧

℘3 (𝑧) = (

Therefore, putting all together we obtain 𝐹(𝑧) =(℘′ )2 − 4 ℘3 + 60𝑆 4 ℘ + 140𝑆 6 24𝑆 9𝑆 4 1 − 2 4 − 80𝑆 6 ) − 4 ( 6 + 24 + 15𝑆 6 ) 𝑧 𝑧 𝑧6 𝑧 1 + 60𝑆 4 2 + 140𝑆 6 + 𝑂(𝑧2 ) = 𝑂(𝑧2 ). 𝑧

=(

Hence 𝐹 has no pole at 𝑧 = 0 and 𝐹(0) = 0, as we wanted.

□

5.5. Elliptic curves

333

Theorem 5.85. Let 𝐶 = ℂ/Λ be an elliptic curve with Λ ⊂ ℂ a lattice. Then 𝐶 is biholomorphic to the smooth plane cubic curve in Weierstrass form 𝐶 ′ = {[𝑧0 , 𝑧1 , 𝑧2 ] ∈ ℂ𝑃 2 | 𝑧0 𝑧22 − 4𝑧31 + 𝑔2 𝑧20 𝑧1 + 𝑔3 𝑧30 = 0} . Proof. We have constructed meromorphic functions 𝑥, 𝑦 ∶ 𝐶 → ℂ, given as 𝑥 = ℘, 𝑦 = ℘′ , with poles at 𝑝0 = 𝜋(0), 𝜋 ∶ ℂ → 𝐶 the projection. These define holomorphic maps 𝑥, 𝑦 ∶ 𝐶 − {𝑝0 } → ℂ, which yield a map 𝜑 ∶ 𝐶 − {𝑝0 } → ℂ2 ,

𝜑(𝑧) = (𝑥(𝑧), 𝑦(𝑧)).

This can be extended to 𝐶 as follows. Note (𝑥(𝑧), 𝑦(𝑧)) = [1, ℘(𝑧), ℘′ (𝑧)] = [𝑧3 , 𝑧3 ℘(𝑧), 𝑧3 ℘′ (𝑧)], where we multiply by 𝑧3 to get rid of the pole. So 𝜑 extends to a map 𝜑 ∶ 𝐶 → ℂ𝑃 2 ,

𝜑(𝑧) = [𝑧3 , 𝑧3 ℘(𝑧), 𝑧3 ℘′ (𝑧)].

This is a holomorphic map on 𝐶 − {𝑝0 }. Also 𝜑 is holomorphic around 𝑝0 because 𝑧3 ℘(𝑧) and 𝑧3 ℘′ (𝑧) are holomorphic functions on a ball around 𝑧 = 0. Moreover, 𝜑(𝑧) = [𝑧3 , 𝑧 + 𝑂(𝑧5 ), −2 + 𝑂(𝑧4 )], so 𝜑(𝑝0 ) = [0, 0, 1]. Let 𝐹(𝑧0 , 𝑧1 , 𝑧2 ) = 𝑧0 𝑧22 − 4𝑧31 + 𝑔2 𝑧20 𝑧1 +𝑔3 𝑧30 and consider the cubic curve 𝐶 ′ = 𝑉(𝐹). The dehomogenized polynomial ̂ 𝑦) = 𝑦2 − (4𝑥3 − 𝑔2 𝑥 − 𝑔3 ). By Proposition 5.84, the meromorphic functions is 𝐹(𝑥, ̂ ̂ 𝑥(𝑧), 𝑦(𝑧) satisfy 𝐹(𝑥(𝑧), 𝑦(𝑧)) ≡ 0. Therefore the image 𝜑(𝐶 − {𝑝0 }) ⊂ 𝐶 ′ ∩ 𝑈0 = 𝑉(𝐹), ′ so 𝜑(𝐶) ⊂ 𝐶 = 𝑉(𝐹). As both are irreducible complex curves, it must be 𝜑(𝐶) = 𝐶 ′ . Finally, let us see that 𝜑 ∶ 𝐶 → 𝐶 ′ is a biholomorphism. Observe that the preimage of [0, 0, 1] ∈ 𝐶 ′ is given by 𝑧 = 0, i.e., the point 𝑝0 = 𝜋(0) ∈ 𝐶. Let us see that 𝜑 is biholomorphic around 𝑝0 = 0. To check this, consider the affine set 𝑈2 and the affine coordinates 𝑥̂ = 𝑧0 /𝑧2 , 𝑦 ̂ = 𝑧1 /𝑧2 . In these coordinates the map 𝜑 is (5.37)

̂ 𝜑(𝑧) = (𝑥(𝑧), ̂ 𝑦(𝑧)) =(

1 ℘′ (𝑧)

,

℘(𝑧) 1 1 ) = (− 𝑧3 + 𝑂(𝑧5 ), − 𝑧 + 𝑂(𝑧3 )) , ′ 2 2 ℘ (𝑧)

which is an immersion. Therefore the degree of 𝜑 is deg 𝜑 = 1. This implies that the genus of 𝐶 ′ is 𝑔 = 1, so 𝐶 ′ must be a smooth cubic, and thus 𝜑 ∶ 𝐶 → 𝐶 ′ is injective. As 𝜑 is a ramified cover (Exercise 5.23), and it has degree 1, item (4) in the proof of Theorem 5.54 says that 𝜑 ∶ 𝐶 → 𝐶 ′ is a biholomorphism everywhere. □ Thus we have an equivalence between elliptic curves and smooth plane cubic curves. Remark 5.86. (1) The tangent at the point at infinity 𝑝 = [0, 0, 1] = 𝜑(𝑝0 ) is given by 𝑥̂ = 0 from (5.37), that is the line 𝑧0 = 0. Moreover, 𝑝 is an inflection point. (2) Take the smooth cubic 𝐶 defined by 𝑦2 = 𝑄(𝑥) in 𝑈0 = {𝑧0 ≠ 0}, where 𝑄(𝑥) is a degree 3 polynomial, with point 𝑝 = [0, 0, 1] at infinity. The map 𝜋(𝑥, 𝑦) = 𝑥 is the projection 𝜋 ∶ ℂ𝑃 2 − {[0, 0, 1]} → ℂ𝑃 1 . Restricted to 𝐶, we have the projection 𝜛 = 𝜋|𝐶 ∶ 𝐶 → ℂ𝑃 1 , which is a ramified cover of degree 2 (Exercise 5.35). Note that 𝜛 is a projection from a point in 𝐶, instead of the maps used in Theorem 5.54, which are projections from a point outside

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𝐶. The map 𝑞 is similar to (2) and (6) of Example 5.55. Note that for a line 𝐿 through 𝑝, 𝐶 ∩ 𝐿 = {𝑝, 𝑝1 , 𝑝2 }, where 𝜛(𝑝1 ) = 𝜛(𝑝2 ). For the line at infinity 𝐿0 = {𝑧0 = 0}, we have 𝐶 ∩ 𝐿0 = {𝑝} but with multiplicity 3 (because it is an inflection point), and 𝑞 can be defined at 𝑝 by taking 𝜛(𝑝) being the point at infinity of ℂ𝑃 1 . (3) The complex numbers 𝑔2 , 𝑔3 only depend on the lattice Λ and actually, they are the unique information needed to obtain the associated cubic curve. For this reason, it is customary to call them the elliptic invariants of 𝐶. (4) Since the cubic plane curve 𝐶 ′ is smooth, its discriminant Δ = 16(𝑔32 −27𝑔23 ) ≠ 0. In particular, this proves that 20𝑆 34 ≠ 49𝑆 26 for any lattice, which is a nonobvious combinatorial relation. (5) It is not necessary to know the value of 𝑔2 , 𝑔3 to obtain the equivalence between elliptic curves and smooth plane cubic curves. As 𝑙(6𝑝0 ) = 6, the set of functions {1, ℘, ℘′ , ℘2 , ℘℘′ , ℘3 , (℘′ )2 } is linearly dependent, so there must be a linear relation between them. This gives a polynomial relation 𝐹(℘, ℘′ ) = 0 of degree 3 in ℘ and degree 2 in ℘′ . What we have done is to compute 𝐹 explicitly. (6) For a compact connected complex curve 𝐶 of any genus, the field of meromorphic functions is denoted ℂ(𝐶). It can be proved that it is always generated by two meromorphic functions 𝑥, 𝑦 ∈ ℂ(𝐶) and that they satisfy some polynomial equation 𝐹(𝑥, 𝑦) = 0 defining the plane model of 𝐶. Any 𝑓 ∈ ℂ(𝐶) is a rational function of these 𝑥, 𝑦. The field ℂ(𝐶) defines the algebraic structure of 𝐶 (Remark 1.23). (7) Let 𝐶 = ℂ/Λ. The map 𝜋 ∶ ℂ → 𝐶 is obviously holomorphic, but it is not an algebraic map. This is a subtle point, with important implications in algebraic geometry. It is due to the fact that the pullback of the functions 𝑥, 𝑦 ∈ ℂ(𝐶) are 𝜋∗ 𝑥 = ℘, 𝜋∗ 𝑦 = ℘′ , which are not algebraic for ℂ (the algebraic structure of ℂ is defined by the rational functions ℂ(𝑧)). The map 𝜑 ∶ 𝐶 = ℂ/Λ → 𝐶 ′ is an algebraic isomorphism, by construction. Hence there is a holomorphic non-algebraic map 𝜑 ∘ 𝜋 ∶ ℂ → 𝐶 ′ ⊂ ℂ𝑃 2 . (8) Any holomorphic map between compact complex curves is algebraic (this is a consequence of the Chow theorem). Also an algebraic map between affine curves extends to the projective curves. Moreover, an algebraic map between curves 𝑓 ∶ 𝐶 → 𝐶 ′ is always of finite degree, and it is always a ramified cover. In particular, the genus satisfy 𝑔𝐶 ≥ 𝑔𝐶 ′ (Exercise 5.23). Therefore for an elliptic curve 𝐶, there are not (non-constant) algebraic maps ℂ𝑃 1 → 𝐶 or ℂ → 𝐶 but there are algebraic maps 𝐶 → ℂ𝑃 1 (e.g. the projection 𝜋 of item (2)). (9) Weierstrass functions serve to compute primitives of the type ∫

𝑑𝑥 . √4𝑥3 −𝐴𝑥−𝐵

For this, take 𝑦2 = 4𝑥3 − 𝐴𝑥 − 𝐵, which defines an elliptic curve. Let ℘(𝑧) be the associated Weierstrass function, so that 𝑥 = ℘(𝑧), 𝑦 = ℘′ (𝑧). Using the

5.5. Elliptic curves

335

change of variables rule, ∫

𝑑𝑥

=∫

℘′ (𝑧)𝑑𝑧 𝑑𝑥 =∫ = ∫ 𝑑𝑧 = 𝑧. 𝑦 ℘′ (𝑧)

√4𝑥3 − 𝐴𝑥 − 𝐵 Then, in order to calculate the primitive, we have to invert the change of variables (𝑥, 𝑦(𝑥)) = (℘(𝑧), ℘′ (𝑧)) as 𝑧 = ℘−1 (𝑥). This function however is multiply defined, since its value is determined modulo the lattice Λ. 𝑑𝑥 Observe the parallelism with the integral ∫ = cos−1 (𝑥). This comes 2 √1−𝑥

from the algebraic curve 𝑥2 + 𝑦2 = 1 and the holomorphic parametrization (𝑥(𝑧), 𝑦(𝑧)) = (cos 𝑧, sin 𝑧). 5.5.3. The 𝐽-invariant. The elliptic invariants 𝑔2 = 𝑔2 (Λ) = 60 𝑆 4 (Λ) and 𝑔3 = 𝑔3 (Λ) = 140 𝑆 6 (Λ) produce functions on the set of lattices (𝑔2 , 𝑔3 ) ∶ ℛ → ℂ2 . If we have two equivalent lattices Λ, Λ′ , with Λ′ = 𝜇Λ, with 𝜇 ∈ ℂ∗ , then for any integer 𝑟 > 1, it holds 1 = 𝜇−𝑟 𝑆𝑟 (Λ). 𝑆𝑟 (Λ′ ) = 𝑆𝑟 (𝜇Λ) = ∑ 𝑟 (𝜇𝑤) 𝑤∈Λ−{0} Hence 𝑔2 (Λ′ ) = 𝜇−4 𝑔2 (Λ), 𝑔3 (Λ′ ) = 𝜇−6 𝑔3 (Λ). This implies that the map (𝑔2 , 𝑔3 ) descends to a map 1 𝐺 = [𝑔2 , 𝑔3 ] ∶ ℳ𝑇ℂ2 = ℂ∗ \ℛ ⟶ 𝒩 𝑊 = ℂ𝑃[2,3] − {[3, 1]}.

Theorem 5.87. The map 𝐺 gives an isomorphism 𝐺 ∶ ℳ𝑇ℂ2 → 𝒩 𝑊 . Proof. The map 𝐺 associates, to each complex torus 𝐶 = ℂ/Λ, the smooth cubic curve 𝐶 ′ with Weierstrass equation 𝑦2 = 4𝑥3 − 𝑔2 𝑥 − 𝑔3 . The embedding 𝜑 ∶ 𝐶 → 𝐶 ′ ⊂ ℂ𝑃 2 says that 𝐶 ≅ 𝐶 ′ . Let us see that 𝐺 is injective. Let 𝐶1 , 𝐶2 be two complex tori, and let 𝐶1′ = 𝐺(𝐶1 ), 𝐶2′ = 𝐺(𝐶2 ) be the corresponding cubic curves. If 𝐶1′ ≅ 𝐶2′ are isomorphic as cubic curves in Weierstrass form, then as there are biholomorphisms 𝐶1 ≅ 𝐶1′ and 𝐶2 ≅ 𝐶2′ , we have that 𝐶1 and 𝐶2 are biholomorphic. This means that they represent the same point in ℳ𝑇ℂ2 . We define the 𝐽-invariant as 𝐽 = 𝐽𝑊 ∘ 𝐺 ∶ ℳ𝑇ℂ2 → ℂ, where 𝐽𝑊 ∶ 𝒩 𝑊 → ℂ is the isomorphism (5.32). That is, for an elliptic curve 𝐶 = ℂ/Λ, it is 𝐽(𝐶) =

𝑔32 , 𝑔32 − 27𝑔23

where 𝑔2 = 𝑔2 (Λ), 𝑔3 = 𝑔3 (Λ). This is injective, since both 𝐽𝑊 and 𝐺 are so. Let us see that 𝐽 is holomorphic. Recall that ℳ𝑇ℂ2 = 𝐻/ PSL(2, ℤ) has the induced complex structure from that of 𝐻 (Proposition 5.61). Let 𝜏 ∈ 𝐻. Take the standard lattice Λ𝜏 = ⟨1, 𝜏⟩ and the associated complex torus 𝐶𝜏 = ℂ/Λ𝜏 . Then we define 1 𝑔2 (𝜏) = 𝑔2 (Λ𝜏 ) = 60 ∑ , (𝑛 + 𝑚𝜏)4 (𝑛,𝑚)≠(0,0) 1 . (𝑛 + 𝑚𝜏)6 (𝑛,𝑚)≠(0,0)

𝑔3 (𝜏) = 𝑔3 (Λ𝜏 ) = 140 ∑

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The corresponding 𝐽-invariant is 𝐽(𝜏) = 𝐽(𝐶𝜏 ) =

𝑔32 (𝜏) . 𝑔32 (𝜏) − 27𝑔23 (𝜏)

Observe that both series 𝑔2 , 𝑔3 converge absolutely and uniformly for 𝜏 in the upper half-plane (Lemma 5.74), thereby defining holomorphic functions there. Therefore 𝐽 ∶ 𝐻 → ℂ is a holomorphic function. This descends to a continuous function 𝐽 ∶ ℳ𝑇ℂ2 = 𝐻/ PSL(2, ℂ) → ℂ. With the notations of the proof of Proposition 5.61, 𝐽 is holomorphic on ℳ𝑇ℂ2 − {𝑣 1 , 𝑣 2 }, since 𝐻 − (𝑇1 ∪ 𝑇2 ) → ℳ𝑇ℂ2 − {𝑣 1 , 𝑣 2 } is a cover. By Proposition 5.61, ℳ𝑇ℂ2 is biholomorphic to ℂ, hence we can interpret 𝐽 as a continuous function 𝐽 ∶ ℂ → ℂ which is holomorphic on ℂ minus two points. As it is continuous on these points, it is also holomorphic on them. So 𝐽 ∶ ℳ𝑇ℂ2 → ℂ is holomorphic and injective. It cannot be that 𝐽 ′ (𝑥) = 0 for some 𝑥 ∈ ℳ𝑇ℂ2 , since then 𝐽 would not be injective around 𝑥. So 𝐽 is a biholomorphism to its image 𝐽(ℳ𝑇ℂ2 ) ⊂ ℂ. If 𝐽(ℳ𝑇ℂ2 ) ≠ ℂ, then the Riemann mapping theorem (Remark 5.57(2)) would imply that 𝐽(ℳ𝑇ℂ2 ) ≅ 𝔻, which is absurd since ℂ ≅ ℳ𝑇ℂ2 ≅ 𝐽(ℳ𝑇ℂ2 ). So −1 𝐽 ∶ ℳ𝑇ℂ2 → ℂ is a biholomorphism, giving that 𝐺 = 𝐽𝑊 ∘ 𝐽 is also a biholomorphism. This completes the proof. □ So there are isomorphisms ℳ𝑇ℂ2 ≅ 𝒩 𝑊 ≅ 𝒩𝑠𝑡𝑑 ≅ ℂ, via the maps in Theorem 5.87, (5.31), and (5.32). As a consequence, • Let 𝐶, 𝐶 ′ be two elliptic curves. Then 𝐶 ≅ 𝐶 ′ if and only if 𝐽(𝐶) = 𝐽(𝐶 ′ ). • If 𝑦2 = 4𝑥3 − 𝐴𝑥 − 𝐵 and 𝑦2 = 4𝑥3 − 𝐴′ 𝑥 − 𝐵 ′ are isomorphic as complex curves, then (𝐴′ , 𝐵 ′ ) = (𝜇2 𝐴, 𝜇3 𝐵) for some 𝜇 ∈ ℂ∗ . The isomorphism is given by (𝑥, 𝑦) ↦ (𝜇𝑥, 𝜇3/2 𝑦). • Let 𝐶 𝜆 = {𝑦2 = 𝑥(𝑥 −1)(𝑥 −𝜆)} be the cubic in standard form. Then 𝐶 𝜆 ≅ 𝐶 𝜆′ if and only if 𝜆′ = 𝜎 ⋅ 𝜆, for some 𝜎 ∈ 𝑆 3 . Remark 5.88. As we saw in Proposition 5.61, there are two special points in 𝐻/ PSL(2, ℂ). The point 𝜏1 = i corresponds to the square lattice. It has 𝐽-invariant 𝐽(𝜏1 ) = 1 since 𝑆 6 (i) = 0, and corresponds to the cubic 𝑦2 = 4𝑥3 − 𝐴𝑥. The point 𝜏2 = 𝑒𝜋i/3 corresponds to the hexagonal lattice. It has 𝐽-invariant 𝐽(𝜏2 ) = 0, since 𝑆 4 (𝑒𝜋i/3 ) = 0, and corresponds to the cubic 𝑦2 = 4𝑥3 − 𝐵 (see Exercise 5.38). Remark 5.89. It is interesting to note what happens to the point 𝜏∞ that we added to ℳ𝑇ℂ2 = 𝐻/ PSL(2, ℂ) to compactify it. Let 𝑟 > 1 be an integer. The series defining 𝑆 2𝑟 (𝜏) converges absolutely and uniformly so, in order to compute lim𝜏→i∞ 𝑆 2𝑟 (𝜏) we can take the term-by-term limit. So 𝑆 2𝑟 (i𝛼) =

1 2𝑟 (𝑛 + i𝛼𝑚) (𝑛,𝑚)≠(0,0) ∑

∞

=

∞

𝛼→∞ 1 1 1 ∑ + ∑ ⟶ 2 ∑ 2𝑟 = 2𝜁(2𝑟), 2𝑟 2𝑟 𝑛 𝑛 (𝑛 + i𝛼𝑚) 𝑛∈ℤ−{0} 𝑚∈ℤ−{0} 𝑛=−∞ 𝑛=1

∑

Problems

337

where 𝜁(𝑟) is the Riemann zeta function. Therefore 𝑔2 (i𝛼) = 60 𝑆 4 (i𝛼) → 120 𝜁(4) = 4𝜋4 8𝜋6 and 𝑔3 (i𝛼) = 140 𝑆 6 (i𝛼) → 280 𝜁(6) = 27 , using the known values of the Riemann 3 zeta function 𝜁(4) =

𝜋4 90

and 𝜁(6) =

𝜋6 . 945

In particular, the discriminant satisfies that 𝛼→∞

Δ(i𝛼) = 16 (𝑔2 (i𝛼)3 − 27𝑔3 (i𝛼)2 ) ⟶ 0. 16𝑔3 (i𝛼)

2 Therefore, we have that 𝐽(i𝛼) = ∆(i𝛼) → ∞ when 𝛼 → ∞. This implies that 𝐽 extends to a homeomorphism (and hence to a biholomorphism)

(5.38)

𝐽 ∶ ℳ ℂ𝑇 2 = ℳ𝑇ℂ2 ∪ {𝜏∞ } → ℂ .

This gives an explicit biholomorphism for Proposition 5.61. The curve corresponding 4 8 to 𝜏∞ is 𝑦2 = 4𝑥3 − 3 𝜋4 𝑥 − 27 𝜋6 , which is a nodal cubic. It is singular since Δ = 0. So the compactified map (5.38) sends the cylinder ℂ/ℤ ≅ ℂ∗ to the nodal cubic curve. It is customary to define the 𝐽-invariant with a factor 1728 in front, so that 𝐽(i) = 1728. This is relevant for arithmetic questions regarding elliptic curves [Sil]. Remark 5.90. In higher dimensions, a complex manifold of the form 𝑀 = ℂ𝑛 /Λ, where Λ ⊂ ℂ𝑛 is a rank 2𝑛 lattice, is called a complex torus. When a complex torus is a Hodge manifold, it is called an Abelian variety. These are very important in arithmetic algebraic geometry.

Problems Exercise 5.1. Let 𝑓 ∶ 𝑈 ⊂ ℂ → ℂ be a smooth function defined on an open subset 𝑈. Using the Stokes theorem, prove the Cauchy theorem: 𝑓 is holomorphic if and only if ∫𝛾 𝑓 𝑑𝑧 = 0 for every piecewise 𝐶 1 loop 𝛾 ∶ 𝑆 1 → 𝑈 bounding a disc 𝐷2 . Use this to prove the following: (1) For 𝑓 ∈ Hol(𝑈), 𝑓(𝑛) (𝑎) =

𝑛! 2𝜋i

∫𝛾

𝑓(𝑧) 𝑑𝑧, (𝑧−𝑎)𝑛+1

for 𝑛 ≥ 0 and 𝑎 ∈ Int(𝐷).

(2) Maximum principle: if 𝑈 is connected, 𝑓 ∈ Hol(𝑈), and |𝑓(𝑧)| has a local maximum at some point of 𝑈, then 𝑓 is constant. (3) Liouville theorem: if 𝑓 ∶ ℂ → ℂ is holomorphic on the whole of ℂ and |𝑓(𝑧)| ≤ 𝐶(1 + |𝑧|𝑁 ), for some constant 𝐶 > 0 and integer 𝑁 ≥ 0, then 𝑓(𝑧) is a polynomial of degree deg(𝑓) ≤ 𝑁. +

Exercise 5.2. Prove that GL(𝑛, ℂ) is connected. Use it to prove that 𝚤(GL(𝑛, ℂ)) < GL (2𝑛, ℝ), and conclude that any biholomorphism preserves orientation. Exercise 5.3. Given (𝑉, 𝐽) a complex vector space, we define the conjugate vector space as 𝑉 = (𝑉, −𝐽). Show that 𝑉 is also a complex vector space. Prove that a map 𝑓 ∶ 𝑉1 → 𝑉2 is ℂ-antilinear (i.e., 𝑓(𝜆𝑣) = 𝜆𝑣, for 𝑣 ∈ 𝑉1 and 𝜆 ∈ ℂ) if and only if 𝑓 ∶ 𝑉1 → 𝑉 2 is ℂ-linear. Exercise 5.4. Let h be a Hermitian product on a complex vector space. Prove that the GramSchmidt method produces h-orthonormal basis. Develop analogues of the results in section 4.1.1 for the group U(𝑛). Exercise 5.5. Prove that for any complex manifold 𝑀 there are always Hermitian metrics, and that these form a convex set.

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Exercise 5.6. Show that if 𝑀 is a complex manifold and 𝜋 ∶ 𝑀 ′ → 𝑀 is a cover, then 𝑀 ′ admits the structure of a complex manifold such that 𝜋 is holomorphic and the deck transformations act by biholomorphisms. Let 𝑀 ′ be a complex manifold, and let Γ < Bihol(𝑀 ′ ) be a group of biholomorphisms of 𝑀 acting freely and properly on 𝑀 ′ . Show that 𝑀 = 𝑀 ′ /Γ admits the structure of a complex manifold such that 𝜋 ∶ 𝑀 ′ → 𝑀 is a holomorphic cover. ′

Exercise 5.7. Let (𝑀, 𝐽) be an almost complex manifold. Define (𝑝, 𝑞)-forms on 𝑀, and prove that 𝑑 ∶ Ω𝑘 (𝑀, ℂ) → Ω𝑘+1 (𝑀, ℂ) sends Ω𝑝,𝑞 (𝑀) → Ω𝑝+2,𝑞−1 (𝑀) ⊕ Ω𝑝+1,𝑞 (𝑀) ⊕ Ω𝑝,𝑞+1 (𝑀) ⊕ Ω𝑝−1,𝑞+2 (𝑀). Exercise 5.8. Let 𝐽 be an almost complex structure on a smooth 2𝑛-manifold. Show that 𝑁𝐽 is a tensor. Relate 𝑁𝐽 with the component 𝜋0,2 ∘ 𝑑 ∶ Ω1,0 (𝑀) → Ω0,2 (𝑀) of the differential. Use this to prove that 𝜕2̄ = 0 if and only if 𝑁𝐽 = 0. Exercise 5.9. Let ∇ be a symmetric connection, and let 𝐽 be an almost complex structure. Show that if ∇𝐽 = 0, then 𝐽 is integrable. Use this to prove that a manifold is Kähler if and only if the 𝑝 Riemannian metric satisfies that its holonomy Hol∇ < U(𝑛) < SO(2𝑛). Exercise 5.10. Prove that the wedge product gives a map on Dolbeault cohomology 𝐻 𝑝,𝑞 (𝑀) × 𝐻 𝑟,𝑠 (𝑀) → 𝐻 𝑝+𝑟,𝑞+𝑠 (𝑀) for any complex manifold 𝑀. Exercise 5.11. Let (𝑀, 𝐽, h) be a Hermitian manifold, with the local expression for the Hermitian metric given by h = ∑𝑗,𝑘 ℎ𝑗𝑘 𝑑𝑧𝑗 ⊗𝑑𝑧𝑘 . Check in coordinates that, as tensors, there is an equality h = g − i𝜔. Exercise 5.12. Suppose that 𝑉 is a real vector space of dimension 𝑚, and let 𝜔 be an antisymmetric map on 𝑉. Prove that there is a basis such that 𝜔 has the form 0 ⎛ −1 ⎜ 𝜔=⎜ ⎜ ⎜ ⎝

1 0 0 −1

1 0 ⋱

⎞ ⎟ ⎟, ⎟ ⎟ 0 ⎠

with 𝑟 blocks 2 × 2, rk 𝜔 = 2𝑟, and the rest of the diagonal entries are zero. Conclude that if 𝜔 is symplectic, then 𝑚 is even. Note that (𝐻 1 (Σ𝑔 ), 𝑄Σ𝑔 ) is a symplectic vector space, where 𝑄Σ𝑔 is the intersection form (see (2.26)). Exercise 5.13. Let 𝑀 be a smooth manifold of dimension 2𝑛, and let 𝜔 ∈ Ω2 (𝑀). Prove that 𝜔 is non-degenerate if and only if 𝜔𝑛 is a volume form. Exercise 5.14. Let 𝛼 ∈ Ω𝑘 (ℂ𝑃 𝑛 ) which is invariant under PU(𝑛 + 1). Prove that 𝑑𝛼 = 0. Exercise 5.15. Let (𝑀, h) be a complex manifold with a Hermitian metric. Show that (𝑀, h) is Kähler if and only if for every 𝑝 ∈ 𝑀, there is a chart 𝜑 ∶ 𝑈 𝑝 → ℂ𝑛 , 𝜑(𝑝) = 0, such that ℎ𝑗𝑘 (𝑧) = Id +𝑂(|𝑧|2 ) (we say that h osculates to order 2 with the standard metric). Exercise 5.16. Prove that (ℂ𝑃𝑁 , g𝐹𝑆 ) has constant holomorphic sectional curvature, that is, for all 𝑝 ∈ ℂ𝑃 𝑁 and 𝜎 ⊂ 𝑇𝑝 ℂ𝑃𝑁 a complex plane (i.e., such that 𝐽(𝜎) = 𝜎), we have that 𝐾𝑝 (𝜎) is constant. Compute the sectional curvature of ℂ𝑃𝑁 . Exercise 5.17. Compute the volume of ℂ𝑃 𝑛 , and justify it geometrically. Exercise 5.18. Give a cellular decomposition of ℝ𝑃 𝑛 , and compute the cellular homology 𝐻𝑘𝐶𝑊 (ℝ𝑃 𝑛 ) and 𝐻𝑘𝐶𝑊 (ℝ𝑃 𝑛 , ℤ2 ).

Problems

339

Exercise 5.19. We denote ℂ𝑃 2 as the manifold ℂ𝑃2 with the opposite orientation to the natural one as a complex manifold. Show that ℂ𝑃 2 and ℂ𝑃2 are not homeomorphic as oriented − manifolds. Therefore Homeo (ℂ𝑃 2 ) = ∅, and hence ℂ𝑃 2 is not the oriented cover of any other manifold. (𝑟)

(𝑠)

Prove that all ℂ𝑃 2 # ⋯ #ℂ𝑃 2 #ℂ𝑃 2 # ⋯ #ℂ𝑃2 are different oriented manifolds for 𝑟, 𝑠 ≥ 0. Exercise 5.20. Let 𝐶 be a complex curve, and let 𝑓 ∶ 𝐶 → ℂ be a meromorphic function. Show that if 𝑓 ≠ 0, then the set of zeros of 𝐶 is discrete. Exercise 5.21. Let 𝑈 ⊂ ℂ be an open subset, and let 𝜓 ∶ 𝑈 → 𝑈 be a biholomorphism such that 𝜓(𝑝) = 𝑝 with 𝑝 ∈ 𝑈, and 𝜓𝑑 = Id, where 𝑑 > 0. Prove that there is a biholomorphism 𝑓 ∶ 𝑉 𝑝 → 𝐵𝜖 (0) from a neighbourhood of 𝑝, such that 𝜓′ = 𝑓 ∘ 𝜓 ∘ 𝑓−1 is of the form 𝜓′ (𝑤) = 𝑒2𝜋i/𝑑 𝑤. (It is said that 𝜓 is holomorphically linearizable). Exercise 5.22. Show that there are no holomorphic maps (non-constant) 𝑓 ∶ ℂ𝑃 1 → ℂ/Λ. • Topologically, using that 𝑓 is a ramified cover. • Analytically, lifting 𝑓 to the universal cover and using the Liouville theorem. Exercise 5.23. Let 𝑓 ∶ 𝐶 → 𝐶 ′ be a non-constant holomorphic map between compact connected complex curves. Prove that it is a ramified cover. Conclude that the genus of 𝐶 is bigger than or equal to the genus of 𝐶 ′ . Exercise 5.24. Take a plane complex curve 𝐶 ⊂ ℂ𝑃 2 . Compute the metric of 𝐶 induced by the Fubini-Study metric, in affine coordinates (𝑧, 𝑤) with the curve described locally as 𝑤 = 𝑓(𝑧). Conclude that in general the embedding given by the Kodaira embedding theorem 𝐶 ↪ ℂ𝑃 𝑁 is not an isometry. Exercise 5.25. Let 𝑔 ∶ 𝑈 ⊂ ℂ𝑛 → ℂ be a holomorphic function which is not the power of 𝜕𝑔 another holomorphic function. Show that there exists 𝑤 ∈ 𝑈 such that 𝑔(𝑤) = 0 and 𝜕𝑤 (𝑤) ≠ 0 𝑗

for some 1 ≤ 𝑗 ≤ 𝑛. Show that the ideal 𝐼 = (𝑧0 , 𝑧21 ) ⊂ ℂ[𝑧0 , 𝑧1 , 𝑧2 , 𝑧3 ] is not radical and satisfies the conditions of Proposition 5.42 with 𝑑 = 2. However 𝑉(𝐼) ⊂ ℂ𝑃 3 is a smooth projective variety of complex dimension 1. Exercise 5.26. Prove that a smooth connected projective variety is irreducible, and that an irreducible projective variety is connected (in the usual topology). Exercise 5.27. Prove that a hypersurface 𝑋 ⊂ ℂ𝑃 𝑛 is defined by a single homogeneous polynomial 𝐹, i.e., 𝑋 = 𝑉(𝐹). Exercise 5.28. Give a method to parametrize a plane curve of degree 𝑑 = 4 with 𝛿 = 3 nodes. Exercise 5.29. Prove that given 𝑔 ≥ 0, there are irreducible plane curves of some degree 𝑑 ≥ 1 with 𝛿 nodes such that 𝑔 = (𝑑 − 1)(𝑑 − 2)/2 − 𝛿. Exercise 5.30. Let 𝐶 ′ = 𝑉(𝐹) be a plane model of a compact connected complex curve 𝐶 of genus 𝑔 with possibly higher order nodes. Let 𝑁 be the set of nodes, let 𝑟𝑝 ≥ 2 be the order of 𝑝 ∈ 𝑁, and let 𝛿𝑝 = 𝑟𝑝 (𝑟𝑝 − 1)/2. Prove that 𝑔 = (𝑑 − 1)(𝑑 − 2)/2 − ∑𝑝∈𝑁 𝛿𝑝 . Exercise 5.31. Let 𝐹 ∈ ℂ[𝑧0 , 𝑧1 , 𝑧2 ] be a homogeneous polynomial of degree 3 such that 𝑉(𝐹) is a smooth cubic. Prove that 𝑉(𝐹) and 𝑉(det Hess 𝐹) intersect transversally, and conclude that 𝑉(𝐹) has exactly nine inflection points. Exercise 5.32. Show that the map on the moduli space ℳ ℂ𝑇 2 that reverses the orientation of a complex torus, 𝐶 ↦ 𝐶, is given by 𝜏 ↦ −𝜏. Give the expressions of this map in the different models of ℳ ℂ𝑇 2 .

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Exercise 5.33. Let 𝑞 ∈ ℂ∗ be a non-zero complex number with |𝑞| ≠ 1. Consider the action 𝑓(𝑧) = 𝑞𝑧 on ℂ − {0}. Show that the quotient 𝑀 = (ℂ − {0})/⟨𝑓⟩ is a compact connected complex curve of genus 𝑔 = 1. Find the value 𝜏 ∈ 𝐻 that characterizes it. Exercise 5.34. Let Λ ⊂ ℂ be a rank 1 lattice. Using that ℂ/Λ ≅ ℂ∗ , construct all holomorphic functions which are Λ-periodic (they are called simply periodic). ∞

1

Prove that ∑𝑛=−∞ (𝑧−𝑛)𝑑 converges for 𝑑 ≥ 2, and give a renormalization method to make it convergent for 𝑑 = 1. Determine the functions obtained in this way. Exercise 5.35. Let 𝐶 be a smooth plane cubic defined by the affine equation 𝑦2 − 𝑎(𝑥 − 𝜆1 )(𝑥 − 𝜆2 )(𝑥 − 𝜆3 ) = 0 Prove that the projection 𝜋 ∶ 𝐶 → ℂ𝑃 1 , 𝜋(𝑥, 𝑦) = 𝑥, is a ramified cover of degree 2 ramified at 𝜆1 , 𝜆2 , 𝜆3 , and ∞. Exercise 5.36. Let 𝑑0 , 𝑑1 > 0 be coprime integers. Prove that the weighted projective line ℂ𝑃[𝑑1 0 ,𝑑1 ] = (ℂ2 − {0})/ℂ∗ , (𝑧0 , 𝑧1 ) ∼ (𝜇𝑑0 𝑧0 , 𝜇𝑑1 𝑧1 ) for 𝜇 ∈ ℂ∗ , has an orbifold structure. Prove that it also has a natural complex structure and, as a complex manifold, ℂ𝑃[𝑑1 0 ,𝑑1 ] ≅ ℂ𝑃 1 . Exercise 5.37. Let Λ = ℤ ⊂ ℂ be the standard lattice of rank 1 (that formally corresponds to 𝜏∞ ). Find the ℘-function associated to it, and describe it in terms of trigonometric functions. Determine the embedding 𝜑 ∶ ℂ/ℤ → ℂ𝑃 2 , with image the nodal cubic curve. Exercise 5.38. Prove that 𝑆 6 (i) = 0 and 𝑆 4 (𝑒𝜋i/3 ) = 0.

References and extra material Basic reading. The basic facts of complex analysis can be found in [Ahl]. Complex manifolds are widely studied in [Huy] and [Wel]. Moving to complex curves, we recommend the book [Kir]. For basic concepts of algebraic geometry, the student can read the extensive monograph [G-H]. The last part of the chapter focuses on elliptic curves, which appear in [Sil]. [Ahl] L. Ahlfors, Complex Analysis, 3rd Edition, McGraw-Hill, 1979. [G-H] P. Griffiths, J. Harris, Principles of Algebraic Geometry, John Wiley and Sons Inc., 1978. [Huy] D. Huybrechts, Complex Geometry: An Introduction, Universitext, Springer, 2004. [Kir] F. Kirwan, Complex Algebraic Curves, London Mathematical Society Student Texts, Vol. 23, Cambridge University Press, 1992. [Sil] J.H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, Vol. 106, Springer-Verlag, 1986. [Wel] R.O. Wells, Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, 3rd Edition, 2008. Further reading. We propose the following topics for further study. • Uniformization theorem. We propose to give a proof of this important theorem using techniques of complex variables. This can be taken from: [F-K] H.M. Farkas, I. Kra, Riemann surfaces, 2nd Edition, Springer, 1992. • Frobenius theorem. For this result (Theorem 5.11), the student can look at: [War] F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, Vol. 94, 1983. • Newlander-Nirenberg theorem. This can be found in chapter 8 of: [Dem] J. Demailly, Complex Analytic and Differential Geometry, Université de Grenoble I, www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

References and extra material

341

• Darboux theorem. For the study of symplectic manifolds we refer to: [MDS] D. McDuff, D. Salamon, Introduction to Symplectic Topology. Oxford University Press, 1998. • Kodaira embedding theorem. A proof of Theorem 5.43 can be found in [Huy] or [Wel]. • Algebraic curves. A study of plane curves from the point of view of algebraic geometry, the local study of branches of curves, and the intersection theory of two plane curves. We recommend: [Ful] W. Fulton, Algebraic Curves: An Introduction to Algebraic Geometry, Addison Wesley Longman, 1974. • Belyi theorem. This theory characterizes the smooth algebraic curves defined by a polynomial with coefficients in ℚ (Example 5.55(9)). [GGD] E. Girondo, G. González-Díez, Introduction to Compact Riemann Surfaces and Dessins d’Enfants, London Mathematical Society Student Texts, Vol. 79, Cambridge University Press, 2011. • Sheaf cohomology. This can be found in chapters I and II of the book: [Iv2] B. Iversen, Cohomology of Sheaves, Universitext, Springer, 1986. • Octonions. These appear in Example 5.13. [Bae] J. Baez, The octonions, Bull. Amer. Math. Soc. 39, 145-205, 2001. • Riemann-Roch theorem. The study of meromorphic forms for Riemann surfaces and a proof of the Riemann-Roch theorem for curves can be found in [Do3,§8.2] or [Mir]. [Do3] S.K. Donaldson, Riemann Surfaces, Oxford Graduate Texts in Mathematics, Vol. 22, Oxford University Press, 2011. [Mir] R. Miranda, Algebraic Curves and Riemann Surfaces, Graduate Studies in Mathematics, Vol. 5, AMS, 1995. References. [C-R] E. Calabi, M. Rosenlicht, Complex analytic manifolds without countable base, Proc. Amer. Math. Soc. 4, 335-340, 1953. [C-S] J. Conway, D. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A. K. Peters, 2003. [Har] J. Harris, Algebraic Geometry: A First Course, Graduate Texts in Mathematics, Vol. 133, Springer, 1995. [Jo2] J. Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics, 3rd Edition, Universitext, Springer, 2006. [Nar] R. Narasimhan, Several Complex Variables, Chicago Lectures in Mathematics, University of Chicago Press, 1995. [N-N] A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Annals Math. (2), 65, 391-404, 1957. [New] I. Newton, The method of fluxions and infinite series; with its application to the geometry of curve-lines, translated by J. Colson, London: Henry Woodfall, p. 378, 1736. [O-T] J. Oprea, A. Tralle, Symplectic Manifolds with no Kähler Structure, Lecture Notes in Mathematics, 1661, Springer-Verlag, 1997. [Sha] I.R. Shafarevich, Basic Algebraic Geometry 1: Varieties in Projective Space, 3rd Edition, Springer, 2013.

Chapter 6

Global analysis

In this chapter we deal with some problems that have been left uncompleted in Chapter 5. The main focus is to show different methods of partial differential equations that can be successfully applied to solve deep geometrical questions. The area that deals with differential equations on manifolds is called global analysis, since whereas mathematical analysis is in nature “local” (it deals with problems on open sets of ℝ𝑛 ), here a key ingredient is the “global” nature of manifolds (defined as a collection of charts glued together). In global analysis, several features, such as dealing with compact manifolds, produce improvements for solving differential equations. We start by introducing conformal structures and seeing that they are equivalent to complex structures in the case of surfaces (the focus of this book). We then review the main ingredients of Hodge theory for compact manifolds, and how the solutions of the elliptic problem of the Laplacian links directly to de Rham cohomology, and hence to purely topological properties of the manifold. Then we move to the problem of finding metrics of constant curvature in a conformal class for surfaces, for which we give several proofs. This is a non-linear elliptic problem with nice geometric features. This yields a proof of the Newlander-Nirenberg theorem, the existence of conformal coordinates for surfaces, and the uniformization theorem for the universal cover of a compact complex curve. Our proof in the case of positive constant curvature is a simplification with respect to the usual proofs found in the literature. We finish with a brief discussion on the curvature flow, for its deep implications to current research in geometry.

6.1. Conformal structures A Riemannian metric on a smooth manifold allows us to measure lengths and properties derived from them, such as the curvature. Riemannian geometry can be understood as the study of those properties which are invariant under length preserving maps (i.e., isometries). In an analogous manner, a conformal structure on a manifold allows us to measure angles, and can be understood as the study of properties which are invariant by conformal maps. Conformal maps preserve angles but may distort 343

344

6. Global analysis

lengths and volumes. We will see a deep relation of conformal geometry and complex geometry in dimension 2, although it no longer holds in higher dimension. We start by recalling Definition 4.11. Definition 6.1. Let (𝑀1 , g1 ) and (𝑀2 , g2 ) be two Riemannian manifolds. A smooth map 𝑓 ∶ 𝑀1 → 𝑀2 is locally conformal if 𝑓∗ g2 = 𝜇 g1 for some positive smooth function 𝜇 ∶ 𝑀1 → ℝ>0 , which we call dilation factor of 𝑓. The map 𝑓 is conformal if it is locally conformal and a diffeomorphism. By Lemma 4.13, if 𝑓 ∶ 𝑀1 → 𝑀2 is a locally conformal map, then, for every 𝑝 ∈ 𝑀1 , the map 𝑑𝑝 𝑓 ∶ 𝑇𝑝 𝑀1 → 𝑇 𝑓(𝑝) 𝑀2 is a dilation. This means that 𝑑𝑝 𝑓 expands the norm by a factor 𝜇(𝑝)1/2 , while it preserves angles and orthogonality. In particular, if dim 𝑀1 = dim 𝑀2 , then 𝑓 is a local diffeomorphism. The composition of locally conformal maps is locally conformal. If 𝑓 is conformal, then the inverse 𝑓−1 is also conformal. Hence the conformal maps of a Riemannian manifold (𝑀, g) form a group (6.1)

Conf(𝑀, g) = {𝑓 ∶ (𝑀, g) → (𝑀, g)| 𝑓 is conformal}.

Definition 6.2. Let 𝑀 be a manifold, and let g1 , g2 be two Riemannian metrics of 𝑀. We say that g1 and g2 are conformal metrics if Id ∶ (𝑀, g1 ) → (𝑀, g2 ) is a conformal map, i.e., if there exists a smooth function 𝜇 ∶ 𝑀 → ℝ>0 so that g2 = 𝜇 g1 . In the set of Riemannian metrics ℳet(𝑀) on 𝑀 we can define an equivalence relation as g1 ∼ g2 if and only if g1 and g2 are conformal metrics. We denote by [g] the equivalence relation of a metric g under ∼, and we call [g] a conformal class. A choice of a equivalence class [g] of Riemannian metrics is called a conformal structure on 𝑀. Remark 6.3. Let g, ĝ be two conformal metrics, and write ĝ = 𝑒2ᵆ g, for a smooth function 𝑢. Then locally 𝑔𝑖𝑗 ̂ = 𝑒2ᵆ 𝑔𝑖𝑗 , and for the inverse matrix 𝑔𝑖𝑗 ̂ = 𝑒−2ᵆ 𝑔𝑖𝑗 . The 𝑘 ̂𝑘 Christoffel symbols of both metrics, denoted Γ𝑖𝑗 , Γ𝑖𝑗 respectively, are related by 1 𝜕𝑢 𝜕𝑢 𝜕𝑢 Γ𝑖𝑗̂𝑘 = Γ𝑖𝑗𝑘 + ∑ (𝛿𝑗𝑘 + 𝛿 𝑖𝑘 − 𝑔𝑘𝑙 𝑔𝑖𝑗 ), 2 𝜕𝑥𝑖 𝜕𝑥𝑗 𝜕𝑥𝑙 using (3.19). 𝑛

We define a category 𝐂𝐨𝐧𝐟 whose objects (𝑀, [g]) are smooth manifolds 𝑀 of dimension 𝑛 endowed with a conformal class [g] of Riemannian metrics. The morphisms are smooth maps 𝑓 ∶ (𝑀1 , [g1 ]) → (𝑀2 , [g2 ]) so that 𝑓∗ ([g2 ]) = [g1 ]. It is straightforward to check that the morphisms 𝑓 ∶ (𝑀1 , [g1 ]) → (𝑀2 , [g2 ]) in this category are identified with locally conformal maps 𝑓 ∶ (𝑀1 , g1 ) → (𝑀2 , g2 ), for any choice of representatives g1 and g2 of the respective conformal classes of metrics. Therefore the group (6.1) can also be denoted Conf(𝑀, [g]). Definition 6.4. Let 𝑀 be a smooth 𝑛-manifold. A conformal atlas on 𝑀 is a family 𝒜 = {(𝑈𝛼 , 𝜑𝛼 , g𝛼 ) | 𝛼 ∈ Λ} where {𝑈𝛼 } is an open covering of 𝑀, 𝜑𝛼 ∶ 𝑈𝛼 → 𝑉𝛼 ⊂ ℝ𝑛 are smooth charts of 𝑀, g𝛼 is a Riemannian metric on 𝑉𝛼 , and for each pair 𝛼, 𝛽 ∈ Λ, the change of coordinates 𝜑𝛼 ∘ 𝜑−1 𝛽 ∶ (𝜑 𝛽 (𝑈𝛼 ∩ 𝑈 𝛽 ), g𝛽 ) → (𝜑 𝛼 (𝑈𝛼 ∩ 𝑈 𝛽 ), g𝛼 ) is a conformal map.

6.1. Conformal structures

345

Remark 6.5. Abusing slightly the notation, we can denote g𝛼 to the corresponding metric 𝜑∗𝛼 (g𝛼 ) on 𝑈𝛼 . In Definition 6.4, the fact that 𝜑𝛼 ∘ 𝜑−1 𝛽 is a conformal map translates to Id ∶ (𝑈𝛼 ∩ 𝑈 𝛽 , g𝛽 ) → (𝑈𝛼 ∩ 𝑈 𝛽 , g𝛼 ) being a conformal map. In other words, the metrics g𝛼 on each 𝑈𝛼 belong to the same conformal class when restricted to the intersections of the cover {𝑈𝛼 }. So a conformal atlas allows us to measure angles in any chart, and the result is independent of the chart. Proposition 6.6. Conformal structures on 𝑀 are in one-to-one correspondence with conformal atlases on 𝑀. Proof. In one direction the result is easy. Suppose that we are given a smooth manifold endowed with a conformal structure (𝑀, [g]). Take {(𝑈𝛼 , 𝜑𝛼 )} any atlas for 𝑀, with ∗ 𝜑𝛼 ∶ 𝑈𝛼 → 𝑉𝛼 ⊂ ℝ𝑛 . Define g𝛼 = (𝜑−1 𝛼 ) (g), being g a fixed representative of [g]. Since g is a well defined Riemannian metric on 𝑀, it follows immediately that {(𝑈𝛼 , 𝜑𝛼 , g𝛼 )} is a conformal atlas for 𝑀. Now suppose that we are given a conformal atlas 𝒜 = {(𝑈𝛼 , 𝜑𝛼 , g𝛼 )| 𝛼 ∈ Λ} for a manifold 𝑀. We can assume that {𝑈𝛼 } is a locally finite covering. On each intersection 𝑈𝛼 ∩ 𝑈 𝛽 ≠ ∅ we have that g𝛽 = 𝜇𝛼𝛽 g𝛼 for some function 𝜇𝛼𝛽 ∶ 𝑈𝛼 ∩ 𝑈 𝛽 → ℝ>0 . Take a smooth partition of unity 𝜌𝛼 subordinated to the open covering 𝑈𝛼 , and define g = ∑ 𝜌𝛼 g𝛼 . We already know that g is a Riemannian metric on 𝑀 (Proposition 3.22). Let us see that g gives locally the same conformal structure as the one defined by the atlas 𝒜. Let 𝑝 ∈ 𝑀, and take a small neighbourhood 𝑉 𝑝 such that Λ′ = {𝛼 ∈ Λ|𝑉 ∩ 𝑈𝛼 ≠ ∅} is a finite set. Fix 𝛼0 ∈ Λ′ and consider the functions 𝜇𝛼0 𝛼 such that g𝛼 = 𝜇𝛼0 𝛼 g𝛼0 on the intersections 𝑈𝛼0 ∩ 𝑈𝛼 . Therefore g|𝑉 = ∑ 𝜌𝛼 g𝛼 |𝑉 = ∑ 𝜌𝛼 g𝛼 |𝑉 = ( ∑ 𝜇𝛼0 𝛼 𝜌𝛼 ) g𝛼0 |𝑉 , 𝛼∈Λ

𝛼∈Λ′

𝛼∈Λ′

and so [g|𝑉 ] = [g𝛼0 |𝑉 ]. Hence the restriction of g to each 𝑈𝛼 belongs to the same conformal class as g𝛼 . □ 𝑛

The category 𝐂𝐨𝐧𝐟 gives rise to a classification problem. This is certainly related to the classification problem for Riemannian manifolds, in particular there are local and global invariants. Surprisingly, in dimension 2 this problem turns out to be equivalent to the classification problem for complex structures. To see it, we need to introduce an important class of conformal structures. Definition 6.7. A conformally flat structure on a manifold 𝑀 is a conformal atlas 𝒜 𝑛 such that the local metrics g𝛼 on the sets 𝑉𝛼 ⊂ ℝ𝑛 are g𝛼 = ∑𝑖=1 𝑑𝑥𝑖2 , that is g𝛼 is the standard Euclidean metric. Remark 6.8. The conformally flat structure means that the changes of coordinates 𝜑𝛼 ∘ 𝜑−1 𝛽 ∶ (𝜑 𝛽 (𝑈𝛼 ∩ 𝑈 𝛽 ), g𝑠𝑡𝑑 ) → (𝜑 𝛼 (𝑈𝛼 ∩ 𝑈 𝛽 ), g𝑠𝑡𝑑 ) are conformal maps with respect to the Euclidean metric g𝑠𝑡𝑑 = ∑ 𝑑𝑥𝛽2 . As a consequence of Proposition 6.6, a conformally flat structure on a manifold is equivalent to having a Riemannian metric g on 𝑀 such that for any point 𝑝 ∈ 𝑀 there exists coordinates (𝑥1 , . . . , 𝑥𝑛 ) around 𝑝 such that g = 𝜇 ∑ 𝑑𝑥𝑖2 for some positive function 𝜇. Following Definition 4.14, these coordinates are called conformal coordinates.

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A metric g, such that it admits conformal coordinates everywhere (i.e., that locally it is conformal to a flat metric), is called locally conformally flat. Therefore a conformally flat structure is equivalent to giving a locally conformally flat metric. Example 6.9. The round sphere 𝕊𝑛 = (𝑆 𝑛 , g𝑆𝑛 ) is locally conformally flat. The metric in the chart given by the stereographic projection is 4 g= (𝑑𝑢21 + ⋯ + 𝑑𝑢2𝑛 ). 2 (1 + 𝑢1 + ⋯ + 𝑢2𝑛 )2 Remark 6.10. For surfaces, we will prove that any metric is locally conformally flat (see Theorem 6.51). For dimension 𝑛 ≥ 3 it is not true that any metric g on a manifold 𝑀 is locally conformally flat. There are tensors which measure the local obstructions for a metric g to be locally conformally flat, and they are associated to the curvature tensor of g. This is true because g is locally conformally flat if and only if there exists (locally) a function 𝜇 so that the curvature tensor of 𝜇g vanishes. Writing the curvature tensor 𝑅𝜇g of 𝜇g in terms of the curvature tensor 𝑅g of g, we get a system of equations 𝑅𝜇g = 0 (Exercise 6.1). The integrability conditions for this system to admit a solution 𝜇 are the components of the Cotton tensor 𝐶𝑖𝑗𝑘 in dimension 3 and the components of the Weyl tensor 𝑊 𝑖𝑗𝑘𝑙 in dimension ≥ 4. These tensors depend on the curvature. The Weyl tensor is one of the components of the Riemannian curvature, mentioned in Remark 3.46(3). The Cotton tensor is defined as 𝐶 = 𝒜(∇𝑆), where 𝒜 denotes 𝑛 the antisymmetrization of tensors (cf. Exercise 3.10), and 𝑆 = Ric − 2(𝑛−1) Scal ⋅g is the Schouten tensor, where Ric is the Ricci tensor and Scal is the scalar curvature. By questions of symmetry, the Weyl tensor vanishes in dimension 𝑛 ≤ 3, and the Cotton tensor vanishes in dimension 𝑛 ≤ 2. In the terminology of Remark 5.12(2), conformal structures are non-soft. See [Laf] for conformal geometry in higher dimensions. 6.1.1. Conformal structures on surfaces. Let us now turn our attention to the case of surfaces, 𝑛 = 2. In this case there is a close relationship between conformal geometry and complex geometry. Proposition 6.11. Let 𝑆 be an oriented surface. Conformal structures [g] on 𝑆 are in bijective correspondence with almost complex structures 𝐽. Proof. ⇒ Let [g] be a conformal structure on 𝑆. Take a representative g ∈ [g]. We want to define the tensor 𝐽. For 𝑝 ∈ 𝑆 and 𝑢 ∈ 𝑇𝑝 𝑆 with 𝑢 ≠ 0, we define 𝐽𝑢 ∈ 𝑇𝑝 𝑆 as the unique vector so that: (1) g(𝑢, 𝐽𝑢) = 0, (2) (𝑢, 𝐽𝑢) is an oriented basis, (3) ||𝑢|| = ||𝐽𝑢||. That is, 𝐽𝑢 is the rotation of the vector 𝑢 an angle

𝜋 2

in the positive direction.

Clearly, 𝐽𝑢 does not depend on the choice of representative g, so 𝐽 is well defined and determined uniquely from [g]. It is easy to see that it is a smooth tensor. Finally the fact that 𝐽 2 = − Id follows since as g(𝐽𝑢, −𝑢) = 0, (𝐽𝑢, −𝑢) is oriented, and ||𝐽𝑢|| = || − 𝑢||, then 𝐽(𝐽𝑢) = −𝑢.

6.1. Conformal structures

347

S Tp S

u

Ju

⇐ Suppose that 𝐽 is an almost complex structure on 𝑆. Let us construct a conformal atlas 𝒜. Take positive charts 𝜑 ∶ 𝑈𝛼 → 𝑉𝛼 ⊂ ℝ2 on 𝑆. Take a non-vanishing vector field 𝐸1𝛼 on 𝑉𝛼 , e.g., 𝐸1𝛼 = 𝜕ᵆ . Let 𝐸2𝛼 = 𝐽(𝐸1𝛼 ). Then consider the metric g𝛼 which makes the basis (𝐸1𝛼 , 𝐸2𝛼 ) orthonormal. The collection of {(𝑈𝛼 , 𝜑𝛼 , g𝛼 )} gives the required atlas. To see that this is a conformal atlas, we look at an intersection 𝑈𝛼 ∩ 𝑈 𝛽 . Then we have 𝛽

𝛽

𝛽

two bases 𝐵𝛼 = (𝐸1𝛼 , 𝐽(𝐸1𝛼 )) and 𝐵𝛽 = (𝐸1 , 𝐽(𝐸1 )). If we write 𝐸1 = 𝑎𝐸1𝛼 + 𝑏𝐽(𝐸1𝛼 ), for 𝛽

some functions 𝑎, 𝑏 on 𝑈𝛼 ∩ 𝑈 𝛽 , then 𝐽(𝐸1 ) = −𝑏𝐸1𝛼 + 𝑎𝐽(𝐸1𝛼 ), and hence the change 𝑎 −𝑏 of basis from 𝐵𝛼 to 𝐵𝛽 is given by the matrix ( ). Writing 𝜇 = 𝑎2 + 𝑏2 > 0, then 𝑏 𝑎 the metric g𝛼 making 𝐵𝛼 orthonormal, and the metric g𝛽 making 𝐵𝛽 orthonormal, are related by g𝛼 = 𝜇 g𝛽 . So 𝒜 is a conformal atlas. □ To relate conformally flat structures with complex structures, we use that for a smooth map 𝑓 ∶ 𝑈 ⊂ ℂ → 𝑉 ⊂ ℂ, 𝑤 = 𝑓(𝑧), interpreted as a map 𝑓 ∶ (𝑈, g𝑠𝑡𝑑 ) → (𝑉, g𝑠𝑡𝑑 ), 𝑓 is locally conformal and preserves the orientation if and only if it is holomorphic and 𝑓′ (𝑧) ≠ 0, for all 𝑧 ∈ 𝑈 (Lemma 4.16). Proposition 6.12. Let 𝑆 be an oriented surface. Complex structures on 𝑆 are in bijective correspondence with conformally flat structures on 𝑆. Proof. Let 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )} be a complex atlas for 𝑆. The change of coordinates 𝜑𝛽 ∘ 𝜑−1 𝛼 ∶ 𝜑 𝛼 (𝑈𝛼 ∩𝑈 𝛽 ) → 𝜑 𝛽 (𝑈𝛼 ∩𝑈 𝛽 ) are biholomorphic maps between open sets of ℂ. By Lemma 4.16, they are orientation preserving conformal maps between open sets of ℝ2 with the Euclidean metric g𝑠𝑡𝑑 . This means that 𝒜 = {(𝑈𝛼 , 𝜑𝛼 , g𝑠𝑡𝑑 )} is a conformally flat atlas for 𝑆. The converse is given by reversing the above argument.

□

Putting it all together, we have the following result. Theorem 6.13. Let 𝑆 be an oriented surface. There is a bijective correspondence between: (1) almost complex structures on 𝑆, (2) complex structures on 𝑆, (3) conformal structures on 𝑆, (4) conformally flat structures on 𝑆.

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6. Global analysis

Proof. The equivalence (1) ⇔ (3) is given by Proposition 6.11, and the equivalence (2) ⇔ (4) is given by Proposition 6.12. Trivially (2) ⇒ (1). Finally, the implication (1) ⇒ (2) follows from the Newlander-Nirenberg Theorem 5.10 since the Nijenhuis tensor vanishes (cf. Remark 5.49(3)). □ 1

2

Therefore the categories 𝐂𝐌𝐚𝐧 and 𝐂𝐨𝐧𝐟𝑜𝑟 of complex curves and conformal oriented surfaces coincide. In particular, 𝕃𝐂𝐌𝐚𝐧1 = 𝕃𝐂𝐨𝐧𝐟2 . 𝑜𝑟

Remark 6.14. (1) A consequence of Theorem 6.13 is that a conformal structure on an oriented surface is automatically conformally flat. Otherwise said, if (𝑆, g) is an oriented Riemannian surface, then around every point 𝑝 ∈ 𝑆 there are conformal coordinates. Note that a chart is conformal if and only if it is a complex chart. (2) In Theorem 6.13, we have used the Newlander-Nirenberg theorem, which we have not proved. Note that if we prove (3) ⇒ (4), then we have a proof of (1) ⇒ (2). So to prove the Newlander-Nirenberg theorem for surfaces, it is enough to prove that a Riemannian surface (𝑆, g) has always conformal coordinates. We shall do that in Corollary 6.53 (3) If a surface 𝑆 is non-orientable, it is also true that every conformal structure [g] is conformally flat. In other words, in non-orientable surfaces 𝑆, conformal coordinates also exist. This follows by taking a small chart (which is always orientable) and an orientation on it, and applying the above result to it. (4) Combining Theorem 6.13 with Remark 5.49(5), we have that a conformal oriented surface (𝑆, [g]) is the same as a complex structure (𝑆, 𝐽), whereas a Riemannian oriented surface (𝑆, g) is the same as a complex structure (𝑆, 𝐽) together with a Hermitian metric h. In this correspondence, on a complex chart with coordinate 𝑧 = 𝑥 + 𝑖𝑦, the Hermitian metric is h = ℎ(𝑧)𝑑𝑧 ⊗ 𝑑𝑧, and g = ℎ(𝑥, 𝑦)(𝑑𝑥2 + 𝑑𝑦2 ) is the Riemannian metric, where ℎ is a positive real valued function. So a Riemann surface (that is, a complex curve) can be thought of as a surface with a conformal class of Riemannian metrics. (5) Observe that h is a Hermitian metric on a one-dimensional complex vector ′ space. So any two Hermitian metrics h, h differ by multiplication by a posi′ tive real function 𝜇 > 0, that is h = 𝜇 h, which implies that the corresponding ′ Riemannian metrics g, g are conformal. This reproves Proposition 6.12. (6) For 𝑛 ≥ 3 there is no correspondence between conformal structures and complex structures. Not the least because the complex structures only happen in even dimension. In even dimensions 𝑛 ≥ 4, these structures also differ. This is due to the algebraic difference between the conformal group ℝ∗ ⋅O(2𝑛) (the group of dilations) and the complex group GL(𝑛, ℂ). (7) Let (𝑆, [g]), (𝑆 ′ , [g′ ]) be two conformal oriented surfaces, with associated complex structures 𝐽, 𝐽 ′ . A map 𝑓 ∶ 𝑆 → 𝑆 ′ is locally conformal and orientation

6.1. Conformal structures

349

preserving if and only if 𝑓 ∶ (𝑆, 𝐽) → (𝑆 ′ , 𝐽 ′ ) is holomorphic with non-zero derivative at every point (that is, a local biholomorphism). The map 𝑓 ∶ 𝑆 → 𝑆 ′ is locally conformal and orientation reversing if and only if it is antiholomorphic (with non-zero derivative at every point). (8) A Klein surface is defined as a non-orientable surface with an atlas whose changes of charts are either biholomorphisms or anti-biholomorphisms. If 𝑆 is a non-orientable surface, then a conformal structure is equivalent to a Klein surface structure (Exercise 6.3). 6.1.2. Conformal structures on orbifolds. We extend the discussion on conformal structures to orbifolds (section 3.2.4). Let 𝑋 be a differentiable 𝑛-orbifold (Definition 3.75) with an orbifold atlas 𝒜 = {(𝑈𝑝 , 𝑈𝑝̃ , Γ𝑝 , 𝜑𝑝 )}. In analogy with Definition 6.4, a conformal atlas on 𝑋 is a collection of Γ𝑝 -equivariant Riemannian metrics g𝑝 on 𝑈𝑝̃ , such that the changes of charts 𝐹 ∶ (𝜑𝑝−1 (𝑈𝑝 ∩ 𝑈𝑝′ ), g𝑝 ) → ((𝜑𝑝′ )−1 (𝑈𝑝 ∩ 𝑈𝑝′ ), g′𝑝 ) are conformal maps. Let ℳet(𝑋) be the set of orbifold Riemannian metrics on 𝑋, which is non-empty by Exercise 3.26. We say that g1 , g2 ∈ ℳet(𝑋) are conformal orbifold metrics if there exists an orbifold function 𝜇 > 0 such that g2 = 𝜇 g1 . Analogously to Proposition 6.6, a conformal atlas on a differentiable orbifold is equivalent to a conformal class [g]. This is called a conformal structure on 𝑋. Now we focus on the case of oriented orbisurfaces. We have the following equivalence. Proposition 6.15. An oriented conformal orbisurface (𝑋, [g]) is equivalent to a complex curve (𝑋, 𝐽) with a discrete set of points 𝑆 and integer numbers 𝑠𝑝 > 1 for each 𝑝 ∈ 𝑆. Proof. ⇒ Let us have an oriented conformal orbisurface (𝑋, [g]). By Lemma 3.77, 𝑋 is a topological surface with a discrete set of singular points 𝑆, and for each 𝑝 ∈ 𝑆, an index 𝑠𝑝 > 1. The set 𝑋 − 𝑆 is a differentiable oriented surface, with a conformal structure [g] given by a smooth Riemannian metric g. By Theorem 6.13, 𝑋 − 𝑆 admits a complex structure, given by the almost complex structure 𝐽 of Proposition 6.11. We have to extend the complex atlas to the points of 𝑆. Let 𝑝 ∈ 𝑆 and consider the chart (𝑈𝑝 , 𝑈𝑝̃ = 𝐵𝜖 (0), Γ𝑝 = ⟨𝜌⟩, 𝜑𝑝 ), where 𝜌(𝑧) = 𝜌2𝜋/𝑠 (𝑧) = 𝑒2𝜋/𝑠 𝑧, 𝑠 = 𝑠𝑝 , and 𝜑𝑝 ∶ 𝐵𝜖 (0) → 𝑈𝑝 = 𝐵𝜖 (0)/Γ𝑝 is the quotient map. The orbifold Riemannian metric g is given by a Γ𝑝 -invariant metric g𝑝 on 𝑈𝑝̃ = 𝐵𝜖 (0). This metric g𝑝 gives a conformal structure on 𝐵𝜖 (0), and hence a complex structure 𝐽 by Theorem 6.13 again. Clearly 𝐽 is Γ𝑝 -equivariant. Therefore there is a diffeomorphism 𝜓 ∶ 𝐵𝜖 (0) → 𝑉 ⊂ ℂ such that 𝜓∗ 𝐽 = 𝐽0 , where 𝐽0 is the standard complex structure on ℂ. We can assume that 𝜓(0) = 0, so the map 𝜌 translates to a biholomorphism 𝑓 = 𝜓∘𝜌∘𝜓−1 on 𝑉 with 𝑓(0) = 0 and 𝑓𝑠 = Id. By Exercise 5.21, there is a biholomorphism 𝜛 ∶ 𝑉 → 𝑊 ⊂ ℂ such that 𝜛 ∘ 𝑓 ∘ 𝜛−1 = 𝜌 and 𝜛(0) = 0. Then ℎ = 𝜛 ∘ 𝜓 ∶ (𝐵𝜖 (0), 𝐽) → (𝑊, 𝐽0 ) is a biholomorphism that commutes with 𝜌. Our complex chart will be 𝜙 ∶ 𝑈𝑝 = 𝐵𝜖 (0)/Γ𝑝 → ℂ, 𝜙(𝑧) = (ℎ(𝑧))𝑠 . ⇐ Let us have a complex curve (𝑋, 𝐽) with a discrete set of points 𝑆 and integer numbers 𝑠𝑝 > 1 for each 𝑝 ∈ 𝑆. By Lemma 3.77, we reconstruct an oriented orbifold whose

350

6. Global analysis

singular points are given by 𝑆. On 𝑋 − 𝑆 we have a conformal structure given by the complex structure. Let 𝑝 ∈ 𝑆, and consider a complex chart 𝜙 ∶ 𝑈𝑝 → 𝐵𝛿 (0) ⊂ ℂ. We define 𝑈𝑝̃ = 𝐵𝜖 (0), where 𝜖 = 𝛿1/𝑠 , 𝑠 = 𝑠𝑝 , and 𝜑𝑝 ∶ 𝐵𝜖 (0) → 𝑈𝑝 by 𝜑𝑝 (𝑧) = 𝜙−1 (𝑧𝑠 ). So (𝑈𝑝 , 𝐵𝜖 (0), Γ𝑝 = ⟨𝜌2𝜋/𝑠 ⟩, 𝜑𝑝 , g𝑠𝑡𝑑 ) gives a conformal orbifold chart at 𝑝. □ Remark 6.16. The above construction is equivalent to checking that 𝐽 is actually an orbifold almost-complex structure on the orbifold 𝑋, and putting an orbifold Hermitian metric on it. Remark 6.17. In Remark 4.78, we have given an orbifold structure to the space 𝐻/ PSL(2, ℤ). By Proposition 6.15, this gives a complex structure to 𝐻/ PSL(2, ℤ). This is the complex structure constructed in Theorem 5.60.

6.2. Hodge theory We review now Hodge theory for the Laplacian on compact manifolds. This gives an application of elliptic differential operators to geometrical and topological questions. For this, it is necessary to endow a manifold with a Riemannian structure, although this is in a sense accessory for the topological implications (like the smooth structure is for de Rham cohomology). We are not providing full proofs. See [Eva], [Wel] for the theory of functional analysis, elliptic operators, or a complete treatment of Hodge theory. 6.2.1. Hodge star operator. Let (𝑀, g) be an oriented Riemannian manifold. We denote 𝜈 the Riemannian volume form (3.16) of 𝑀. For any 𝑝 ∈ 𝑀, g defines a metric on 𝑇𝑝 𝑀, and this induces a metric on 𝑇𝑝∗ 𝑀 under the isomorphism (3.17). Observe that if, in coordinates g = (𝑔𝑖𝑗 ), then the matrix of the scalar product of 𝑇𝑝∗ 𝑀 𝑠 is the inverse (𝑔𝑖𝑗 ) (Exercise 6.8). This also defines a metric on Ω𝑝𝑠 (𝑀) = ⋀ 𝑇𝑝∗ 𝑀 as follows. Let (𝑤 1 , . . . , 𝑤 𝑛 ) be a positive orthonormal basis of 𝑇𝑝∗ 𝑀, then the elements 𝑤 𝐼 = 𝑤 𝑖1 ∧ ⋯ ∧ 𝑤 𝑖𝑠 , 𝑖1 < ⋯ < 𝑖𝑠 , are declared orthonormal. Note that the volume form 𝜈 = 𝑤 1 ∧ ⋯ ∧ 𝑤 𝑛 is unitary. This defines a duality, called the Hodge star operator, ⋆ ∶ Ω𝑝𝑠 (𝑀) ⟶ Ω𝑝𝑛−𝑠 (𝑀), which is determined by 𝛼 ∧ ⋆𝛽 = ⟨𝛼, 𝛽⟩𝜈, Ω𝑝𝑠 (𝑀).

for all 𝛼 ∈ Then ⋆𝑤 𝐼 = (−1)𝜍𝐼 𝑤 𝐼 𝑐 , where 𝐼 𝑐 = {1, . . . , 𝑛} − 𝐼, and (−1)𝜍𝐼 is the sign of the permutation (𝐼, 𝐼 𝑐 ). It follows easily that ⋆ is an isometry and that ⋆⋆ = (−1)𝑠(𝑛−𝑠) . Remark 6.18. • For an oriented surface 𝑆 with metric g = ( 𝐸 𝐹 1 form is 𝜈 = √𝐸𝐺 − 𝐹 2 𝑑𝑢 ∧ 𝑑𝑣 and g−1 =

𝐹 𝐺 ),

𝐸𝐺−𝐹 2

⋆𝑑𝑢 = ⋆𝑑𝑣 =

1 √𝐸𝐺 − 𝐹 2 1 √𝐸𝐺 − 𝐹 2

(−𝐹 𝑑𝑢 + 𝐺 𝑑𝑣), (−𝐸 𝑑𝑢 + 𝐹 𝑑𝑣).

we have that the volume

𝐺 ( −𝐹

−𝐹 ). 𝐸

Hence

6.2. Hodge theory

351

Note that, under the isomorphism (−)♯ ∶ 𝑇𝑝∗ 𝑀 → 𝑇𝑝 𝑀 (cf. (3.17)), the Hodge star ⋆ corresponds to the almost complex structure 𝐽 defined by the conformal structure [g]. • Always ⋆1 = 𝜈, so ⋆𝑓 = 𝑓 𝜈 for a function 𝑓, where 𝜈 is the Riemannian volume form. • Consider ℝ3 with the standard metric. Then ⋆𝑑𝑥 = 𝑑𝑦 ∧ 𝑑𝑧, ⋆𝑑𝑦 = 𝑑𝑧 ∧ 𝑑𝑥, and ⋆𝑑𝑧 = 𝑑𝑥 ∧ 𝑑𝑦. Therefore, for a function 𝑓 ∈ 𝐶 ∞ (ℝ3 ) and a vector field 𝑋 ∈ 𝔛(ℝ3 ), we have grad (𝑓) = (𝑑𝑓)♯ ,

rot(𝑋) = (⋆𝑑(𝑋♭ ))♯ ,

div(𝑋) = ⋆𝑑 ⋆ (𝑋♭ ),

by rewriting Remark 2.117. Remark 6.19. We extend the classical operators gradient and divergence to any Riemannian manifold (𝑀, g). These were defined in Remark 2.117 for (ℝ𝑛 , g𝑠𝑡𝑑 ). 𝜕𝑓

(1) We define grad 𝑓 = (𝑑𝑓)♯ = ∑ 𝑔𝑖𝑗 𝜕𝑥

𝑖

𝜕 . 𝜕𝑥𝑗

(2) In general, the operator ⋆ ∶ Ω1 (𝑀) → Ω𝑛−1 (𝑀) goes as follows. First, we have that ⟨𝑑𝑥𝑖 , 𝑑𝑥𝑗 ⟩ = 𝑔𝑖𝑗 , by Exercise 6.8. Then (6.2)

ˆ 𝑗 ∧ ⋯ ∧ 𝑑𝑥𝑛 . ⋆𝑑𝑥𝑖 = ∑(−1)𝑗−1 √det(𝑔𝑘𝑙 )𝑔𝑖𝑗 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥 𝑗

(3) The divergence of a vector field 𝑋 is defined by the equation 𝐿𝑋 (𝜈) = div(𝑋) 𝜈, where 𝐿𝑋 is the Lie derivative defined in page 143. By Exercise 3.7, div(𝑋)𝜈 = 𝑑 | 𝜙∗ 𝜈, where 𝜙𝑡 is the flow of 𝑋. So div(𝑋) controls the variation of the 𝑑𝑡 𝑡=0 𝑡 volume by the flow of 𝑋 (Exercise 6.4). (4) Cartan formula (Exercise 6.5) says that 𝐿𝑋 = 𝑑 𝑖𝑋 +𝑖𝑋 𝑑, where 𝑖𝑋 ∶ Ω𝑠 (𝑀) → Ω𝑠−1 (𝑋) is the contraction operator given by 𝑖𝑋 (𝛼)(𝑋2 , . . . , 𝑋𝑠 ) = 𝛼(𝑋, 𝑋2 , . . . , 𝑋𝑠 ), i.e., 𝑖𝑋 𝛼 = 𝐶(𝑋 ⊗ 𝛼) in the notation of Remark 3.4(2). Then, div(𝑋)𝜈 = 𝐿𝑋 (𝜈) = 𝑑𝑖𝑋 𝜈. Since 𝑖𝑋 𝜈 = ⋆𝑋♭ by (6.2), div(𝑋)𝜈 = 𝑑 ⋆ (𝑋♭ ), so div(𝑋) = ⋆𝑑 ⋆ (𝑋♭ ). 𝜕

(5) Locally, for a vector field 𝑋 = ∑ 𝑋 𝑖 𝜕𝑥 , the divergence is computed as 𝑖

𝑖

div(𝑋) = ∑ ⋆𝑑 ⋆ (𝑋 𝑔𝑖𝑗 𝑑𝑥𝑗 ) ˆ 𝑘 ∧ ⋯ ∧ 𝑑𝑥𝑛 ) = ∑(−1)𝑘−1 ⋆ 𝑑(𝑋 𝑖 𝑔𝑖𝑗 √det(𝑔𝑟𝑠 )𝑔𝑗𝑘 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥 ˆ 𝑖 ∧ ⋯ ∧ 𝑑𝑥𝑛 ) = ∑(−1)𝑖−1 ⋆ 𝑑(𝑋 𝑖 √det(𝑔𝑟𝑠 )𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥 = ∑⋆ =

𝜕(𝑋 𝑖 √det(𝑔𝑟𝑠 )) 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 𝜕𝑥𝑖

1 √det(𝑔𝑟𝑠 )

∑

𝜕(𝑋 𝑖 √det(𝑔𝑟𝑠 )) 𝜕𝑋 𝑖 1 𝜕𝑔 =∑ + 𝑋 𝑖 ( 𝑔𝑘𝑙 𝑙𝑘 ) . 𝜕𝑥𝑖 𝜕𝑥𝑖 2 𝜕𝑥𝑖

352

6. Global analysis

In the last line we use 𝜕𝑔 𝜕 (det(𝑔𝑟𝑠 )) = det(𝑔𝑟𝑠 ) tr(∑ 𝑔𝑘𝑙 𝑙𝑚 ) 𝜕𝑥𝑖 𝜕𝑥𝑖 𝑙

= det(𝑔𝑟𝑠 ) ∑ 𝑔𝑘𝑙 𝑘,𝑙

𝑘,𝑚

𝜕𝑔𝑙𝑘 , 𝜕𝑥𝑖

from Exercise 6.11. 6.2.2. Review of functional analysis. Within this section, we will work on a compact oriented Riemannian manifold (𝑀, g). Definition 6.20. Let (𝑀, g) be a compact oriented Riemannian manifold. We define the 𝐿2 metric on the functional spaces of 𝑠-forms Ω𝑠 (𝑀) by ⟨𝛼, 𝛽⟩𝐿2 = ∫ ⟨𝛼, 𝛽⟩𝜈 = ∫ 𝛼 ∧ ⋆𝛽. 𝑀

𝑀

The norm corresponding to the 𝐿2 metric is given by ||𝛼||𝐿2 = √⟨𝛼, 𝛼⟩𝐿2 . The spaces Ω (𝑀) are not complete, but we can consider their completion with respect to this norm, which are the spaces of 𝐿2 forms, denoted Ω𝑠𝐿2 (𝑀). Locally, the elements 𝛼 ∈ Ω𝑠𝐿2 (𝑀) are 𝑠-forms whose coefficients (locally) are 𝐿2 functions (that is, measurable functions 𝑓 such that ∫ |𝑓|2 < ∞). 𝑠

The spaces Ω𝐿𝑠 2 (𝑀) are real Hilbert spaces. A Hilbert space ℋ is a vector space with a scalar product ⟨−, −⟩ and a norm ||𝑢|| = √⟨𝑢, 𝑢⟩ such that ℋ is complete with respect to the topology induced by the norm (that is, Cauchy sequences are convergent). If ℋ is a Hilbert space which is separable (that is, it has a dense countable set) then it has a Hilbert basis (𝑒 𝑖 )∞ 𝑖=1 . This consists of orthonormal vectors such that any 𝑢 ∈ ℋ can be uniquely written as ∞

𝑢 = ∑ 𝑥𝑖 𝑒 𝑖 , 𝑖=1 𝑁

with some 𝑥𝑖 ∈ ℝ, 𝑖 ≥ 1. This means that 𝑢 = lim𝑁→∞ ∑𝑖=1 𝑥𝑖 𝑒 𝑖 , where the series ∞ is convergent in ℋ. Note that ||𝑢||2 = ∑𝑖=1 𝑥𝑖2 < ∞. We remark that (𝑒 𝑖 )∞ 𝑖=1 is not a basis of ℋ as a vector space. Hilbert spaces are suited to use techniques of functional analysis, needed to deal with differential operators and to solve differential equations [Eva]. Remark 6.21. Other useful functional spaces are defined in terms of the Sobolev norms 𝑊 𝑘,2 , 𝑘 ≥ 0. The norm is defined, for 𝛼 ∈ Ω𝑠 (𝑀), as 1/2

𝑘 𝑖

||𝛼||𝑊 𝑘,2 = ( ∑ ||∇

(6.3)

𝛼||2𝐿2 )

,

𝑖=0

where ∇ is a fixed covariant derivative (for instance, the Levi-Civita connection of g), (𝑖)

and ∇𝑖 = ∇∘ ⋯ ∘∇. We complete Ω𝑠 (𝑀) to obtain Hilbert spaces Ω𝑠𝑊 𝑘,2 (𝑀), called Sobolev spaces of 𝑠-forms. The norm is dependent on the choice of connection, but the Sobolev spaces are not, since for any two connections ∇, ∇′ , we have that the norms (6.3) for ∇, ∇′ are comparable: 𝐶1 ||𝛼||𝑊 𝑘,2 ≤ ||𝛼||′𝑊 𝑘,2 ≤ 𝐶2 ||𝛼||𝑊 𝑘,2 , for some constants 𝐶1 , 𝐶2 > 0 (Exercise 6.18).

6.2. Hodge theory

353

We have that 𝛼 ∈ Ω𝑠𝑊 𝑘,2 (𝑀) if the coefficients of 𝛼 have (weak) derivatives up to order 𝑘 and all of them are in 𝐿2 . We recall that a function 𝑓 ∶ 𝑈 ⊂ ℝ𝑛 → ℝ has a weak (partial) derivative 𝐷𝑥𝑖 𝑓 ∈ 𝐿2 (𝑈) if for all (compactly supported smooth) functions 𝑔 on 𝜕𝑔 𝑈, one has ∫𝑈 𝑓 𝜕𝑥 = − ∫𝑈 (𝐷𝑥𝑖 𝑓)𝑔. If 𝑓 is smooth, then 𝐷𝑥𝑖 𝑓 coincides with the usual 𝜕𝑓

𝑖

derivative 𝜕𝑥 (by the integration by parts formula), so we will use the last notation also 𝑖 for weak derivatives. Remark 6.22. The Sobolev embedding theorem [Wel] says that 𝑊 𝑘,2 (ℝ𝑛 ) ⊂ 𝐶 𝑟 (ℝ𝑛 ) 𝑛 𝑛 for 𝑟 ≤ 𝑘 − 2 . In particular, for 𝑘 ≥ 2 , the functions in 𝑊 𝑘,2 are continuous. This allows us to define manifolds whose atlases have changes of charts in Sobolev spaces (cf. Remark 1.20). Definition 6.23. Let 𝑀 be a smooth manifold. A differential (linear) operator is a linear map 𝑃 ∶ 𝒯(𝑀) → 𝒯 ′ (𝑀), where 𝒯(𝑀), 𝒯 ′ (𝑀) are spaces of tensors of some type,1 that in a local chart (𝑈, (𝑥1 , . . . , 𝑥𝑛 )) is written as an operator 𝑃 ∶ 𝒯(𝑈) ≅ 𝐶 ∞ (𝑈)𝑚 → ′ 𝒯 ′ (𝑈) ≅ 𝐶 ∞ (𝑈)𝑚 (the isomorphisms are given by taking a basis of tensors over 𝑈 and taking the coordinates with respect to the basis), with 𝑃 = ∑ 𝐴𝐼 (𝑥)

(6.4)

|𝐼|≤𝑘

𝜕|𝐼| , 𝜕𝑥𝐼

′

where 𝐴𝐼 (𝑥) are 𝑚 × 𝑚-matrices of functions and 𝐼 = (𝑖1 , . . . , 𝑖𝑝 ) is a multi-index, with 1 ≤ 𝑖𝑗 ≤ 𝑛, |𝐼| = 𝑝, 0 ≤ 𝑝 ≤ 𝑘. Here 𝜕|𝐼|

so that 𝜕𝑥 (𝑓1 , . . . , 𝑓𝑚 ) = ( 𝐼 |𝐼| = 𝑘.

𝜕|𝐼| 𝜕𝑥𝐼

=

𝜕|𝐼| 𝑓1 𝜕|𝐼| 𝑓 , . . . , 𝜕𝑥 𝑚 ). 𝜕𝑥𝐼 𝐼

𝜕𝑝 𝜕𝑥𝑖1 ⋯ 𝜕𝑥𝑖𝑝

acts on 𝐶 ∞ (𝑈)𝑚 componentwise

The order of 𝑃 is 𝑘 if some 𝐴𝐼 (𝑥) ≠ 0 with

The exterior differential (6.5)

𝑑 ∶ Ω𝑠 (𝑀) → Ω𝑠+1 (𝑀)

is a differential operator of order 1. The exterior differential extends to the Hilbert spaces of 𝐿2 forms as [Wel] (6.6)

𝑑 ∶ Ω𝑠𝐿2 (𝑀) → Ω𝑠+1 𝐿2 (𝑀).

Note that (6.6) is actually not defined (as a map between vector spaces) on the whole of Ω𝑠𝐿2 (𝑀). Its domain is Ω𝑠𝑊 1,2 (𝑀) . Moreover, (6.6) is an unbounded linear operator, i.e., there does not exist a constant 𝐶 > 0 such that ||𝑑𝛼||𝐿2 ≤ 𝐶 ||𝛼||𝐿2 (bounded linear operators can always be extended to the whole of the source space). The map (6.6) does define a bounded operator 𝑑 ∶ Ω𝑠𝑊 1,2 (𝑀) → Ω𝑠𝐿2 (𝑀), that is there is some constant 𝐶 > 0 such that ||𝑑𝛼||𝐿2 ≤ 𝐶 ||𝛼||𝑊 1,2 . Remark 6.24. The exterior differential can be defined in the spaces 𝒱 𝑘 = {𝛼 ∈ Ω𝑘𝐿2 (𝑀)|𝑑𝛼 ∈ Ω𝑘+1 𝐿2 (𝑀)}, and there is a differential complex (𝒱 • , 𝑑). This actually has cohomology isomorphic to the de Rham cohomology (Exercise 6.19). 1

The correct setting is that of vector bundles, and differential operators are defined on sections of vector bundles.

354

6. Global analysis

The map (6.5) has a formal adjoint, that is, an operator 𝑑 ∗ ∶ Ω𝑠+1 (𝑀) → Ω𝑠 (𝑀) defined by the equality ⟨𝛼, 𝑑𝛽⟩𝐿2 = ⟨𝑑 ∗ 𝛼, 𝛽⟩𝐿2 , for any 𝛼 ∈ Ω𝑠+1 (𝑀), and 𝛽 ∈ Ω𝑠 (𝑀). Any differential operator 𝑃 of order 𝑘 has a formal adjoint 𝑃 ∗ which is again another differential operator of order 𝑘. Lemma 6.25. For 𝛼 ∈ Ω𝑠 (𝑀), we have 𝑑 ∗ 𝛼 = (−1)𝑛(𝑠−1)+1 ⋆ 𝑑 ⋆ 𝛼. Proof. Let 𝛽 ∈ Ω𝑠−1 (𝑀). Then we compute ⟨𝛽, 𝑑 ∗ 𝛼⟩𝐿2 = ⟨𝑑𝛽, 𝛼⟩𝐿2 = ∫ 𝑑𝛽 ∧ ⋆𝛼 = (−1)𝑠 ∫ 𝛽 ∧ 𝑑 ⋆ 𝛼 𝑠

= (−1) (−1)

𝑀 (𝑠−1)(𝑛−𝑠+1)

𝑀

⟨𝛽, ⋆𝑑 ⋆ 𝛼⟩𝐿2 ,

where we have used 𝑑(𝛽 ∧ ⋆𝛼) = 𝑑𝛽 ∧ ⋆𝛼 + (−1)𝑠−1 𝛽 ∧ 𝑑(⋆𝛼). Finally 𝑠 + (𝑠 − 1)(𝑛 − 𝑠 + 1) ≡ 𝑛(𝑠 − 1) + 1 for simplifying the exponent.

(mod 2), □

With the description of Lemma 6.25, the operator 𝑑 ⋆ makes sense for any oriented manifold, even if it is not compact. Remark 6.26. The formal adjoint is modelled on the finite dimensional situation as follows. Let 𝑉, 𝑊 be finite dimensional real vector spaces with scalar products, and let 𝑓 ∶ 𝑉 → 𝑊 be a linear map. The adjoint of 𝑓 is the linear map 𝑓∗ ∶ 𝑊 → 𝑉 defined by the equality ⟨𝑤, 𝑓(𝑣)⟩ = ⟨𝑓∗ (𝑤), 𝑣⟩, for all 𝑣 ∈ 𝑉 and 𝑤 ∈ 𝑊. If 𝐴 is the matrix of 𝑓 with respect to orthonormal basis of 𝑉 and 𝑊, then the matrix of 𝑓∗ is 𝐴𝑡 , since ⟨𝑓∗ (𝑤), 𝑣⟩ = ⟨𝑤, 𝑓(𝑣)⟩ = ⟨𝑤, 𝐴𝑣⟩ = 𝑤𝑡 (𝐴𝑣) = (𝐴𝑡 𝑤)𝑡 𝑣 = ⟨𝐴𝑡 𝑤, 𝑣⟩ implies that 𝑓∗ (𝑤) = 𝐴𝑡 𝑤. Note that algebraically, any linear map 𝑓 ∶ 𝑉 → 𝑊 defines a transpose 𝑓𝑡 ∶ 𝑊 ∗ → 𝑉 , via 𝑓𝑡 (𝛼) = 𝛼∘𝑓, and 𝑓∗ equals 𝑓𝑡 under the isomorphisms 𝑉 ≅ 𝑉 ∗ , 𝑊 ≅ 𝑊 ∗ given by the scalar products. ∗

If 𝑉, 𝑊 are complex vector spaces with Hermitian metrics, then the adjoint of a complex linear map 𝑓 ∶ 𝑉 → 𝑊 can be defined by the same formula ⟨𝑤, 𝑓(𝑣)⟩ = ⟨𝑓∗ (𝑤), 𝑣⟩, and 𝑓∗ ∶ 𝑊 → 𝑉 is a complex linear map. If the matrix of 𝑓 with respect to orthonormal basis is 𝐴, then the matrix of 𝑓∗ is 𝐴∗ = 𝐴𝑡 . This justifies the name “adjoint”. 6.2.3. Laplacian and harmonic forms. Let us now move forward to analyse the de 𝑠 Rham cohomology 𝐻𝑑𝑅 (𝑀). We start by giving the following heuristic argument. A 𝑠 cohomology class 𝑎 = [𝛼0 ] ∈ 𝐻𝑑𝑅 (𝑀) is the same as an affine subspace 𝐻𝑎 = 𝛼0 + 𝑠 im 𝑑 ⊂ 𝑍𝑑𝑅 (𝑀). In a finite dimensional vector space 𝑉 with a scalar product, an affine subspace 𝐻 ⊂ 𝑉 is determined by its direction and the point of minimum distance to the origin. In infinite dimensional vector spaces, life is not so easy, but still we can do the same when 𝑉 is a Hilbert space and 𝐻 is a closed affine subspace. Therefore, to

6.2. Hodge theory

355

represent 𝐻𝑎 in this way, first we complete Ω𝑠 (𝑀) with respect to the 𝐿2 norm and then we look for the element 𝛼 ∈ 𝐻𝑎 such that ||𝛼||2𝐿2 = min {||𝛼′ ||2𝐿2 | 𝛼′ = 𝛼0 + 𝑑𝛽, 𝛽 ∈ Ω𝑠−1 𝑊 1,2 (𝑀)}.

(6.7)

Let us assume for the moment that im 𝑑 ⊂ Ω𝑠𝐿2 (𝑀) is closed. Then by Exercise 6.16 there is a unique element 𝛼 satisfying (6.7) and 𝛼 ⟂ im 𝑑. So ⟨𝛼, 𝑑𝛽⟩𝐿2 = 0 , and thus ⟨𝑑 ∗ 𝛼, 𝛽⟩𝐿2 = 0, for all 𝛽 ∈ Ω𝑠−1 (𝑀). This implies that 𝑑 ∗ 𝛼 = 0. So the cohomology class 𝑎 has a unique representative 𝛼 satisfying the two equations (6.8)

𝑑𝛼 = 0,

𝑑 ∗ 𝛼 = 0.

𝑠 This justifies the existence of an isomorphism 𝐻𝑑𝑅 (𝑀) ≅ {𝛼 ∈ Ω𝑠 (𝑀) | 𝑑𝛼 = 0, 𝑑 𝛼 = 0}, which is known as Hodge theorem and will be proved in Theorem 6.37. However the assumption that im 𝑑 is closed as a subspace of Ω𝑠𝐿2 (𝑀) is by no means trivial (Remark 6.39). We will introduce the Laplacian and the theory of elliptic operators, which will eventually prove the Hodge theorem and the closeness of im 𝑑 will follow. See however Exercise 6.21. ∗

Definition 6.27. We define the Laplacian on forms (also called the Laplace-Beltrami operator) Δg ∶ Ω𝑠 (𝑀) → Ω𝑠 (𝑀) (with respect to the metric g) as the differential operator Δg = 𝑑𝑑 ∗ + 𝑑 ∗ 𝑑. A solution to Δg 𝛼 = 0 is called a harmonic form. Remark 6.28. (1) On functions, 𝑑 ∗ 𝑓 = 0 automatically. Since div(𝑋) = ⋆𝑑 ⋆ (𝑋♭ ) = −𝑑 ∗ 𝑋♭ (Lemma 6.25), the Laplacian is given by Δg 𝑓 = 𝑑 ∗ 𝑑𝑓 = − div(grad 𝑓). Using Remark 6.19(1) and (5), we have in coordinates, Δg 𝑓 = − ∑ (6.9)

𝜕𝑓 𝜕𝑓 𝑘𝑙 𝜕𝑔𝑙𝑘 𝜕 1 (𝑔𝑖𝑗 ) + (𝑔𝑖𝑗 )𝑔 𝜕𝑥𝑖 𝜕𝑥𝑗 2 𝜕𝑥𝑗 𝜕𝑥𝑖

= − ∑ 𝑔𝑖𝑗

𝜕2 𝑓 𝜕𝑓 1 𝑖𝑗 𝑘𝑙 𝜕𝑔𝑙𝑘 𝜕𝑔 + + 𝑔𝑖𝑟 𝑟𝑠 𝑔𝑠𝑗 ) ( 𝑔 𝑔 𝜕𝑥𝑖 𝜕𝑥𝑗 𝜕𝑥𝑗 2 𝜕𝑥𝑖 𝜕𝑥𝑖

= − ∑ 𝑔𝑖𝑗

𝜕2 𝑓 𝑗 𝜕𝑓 − 𝑔𝑖𝑙 Γ𝑖𝑙 . 𝜕𝑥𝑖 𝜕𝑥𝑗 𝜕𝑥𝑗

The derivative of 𝑔𝑖𝑗 in the second line is as follows. From g−1 g = Id, we get 𝜕g−1 𝜕g 𝜕g−1 𝜕g 𝜕𝑔𝑖𝑗 𝜕𝑔 g + g−1 𝜕𝑥 = 0, so 𝜕𝑥 = −g−1 𝜕𝑥 g−1 , In particular, 𝜕𝑥 = −∑𝑔𝑖𝑎 𝜕𝑥𝑎𝑏 𝑔𝑏𝑗 . 𝜕𝑥𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 The last line in (6.9) follows from (3.19). (2) The standard Laplacian on ℝ𝑛 is Δ𝑓 = ∑ ∑ 𝑑𝑥𝑖2 ,

𝜕2 𝑓 . 𝜕𝑥2𝑖

However, for the standard met𝜕2 𝑓

ric g𝑠𝑡𝑑 = the Laplacian comes with a minus sign, Δg𝑠𝑡𝑑 𝑓 = − ∑ 𝜕𝑥2 . 𝑖 For this reason, Δg is sometimes called the geometric Laplacian, whereas the standard Laplacian is more often used in mathematical analysis. (3) Let us relate the Laplacian of two conformal metrics. Let ĝ = 𝑒2𝜑 g be two conformal metrics. Using the Christoffel symbols given in Remark 6.3, we

356

6. Global analysis

have that 𝜕𝜑 𝜕𝜑 𝜕𝜑 𝜕𝑓 1 ∑ (𝑔𝑖𝑘 + 𝑔𝑗𝑘 − 𝑛𝑔𝑘𝑙 ) 2 𝜕𝑥𝑖 𝜕𝑥𝑗 𝜕𝑥𝑙 𝜕𝑥𝑘 𝑛−2 = 𝑒−2𝜑 Δg 𝑓 + ⟨𝑑𝜑, 𝑑𝑓⟩g , 2

Δĝ 𝑓 = 𝑒−2𝜑 Δg 𝑓 −

where we have used ∑ 𝑔𝑖𝑗 𝑔𝑖𝑗 = 𝑛. Therefore, for surfaces, we have Δĝ 𝑓 = 𝑒−2𝜑 Δg 𝑓 .

(6.10)

Hence the harmonic functions are the same for conformal metrics. (4) Let 𝑈 ⊂ ℂ = ℝ2 , and let 𝑓 ∶ 𝑈 → ℂ be a holomorphic function. Write 𝑓 = 𝑢 + i𝑣, where 𝑢 = Re 𝑓, 𝑣 = Im 𝑓. Then the Cauchy-Riemann relations (5.1) imply that Δ𝑢 =

𝜕2 𝑢 𝜕2 𝑢 𝜕 𝜕𝑣 𝜕 𝜕𝑣 + = ( )− ( ) = 0, 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝜕𝑥2 𝜕𝑦2

since the crossed derivatives coincide for 𝐶 2 functions. So 𝑢 = Re 𝑓 is harmonic. Analogously 𝑣 = Im 𝑓 is harmonic. There is a converse: for 𝑈 ⊂ ℂ simply connected and a harmonic function 𝑢 ∶ 𝑈 → ℝ, there is a holomorphic function 𝑓 ∶ 𝑈 → ℂ with 𝑢 = Re 𝑓 (Exercise 6.13). Note that in particular, harmonic functions are real analytic. Given 𝑢 harmonic, the function 𝑣 such that 𝑓 = 𝑢 + i𝑣 is holomorphic, is called its harmonic conjugate. Moreover, 4 (6.11)

𝜕2 𝑓 𝜕 𝜕 𝜕 𝜕 =( − i )( + i ) (𝑓) 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝜕𝑧𝜕𝑧 𝜕2 𝜕2 = ( 2 + 2 ) (𝑓) = Δ𝑓 , 𝜕𝑥 𝜕𝑦

for any (complex valued) smooth function 𝑓. (5) Let 𝑈 ⊂ ℂ be an open set, and let 𝑓 ∶ 𝑈 → ℂ be holomorphic. By Exercise 1 𝑓(𝑧) 5.1, 𝑓(𝑎) = 2𝜋i ∫𝛾 𝑧−𝑎 𝑑𝑧, for 𝛾 a curve encircling 𝑎. We take a small circle 𝛾(𝑡) = 𝑎 + 𝑟𝑒2𝜋i𝑡 , 𝑡 ∈ [0, 1], with 𝑟 > 0 small. Then 1

𝑓(𝑎) =

1

𝑓(𝑎 + 𝑟𝑒2𝜋i𝑡 ) 1 ∫ 2𝜋i𝑟𝑒2𝜋i𝑡 𝑑𝑡 = ∫ 𝑓(𝑎 + 𝑟𝑒2𝜋i𝑡 )𝑑𝑡, 2𝜋i 0 𝑟𝑒2𝜋i𝑡 0

that is the value 𝑓(𝑎) is the average of 𝑓 on a circle around 𝑎. If 𝑢 is harmonic, then locally 𝑢 = Re 𝑓 for some holomorphic function, and so 𝑢(𝑎) = 1 ∫0 𝑢(𝑎+𝑟𝑒2𝜋i𝑡 )𝑑𝑡. Harmonic functions are precisely those for which the value at any point 𝑎 is the average of the values on small circles around 𝑎, a property called the mean value property (for the converse, use Exercise 5.1). (6) Let us give the Laplacian on functions for a surface with orthogonal coordi1 1 nates g = 𝐸 𝑑𝑢2 + 𝐺 𝑑𝑣2 . Here 𝑔11 = 𝐸, 𝑔22 = 𝐺, 𝑔11 = 𝐸 , 𝑔22 = 𝐺 . We use

6.2. Hodge theory

357

the formula in the fifth line of Remark 6.19(5) to compute 1 𝜕 1 𝜕 Δg 𝑓 = − div(grad 𝑓) = − div( 𝑓ᵆ + 𝑓 ) 𝐸 𝜕𝑢 𝐺 𝑣 𝜕𝑣 =− =−

1 √𝐸𝐺 1 √𝐸𝐺

1

1

(

𝜕( 𝐸 𝑓ᵆ √𝐸𝐺) 𝜕( 𝐺 𝑓𝑣 √𝐸𝐺) + ) 𝜕𝑢 𝜕𝑣

((

𝐸𝑓𝑣 √𝐸𝐺

) +( 𝑣

𝐺𝑓ᵆ √𝐸𝐺

In particular this reproves that Δĝ 𝑓 = 𝑒

−2𝜑

) ). ᵆ

Δg 𝑓 for ĝ = 𝑒2𝜑 g in item (3).

(7) Let (𝑆, 𝐽) be a complex curve, which is equivalent to a conformal surface (𝑆, [g]) by Theorem 6.13. Take a Hermitian metric for 𝑆, or equivalently, fix a Riemannian metric g ∈ [g]. Let 𝑓 ∈ 𝐶 ∞ (𝑆) and take a holomorphic chart 𝜑 ∶ 𝑈 → ℂ = ℝ2 . By Remark 6.14(4), this is a conformal chart, so the metric 𝜕2 𝑓 is of the form g = 𝑒2𝜑 g𝑠𝑡𝑑 . By (6.11), we have that Δg𝑠𝑡𝑑 𝑓 = −4 𝜕𝑧𝜕𝑧 , so by 𝜕2 𝑓

(6.10), we get Δg 𝑓 = −4𝑒−2𝜑 𝜕𝑧𝜕𝑧 , for a smooth function 𝑓. Now observe that i

the volume form is 𝜈g = 𝑒2𝜑 𝑑𝑥 ∧ 𝑑𝑦 = 𝑒2𝜑 2 𝑑𝑧 ∧ 𝑑𝑧. Hence 𝜕2 𝑓 𝑑𝑧 ∧ 𝑑𝑧 = −2i𝜕𝜕𝑓 . 𝜕𝑧𝜕𝑧 Thus the Laplacian on a complex curve has a holomorphic nature. ⋆Δg 𝑓 = −2i

(8) If 𝑓 is a holomorphic function on a complex curve (𝑆, 𝐽), then 𝑢 = Re 𝑓 is harmonic (see Exercise 6.13 for the converse). Note that the condition of being harmonic only depends on the conformal structure, and not on the particular metric representing the conformal class. The Laplacian is a differential operator of order 2, so Δg ∶ Ω𝑠𝑊 2,2 (𝑀) → Ω𝑠𝐿2 (𝑀). The operator Δg is self-adjoint, that is, for 𝛼, 𝛽 ∈ Ω𝑠 (𝑀), ⟨Δg 𝛼, 𝛽⟩𝐿2 = ⟨(𝑑𝑑 ∗ + 𝑑 ∗ 𝑑)𝛼, 𝛽⟩𝐿2 = ⟨𝑑𝑑 ∗ 𝛼, 𝛽⟩𝐿2 + ⟨𝑑 ∗ 𝑑𝛼, 𝛽⟩𝐿2 = ⟨𝛼, 𝑑𝑑 ∗ 𝛽⟩𝐿2 + ⟨𝛼, 𝑑 ∗ 𝑑𝛽⟩𝐿2 = ⟨𝛼, (𝑑 ∗ 𝑑 + 𝑑𝑑 ∗ )𝛽⟩𝐿2 = ⟨𝛼, Δg 𝛽⟩𝐿2 . Otherwise said, Δ∗g = Δg . Remark 6.29. Let 𝑃 be a self-adjoint differential operator 𝑃 ∶ 𝒯(𝑀) → 𝒯(𝑀), so 𝑃 ∗ = 𝑃. We extend 𝑃 to the complexified vector space 𝑃 ∶ 𝒯(𝑀, ℂ) → 𝒯(𝑀, ℂ), where 𝒯(𝑀, ℂ) = 𝒯(𝑀) ⊗ ℂ. We consider on 𝒯(𝑀, ℂ) the Hermitian metric 𝐿2ℂ induced from the 𝐿2 scalar product on 𝒯(𝑀), that is ⟨𝑓, 𝑔⟩𝐿2ℂ = ⟨𝑓, 𝑔⟩𝐿2 , for 𝑓, 𝑔 ∈ 𝒯(𝑀, ℂ). Then the eigenvalues of 𝑃 are real. Let 𝜆 ∈ ℂ be an eigenvalue, with 𝑓 ∈ 𝒯(𝑀, ℂ) an eigenfunction, 𝑃𝑓 = 𝜆𝑓. Therefore 𝜆||𝑓||2𝐿2 = ⟨𝑃𝑓, 𝑓⟩𝐿2ℂ = ⟨𝑓, 𝑃𝑓⟩𝐿2ℂ = 𝜆||𝑓||2𝐿2 , from ℂ

where 𝜆 = 𝜆. The operator Δg is semipositive, that is, for 𝛼 ∈ Ω𝑠 (𝑀), (6.12)

⟨Δg 𝛼, 𝛼⟩𝐿2 = ⟨𝑑𝑑 ∗ 𝛼, 𝛼⟩𝐿2 + ⟨𝑑 ∗ 𝑑𝛼, 𝛼⟩𝐿2 = ⟨𝑑 ∗ 𝛼, 𝑑 ∗ 𝛼⟩𝐿2 + ⟨𝑑𝛼, 𝑑𝛼⟩𝐿2 = ||𝑑 ∗ 𝛼||2𝐿2 + ||𝑑𝛼||2𝐿2 ≥ 0.

ℂ

358

6. Global analysis

This implies that its eigenvalues are in [0, ∞). Note that this is a strong reason to prefer the sign of Δg as we are doing. The analytical Laplacian Δ is seminegative. By (6.12), we have that (6.13)

Δg 𝛼 = 0 ⟺ 𝑑𝛼 = 0,

𝑑 ∗ 𝛼 = 0.

So the kernel of Δg (the harmonic forms) consists of the solutions to the equations 𝑑𝛼 = 𝑑 ∗ 𝛼 = 0 (the solutions to (6.8)). Definition 6.30. We define the space of harmonic forms as ℋ∆𝑠 g (𝑀) = {𝛼 ∈ Ω𝑠𝑊 2,2 (𝑀) | Δg 𝛼 = 0}. Remark 6.31. In general, if 𝑃 ∶ 𝒯(𝑀) → 𝒯 ′ (𝑀) is a differential operator, then the rolled up operator 𝑄 = 𝑃𝑃 ∗ + 𝑃 ∗ 𝑃 is automatically self-adjoint and semipositive. Moreover, the solutions to 𝑄𝑓 = 0 are the solutions to the equations 𝑃𝑓 = 0, 𝑃 ∗ 𝑓 = 0. Definition 6.32. Let 𝑃 ∶ 𝒯(𝑀) → 𝒯(𝑀) be a differential operator of order 𝑘. We 𝜕|𝐼| define the symbol of 𝑃 as follows. Locally write 𝑃 = ∑|𝐼|≤𝑘 𝐴𝐼 (𝑥) 𝜕𝑥 as in (6.4). Then 𝐼

𝐼

𝜎𝑃 (𝑥, 𝜉) = ∑ 𝐴𝐼 (𝑥)𝜉 , |𝐼|=𝑘

for 𝑥 ∈ 𝑈, 𝜉 = (𝜉1 , . . . , 𝜉𝑛 ) ∈ 𝑇𝑥∗ 𝑈, and 𝜉𝐼 = 𝜉𝑖1 ⋯ 𝜉𝑖𝑝 . This is well defined independently of charts. We say that the differential operator 𝑃 is elliptic if 𝜎𝑃 (𝑥, 𝜉) is an isomorphism for all 𝑥 ∈ 𝑀, 𝜉 ≠ 0. Remark 6.33. The condition of being elliptic is natural from the point of view of harmonic analysis. The Fourier transform of a function 𝑓 ∶ ℝ𝑛 → ℂ is defined as the funĉ = ∫ 𝑛 𝑓(𝑥)𝑒−2𝜋i⟨𝜉,𝑥⟩ 𝑑𝑥. The inverse Fourier transform allows tion 𝑓 ̂ ∶ ℝ𝑛 → ℂ, 𝑓(𝜉) ℝ 𝜕 2𝜋i⟨𝜉,𝑥⟩ ̂ us to recover 𝑓(𝑥) = ∫ℝ𝑛 𝑓(𝜉)𝑒 𝑑𝜉. The partial derivative 𝜕𝑥 gets transformed to 𝑗

multiplication by 2𝜋i𝜉𝑗 , so differential operators such as (6.4) are transformed to multiplication by polynomials 𝑝 = ∑ 𝐴𝐼 𝜉𝐼 , after doing a Fourier transform (at least when the coefficients 𝐴𝐼 are constant). Now solving a differential equation consists of inverting the differential operator, which corresponds to dividing by 𝑝. So the invertibility of 𝑝 is the natural condition to impose. The operator Δg ∶ Ω𝑠 (𝑀) → Ω𝑠 (𝑀) is an elliptic operator. Let us look first at the Laplacian on functions. By (6.9) the symbol is 𝜎 ∆g (𝑥, 𝜉) = − ∑ 𝑔𝑖𝑗 (𝑥)𝜉𝑖 𝜉𝑗 . Since 𝜎 ∆g (𝑥, 𝜉) ≠ 0, for 𝜉 ≠ 0, Δg is elliptic. The symbol of the Laplacian on 𝑠-forms is also invertible (Exercise 6.17), so Δg is elliptic. Theorem 6.34 (Spectral theorem). Let 𝑃 ∶ 𝒯(𝑀) → 𝒯(𝑀) be a self-adjoint, semipositive, elliptic differential operator on a compact manifold 𝑀, and consider it on the Hilbert space of 𝐿2 tensors. Then the eigenvalues of 𝑃 form a sequence of real numbers 𝜆0 = 0 < 𝜆1 < 𝜆2 < ⋯ which tends to +∞, and the eigenspaces are finite dimensional orthogonal subspaces 𝐻0 , 𝐻1 , 𝐻2 , . . . that give a (Hilbert) orthogonal decomposition ∞ 𝒯𝐿2 (𝑀) = ⨁𝑠=0 𝐻𝑠 .

6.2. Hodge theory

359

Proof. Let 𝑘 be the order of 𝑃 so that 𝑃 ∶ 𝒯𝑊 𝑘,2 (𝑀) → 𝒯𝐿2 (𝑀) is a bounded operator. We already know that the eigenvalues of 𝑃 are real and positive. Suppose for a while that 𝐻0 = ker 𝑃 ⊂ 𝒯𝑊 𝑘,2 (𝑀) is finite dimensional. Then we have an orthogonal decomposition 𝒯𝑊 𝑘,2 (𝑀) = 𝐻0 ⊕ 𝐻0⟂ . By definition 𝑃 ∶ 𝐻0⟂ → 𝐻0⟂ is invertible. Its inverse 𝐺 = (𝑃|𝐻0⟂ )−1 can be extended by zero on 𝐻0 . As 𝑃 is elliptic, 𝐺 is a bounded operator. This is called the Green operator 𝐺 ∶ 𝒯𝐿2 (𝑀) → 𝒯𝑊 𝑘,2 (𝑀), which satisfies that (6.14)

𝑃 ∘ 𝐺 = 𝐺 ∘ 𝑃 = 𝜋𝐻0⟂ = Id −𝜋𝐻0

is the orthogonal projection onto 𝐻0⟂ , where 𝜋𝐻0 is the projection onto 𝐻0 . The embedding of functional spaces 𝒯𝑊 𝑘,2 (𝑀) ⊂ 𝒯𝐿2 (𝑀) is compact (meaning that bounded sequences get mapped into sequences which have convergent subsequences) by the Kondrachov-Rellich lemma [Wel], implying that 𝐺 ∶ 𝒯𝐿2 (𝑀) → 𝒯𝐿2 (𝑀) is a compact operator (meaning that it sends bounded sets to sets whose closure is compact). There is a well known spectral theorem for compact operators [Eva], saying that the eigenvalues form a countable set {𝜇𝑖 | 𝑖 ≥ 1} ⊂ ℝ which at most accumulate at 0, and that the eigenspace 𝐻𝑖 associated to 𝜇𝑖 is finite dimensional. Moreover, it gives an orthog∞ onal decomposition 𝒯𝐿2 (𝑀) = ⨁𝑖=0 𝐻𝑖 (the orthogonality is clear, as 𝑃 is self-adjoint). As 𝐺 is semipositive, the eigenvalues are 𝜇𝑖 ≥ 0, so they form a decreasing sequence 𝜇1 > 𝜇2 > ⋯ > 0. Going back to 𝑃, its eigenvalues are 𝜆0 = 0 and 𝜆𝑖 = 𝜇−1 𝑖 for 𝑖 > 0. We have the ∞ same decomposition 𝒯𝐿2 (𝑀) = ⨁𝑠=0 𝐻𝑠 in eigenspaces. Finally, the finite dimensionality of 𝐻0 follows by applying the same argument to the operator 𝑃 ̃ = 𝑃 + Id and its (possible) eigenvalue 𝜆 ̃ = 1. □ ∞

Corollary 6.35. In the situation of Theorem 6.34, we have im 𝑃 = 𝐻0⟂ = ⨁𝑠=1 𝐻𝑠 . Remark 6.36. The spectral theorem is an analogue of the result in the finite dimensional case that says that a symmetric endomorphism 𝑓 ∶ 𝑉 → 𝑉 on an Euclidean vector space (a self-adjoint linear map in the terminology of Remark 6.26) can be diagonalized on an orthonormal basis, and it has real eigenvalues. Finally, the application of the spectral Theorem 6.34 to the Laplacian gives the main result that we were aiming at. Theorem 6.37 (Hodge theorem). Let (𝑀, g) be a compact oriented Riemannian manifold. In every de Rham cohomology class there is exactly one harmonic form. In other 𝑠 words, there is an isomorphism 𝐻𝑑𝑅 (𝑀) ≅ ℋ∆𝑠 g (𝑀), for all 𝑠 ≥ 0. Proof. By Theorem 6.34, the space of harmonic forms is ℋ∆𝑠 g (𝑀) = ker Δg = 𝐻0 and it is finite dimensional. Let 𝐺 ∶ Ω𝑠𝐿2 (𝑀) → Ω𝑠𝐿2 (𝑀) be the associated Green operator. By (6.14), we have that for any form 𝛼 ∈ Ω𝑠𝐿2 (𝑀), (6.15)

𝛼 = 𝛼0 + Δg (𝐺𝛼) = 𝛼0 + 𝑑(𝑑 ∗ 𝐺𝛼) + 𝑑 ∗ (𝑑𝐺𝛼),

where 𝛼0 = 𝜋𝐻0 (𝛼), being 𝜋𝐻0 ∶ Ω𝑠𝐿2 (𝑀) → 𝐻0 the orthogonal projection onto the harmonic forms. The operators 𝑑 and 𝑑 ∗ commute with Δg , and thereby they also 𝑠 commute with 𝐺. Now if [𝛼] ∈ 𝐻𝑑𝑅 (𝑀), then 𝑑𝛼 = 0 and 𝑑𝐺𝛼 = 𝐺(𝑑𝛼) = 0. So

360

6. Global analysis

𝛼 = 𝛼0 + 𝑑(𝑑 ∗ 𝐺𝛼) and thus [𝛼] = [𝛼0 ]. This implies that the map 𝜋𝐻0 hence descends to a map 𝑠 𝜋𝐻0 ∶ 𝐻𝑑𝑅 (𝑀) → ℋ∆𝑠 g (𝑀).

(6.16)

The map (6.16) is well defined, since 𝜋𝐻0 (𝑑𝛽) = 0 as 𝑑𝛽 ∈ 𝐻0⟂ for all 𝛽 ∈ Ω𝑠 (𝑀). Certainly, if 𝛼 ∈ 𝐻0 , then 𝑑𝛼 = 𝑑 ∗ 𝛼 = 0 by (6.13), hence ⟨𝑑𝛽, 𝛼⟩ = ⟨𝛽, 𝑑 ∗ 𝛼⟩ = 0. By Proposition 6.38, the harmonic forms are actually smooth, so (6.16) is surjective. It is injective since if 𝜋𝐻0 ([𝛼]) = 𝛼0 = 0, then 𝛼 = 𝑑(𝑑 ∗ 𝐺𝛼) and [𝛼] = 0. □ Proposition 6.38. If Δg 𝛼 = 𝛽, and 𝛽 is smooth, then 𝛼 is smooth. Proof. Let 𝛼 be an eigenfunction of Δg with eigenvalue 𝜆 > 0. Then as Δg ∶ Ω𝑠𝑊 𝑘,2 (𝑀) → Ω𝑠𝑊 𝑘−2,2 (𝑀), we have that 𝐺 ∶ Ω𝑠𝑊 𝑘−2,2 (𝑀) → Ω𝑠𝑊 𝑘,2 (𝑀). So if 𝛼 ∈ Ω𝑠𝑊 𝑘−2,2 (𝑀), then 1 𝐺(𝛼) = 𝜆 𝛼 ∈ Ω𝑠𝑊 𝑘,2 (𝑀). This means that 𝛼 can be differentiated two times more. Repeating the argument 𝛼 admits infinitely many (weak) derivatives in 𝐿2 . By the Sobolev embedding theorem (Remark 6.22), this implies that 𝛼 is smooth. This does not work for 𝐻0 . But one can use the operator 𝐿 = Δg + Id, which has no kernel, and where 𝐻0 is now the eigenspace of eigenvalue 1. Therefore the harmonic forms are smooth. To conclude the statement, note that 𝛼 = 𝜋0 (𝛼) + 𝐺(𝛽). This is smooth because the elements in 𝐻0 are smooth by the argument above, and 𝐺 is a regularizing operator: it increases the order of differentiation by two units. Hence 𝐺(Ω𝑠 (𝑀)) ⊂ Ω𝑠 (𝑀), and thus 𝛼 ∈ Ω𝑠 (𝑀). □ Remark 6.39. • Theorem 6.37 also holds for compact non-orientable manifolds (Exercise 6.20). • By (6.15), we have an orthogonal decomposition Ω𝑠𝐿2 (𝑀) = 𝐻0 ⊕im 𝑑⊕im 𝑑 ∗ , where 𝐻0 are the harmonic forms. The orthogonality im 𝑑 ⟂ im 𝑑 ∗ follows from ⟨𝑑𝛽, 𝑑 ∗ 𝛼⟩ = ⟨𝑑𝑑𝛽, 𝛼⟩ = 0, for all 𝛼, 𝛽. In particular, this proves that im 𝑑 is a closed subspace of Ω𝑠𝐿2 (𝑀), as was mentioned before. Note that by Theorem 6.37, ker 𝑑 = 𝐻0 ⊕ im 𝑑, ker 𝑑 ∗ = 𝐻0 ⊕ im 𝑑 ∗ , ker 𝑑 ∩ ker 𝑑 ∗ = 𝐻0 , and im Δg = im 𝑑 ⊕ im 𝑑 ∗ . • The isomorphism (6.16) is a vector space isomorphism, but it is in general not an isomorphism of algebras. Remark 6.40. 𝑠 (1) We reprove that 𝐻𝑑𝑅 (𝑀) is finite dimensional for a compact manifold 𝑀 (known before by Exercise 3.15). It is a consequence of the finite dimensionality of 𝐻0 = ker Δg stated in Theorem 6.34.

(2) Let 𝑀 be a compact and connected manifold. Then (global) harmonic functions are constant. This is a consequence of the isomorphism ℋ∆0 g (𝑀) ≅ 0 𝐻𝑑𝑅 (𝑀) = ℝ. Let us prove it by a direct computation on surfaces. Let 𝑓 ∈ ℋ∆0 g (𝑀). By Remark 6.28(5), we have that the value 𝑓(𝑝) for 𝑝 ∈ 𝑀,

6.2. Hodge theory

361

is the average of the values of 𝑓 in small circles around 𝑝. Let 𝑝 be a maximum of 𝑓, that exists as 𝑀 is compact. Then 𝑓 is constant in a ball around 𝑝. Since two points of 𝑀 can always be encircled in an open set diffeomorphic to a ball, 𝑓 will be constant everywhere. (3) By the previous item, if 𝑀 is connected, then 𝐻0 = ℝ are the constant functions. So by Theorem 6.34, we have that Δg ∶ 𝐿2 (𝑀) → 𝐿2 (𝑀) has image im Δg = 𝐻0⟂ = {𝑓| ∫𝑀 𝑓 = ⟨𝑓, 1⟩𝐿2 = 0}. In particular ∫𝑀 Δg 𝑓 = 0, for all smooth functions 𝑓. Note that ∫𝑀 Δg 𝑓 = 0 can also be proved by the selfadjointness of Δg , as ⟨Δg 𝑓, 1⟩ = ⟨𝑓, Δg 1⟩ = 0. Another proof is by Stokes theorem, ∫𝑀 Δg 𝑓 = ∫𝑀 (𝑑 ∗ 𝑑𝑓)𝜈 = ± ∫𝑀 (⋆𝑑 ⋆ 𝑑𝑓)𝜈 = ± ∫𝑀 𝑑 ⋆ 𝑑𝑓 = 0. (4) For a smooth function 𝑓, let 𝑝 ∈ 𝑀 be a local maximum. We have that 𝜕𝑓 (𝑝) 𝜕𝑥𝑖

𝜕2 𝑓

= 0, 𝑖 = 1, . . . , 𝑛, and the Hessian Hess 𝑓(𝑝) = ( 𝜕𝑥 𝜕𝑥 (𝑝)) is semi𝑖

𝑗

definite negative. Take coordinates which are orthonormal at 𝑝, so that (6.9) says that Δg 𝑓(𝑝) = − tr(Hess 𝑓(𝑝)) ≥ 0. Analogously, if 𝑞 ∈ 𝑀 is a local minimum of 𝑓, then Δg 𝑓(𝑞) ≤ 0. See Exercise 6.10. (5) The operator ⋆ ∶ Ω𝑠 (𝑀) → Ω𝑛−𝑠 (𝑀) commutes with the Laplacian. ⋆ Δg = ⋆ (𝑑 ∗ 𝑑 + 𝑑𝑑 ∗ ) = ⋆((−1)𝑛𝑠+1 ⋆ 𝑑 ⋆ 𝑑 + (−1)𝑛(𝑠−1)+1 𝑑 ⋆ 𝑑⋆) = (−1)(𝑛−𝑠)𝑠 (−1)𝑛𝑠+1 𝑑 ⋆ 𝑑 + (−1)𝑛(𝑠−1)+1 ⋆ 𝑑 ⋆ 𝑑⋆ = ((−1)𝑛𝑠+1 𝑑 ⋆ 𝑑 ⋆ +(−1)𝑛(𝑠−1)+1 ⋆ 𝑑 ⋆ 𝑑)⋆ = ((−1)𝑛𝑠+1 (−1)𝑛(𝑛−𝑠−1)+1 𝑑𝑑 ∗ + (−1)𝑛(𝑠−1)+1 (−1)𝑛(𝑛−𝑠)+1 𝑑 ∗ 𝑑)⋆ = Δg ⋆ . In particular, the Hodge operator ⋆ sends eigenspaces of Δg to eigenspaces of Δg . This gives an isomorphism ≅

⋆ ∶ ℋ∆𝑠 g (𝑀) ⟶ ℋ∆𝑛−𝑠 (𝑀). g This reproves Poincaré duality (Example 2.151(3)). (6) For a compact Hermitian manifold (𝑀, 𝐽, ℎ), we can define in an analogous fashion the 𝜕-Laplacian, as Δ𝜕 = 𝜕∗ 𝜕 + 𝜕 𝜕∗ . We have 𝜕-harmonic forms 𝑝,𝑞 𝑝,𝑞 ℋ∆ (𝑀), and an isomorphism ℋ∆ (𝑀) ≅ 𝐻 𝑝,𝑞 (𝑀), with Dolbeault coho𝜕 𝜕 mology (Definition 5.15). (7) For a Kähler manifold and complex valued forms, one has that Δg = 2Δ𝜕 . The reason is that, when 𝑀 is Kähler, at each point 𝑝 ∈ 𝑀, we can choose complex coordinates which osculate to the flat metric at order 2, that is ℎ𝑖𝑗 = Id +𝑂(|𝑧|2 ) (Exercise 5.11). Then the computation Δg = 2Δ𝜕 follows from the same computation on the flat space (ℂ𝑛 , g𝑠𝑡𝑑 ). As a consequence, Δg preserves the spaces Ω𝑝,𝑞 (𝑀, ℂ) since Δ𝜕 does. If 𝑀 is compact, this implies that ℋ∆𝑠 g (𝑀, ℂ) =

⨁

𝑝+𝑞=𝑠

ker(Δg |Ω𝑝,𝑞 (𝑀,ℂ) ) =

𝑝,𝑞

⨁

𝑝+𝑞=𝑠

ℋ∆ (𝑀). 𝜕

362

6. Global analysis

The Hodge Theorem 5.31 follows from here, using the isomorphism 𝑠 ℋ∆𝑠 g (𝑀, ℂ) ≅ 𝐻𝑑𝑅 (𝑀, ℂ) 𝑝,𝑞

of Theorem 6.37 and the analogous isomorphism ℋ∆ (𝑀) ≅ 𝐻 𝑝,𝑞 (𝑀) for 𝜕 Dolbeault cohomology. 𝑝,𝑞

𝑞,𝑝

𝑞,𝑝

1

Moreover, ℋ∆ = ℋ∆𝜕 = ℋ∆ , since Δ𝜕 = Δ𝜕 = 2 Δg . 𝜕

𝜕

(8) For a compact Kähler manifold 𝑀, we have that the Hodge operator sends ⋆ ∶ Ω𝑝,𝑞 (𝑀) → Ω𝑛−𝑝,𝑛−𝑞 (𝑀) and commutes with the Laplacian by (5). Then there is an isomorphism 𝐻 𝑛−𝑝,𝑛−𝑞 (𝑀) ≅ 𝐻 𝑝,𝑞 (𝑀)∗ , which is known as Serre duality (cf. Remark 5.32(4)). Remark 6.41. Let 𝑋 be a differentiable compact oriented orbifold, and endow it with an orbifold Riemannian metric. Hodge theory can be extended to this setting [BBF]. The Hodge star operator is an orbifold operator, and it acts on orbifold forms. The exterior differential also acts on orbifold forms, and we can define a de Rham coho• mology 𝐻𝑑𝑅 (𝑋) by using orbifold forms, for which it holds a de Rham isomorphism • 𝐻𝑑𝑅 (𝑋) ≅ 𝐻 • (𝑋, ℝ). For Hodge theory, the spaces of orbifold functions and orbifold forms can be completed with respect to the Sobolev norm (which is defined as in (6.3) with respect to the orbifold Levi-Civita connection). The Laplacian is defined again as Δg = 𝑑 ∗ 𝑑 + 𝑑𝑑 ∗ , and all the local formulas hold verbatim on the orbifold charts. The space of harmonic forms ℋ∆𝑠 g (𝑋) is as in Definition 6.27, and satisfies the isomorphism of Theorem 6.37. Finally, the spectral Theorem 6.34 and the regularity result of Proposition 6.38 hold with orbifold smooth forms.

6.3. Metrics of constant curvature One of the areas of main focus in geometry is constructing geometric structures on manifolds. There are geometric structures of many sorts (such as Riemannian structures, complex structures, or symplectic structures), but there are many others with physical and geometric meaning which have not been treated in this book. A large number of geometric structures are defined by tensors (although this is not the only way), and very often these tensors have to satisfy some equations that can be written as a differential equation. The differential equations coming from geometrical problems have to be defined globally on a smooth manifold, and therefore they must be compatible with changes of charts (i.e., they can be written in coordinate-free terms). For Riemannian structures, it is natural to search for metrics with “good” geometrical properties. As we explained in section 3.3, these are typically homogeneous and isotropic geometries. We shall thus focus on the case of metrics with constant curvature on surfaces (which were studied in Chapter 4). We shall deal with the question: Given a Riemannian metric g0 on a connected surface 𝑆, does there exist a metric g conformally equivalent to g0 which has constant curvature?

6.3. Metrics of constant curvature

363

We can rephrase this in terms of complex geometry. Let (𝑆, 𝐽) be a connected complex surface. By Remark 6.14, this is equivalent to a conformal structure, and the metrics in the conformal class correspond to Hermitian metrics for (𝑆, 𝐽). Hence the question can be restated: Given a connected complex surface (𝑆, 𝐽), does there exist a Hermitian metric h such that g = Re h has constant curvature? The set up is the following. We start with a compact connected oriented Riemannian surface (𝑆, g0 ), and we look for a metric g ∈ [g0 ] such that the curvature of g is 𝜅g ≡ 𝑘0 , where 𝑘0 ∈ ℝ. (As there will be different metrics involved, we will denote the curvature for the metric g as 𝜅g instead of the earlier notation 𝜅𝑆 in Definition 3.48.) The metrics conformally equivalent to g0 are of the form g = 𝑒2𝜑 g0 , where 𝜑 ∈ 𝐶 ∞ (𝑆) is a smooth function. To write down the differential equation that we want 𝜑 to satisfy, we need to relate the curvatures of g0 and g. Lemma 6.42. If g = 𝑒2𝜑 g0 , then 𝜅g = 𝑒−2𝜑 Δg0 𝜑 + 𝑒−2𝜑 𝜅g0 . Proof. We can prove this locally. We shall use orthogonal coordinates for (𝑆, g0 ), whose existence is guaranteed by Proposition 3.52. In these coordinates, g0 = 𝐸 𝑑𝑢2 + 𝐺 𝑑𝑣2 , and g = 𝑒2𝜑 𝐸 𝑑𝑢2 + 𝑒2𝜑 𝐺 𝑑𝑣2 . Then Proposition 3.54 says that 𝜅g = −

1

((

2𝑒2𝜑 √𝐸𝐺

= 𝑒−2𝜑 𝜅g0 −

(𝑒2𝜑 𝐸)𝑣 𝑒2𝜑 √𝐸𝐺

−2𝜑

𝑒

√𝐸𝐺

((

) +(

𝐸𝜑𝑣 √𝐸𝐺

𝑣

(𝑒2𝜑 𝐺)ᵆ 𝑒2𝜑 √𝐸𝐺

) +( 𝑣

𝐺𝜑ᵆ √𝐸𝐺

) ) ᵆ

) ) ᵆ

= 𝑒−2𝜑 𝜅g0 + 𝑒−2𝜑 Δg0 𝜑 , using Remark 6.28, items (3) and (6).

□

Remark 6.43. We recover the formula for the curvature of a surface (𝑆, g) when we have conformal coordinates (Corollary 3.56). Let (𝑈, 𝜙 = (𝑢, 𝑣)) be a conformal chart, i.e., g = 𝑒2𝜑 (𝑑𝑢2 + 𝑑𝑣2 ). This means that g is conformally equivalent to the standard metric g0 = 𝑑𝑢2 + 𝑑𝑣2 . Then 𝜅g0 = 0 and Δg0 = −Δ is (minus) the standard Laplacian (Remark 6.28(2)). Therefore 𝜅g = −𝑒−2𝜑 Δ𝜑 by Lemma 6.42. Therefore, to find a metric g = 𝑒2𝜑 g0 of constant curvature 𝑘0 , we have to solve the differential equation 𝑘0 = 𝑒−2𝜑 Δg0 𝜑 + 𝑒−2𝜑 𝜅g0 . Equivalently, (6.17)

Δg0 𝜑 − 𝑒2𝜑 𝑘0 + 𝜅g0 = 0.

This is a non-linear elliptic equation of order 2 on 𝑆. The ellipticity makes it retain some of the good properties of the equations in section 6.2. However, the non-linearity forces us to study the equation by hand. It is remarkable how hard a mildly non-linear term can make an equation. We aim for the following result.

364

6. Global analysis

Theorem 6.44 (Theorem CC). Let (𝑆, g0 ) be a compact connected oriented Riemannian surface of genus 𝑔. Then there exists a metric g ∈ [g0 ] of constant curvature 𝜅g ≡ 𝑘0 . • If 𝑔 = 0, then 𝑘0 > 0. We can arrange 𝑘0 = 1, but the metric g is not unique. • If 𝑔 = 1, then 𝑘0 = 0. The metric g is unique if we fix the area of (𝑆, g). • If 𝑔 ≥ 2, then 𝑘0 < 0. The metric g is unique if we fix the area of (𝑆, g). Alternatively, we can fix 𝑘0 = −1 and the metric g is unique. By Remark 3.70(5), these cases are exclusive. Again, this adds a new reason to the fact that we call Riemann surface to a complex curve (that is, a conformal surface), cf. Remark 5.49(6). At least for 𝑔 ≥ 2, a (compact, connected) complex curve admits a unique Riemannian metric of constant curvature −1. So a Riemann surface can be interpreted as a complex curve with such a special Riemannian metric. Remark 6.45. If (𝑆, g0 ) is non-orientable, take the oriented cover 𝜋 ∶ 𝑆 ̂ → 𝑆 with metric g0̂ = 𝜋∗ g0 . Let 𝜓 be the non-trivial deck transformation of 𝜋. Solve the problem (6.17) for (𝑆,̂ g0̂ ), so there is ĝ = 𝑒2𝜑 g0̂ of constant curvature and conformal to g0̂ . Then 𝜓∗ ĝ also solves the problem (6.17). In the case that the genus is 𝑔 ≥ 1, the uniqueness part of Theorem 6.44 guarantees that 𝜓∗ ĝ = g.̂ Thus ĝ induces a metric g on 𝑆, which has constant curvature and it is conformal to g0 . The case of genus 𝑔 = 0 is left to Exercise 6.22. Also, the same question can be posed for a non-connected surface 𝑆. In this case, we can work on the connected components of 𝑆 to get the desired metric of constant curvature. However, observe that in general, this constant may vary between connected components (e.g., if the components have different genus). Hence, the results of this section are valid in the non-connected setting with suitable adaptations. The proof of Theorem 6.44 is divided into several steps. First we prove the uniqueness part, and then we move to the existence. For existence, we shall give different proofs using important methods in global analysis to solve partial differential equations on compact manifolds. The proofs work in the cases 𝑘0 ≤ 0. A final trick using orbifolds will do the case 𝑘0 > 0 indirectly. 6.3.1. Uniqueness on Theorem CC. Let us check the uniqueness assertions in Theorem 6.44. First observe that if g has constant curvature 𝑘0 , then ĝ = 𝜆2 g, for 𝜆 > 0 a real number, and it also has constant curvature equal to 𝜆−2 𝑘0 (Remark 4.9). This means that the metrics g and ĝ are homothetic (Remark 4.9). Thus if 𝑘0 > 0, then we can find a metric ĝ with 𝜅ĝ ≡ 1; and if 𝑘0 < 0, then we can find a metric ĝ with 𝜅ĝ ≡ −1. On the other hand, area(𝑆, g)̂ = 𝜆2 area(𝑆, g), so homotheties allow us to normalize 𝑘0 ∈ {0, ±1} (Remark 4.9) and also they allow us to change the area. Second, if g has constant curvature 𝑘0 , the Gauss-Bonnet theorem (see Remark 3.70(5)) implies that 𝑘0 area(𝑆, g) = ∫𝑆 𝜅g = 2𝜋𝜒(𝑆). So if the genus is 𝑔 = 0, then 𝑘0 > 0, and fixing the area is equivalent to fixing 𝑘0 . Analogously if the genus is 𝑔 ≥ 2, then 𝑘0 < 0, and fixing the area is equivalent to fixing 𝑘0 . In the case of genus 𝑔 = 1, we always have 𝑘0 = 0, and we are only allowed to fix the area.

6.3. Metrics of constant curvature

365

Suppose that g1 , g2 are two conformal metrics of the same constant curvature 𝑘0 . As they are conformally equivalent, we have that g2 = 𝑒2𝜑 g1 , for some 𝜑 ∈ 𝐶 ∞ (𝑆). So by Lemma 6.42, we have (6.18)

Δg1 𝜑 = 𝑒2𝜑 𝑘0 − 𝑘0 .

Let us discuss the cases according to the value of 𝑘0 . (1) If 𝑘0 = 0, then (6.18) says that Δg1 𝜑 = 0. This means that 𝜑 is constant (Remark 6.40(2)). Then g2 = 𝜆2 g1 , for some constant 𝜆 > 0. If we fix the area, then g2 = g1 . (2) If 𝑘0 < 0, consider a point 𝑝 ∈ 𝑆 where 𝜑 achieves its maximum. Then Δg1 𝜑(𝑝) ≥ 0 by Remark 6.40(4). Using this in (6.18), we get that 𝑒2𝜑(𝑝) −1 ≤ 0, so 𝜑(𝑝) ≤ 0. This means that 𝜑 ≤ 0 everywhere. Next consider a point 𝑞 ∈ 𝑆 where 𝜑 achieves its minimum. Then Δg1 𝜑(𝑞) ≤ 0, which implies that 𝑒2𝜑(𝑞) − 1 ≥ 0, so 𝜑(𝑞) ≥ 0. Thus 𝜑 ≥ 0 everywhere, and hence 𝜑 ≡ 0, and finally g2 = g1 . (3) If 𝑘0 > 0, the manifold 𝑆 must be homeomorphic to the 2-sphere. By uniqueness of the simply connected space form of positive constant curvature (Theorem 3.108), we have that (𝑆, g1 ), (𝑆, g2 ) are isometric. So there exists an isometry 𝐹 ∶ (𝑆, g2 ) → (𝑆, g1 ). Hence 𝐹 ∗ (g1 ) = g2 = 𝑒2𝜑 g1 . This means that 𝐹 ∶ (𝑆, g1 ) → (𝑆, g1 ) is a conformal map, that is 𝐹 is a Möbius transformation (Remark 4.17). So there is no uniqueness in the solution to (6.17), but it is unique up to PGL(2, ℂ)/PU(2) (conformal maps modulo isometries), see Remark 5.64. 6.3.2. Existence on Theorem CC for 𝑘0 ≤ 0. In this section we prove that equation (6.17) has a solution for 𝑘0 ≤ 0. First we start with the case 𝑘0 = 0. So (6.17) is the equation (6.19)

Δg0 𝜑 + 𝜅g0 = 0.

This is a linear elliptic equation. Since Δg0 ∶ 𝐻0⟂ → 𝐻0⟂ is an isomorphism, it is enough 0 to prove that −𝜅g0 ∈ 𝐻0⟂ . Note that 𝐻0 = ℋ∆0 g (𝑆) = 𝐻𝑑𝑅 (𝑆) consists only of the con⟂ 2 stant functions, and 𝐻0 = {𝑓 ∈ 𝐿 (𝑆) | ∫𝑆 𝑓 = 0} (Remark 6.40(2) and (3)). Equation (6.19) has a solution 𝜑 ∈ 𝐿2 (𝑆) because ∫𝑆 𝜅g0 = 2𝜋𝜒(𝑆) = 0 and so −𝜅g0 ∈ 𝐻0⟂ . Finally by elliptic regularity, we have that 𝜑 ∈ 𝐶 ∞ (𝑆) (Proposition 6.38). In the case 𝑘0 ≠ 0, the equation (6.17) is genuinely non-linear. We make a simplification to start with. We can assume that 𝜅g0 has constant sign on the surface 𝑆. Lemma 6.46. There is a metric g ∈ [g0 ] such that 𝜅g has constant sign. Proof. Consider the initial metric g0 and the constant 𝜆0 = 𝐻0⟂ . 2𝜑

1 area(𝑆,g0 )

∫𝑆 𝜅g0 . Then

𝜆0 − 𝜅g0 has zero mean, that is 𝜆0 − 𝜅g0 ∈ This means that there is some function 𝜑 ∈ 𝐻0⟂ such that Δg0 𝜑 = 𝜆0 −𝜅g0 . Take g = 𝑒 g0 , then 𝜅g = 𝑒−2𝜑 (Δg0 𝜑+𝜅g0 ) = 𝜆0 𝑒−2𝜑 , which has constant sign, as required. □

366

6. Global analysis

We shall give two different proofs to solve equation (6.17) for 𝑘0 < 0, where we assume that the curvature 𝜅g0 < 0 everywhere on 𝑆, possible by Lemma 6.46. Variational method. Variational methods consist of defining a functional (sometimes called energy functional) which has to be minimized or maximized. This is often done in a functional space where techniques of functional analysis are useful. The proof that we give is due to Berger, and also appears in the book [Tay]. We start with a metric g0 which satisfies 𝜅g0 < 0 everywhere on 𝑆. The integration on 𝑆 will always be considered with respect to this metric. Let 𝐶1 , 𝐶2 > 0 such that −𝐶2 ≤ 𝜅g0 (𝑝) ≤ −𝐶1 < 0 for all 𝑝 ∈ 𝑆. Consider the space 𝒮 𝑘 = {𝑢 ∈ 𝑊 𝑘,2 (𝑆) || ∫ 𝑒2ᵆ = area(𝑆)} , 𝑆

for an integer 𝑘 ≥ 1. A first observation is that there exists a constant 𝐶 > 0 such that for any 𝑢 ∈ 𝒮 𝑘 , ∫𝐴 𝑢 < 𝐶, where 𝐴 = {𝑢 > 0}. For that, note that on 𝐴, 𝑒2ᵆ ≥ 1+2𝑢 ≥ 𝑢, 1 1 so ∫𝐴 𝑢 ≤ 2 ∫𝑆 𝑒2ᵆ = 2 area(𝑆). Consider the functional on 𝒮 𝑘 1 ℱ(𝑢) = ∫ ( |𝑑𝑢|2 + 𝜅g0 𝑢) . 2 𝑆 Then ℱ is bounded below since for all 𝑢 ∈ 𝒮 𝑘 , take 𝐴 = {𝑢 > 0} and then ℱ(𝑢) =

1 ||𝑑𝑢||2𝐿2 + ∫ 𝜅g0 𝑢 + ∫ 𝜅g0 𝑢 ≥ −𝐶2 𝐶, 2 𝐴 𝑆−𝐴

since ||𝑑𝑢||𝐿2 ≥ 0, 𝜅g0 ≥ −𝐶2 on 𝐴 (actually, everywhere) and 𝜅g0 𝑢 ≥ 0 on 𝑆 − 𝐴. Now take 𝑢𝑛 ∈ 𝒮 1 such that ℱ(𝑢𝑛 ) tends to the infimum of ℱ on 𝒮 1 (this is called a minimizing sequence). Let us see that 𝑢𝑛 is bounded in 𝑊 1,2 . We need the following result, called the Poincaré estimate. 2

Lemma 6.47. For all 𝑢 ∈ 𝑊 1,2 (𝑆), we have ||𝑢||2𝐿2 ≤ 𝑐 (||𝑑𝑢||2𝐿2 + |∫𝑆 𝑢| ), for some constant 𝑐 > 0. Proof. By the spectral Theorem 6.34, Δg ∶ 𝐻0⟂ → 𝐻0⟂ has a minimum eigenvalue 𝜆1 > 0. Then (6.20)

𝜆1 ||𝑢||2𝐿2 ≤ ⟨Δg 𝑢, 𝑢⟩𝐿2 = ⟨𝑑 ∗ 𝑑𝑢, 𝑢⟩𝐿2 = ⟨𝑑𝑢, 𝑑𝑢⟩𝐿2 = ||𝑑𝑢||2𝐿2

for 𝑢 ∈ 𝐻0⟂ . For general 𝑢, decompose 𝑢 = 𝑢1 + 𝑢2 , where 𝑢1 =

1 area(𝑆)

∫𝑆 𝑢 ∈ 𝐻0 = ℝ,

𝑢2 ∈ 𝐻0⟂ . As 𝑑𝑢 = 𝑑𝑢2 , we apply (6.20) to 𝑢2 , to get the estimate 2

||𝑢||2𝐿2 = ||𝑢1 ||2𝐿2 + ||𝑢2 ||2𝐿2 ≤

1 1 (∫ 𝑢) + ||𝑑𝑢||2𝐿2 . 𝜆1 area(𝑆) 𝑆

□

We continue with our argument. The sequence ℱ(𝑢𝑛 ) is bounded above, since it is a minimizing sequence, i.e., ℱ(𝑢𝑛 ) ≤ 𝐶3 , for all 𝑛 and some 𝐶3 > 0. Writing

6.3. Metrics of constant curvature

367

𝐴𝑛 = {𝑢𝑛 > 0}, we estimate 𝐶3 ≥ ℱ(𝑢𝑛 ) = ||𝑑𝑢𝑛 ||2𝐿2 + ∫ 𝜅g0 𝑢𝑛 + ∫ 𝐴𝑛

(6.21)

≥ ||𝑑𝑢𝑛 ||2𝐿2 − 𝐶2 𝐶 + 𝐶1 ∫

𝑆−𝐴𝑛

𝜅g0 𝑢𝑛

(−𝑢𝑛 ),

𝑆−𝐴𝑛

using that 𝜅g0 ≥ −𝐶2 on 𝐴𝑛 and −𝜅g0 ≥ 𝐶1 on 𝑆 − 𝐴𝑛 . Therefore ∫𝑆−𝐴𝑛 𝑢𝑛 is uniformly bounded (above and below). As ∫𝐴𝑛 𝑢𝑛 is also uniformly bounded, we have that ∫𝑆 𝑢𝑛 is uniformly bounded. From (6.21), ||𝑑𝑢𝑛 ||𝐿2 is also uniformly bounded, and so Lemma 6.47 implies that {𝑢𝑛 } is a uniformly bounded sequence in 𝑊 1,2 . The embedding of functional spaces 𝑊 1,2 (𝑆) ⊂ 𝐿2 (𝑆) is compact. This is again the Kondrachov-Rellich lemma [Wel], which means that bounded sequences get mapped into sequences which have convergent subsequences. Therefore there exists a subsequence of {𝑢𝑛 } that converges in 𝐿2 . To avoid burdening the notation, we relabel the subsequence and call it again {𝑢𝑛 }, so 𝑢𝑛 → 𝑢0 in 𝐿2 for some function 𝑢0 . Working similarly with 𝒮 𝑘 , we get a convergent sequence 𝑢𝑛 → 𝑢0 in 𝑊 𝑘−1,2 . In particular, if 𝑘 ≥ 2, then the functional ℱ(𝑢𝑛 ) → ℱ(𝑢0 ), so ℱ(𝑢0 ) gives the minimum of ℱ. Moreover 𝑢0 ∈ ⋂𝑘≥1 𝑊 𝑘,2 (𝑆), so by the Sobolev embedding theorem, 𝑢0 is smooth. Let us see now that 𝑢0 solves (6.17). The set 𝒮 𝑘 ⊂ 𝑊 𝑘,2 is a (Hilbert) hypersurface. Let us compute the tangent space at 𝑢0 . Write 𝐺(𝑢) = ∫𝑆 𝑒2ᵆ − area(𝑆), so that 𝒮 𝑘 = 𝐺 −1 (0). Hence 𝑇ᵆ0 𝒮 𝑘 = {ℎ ∈ 𝑊 𝑘,2 (𝑆)|𝐷ᵆ0 𝐺(ℎ) = 0}. The operator differential of 𝐺 is given by 𝐷ᵆ0 𝐺(ℎ) =

𝑑| 𝑑 𝐺(𝑢0 + 𝑡ℎ) = || (∫ 𝑒2ᵆ0 +𝑡ℎ − area(𝑆)) = ∫ 𝑒2ᵆ0 ℎ . 𝑑𝑡 |𝑡=0 𝑑𝑡 𝑡=0 𝑆 𝑆

So if ℎ ∈ 𝑇ᵆ0 𝒮 𝑘 , then 𝑓 = 𝑒2ᵆ0 ℎ ∈ 𝐻0⟂ , that is ℎ = 𝑒−2ᵆ0 𝑓, for some 𝑓 ∈ 𝐻0⟂ . Otherwise said, 𝑊 𝑘,2 (𝑆) = 𝑇ᵆ0 𝒮 𝑘 ⊕ ⟨𝑒2ᵆ0 ⟩ is an orthogonal sum. Take now ℎ ∈ 𝑇ᵆ0 𝒮 𝑘 with 𝑘 ≥ 1 (actually 𝑘 = 1 is enough). Then there is a smooth curve 𝑢(𝑡) ∈ 𝒮 𝑘 such that 𝑢(0) = 𝑢0 and 𝑢′ (0) = ℎ. We can write 𝑢(𝑡) = 𝑢0 +𝑡ℎ +𝑂(𝑡2 ), where the last term has 𝑊 𝑘,2 norm bounded by 𝐶 ′ 𝑡2 , for a constant 𝐶 ′ > 0. We compute ℱ(𝑢(𝑡)) = ℱ(𝑢0 + 𝑡ℎ + 𝑂(𝑡2 )) =

1 𝑡2 ||𝑑𝑢0 ||2𝐿2 + 𝑡⟨𝑑𝑢0 , 𝑑ℎ⟩𝐿2 + ||𝑑ℎ||2𝐿2 + ∫ 𝜅g0 𝑢0 + 𝑡 ∫ 𝜅g0 ℎ + 𝑂(𝑡2 ) 2 2 𝑆 𝑆

= ℱ(𝑢0 ) + 𝑡⟨𝑑 ∗ 𝑑𝑢0 + 𝜅g0 , ℎ⟩𝐿2 + 𝑂(𝑡2 ) = ℱ(𝑢0 ) + 𝑡⟨(Δg0 𝑢0 + 𝜅g0 ), ℎ⟩𝐿2 + 𝑂(𝑡2 ), where the last term has 𝐿2 norm bounded by 𝐶 ″ 𝑡2 for some constant 𝐶 ″ > 0. As ℱ(𝑢(𝑡)) ≥ ℱ(𝑢0 ) for 𝑡 close to 0 and ℎ ∈ 𝑇ᵆ0 𝒮 𝑘 , we have that Δg0 𝑢0 + 𝜅g0 must be equal to 𝑒2ᵆ0 𝑘, for some 𝑘 ∈ 𝐻0 = ℝ. This gives a solution to (6.17). Finally, integrating on the equality Δg0 𝑢0 + 𝜅g0 = 𝑒2ᵆ0 𝑘, we get that ∫𝑆 𝜅g0 = 2𝜋𝜒(𝑆) = ∫𝑆 𝑒2ᵆ0 𝑘 = area(𝑆)𝑘, since 𝑢0 ∈ 𝒮 𝑘 . Therefore we have 𝑘 = 2𝜋𝜒(𝑆)/ area(𝑆) = 𝑘0 .

368

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Remark 6.48. Variational methods are powerful in geometric analysis. An energy functional is ℰ ∶ ℳ → ℝ, where ℳ is some functional space. The critical points of ℰ are those 𝑢 ∈ ℳ such that 𝐷ᵆ ℰ = 0. In general this is rewritten as a differential equation on 𝑢, called the Euler-Lagrange equation of ℰ. In most cases we start with the differential equation and construct the functional ℰ to match our problem. A remarkable instance of this is in the variational study of geodesics in a Riemannian manifold 𝑀, where ℳ = Ω𝑝,𝑞 (𝑀) is the space of piecewise 𝐶 1 paths joining two points 𝑝, 𝑞 ∈ 𝑀. We take the functional ℓ ∶ Ω𝑝,𝑞 (𝑀) → ℝ, the length of the path, 1

or ℰ ∶ Ω𝑝,𝑞 (𝑀) → ℝ, the energy of the path, ℰ(𝛾) = ∫0 ||𝛾′ (𝑡)||2 𝑑𝑡. Then the critical points of ℓ or ℰ give the geodesics (Exercise 6.25). Continuity method. The continuity method consists of introducing a parameter 𝑡 ∈ [0, 1] in the differential equation, so that for 𝑡 = 0 the equation is easy to solve, and for 𝑡 = 1 the equation is the one that we want to solve. Then we have to prove that we can prolong the property of having a solution along the whole interval [0, 1]. As before, take 𝑘0 < 0, and assume that 𝜅g0 < 0 everywhere on 𝑆, by Lemma 6.46. Take 𝐶1 , 𝐶2 > 0 so that −𝐶2 ≤ 𝜅g0 ≤ −𝐶1 . Consider the equation in 𝑊 2,2 (𝑆), 𝑃𝑡 (𝑢) = Δg0 𝑢 − 𝑘0 𝑒2ᵆ + (1 − 𝑡)𝑘0 + 𝑡𝜅g0 = 0,

(6.22)

for 𝑡 ∈ [0, 1]. For 𝑡 = 0 the equation 𝑃0 (𝑢) = 0 has solution 𝑢 ≡ 0. Let us prove that the set 𝐴 = {𝑡 ∈ [0, 1] | 𝑃𝑡 (𝑢) = 0 has solution} is open and closed. Then by connectedness, it must be 𝐴 = [0, 1], and hence 𝑃1 (𝑢) = 0, which is our equation (6.17), has a solution. • 𝐴 is open. Take 𝑡0 ∈ 𝐴. Then there exists 𝑢0 such that 𝑃𝑡0 (𝑢0 ) = 0. The linearization of 𝑃𝑡0 at 𝑢0 is the linear operator 𝐿(ℎ) = 𝐷ᵆ0 𝑃𝑡0 (ℎ) = Δg ℎ − 2𝑘0 𝑒2ᵆ0 ℎ. This operator 𝐿 ∶ 𝑊 2,2 (𝑆) → 𝐿2 (𝑆) is invertible, since it is a self-adjoint, positive, linear elliptic operator. Hence 𝑃𝑡0 ∶ 𝑊 2,2 (𝑆) → 𝐿2 (𝑆) is a local homeomorphism from a neighbourhood of 𝑢0 to a neighbourhood of 0, by the inverse function theorem for Hilbert spaces [Lan]. For 𝜖 > 0 small enough, 𝜖(𝑘0 − 𝜅g0 ) is small in 𝐿2 . Hence 𝑃𝑡0 (𝑢) = 𝜖(𝑘0 − 𝜅g0 ) has a solution 𝑢. Such 𝑢 satisfies 𝑃𝑡0 +𝜖 (𝑢) = 0, and hence 𝑡0 + 𝜖 ∈ 𝐴. So 𝐴 is open. • 𝐴 is closed. Let 𝑢 be a solution to 𝑃𝑡 (𝑢) = 0, for some value of 𝑡. Let us find a priori bounds for 𝑢. At a maximum 𝑝 of 𝑢, we have Δg0 𝑢(𝑝) ≥ 0, so 𝑒2ᵆ(𝑝) ≤ 1 |(1 − 𝑡)𝑘0 + 𝑡𝜅g | ≤ max{1, 𝐶2 /|𝑘0 |}, hence 𝑢 is uniformly bounded above |𝑘0 |

0

independently of 𝑡 ∈ [0, 1]. To get a lower bound, we look at a minimum 𝑞 1 of 𝑢. Then Δg0 𝑢(𝑞) ≤ 0, 𝑒2ᵆ(𝑞) ≥ |𝑘 | |(1 − 𝑡)𝑘0 + 𝑡𝜅g0 | ≥ min{1, 𝐶1 /|𝑘0 |}, and 0 𝑢 is uniformly bounded below independently of 𝑡 ∈ [0, 1]. 1

As 𝑢 is uniformly bounded, we have that 𝜋𝐻0 (𝑢) = area(𝑆) ∫𝑆 𝑢 ∈ 𝐻0 = ℝ is uniformly bounded. (𝜋𝐻0 ∶ 𝐿2 (𝑆) → 𝐻0 = ℝ is the projection onto the harmonic forms 𝐻0 ). Also 𝑒ᵆ is bounded, and using (6.22) we get that Δg0 𝑢 ∈ 𝐻0⟂ is bounded in 𝐿2 .

6.3. Metrics of constant curvature

369

Therefore 𝜋𝐻0⟂ (𝑢) = 𝐺(Δg0 (𝑢)) ∈ 𝐻0⟂ is bounded in 𝑊 2,2 (where 𝜋𝐻0⟂ ∶ 𝐿2 (𝑆) → 𝐻0⟂ is the projection onto 𝐻0⟂ and 𝐺 is the Green operator). Thus 𝑢 is bounded in 𝑊 2,2 , uniformly for all 𝑡. This produces a uniform bound of 𝑒ᵆ in 𝑊 2,2 (using the pointwise bound on 𝑒ᵆ found before, and the Leibniz rule for weak derivatives). Using (6.22) again, we get a bound of 𝑢 in 𝑊 4,2 . This process is called bootstrapping. It ends up having uniform bounds in all Sobolev spaces 𝑊 2𝑘,2 . By the Sobolev embedding theorem, we have 𝐶 ∞ bounds uniformly on all 𝑡 (i.e., all derivatives are uniformly bounded). Take now a sequence of 𝑡 𝑖 ∈ 𝐴, with 𝑡 𝑖 → 𝑡∞ , and solutions 𝑢𝑖 to 𝑃𝑡𝑖 (𝑢𝑖 ) = 0. As we have 𝐶 1 bounds, the family {𝑢𝑖 } is equicontinuous. So by the Arzelá-Ascoli theorem [Eva], there is a subsequence of 𝑢𝑖 convergent to some smooth function 𝑢∞ . Taking limits in 𝑃𝑡𝑖 (𝑢𝑖 ) = 0, we have that 𝑢∞ solves 𝑃𝑡∞ (𝑢∞ ) = 0, and hence 𝑡∞ ∈ 𝐴. Remark 6.49. This argument is taken from [Do4]. The continuity argument is very useful for solving highly non-linear partial differential equations, such as the famous proof of Yau of the Calabi conjecture. This says that, if (𝑀, 𝐽, 𝜔) is a compact Kähler manifold of complex dimension 𝑛 such that it has a nowhere vanishing global holomorphic 𝑛-form 𝜔 ∈ Ω 𝑛 (𝑀) (see Example 5.17 for notation), then there is a metric of holonomy SU(𝑛) (recall that Kähler manifolds have a metric of holonomy U(𝑛), see Exercise 5.9). For this reason, such manifolds are called Calabi-Yau manifolds (Remark 5.66). 6.3.3. Existence on Theorem CC for 𝑘0 > 0. The case of positive curvature is harder because we cannot obtain a priori bounds (either when using the variational or the continuity method). This happens precisely because the conformal group of the sphere acts on the space of solutions, but this group does not respect the bounds on the solutions. In other words, the non-uniqueness of the solutions which appears in section 6.3.1 is responsible for the failure of the arguments of existence in section 6.3.2. We deal with the case 𝑘0 > 0 using a roundabout argument, reducing it to the case 𝑘0 ≤ 0. Let (𝑆, [g0 ]) be a conformal (compact, connected) oriented surface with genus 𝑔 = 0, i.e., 𝑆 is homeomorphic to 𝑆 2 . We are going to choose some points 𝑝1 , . . . , 𝑝𝑚 ∈ 𝑆 and put an orbifold structure around them with indices 𝑠1 , . . . , 𝑠𝑚 ≥ 2, respectively, so that the resulting orbifold has a non-positive orbifold Euler-Poincaré characteristic. Then we use the result of section 6.3.2 applied to orbifolds. Actually, if we adjust to have a zero orbifold Euler-Poincaré characteristic, we can appeal to the easiest case with 𝑘0 = 0. To do this, choose four different points 𝑞1 , 𝑞2 , 𝑞3 , 𝑞4 ∈ 𝑆 and put orbifold indices 𝑠1 = 𝑠2 = 𝑠3 = 𝑠4 = 2. The resulting orbisurface, obtained using Lemma 3.77, will be denoted 𝑋. The orbifold Euler-Poincaré characteristic is given by (3.42) as 4

1 1 𝜒orb (𝑋) = 𝜒(𝑆) − ∑ (1 − ) = 2 − 4 = 0. 2 2 𝑖=1 The conformal structure (𝑆, [g0 ]) produces an almost complex structure (𝑆, 𝐽), using the equivalence (1) ⇔ (3) in Theorem 6.13. The almost complex structure and the orbifold indices give an orbifold conformal structure (𝑋, [g1 ]), by Proposition 6.15. Here

370

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g1 is an orbifold metric on 𝑋, whose Gaussian orbifold curvature 𝜅g1 satisfies ∫𝑋 𝜅g1 = 2𝜋𝜒orb (𝑋) = 0, by Theorem 3.82, and such that [g1 ] = [g0 ] on 𝑆 − {𝑞1 , 𝑞2 , 𝑞3 , 𝑞4 }. Now solve equation (6.19) for g1 and for an orbifold function 𝜑 (see Remark 6.41). The solution gives a conformally equivalent orbifold metric g2 = 𝑒2𝜑 g1 such that 𝜅g2 ≡ 0. Take the orbifold cover 𝜛 ∶ 𝑆 ̂ → 𝑋 of degree 2 with ramification points over 𝑞1 , 𝑞2 , 𝑞3 , 𝑞4 . By diagram (3.43), we have that 𝜛−1 (𝑞𝑗 ) = {𝑝𝑗 }, and 𝑠𝑝𝑗 = 1, 𝑚𝑝𝑗 = 2, for 𝑗 = 1, 2, 3, 4 (with the notations of Proposition 3.79). This implies that 𝑆 ̂ is actually a smooth surface, and 𝜒(𝑆)̂ = 𝜒orb (𝑆)̂ = 2𝜒orb (𝑋) = 0, by Proposition 3.79. So 𝑆 ̂ is topologically a torus. The pull-back metric 𝜛∗ g2 is a smooth Riemannian metric (Exercise 3.28) of zero curvature, so 𝑆 ̂ is a flat torus, say 𝑆 ̂ ≅ 𝔼2 /Λ, for a lattice Λ ⊂ ℝ2 . The associated conformal structure is a complex torus, 𝑆 ̂ ≅ ℂ/Λ. Observe that we can use Corollary 6.53 for proving that the almost complex structure of (𝑆,̂ 𝜛∗ g2 ) is integrable, as the proof for genus 1 does not use the existence of metrics of constant curvature for 𝑘0 > 0. The non-trivial deck transformation 𝐴 of 𝜛 ∶ 𝑆 ̂ → 𝑋 is a biholomorphism of 𝑆.̂ By (5.30), biholomorphisms of the complex torus are isometries of the flat torus. By the discussion of isometries in Remark 4.47, the only isometry of order 2 is given as 𝐴(𝑧) = −𝑧. By Theorem 5.85, the complex torus 𝑆 ̂ is biholomorphic to a plane cubic model 𝐶 ⊂ ℂ𝑃 2 with affine equation 𝑦2 = 4𝑥3 − 𝑔2 𝑥 − 𝑔3 , via 𝜙 ∶ 𝑆 ̂ → 𝐶, 𝜙(𝑧) = (𝑥, 𝑦) = (℘(𝑧), ℘′ (𝑧)), where ℘ is the Weierstrass function of the lattice Λ. As ℘ is an even function and ℘′ is an odd function, the involution 𝐴 on 𝐶 is given as 𝐴(𝑥, 𝑦) = (𝑥, −𝑦). Take the map 𝜋 ∶ 𝐶 → ℂ𝑃 1 , 𝜋(𝑥, 𝑦) = 𝑥 (cf. Exercise 5.35). Clearly 𝜛 factors through 𝜋 ∘ 𝜙, that is, there exists a map 𝑓 ∶ ℂ𝑃 1 → 𝑋 so that the following diagram commutes. 𝑆̂

𝜙 ≅

𝜛

 𝑋o

/𝐶 𝜋

𝑓

 ℂ𝑃 1

The map 𝑓 is bijective, and it is clearly a biholomorphism outside the ramification values. By continuity 𝑓 is also conformal at the missing points, and hence it is a pseudobiholomorphism on the whole of 𝑋. Now the metric 𝑓∗ (4g𝐹𝑆 ), where g𝑆2 = 4g𝐹𝑆 is the round metric of ℂ𝑃 1 ≅ 𝑆 2 , is a metric of constant curvature 1 for 𝑋. It is compatible with the almost complex structure of 𝑋, so it is conformal to both g2 , g1 , and eventually to g0 , as required. Remark 6.50. • As we said before, the reason that the case 𝑘0 > 0 is difficult is that there are conformal transformations which are not isometries, so there is not a unique metric with 𝜅 ≡ 1. By choosing points 𝑝1 , . . . , 𝑝𝑚 , one gets rid of these conformal symmetries. This is due to the fact that it has to be 𝑚 ≥ 3 for having 𝜒orb (𝑋) ≤ 0, and a Möbius map that fixes three points is the identity. • Actually, we can avoid the use of Hodge theory for orbifolds, and use it only for smooth manifolds, as we can solve the equation 𝜅g2̂ ≡ 0 already on the

6.3. Metrics of constant curvature

371

torus 𝑆,̂ to obtain a metric g2̂ that we can push forward to a metric g2 on 𝑋 that solves (6.19) for 𝑋. • The proof above actually is a proof of the uniformization Theorem 5.56 for the genus 𝑔 = 0, as 𝑓 ∶ ℂ𝑃 1 → 𝑆 is the biholomorphism. • The argument can be carried out using orbifolds with 𝜒orb (𝑋) < 0, but without using explicit models for the complex curves. 6.3.4. Existence of conformal coordinates. We left undone the problem of the existence of conformal coordinates on a Riemannian surface (𝑆, g). We devote this section to proving this, and we collect some remarks around it. Theorem 6.51. There exist conformal coordinates for (𝑆, g) around any point. Proof. This is a local question. Take 𝑝 ∈ 𝑆 and coordinates 𝜓 ∶ 𝑈 ⊂ 𝑆 → 𝐵1/2 (0) around 𝑝. To prove the existence of conformal coordinates on 𝑈, we employ the useful technique of inserting the local problem into a compact manifold. Consider the standard torus 𝑇 2 = ℝ2 /ℤ2 with its flat metric g𝑇 2 , and the image of the ball 𝐵1/2 (0) ⊂ 𝑇 2 . Let 𝜌 be a bump function with 𝜌 ≡ 1 on 𝐵1/4 (0) and 𝜌 ≡ 0 on 𝑇 2 − 𝐵1/2 (0). Consider the metric g0 = 𝜌𝜓∗ (g) + (1 − 𝜌)g𝑇 2 . By Theorem 6.44 (actually by the easiest case of 𝑘0 = 0), there is some function 𝜑 ∈ 𝐶 ∞ (𝑇 2 ) such that g′ = 𝑒2𝜑 g0 is a metric of zero curvature. By uniqueness of the simply connected space form, there is an isometry 𝐹 ∶ (𝐵1/4 (0), g′ ) → 𝐵1/4 (0) ⊂ ℝ2 . Let 𝑈 ′ = 𝜓−1 (𝐵1/4 (0)) ⊂ 𝑆 and consider the chart 𝜓′ = 𝐹 ∘ 𝜓 ∶ 𝑈 ′ → 𝐵1/4 (0). Then g = 𝜓∗ (g0 ) = 𝑒−2𝜑 𝜓∗ (g′ ) = 𝑒−2𝜑 (𝜓′ )∗ g𝑠𝑡𝑑 on 𝑈 ′ . This means that (𝑈 ′ , 𝜓′ ) are conformal coordinates. □ Remark 6.52. The 𝑛-torus 𝑇 𝑛 = (𝑆 1 )𝑛 is especially suitable for solving differential equations such as Δg 𝑢 = 𝑓. The reason is that harmonic analysis on 𝑇 𝑛 can be done using Fourier series instead of Fourier transforms. A function 𝑓 ∈ 𝐿2 (𝑇 𝑛 ) can be written as 𝑓(𝑥) = ∑ 𝑎𝐦 𝑒2𝜋i(𝑚1 𝑥1 +⋯+𝑚𝑛 𝑥𝑛 ) where 𝐦 = (𝑚1 , . . . , 𝑚𝑛 ) ∈ ℤ𝑛 , 𝑎𝐦 = 𝜕 ∫𝑇 𝑛 𝑓(𝜃)𝑒−2𝜋i(𝑚1 𝜃1 +⋯+𝑚𝑛 𝜃𝑛 ) 𝑑𝜃1 ∧ ⋯ ∧ 𝑑𝜃𝑛 . As in Remark 6.33, the derivative 𝜕𝑥 gets 𝑖 transformed (via Fourier series) to multiplication by 2𝜋i𝑚𝑖 on the coefficients (𝑎𝐦 ). Therefore differential equations can be tackled through the Fourier series on 𝑇 𝑛 . Finally the trick in the proof of Theorem 6.51 of inserting a local chart into 𝑇 𝑛 allows us to transfer the problem from a general 𝑛-manifold to 𝑇 𝑛 (see [War]). Recall that we have used the powerful Newlander-Nirenberg Theorem 5.10 to prove that the Riemannian surfaces admit the structure of a complex curve (Remark 5.49). This was also used to give conformally flat structures to surfaces in Theorem 6.13. Now that we have conformal coordinates at our disposal (Theorem 6.51), which has been proved by using the theory of the Laplacian, we can reverse the path, and give a proof of Newlander-Nirenberg theorem in dimension 2. Corollary 6.53. The Newlander-Nirenberg theorem holds for compact surfaces. Proof. There is an equivalence between the existence of conformal coordinates and the integrability of the almost complex structure. This is contained in Theorem 6.13. As mentioned in Remark 6.14(2), we can prove the Newlander-Nirenberg theorem by

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checking that (3) ⇒ (4) in Theorem 6.13. To have a conformally flat structure on (𝑆, g) is equivalent to having conformal coordinates at a neighbourhood of every point. These are provided by Theorem 6.51. □ Remark 6.54. Let (𝑆, g) be a Riemannian surface. Fix a chart 𝑈 ⊂ 𝑆 with conformal coordinates (𝑥, 𝑦). To find another conformal coordinate on 𝑈 is the same as finding functions (𝑢, 𝑣) which satisfy that 𝑤 = 𝑢 + i𝑣 is a holomorphic function of 𝑧 = 𝑥 + i𝑦. By Remark 6.28(4), Δ𝑢 = 𝑢𝑥𝑥 + 𝑢𝑦𝑦 = 0. This is (minus) the Laplacian with respect to the flat metric 𝑑𝑥2 + 𝑑𝑦2 . By (6.10), Δg 𝑢 = 0, and analogously Δg 𝑣 = 0, because g is conformally flat. The charts (𝑢, 𝑣) where the coordinate functions are harmonic are called harmonic coordinates. For a surface, conformal coordinates are harmonic, in particular harmonic coordinates always exist (cf. Remark 3.57). Note that if we have a harmonic function 𝑢 on a surface with 𝑑𝑢(𝑝) ≠ 0, then we can add a second function to obtain conformal coordinates. Consider the 1-form 𝛼 = −𝑢𝑦 𝑑𝑥 + 𝑢𝑥 𝑑𝑦. As Δg 𝑢 = 0, we have 𝑑𝛼 = 0. So 𝛼 is closed, hence exact on a ball by the Poincaré lemma. So 𝛼 = 𝑑𝑣, for some function 𝑣. Then 𝑣𝑥 = −𝑢𝑦 , 𝑣𝑦 = 𝑢𝑥 , so Δg 𝑣 = 0. The function 𝑣 is the harmonic conjugate of 𝑢 (cf. Remark 6.28(4)), and 𝑤 = 𝑢 + i𝑣 is a holomorphic function. Finally note that (

𝑢𝑥 𝑣𝑥

𝑢𝑦 ) 𝑣𝑦

has determinant 𝑢𝑥2 + 𝑢𝑦2 , which is positive at the origin. So (𝑢, 𝑣) defines a chart in a small ball around 𝑝. The conformal coordinates are also sometimes called isothermal coordinates. This is because the heat equation on (𝑆, g) is defined as (6.23)

𝜕𝑢 = −Δg 𝑢(𝑥, 𝑡), 𝜕𝑡

for a function 𝑢(𝑥, 𝑡) on 𝑆 × [0, ∞). The static (or stationary) solutions are those which 𝜕ᵆ do not change with time, that is when 𝜕𝑡 ≡ 0. So they are given by the solutions of Δg 𝑢 = 0. The level sets of 𝑢 are the sets with same temperature, called isothermal level sets. For a higher dimensional Riemannian manifold there are always harmonic coordinates, (𝑢1 , . . . , 𝑢𝑛 ), i.e., such that Δg 𝑢𝑘 = 0, 1 ≤ 𝑘 ≤ 𝑛. However, for the existence of conformal coordinates there is an obstruction (see Remark 6.10). As a conformal structure on an oriented surface is the same as a complex structure, Theorem 6.44 can be rephrased as follows: • If (𝑆, 𝐽) is a compact connected complex curve, then there exists a metric compatible with 𝐽 of constant curvature. • If (𝑆, 𝐽) is a compact connected complex curve, then there exists a Hermitian metric of constant curvature. This is the surjectivity of (5.30). On page 321, this surjectivity was proved by using the uniformization Theorem 5.56. Now it is our intention to complete the theory by

6.3. Metrics of constant curvature

373

giving a proof of the uniformization theorem, at least in the case relevant to our purposes, that is for a simply connected complex curve which is the universal cover of a compact connected complex curve. This is the only case that we used in section 5.4. Corollary 6.55 (Uniformization (most relevant case)). Let 𝑆 be a complex compact curve. Let 𝑆 ̃ be its universal cover, with the induced complex structure. Then 𝑆 ̃ is one of the following: • If 𝑔 = 0, then 𝑆 ̃ ≅ ℂ𝑃 1 . • If 𝑔 = 1, then 𝑆 ̃ ≅ ℂ. • If 𝑔 ≥ 2, then 𝑆 ̃ ≅ 𝔻. Proof. Take a Riemannian structure (𝑆, g0 ) in the conformal class determined by (𝑆, 𝐽). Then there exists a conformally equivalent metric g = 𝑒2𝜑 g0 with constant curvature 𝜅g = 𝑘0 ∈ {0, 1, −1} by Theorem 6.44. The universal cover with the induced metric (𝑆,̃ g), ̃ is a simply connected space form, and hence: • If 𝑔 = 0, then (𝑆,̃ g)̃ ≅ 𝕊2 = (𝑆 2 , g𝑆2 ), where ≅ means “isometric to”. Since g𝑆2 is conformal to the Fubini-Study metric on 𝑆2 = ℂ𝑃 1 (see page 296, item (8)), the complex structure on 𝑆 2 induced by [g𝑆2 ] is the one of ℂ𝑃 1 . This means that 𝑆 ̃ is biholomorphic to ℂ𝑃 1 . • If 𝑔 = 1, then (𝑆,̃ g)̃ ≅ 𝔼2 = (ℝ2 , g𝑠𝑡𝑑 ). This means that 𝑆 ̃ is biholomorphic to ℂ. • If 𝑔 ≥ 2, then (𝑆,̃ g)̃ ≅ ℍ2 = (𝐵 2 , g𝑃 ). This means that 𝑆 ̃ is biholomorphic to 𝔻 ⊂ ℂ. Note that 𝑔𝑃 given in (4.27) is conformal to the standard metric, hence the complex structure of 𝐵 2 = 𝔻 is the standard complex structure as open set of ℂ. □ We end up with the classification of smooth structures on compact surfaces. 𝑐𝑜

Theorem 6.56. We have 𝕃𝐃𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 |𝑔 ≥ 0, 𝑘 ≥ 1}. 𝑐

𝑐𝑜

Proof. We already know that 𝕃𝐓𝐌𝐚𝐧2 = {Σ𝑔 , 𝑋𝑘 |𝑔 ≥ 0, 𝑘 ≥ 1} by Theorem 2.29. The 𝑐𝑜 𝕃𝐃𝐌𝐚𝐧2 𝑐

𝑐𝑜 𝕃𝐓𝐌𝐚𝐧2 𝑐

𝑐

map → is surjective because all topological compact surfaces admit smooth structures. Let us see that it is injective. Let 𝑆 be a smooth compact connected oriented surface of some genus 𝑔 ≥ 0. Give 𝑆 a Riemannian metric g0 (Proposition 3.22). By Theorem 6.51, 𝑆 has another metric g (in the same conformal class) of constant curvature. If 𝑔 = 0, then (𝑆, g) is isometric (and in particular diffeomorphic) to (𝑆 2 , g𝑆2 ). If 𝑔 = 1, then (𝑆, g) is a flat torus, hence 𝑆 = ℝ2 /Λ, where Λ is a lattice. By item (3) on page 222, 𝑆 is diffeomorphic to the torus 𝑇 2 = ℝ2 /ℤ2 . If 𝑔 ≥ 2, then 𝑆 = ℍ2 /Γ, for some Γ acting freely and properly by isometries. The fundamental domain 𝐷 ⊂ ℍ2 is a hyperbolic 4𝑔-gon (Proposition 4.85). Suppose that we have two smooth surfaces 𝑆 1 , 𝑆 2 of the same genus 𝑔 ≥ 2 with associated hyperbolic 4𝑔-gons 𝐷1 , 𝐷2 , and let 𝑝1 ∈ 𝑆 1 , 𝑝2 ∈ 𝑆 2 be the basepoints given by the vertices of the polygons. Take local coordinates 𝜑1 ∶ 𝑈 𝑝1 ⊂ 𝑆 1 → 𝐵1 (0) and 𝜑2 ∶ 𝑈 𝑝2 ⊂ 𝑆 2 → 𝐵1 (0). By means of a diffeomorphism of the holed fundamental domains,

374

6. Global analysis

−1 ̄ ̄ it is easy to construct a diffeomorphism 𝑓 ∶ 𝑆 1 − 𝜑−1 1 (𝐵1/2 (0)) → 𝑆 2 − 𝜑 2 (𝐵1/2 (0)) such i𝜃 iℎ(𝜃) ̄ (0), for some increasing function ℎ ∶ [0, 2𝜋] → that 𝑓(𝑟𝑒 ) = 𝑟𝑒 on 𝐵1 (0) − 𝐵1/2 [0, 2𝜋]. In order to extend 𝑓 to the whole of 𝑆 1 , let 𝛿 ∶ [0, 1] → [0, 1] be a cut-off function such that 𝛿(𝑟) = 0 for 𝑟 ∈ [0, 3/5], and 𝛿(𝑟) = 1 for 𝑟 ∈ [4/5, 1]. Now let 𝐻(𝑟, 𝜃) = 𝛿(𝑟)ℎ(𝜃) + (1 − 𝛿(𝑟))𝜃, and define 𝑔 ∶ 𝑆 1 → 𝑆 2 by 𝑔 = 𝑓 off 𝑈 𝑝1 , and −1 i𝜃 i𝐻(𝑟,𝜃) 𝑔(𝜑−1 ) on 𝐵1 (0), to get the desired diffeomorphism. 1 (𝑟𝑒 )) = 𝜑 2 (𝑟𝑒

The non-orientable case is completely analogous.

□

Remark 6.57. • The argument of Theorem 6.56 can be paraphrased by saying that the Teichmüller space is connected. • In the case of 𝑔 = 1, the map between the parallelograms descends automatically to a smooth map between the tori. This is because the corners match to yield a linear map. • We can smooth a map at a point for surfaces, but not in higher dimensions (Exercise 6.24). • There are more direct ways to prove Theorem 6.56, but the proof given is tantamount in geometry: we give extra structure to a smooth manifold, which allows us to infer topological information, which is independent of the added structure.

6.4. The curvature flow A modern method of finding geometric structures is via evolution equations. The idea is simple: start with an initial geometric structure, and let it flow following some differential equation that makes it more homogeneous with time. For instance, if we start with a Riemannian metric g0 in a smooth connected manifold 𝑀, we want to construct a family g𝑡 such that, in the limit lim𝑡→∞ g𝑡 , the metric tends to some Riemannian metric with homogeneous or isotropic structure. For this, we write an evolution equation on the time of the form 𝜕g𝑡 (6.24) = 𝑃(g𝑡 ), 𝜕𝑡 where 𝑃 is a suitable (differential) operator. Observe that, along this section, in order to get in touch with the usual notation of parabolic equations, given a function 𝑢 = 𝑢(𝑥, 𝑡), with 𝑥 ∈ 𝑀, 𝑡 ∈ ℝ, we will denote 𝑢𝑡 = 𝑢(−, 𝑡). In this way, the subscript no longer means differentiation. Remark 6.58. (1) Equations like (6.24) are difficult to deal with. They define a flow (i.e., a vector field) in the space of metrics ℳet(𝑀), which is an infinite dimensional manifold (flows appear in Exercise 3.2 for finite dimensional manifolds). Infinite dimensional manifolds are modelled on open subsets of topological vector spaces of infinite dimension. These topological vector spaces have non-trivial topological properties which are studied in functional analysis.

6.4. The curvature flow

375

(2) The typical way to deal with flows in infinite dimensional functional spaces associated to a finite dimensional manifold 𝑀 (like ℳet(𝑀) above) is to pose the question in the infinite dimensional space (as if it were a finite dimensional manifold), and to restate the problem for the manifold 𝑀. This is then tackled usually with techniques of mathematical analysis. (3) Some flow equations are associated to energy functionals. These are functionals ℰ ∶ ℳ → ℝ, where ℳ is the infinite dimensional space of geometric structures, and 𝑃 = grad ℰ. Formally, we are doing Morse theory [Mi3] on ℳ. As in the classical case, the flow lines of 𝑃 are the gradient lines of ℰ, and the limiting values of the flow should be the critical points of ℰ. (4) The operators 𝑃 appearing in (6.24) have to be independent of the coordinates charts, in order to be defined globally on any manifold. There are not so many choices. For metrics, a second order differential operator of this type must be constructed out of the Riemannian curvature tensor 𝑅. As 𝑃(g𝑡 ) has to be a symmetric (0, 2)-tensor, it has to be a combination of the Ricci tensor and the scalar curvature times the metric (cf. Remark 3.46). (5) The appearance of the Ricci tensor in differential equations involving the metric is very natural. We recall that in general relativity, the field equations of Einstein determine the geometry of spacetime, which is a 4-dimensional manifold endowed with a semi-Riemannian metric (Remark 4.55). These are differential equations involving the metric of spacetime, and this justifies the natural shape that they have, 1 Ric − Scal ⋅g + Λg = 𝑇, 2 where 𝑇 is a symmetric (0, 2)-tensor involving the information of matter and energy in spacetime (the so-called stress-energy tensor), and Λ is a constant (called the cosmological constant). (6) An Einstein manifold is a Riemannian (or semi-Riemannian) manifold such that Ric = 𝜆 𝑔, for some constant 𝜆 ∈ ℝ. In dimension 𝑛 ≥ 3, it can be proven to be equivalent to the vanishing of the trace-free Ricci curvature Ric0 = Ric − Scal ⋅g. These are the possible vacuum universes [Ein], since they are the solutions of the Einstein equation for 𝑇 = 0 (the fact that the scalar curvature is constant is a consequence). For simplicity, we are restricting our attention to oriented connected manifolds. However, for a non-connected manifold a similar analysis can be performed working by components. In a similar vein, the non-orientable case can be tackled by considering the oriented cover. Definition 6.59. Let 𝑆 be a compact oriented connected surface, and let g0 be a Riemannian metric. The curvature flow is the flow of Riemannian metrics g𝑡 starting at g0 and satisfying the evolution equation (6.25) where 𝜅𝑡 is the curvature of g𝑡 .

𝜕g𝑡 = −2𝜅𝑡 g𝑡 , 𝜕𝑡

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Remark 6.60. A geometric justification of the flow (6.25) may be as follows. Fix 𝑝 ∈ 𝑀 and 𝑢 ∈ 𝑇𝑝 𝑆. Under the flow equation, the norm of 𝑢, ||𝑢||𝑡 = √g𝑡 (𝑢, 𝑢) satisfies 𝑑 ||𝑢||𝑡 = −𝜅𝑡 (𝑝)||𝑢||𝑡 . So if 𝜅𝑡 (𝑝) > 0 (resp., 𝜅𝑡 (𝑝) < 0), then ||𝑢||𝑡 decreases (resp., 𝑑𝑡 increases), which means that the space around 𝑝 contracts (resp., expands). This makes the space evolving with (6.25) get rounder and rounder (see Figure 6.1).

Figure 6.1. Evolution under the curvature flow.

The first thing to note is that the flow (6.25) moves the metric in a conformal class. For that, observe that when written in a local chart g = (𝑔𝑖𝑗 ), we have that (6.25) is equivalent to

𝜕 𝜕𝑡

log g𝑡 = −2𝜅𝑡 . Hence 𝑡

g𝑡 = 𝑒−2 ∫0 𝜅𝑠 𝑑𝑠 g0 , proving that g𝑡 ∈ [g0 ]. Writing g𝑡 = 𝑒2ᵆ𝑡 g0 , the function 𝑢𝑡 satisfies the PDE 𝜕𝑢𝑡 = −𝜅𝑡 . 𝜕𝑡 Here we see that the reason for the factor of 2 in the equation (6.25) is just not to have a 1/2 in (6.26). Using Lemma 6.42, 𝜅𝑡 = 𝑒−2ᵆ𝑡 Δg0 𝑢𝑡 + 𝑒−2ᵆ𝑡 𝜅0 . This means that the evolution equation for g𝑡 becomes (6.26)

𝜕𝑢𝑡 = −𝑒−2ᵆ𝑡 Δg0 𝑢𝑡 − 𝑒−2ᵆ𝑡 𝜅0 . 𝜕𝑡 Here 𝜅0 is the curvature of the initial metric g0 , and 𝑢0 ≡ 0. This is a parabolic equation, although it is non-linear. (6.27)

Definition 6.61. A parabolic equation is an equation for functions 𝑢(𝑥, 𝑡), (𝑥, 𝑡) ∈ 𝑀 × 𝐼, where 𝑀 a smooth manifold and 𝐼 ⊂ ℝ is an interval, of the type 𝜕𝑢 = −𝑃𝑡 (𝑢), 𝜕𝑡 where 𝑃𝑡 is a semipositive elliptic operator on 𝑀 of order 2 varying smoothly for 𝑡 ∈ 𝐼. Remark 6.62. The basic example is the heat equation (6.23) on ℝ𝑛 × ℝ given as 𝑛

𝜕𝑢 𝜕2 𝑢 = Δ𝑢 = ∑ 2 , 𝜕𝑡 𝑖=1 𝜕𝑥𝑖

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377

where Δ is the standard Laplacian (Remark 6.28(2)). This describes the evolution of the temperature in ambient space with the basic law that heat transfers from hotter to colder places. For smooth initial condition 𝑢0 (𝑥) it has a (unique) solution 𝑢(𝑥, 𝑡) with 𝑢(𝑥, 0) = 𝑢0 (𝑥). For a solution 𝑢𝑡 (𝑥) = 𝑢(𝑥, 𝑡), the total amount of heat ∫ℝ𝑛 𝑢𝑡 (𝑥) 𝑑𝑥 is 𝜕 constant since 𝜕𝑡 ∫ℝ𝑛 𝑢𝑡 (𝑥) 𝑑𝑥 = ∫ℝ𝑛 (−Δ𝑢)𝑑𝑥 = 0. The fundamental solution of the heat equation is the solution to the equation in ℝ𝑛 × (0, ∞) given by Φ(𝑥, 𝑡) = Φ𝑡 (𝑥) =

1 2 𝑒−||𝑥|| /4𝑡 . (4𝜋𝑡)𝑛/2

It satisfies that Φ0 = 𝛿0 , the Dirac delta at the origin (see Figure 6.2). So this models how a unit of heat at the origin at time 𝑡 = 0 spreads out in space with time 𝑡 > 0. A basic result establishes that the solution for the heat equation on 𝑡 ≥ 0, with 𝑢0 (𝑥) = 𝑓(𝑥), 𝑓 integrable, is given by the convolution 𝑢𝑡 = 𝑓 ∗ Φ𝑡 . The convolution of two functions 𝑔, ℎ ∶ ℝ𝑛 → ℝ is given by (𝑔 ∗ ℎ)(𝑥) = ∫ℝ𝑛 𝑔(𝑥 − 𝑦)ℎ(𝑦)𝑑𝑦, when this can be defined. As a conclusion, 𝑢𝑡 is smooth for all 𝑡 > 0 even if 𝑢0 is highly non-smooth (this is called the regularizing property of parabolic equations). Also 𝑢𝑡 exists for all time 𝑡 ∈ (0, ∞). However the heat equation is non-reversible. We cannot expect to prolong solutions for 𝑡 < 0 to the whole interval (−∞, 0). For instance, take as initial datum 𝑢0 = Φ𝑡0 , 𝑡0 > 0, so 𝑢(𝑥, 𝑡) = Φ(𝑥, 𝑡 + 𝑡0 ) is the solution, and it is not defined for 𝑡 ≤ −𝑡0 , even allowing for the use of distributions (e.g., non-smooth functions). This means that the backward heat equation (obtained by the change 𝑡 ↦ −𝑡), 𝜕𝑢 = −Δ𝑢 , 𝜕𝑡 is ill-posed: it is not regularizing and it does not have solutions for all 𝑡 > 0.

Φ1

2

Φ1

Φ2

Φ3 Φ10

Figure 6.2. Fundamental solution of the heat equation for some values of 𝑡.

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Equation (6.27) is a parabolic equation. The minus sign in (6.25) is the good one to have regularizing properties for 𝑡 ≥ 0, since the geometric Laplacian is minus the analytical Laplacian. It makes the metric behave as in the heat equation, by spreading out the curvature. Equation (6.27) is non-linear. It retains some nice properties of parabolic linear equations (like uniqueness, short time existence of solutions, and regularizing properties), but other properties (like the long time existence of solutions), have to be analysed specifically for the equation at hand. Remark 6.63. The static solutions of (6.27) are those 𝑢(𝑥, 𝑡) independent of time. This means that 𝑒−2ᵆ Δg0 𝑢 + 𝑒−2ᵆ 𝜅0 = 0, so 𝜅𝑡 = 0. Hence the only static solutions are flat metrics for tori. On the other hand, constant curvature metrics of non-zero value do evolve. Let g0 be a metric such that 𝜅0 ≡ 𝑘0 . It is easy to see that in this case, 𝑢𝑡 ∶ 𝑆 → ℝ is constant for fixed 𝑡, and the metric g𝑡 has constant curvature 𝜅𝑡 ≡ 𝑘(𝑡), where 𝑘(𝑡) = 𝑒−2ᵆ(𝑡) 𝑘0 . The differential equation becomes the ordinary differential equation 𝑢′ (𝑡) = −𝑒−2ᵆ(𝑡) 𝑘0 , 1

with 𝑢(0) = 0. The solution is 𝑢(𝑡) = 2 log(1 − 2𝑘0 𝑡). Now it is straightforward to check that g𝑡 = (1 − 2𝑘0 𝑡)g0 is actually a solution, as claimed. So the curvature of g𝑡 evolves as 𝑘0 𝑘(𝑡) = . 1 − 2𝑘0 𝑡 Summing up, a metric of non-zero constant curvature changes but keeps the property of constant curvature. In the case of a surface of genus 𝑔 ≥ 2 with curvature 𝑘0 < 0, we have that 𝑘(𝑡) increases to zero when 𝑡 → ∞ (the metric expands indefinitely). In the case of the sphere 𝑘0 > 0, the curvature 𝑘(𝑡) increases and blows up to infinity when 1 𝑡 → 2𝑘 (the metric shrinks to zero in finite time, i.e., the sphere shrinks to a point). 0

Remark 6.64. It is interesting to point out that we have a parabolic non-linear equation whose solutions blow up at finite time. This is a new property that did not happen for the linear parabolic heat equation (6.23). The problem of the non-stability of the constant curvature metrics is due to the fact that the volume changes with the curvature flow. Let 𝜈𝑡 = √det(g𝑡 ) 𝑑𝑥 ∧ 𝑑𝑦 = 𝑒2ᵆ𝑡 𝜈0 be the volume form of the metric g𝑡 . Then we have 𝑑𝜈𝑡 𝜕𝑢 = 2 𝑡 𝜈𝑡 = −2𝜅𝑡 𝜈𝑡 . 𝑑𝑡 𝜕𝑡 The total area 𝐴𝑡 = area(𝑆, g𝑡 ) = ∫𝑆 𝜈𝑡 depends on 𝑡, and it satisfies the equation 𝑑𝐴𝑡 𝑑𝜈 = ∫ 𝑡 = −2 ∫ 𝜅𝑡 𝜈𝑡 = −4𝜋𝜒(𝑆), 𝑑𝑡 𝑑𝑡 𝑆 𝑆

(6.28)

using the Gauss-Bonnet theorem. So if 𝜒(𝑆) < 0, then the volume increases, and when 𝜒(𝑆) > 0, the volume decreases. We want to normalize the flow so that the volume remains constant, while retaining the other properties. Definition 6.65. The normalized curvature flow is given by the equation 𝜕g𝑡 = −2(𝜅𝑡 − 𝑟𝑡 )g𝑡 , 𝜕𝑡 where 𝑟𝑡 =

1 𝐴𝑡

∫𝑆 𝜅𝑡 𝜈𝑡 =

2𝜋𝜒(𝑆) 𝐴𝑡

is the average of the curvature.

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379

The evolution of the geometric quantities under the normalized curvature flow are given as follows. Write g𝑡 = 𝑒2ᵆ𝑡 g0 . Then

(6.29)

𝜕𝑢𝑡 = −(𝜅𝑡 − 𝑟𝑡 ), 𝜕𝑡 𝜕𝜈𝑡 𝜕𝑢 = 2 𝑡 𝜈𝑡 = −2(𝜅𝑡 − 𝑟𝑡 )𝜈𝑡 , 𝜕𝑡 𝜕𝑡 𝑑𝐴𝑡 = −2 ∫(𝜅𝑡 − 𝑟𝑡 )𝜈𝑡 = 0. 𝑑𝑡 𝑆

In particular, 𝐴𝑡 = 𝐴0 is constant, and hence 𝑟𝑡 = 𝑟0 is constant. In this way, the normalized curvature flow satisfies 𝜕g𝑡 (6.30) = −2(𝜅𝑡 − 𝑟0 )g𝑡 , 𝜕𝑡 where 𝑟0 = 2𝜋𝜒(𝑆)/ area(𝑆, g0 ) is constant. Remark 6.66. The evolution under the normalized curvature flow makes the following. When the curvature exceeds the average, 𝜅𝑡 > 𝑟0 , the metric decreases (the space contracts), and when the curvature is less than the average, 𝜅𝑡 < 𝑟0 , the metric increases (the space expands); cf. Remark 6.60. Lemma 6.67. There is a bijection between solutions to the curvature flow and solutions to the normalized curvature flow. If g𝑡 follows the curvature flow and g𝑠̃ follows the normalized curvature flow, both with starting metric g0 , then g𝑠̃ = 𝜆(𝜓(𝑠))g𝜓(𝑠) , for some change of parameters 𝑡 = 𝜓(𝑠), and some positive function 𝜆(𝑡) > 0. This means that g𝑠̃ and g𝑡 are homothetic. Proof. Let g𝑡 be a solution to the curvature flow, and let us construct a solution to the normalized curvature flow. We take 𝜆(𝑡) = 𝐴−1 𝑡 (so that the area of (𝑆, 𝜆(𝑡)g𝑡 ) is 𝑡 constant, as required by the normalized curvature flow). Let 𝜙(𝑡) = ∫0 𝜆(𝜉)𝑑𝜉. As 𝑠 = 𝜙(𝑡) is strictly increasing, it has an inverse 𝑡 = 𝜓(𝑠). Setting g𝑠̃ = 𝜆(𝑡)g𝑡 , we have 1 1 𝑑𝜆 𝑑𝐴𝑡 𝜅𝑠̃ = 𝜆 𝜅𝑡 , 𝑟𝑠̃ = 𝜆 𝑟𝑡 , and 𝑑𝑡 = −𝐴−2 = 2𝑟𝑡 𝜆, using (6.28). Then we compute 𝑡 𝑑𝑡 𝜕g𝑠̃ 𝜕g 𝑑𝜓 𝑑𝜆 𝑑𝜆 = ( g𝑡 + 𝜆 𝑡 ) = ( g𝑡 − 2𝜆𝜅𝑡 g𝑡 ) 𝜆−1 𝜕𝑠 𝑑𝑡 𝜕𝑡 𝑑𝑠 𝑑𝑡 𝑑𝜆 = (𝜆−2 − 2𝜅𝑠̃ ) g𝑠̃ = 2 (𝑟𝑠̃ − 𝜅𝑠̃ ) g𝑠̃ , 𝑑𝑡 proving that g𝑠̃ satisfies the normalized curvature flow equations. The same argument in the opposite way proves the converse. □ From now on, we focus on the normalized curvature flow which, using Lemma 6.42 and writing g𝑡 = 𝑒2ᵆ𝑡 g0 , is governed by the equation 𝜕𝑢𝑡 = −(𝜅𝑡 − 𝑟0 ) = −𝑒−2ᵆ𝑡 Δg0 𝑢𝑡 − 𝑒−2ᵆ𝑡 𝜅0 + 𝑟0 , 𝜕𝑡 where 𝑟0 is the average of 𝜅0 . The static points of (6.31) are all metrics of constant curvature 𝑟0 . In particular, we expect that the flow g𝑡 → g∞ , where g∞ is a metric of constant curvature 𝑟0 in the conformal class [g0 ]. (6.31)

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Again using Lemma 6.42, the curvature of g𝑡 is given by 𝜅𝑡 = 𝑒−2ᵆ𝑡 Δg0 𝑢𝑡 + 𝑒−2ᵆ𝑡 𝜅0 . Differentiating on 𝑡, we get the evolution of the curvature 𝜕𝜅𝑡 𝜕𝑢 𝜕𝑢 = −2 𝑡 𝜅𝑡 + 𝑒−2ᵆ𝑡 Δg0 ( 𝑡 ) 𝜕𝑡 𝜕𝑡 𝜕𝑡 −2ᵆ𝑡 = −2(𝑟0 − 𝜅𝑡 )𝜅𝑡 − 𝑒 Δg0 𝜅𝑡 .

(6.32)

Lemma 6.68. Let (𝑆, g0 ) be a compact connected oriented Riemannian surface with negative curvature 𝜅0 (𝑝) < 0 for all 𝑝 ∈ 𝑆. If 𝜅𝑡 is a smooth solution of (6.32) for 𝑡 ∈ 𝐼 = [0, 𝑇) (maybe 𝑇 = ∞), then 𝑡 ↦ max𝑝∈𝑆 𝜅𝑡 (𝑝) is a non-increasing function on 𝐼. Analogously, 𝑡 ↦ min𝑝∈𝑆 𝜅𝑡 (𝑝) is non-decreasing. Proof. Take 𝑡0 ∈ 𝐼 small and positive. If max 𝜅𝑡0 ≤ 𝑟0 , since 𝑟0 is the average of 𝜅𝑡0 , it must be 𝜅𝑡0 ≡ 𝑟0 , and the solution is constant. Hence, we can suppose that max 𝜅𝑡 > 𝑟0 for all 𝑡 ∈ 𝐼. Let 𝑝0 ∈ 𝑆 be a local maximum of 𝜅𝑡0 with 𝑟0 < 𝜅𝑡0 (𝑝0 ) < 0. By Remark 6.40(4), we have Δg0 𝜅𝑡0 (𝑝0 ) ≥ 0, so 𝜕𝜅𝑡 | | (𝑝 ) = −2(𝑟0 − 𝜅𝑡0 (𝑝0 ))𝜅𝑡0 (𝑝0 ) − 𝑒−2ᵆ𝑡0 (𝑝0 ) Δg0 𝜅𝑡0 (𝑝0 ) < 0. 𝜕𝑡 |𝑡=𝑡0 0 Hence, 𝑡 ↦ 𝜅𝑡 (𝑝0 ) is a decreasing function on a small interval around 𝑡0 . Now, let 𝐶 = max 𝜅0 < 0 and 𝐴 = {𝑡 ∈ 𝐼 | max 𝜅𝑡 ≤ 𝐶}. If 𝐴 ≠ 𝐼, let 𝑡0 be the infimum of 𝐼 − 𝐴. Since the maximum of a continuous function is continuous, it must be max 𝜅𝑡0 = 𝐶. Let 𝑝0 ∈ 𝑆 such that 𝜅𝑡0 (𝑝0 ) = 𝐶. Then 𝑝0 is a local maximum of 𝜅𝑡0 so, as proved above, 𝑠 ↦ 𝜅𝑠 (𝑝0 ) is decreasing around 𝑡0 . This implies that, for some 𝑡1 < 𝑡0 , 𝜅𝑡1 (𝑝0 ) > 𝜅𝑡0 (𝑝0 ) = 𝐶, contradicting that 𝑡0 is the infimum of 𝐼 − 𝐴. In particular, this proves that 𝜅𝑡 < 0 for all 𝑡 ∈ 𝐼. Now, for any 𝑡′ ∈ 𝐼, consider the set 𝐴𝑡′ = {𝑡 ∈ [𝑡′ , 𝑇) | max 𝜅𝑡 ≤ max 𝜅𝑡′ }. Repeating the argument above, we see that 𝐴𝑡′ = [𝑡′ , 𝑇). This shows that max 𝜅𝑡 is a non-increasing function. The argument for min 𝜅𝑡 is analogous. □ Remark 6.69. • Lemma 6.68 says, in a more accurate way, that g𝑡 tends to get rounder with time, as 𝜅𝑡 reduces the range of variation with time. • Let us forget about the geometric interpretation for a while, and look for solutions 𝑚(𝑡) of (6.32) constant on 𝑆. As Δg0 𝑚 = 0, they have to satisfy the ordinary differential equation 𝑑𝑚 = −2(𝑟0 − 𝑚)𝑚. 𝑑𝑡 This is a Bernoulli differential equation of degree 2, very similar to the logistic equation. A straightforward computation shows that its solutions have the form 𝑟0 𝑚(𝑡) = , 1 + 𝛼𝑒2𝑟0 𝑡 for some 𝛼 ∈ ℝ. For 𝛼 > 0 and 𝑟0 < 0, these solutions exist for all 𝑡 > 0 and decay exponentially to 𝑟0 as 𝑡 → ∞. Observe that 𝑚(𝑡) cannot be the curvature of any solution of the curvature flow, since 𝑚(𝑡) > 𝑟0 . These 𝑚(𝑡)

6.4. The curvature flow

381

can be understood as a kind of barrier solution that control the behaviour of the curvature of genuine solutions of the curvature flow. • A more suitable barrier solution appears when we solve the linear ordinary differential equation 𝑑𝑀 (6.33) = −2(𝑟0 − 𝑀)𝐶, 𝑑𝑡 for some constants 𝑟0 , 𝐶 < 0. In that case, the solution is 𝑀(𝑡) = 𝑟0 + 𝛼𝑒2𝐶𝑡 , for some 𝛼 ∈ ℝ. This solution decays exponentially to 𝑟0 for 𝑡 → ∞. As we will show below, they can be understood as a kind of supersolutions of (6.32). With the coarse estimates of Lemma 6.68 at hand, we can obtain sharper bounds on the decay of the curvature. Proposition 6.70. Let (𝑆, g0 ) be a compact connected oriented Riemannian surface with everywhere negative curvature 𝜅0 , and suppose that −𝐶2 ≤ 𝜅0 ≤ −𝐶1 . If 𝜅𝑡 , for 𝑡 ∈ [0, 𝑇) (maybe 𝑇 = ∞), is the curvature of a smooth solution of the normalized curvature flow (6.30), then there exist constants 𝛼1 , 𝛼2 > 0 such that, for all 𝑡 ∈ [0, 𝑇), 𝑟0 − 𝛼2 𝑒−2𝐶2 𝑡 ≤ 𝜅𝑡 ≤ 𝑟0 + 𝛼1 𝑒−2𝐶1 𝑡 . Proof. We will prove the upper bound, the lower bound being similar. Consider the function 𝑀𝑡 = 𝑟0 + 𝛼1 𝑒−2𝐶1 𝑡 with 𝛼1 = −(𝑟0 + 𝐶1 ) ≥ 0. Since 𝑀𝑡 satisfies equation (6.33) with 𝐶 = −𝐶1 , we have that 𝜕(𝜅𝑡 − 𝑀𝑡 ) = −𝑒−2ᵆ𝑡 Δg0 (𝜅𝑡 − 𝑀𝑡 ) − 2(𝑟0 − 𝜅𝑡 )𝜅𝑡 − 2(𝑟0 − 𝑀𝑡 )𝐶1 𝜕𝑡 ≤ −𝑒−2ᵆ𝑡 Δg0 (𝜅𝑡 − 𝑀𝑡 ) + 2(𝜅𝑡 − 𝑀𝑡 )𝜅𝑡 , using that 𝜅𝑡 ≤ −𝐶1 and 𝑟0 ≤ 𝑀𝑡 . We denote (𝜅𝑡 −𝑀𝑡 )+ = max{𝜅𝑡 − 𝑀𝑡 , 0}. Multiplying the previous inequality by (𝜅𝑡 − 𝑀𝑡 )+ and integrating on 𝑆 with respect to the volume form 𝜈𝑡 , we obtain 𝜕(𝜅𝑡 − 𝑀𝑡 ) 1 𝑑 (𝜅𝑡 − 𝑀𝑡 )+ 𝜈𝑡 (||(𝜅𝑡 − 𝑀𝑡 )+ ||2𝑡 ) = ∫ 2 𝑑𝑡 𝜕𝑡 𝑆 ≤ −∫ 𝑒−2ᵆ𝑡 (Δg0 (𝜅𝑡 − 𝑀𝑡 )) (𝜅𝑡 − 𝑀𝑡 )+ 𝜈𝑡 + 2∫ 𝜅𝑡 (𝜅𝑡 − 𝑀𝑡 )(𝜅𝑡 − 𝑀𝑡 )+ 𝜈𝑡 𝑆

𝑆

= − ∫ (Δg𝑡 (𝜅𝑡 − 𝑀𝑡 )+ ) (𝜅𝑡 − 𝑀𝑡 )+ 𝜈𝑡 + 2 ∫ 𝜅𝑡 (𝜅𝑡 − 𝑀𝑡 )2+ 𝜈𝑡 𝑆

𝑆

= −||𝑑(𝜅𝑡 − 𝑀𝑡 )||2𝑡 + 2 ∫ 𝜅𝑡 (𝜅𝑡 − 𝑀𝑡 )2+ 𝜈𝑡 ≤ 0, 𝑆 2

where || − ||𝑡 denotes the 𝐿 norm induced by g𝑡 , and the derivatives are understood in the weak sense (they are well defined off a set of zero measure). In the third line we use (6.10), and in the fourth line, integration by parts as ∫𝑆 (Δg 𝑓)𝑓 = ⟨𝑑 ∗ 𝑑𝑓, 𝑓⟩𝐿2 = ||𝑑𝑓||2𝐿2 . Hence, ||(𝜅𝑡 −𝑀𝑡 )+ ||2𝑡 is a decreasing non-negative function and, since ||(𝜅0 −𝑀0 )+ ||2g0 = 0, it must be (𝜅𝑡 −𝑀𝑡 )+ = 0 for all 𝑡 ≥ 0. Therefore, 𝜅𝑡 ≤ 𝑀𝑡 , which proves the estimate. □

382

6. Global analysis

Corollary 6.71. In the conditions above, if the solution of the normalized curvature flow g𝑡 converges to a Riemannian metric g∞ for 𝑡 → ∞ (in 𝐶 2 norm), then g∞ has constant curvature 𝑟0 . Remark 6.72. • These kinds of arguments are very common in partial differential equations and are called comparison principles. The first step in the proofs is to show that if 𝑣 𝑡 is a subsolution of the differential equation (i.e., satisfying the equation with ≤ instead of =) and 𝑣 0 ≤ 0, then 𝑣 𝑡 ≤ 0 for all 𝑡 ≥ 0. In the second step one takes a family of supersolutions (i.e., with ≥ instead of =) 𝑀𝑡 as simple as possible giving suitable bounds. Now, if 𝑢𝑡 is a genuine solution of the differential equation and 𝑀𝑡 is one of such supersolutions with 𝑢0 ≤ 𝑀0 , then 𝑣 𝑡 = 𝑢𝑡 − 𝑀𝑡 is a subsolution with 𝑣 0 ≤ 0, so 𝑣 𝑡 ≤ 0 for all 𝑡 > 0, that is 𝑢𝑡 ≤ 𝑀 𝑡 . • The previous argument works in this case because, for the ordinary differential equation (6.33), 𝑟0 is an attractor for 𝑟0 < 0. However, 𝑟0 is a repeller for 𝑟0 > 0 so the barrier functions give no bounds for the solutions of the partial differential equation. This is one of the reasons why curvature flow for the positive curvature is much harder than the negative case. These kinds of estimates are the key for proving long time existence of the curvature flow. Despite that equation (6.31) is parabolic, it is non-linear, so standard results about existence and uniqueness of parabolic equations do not apply directly. However, we can recast the solutions of (6.31) as the fixed points of a certain operator in 𝐿2 (𝑆). Such an operator is a contraction for short times, so the Banach fixed point theorem [Eva] proves the existence and uniqueness of these solutions, at least for short time. Once that is done, the a priori estimates of Proposition 6.70 will prevent the solution from blowing up, providing long time existence. Here, we will just sketch the proof of this result. The whole proof can be found in [IMS]. Theorem 6.73 (Hamilton). Let (𝑆, g0 ) be a compact connected oriented Riemannian surface with 𝜅0 < 0 everywhere. Then equation (6.31) has a unique solution g𝑡 = 𝑒2ᵆ𝑡 g0 , for all 𝑡 ∈ [0, ∞). This has limit g∞ = lim𝑡→∞ g𝑡 (in 𝐶 ∞ ), which is a metric in the conformal class [g0 ] with constant curvature 𝑟0 . Sketch of proof. First of all, we will show short time existence for equation (6.31). That is, we will show that, for 𝑇 > 0 small enough, there exists 𝑢 ∈ 𝐶 ∞ (𝑆 × [0, 𝑇)) such that 𝑃(𝑢) =

𝜕𝑢𝑡 + 𝑒−2ᵆ𝑡 Δg0 𝑢𝑡 + 𝑒−2ᵆ𝑡 𝜅0 − 𝑟0 = 0, 𝜕𝑡

with 𝑢0 = 0. For that, rewrite 𝑃(𝑢) = 𝐿(𝑢) − 𝑓(𝑢), where 𝐿(𝑢) =

𝜕𝑢𝑡 + Δg0 𝑢𝑡 , 𝜕𝑡

𝑓(𝑢) = (1 − 𝑒−2ᵆ𝑡 )Δg0 𝑢𝑡 − 𝑒−2ᵆ𝑡 𝜅0 + 𝑟0 .

6.4. The curvature flow

383

Fixed 𝑢 ∶ [0, 𝑇) → 𝑊 2,2 (𝑆) continuous, we define 𝐴(𝑢) = 𝑤, where 𝑤 is the solution to the linear parabolic problem (6.34)

{

𝐿(𝑤) = 𝑤 0 = 0.

𝜕𝑤𝑡 𝜕𝑡

+ Δg0 𝑤 𝑡 = 𝑓(𝑢),

By the usual theory of linear parabolic equations (see [Eva, chapter 7]) the problem (6.34) has a unique continuous (weak) solution 𝑤 ∶ [0, 𝑇) → 𝑊 2,2 (𝑆). So 𝐴 can be seen as a (non-linear) map 𝐴 ∶ 𝐶 0 ([0, 𝑇), 𝑊 2,2 (𝑆)) → 𝐶 0 ([0, 𝑇), 𝑊 2,2 (𝑆)). The uniform bounds on the curvature given by Proposition 6.70 produce uniform bounds for 𝑢𝑡 by integration of (6.29), and thus, for 𝑒2ᵆ𝑡 . Hence, the equation Δg0 𝑢𝑡 − 𝑒2ᵆ𝑡 𝜅𝑡 + 𝜅0 = 0 shows that Δg0 𝑢𝑡 is uniformly bounded. These estimates, together with standard properties of linear parabolic equations show that 𝐴 is a contraction between Banach spaces 𝐶 0 ([0, 𝑇), 𝑊 2,2 (𝑆)) with the uniform norm, for 𝑇 small enough. Hence by the Banach fixed point theorem, there exists a unique 𝑢 ∈ 𝐶 0 ([0, 𝑇), 𝑊 2,2 (𝑆)) such that 𝐴(𝑢) = 𝑢, which precisely is the unique solution of 𝑃(𝑢) = 𝐿(𝑢) − 𝑓(𝑢) = 0. Using a bootstrapping argument, we see that such a weak solution 𝑢 is smooth, so it is a solution in the classical sense. Long time existence essentially follows from Proposition 6.70, as the bounds on the curvature prevent the solution to blow up. Suppose that we have a solution on a maximal interval [0, 𝑇). Then the functions 𝑢𝑡 , 𝑡 ∈ (𝑇 − 𝜖, 𝑇), are equicontinuous, that is, continuous, uniformly bounded and uniformly Lipschitz. This follows from a uniform bound in the first derivative 𝑑𝑢𝑡 , which is obtained arguing as in the proof of Proposition 6.70 (with bounds on 𝑢𝑡 and Δg0 𝑢𝑡 and using ∫𝑆 𝑢𝑡 Δg0 𝑢𝑡 = ∫𝑆 ||𝑑𝑢𝑡 ||2g0 ). By the Arzelá-Ascoli theorem, there is a limit 𝑢𝑡 → 𝑢𝑇 , when 𝑡 → 𝑇, which is a continuous function. To prove that 𝑢𝑇 is smooth, one has to write the differential equations satisfied by the derivatives ∇𝑘 𝑢𝑡 , and apply similar arguments. Hence, by the short time existence, we can prolong 𝑢𝑡 for 𝑡 ∈ [𝑇, 𝑇 + 𝜖) by solving the differential equation with initial datum 𝑢𝑇 . This proves that the solution 𝑢𝑡 exists for all 𝑡 ≥ 0. Finally, since lim 𝜅𝑡 → 𝑟0 , for 𝑡 → ∞, uniformly, the limit 𝑢∞ = lim𝑡→∞ 𝑢𝑡 exists and another bootstrapping argument serves to prove that it is smooth. Hence, g𝑡 converges to g∞ = 𝑒2ᵆ∞ g0 , which is a metric of constant curvature 𝑟0 . □ In the general case, with positive curvature (or, even worse, with curvature of varying sign), the previous result holds but its proof is much harder. Actually, the curvature flow was introduced by Hamilton in [Ham] in which the previous situation was accomplished, but the general case remained unsolved until the paper of Chow [Cho]. Theorem 6.74 (Chow-Hamilton). Let (𝑆, g0 ) be any compact connected oriented Riemannian surface. Then, the curvature flow has a unique solution g𝑡 , for all 𝑡 ∈ [0, ∞). This has limit g∞ = lim𝑡→∞ g𝑡 which is a metric in the conformal class [g0 ] with constant curvature 𝑟0 . Remark 6.75. • Theorem 6.74 reproves Theorem 6.44 on the existence of metrics of constant curvature on compact connected surfaces. Actually, Theorem 6.73 does the work in the case of genus 𝑔 ≥ 2.

384

6. Global analysis

• The case of 𝑟0 = 0 has the characteristic that the rate of convergence is not exponential but polynomial. • In the case 𝑟0 > 0, the limit g∞ is well defined, although there is no uniqueness for the metrics of constant curvature. In the general case, for a connected manifold 𝑀 of dimension 𝑛 ≥ 3, the natural generalization of the curvature flow is the Ricci flow. The Ricci flow is the differential equation for an evolution of metrics g𝑡 given by (6.35)

𝜕g𝑡 = −2 Ric𝑡 , 𝜕𝑡

where Ric𝑡 is the Ricci tensor of g𝑡 . There is also a normalized Ricci flow given by 𝜕g𝑡 = −2 (Ric𝑡 −𝑟𝑡 g𝑡 ) , 𝜕𝑡 where 𝑟𝑡 = ∫𝑀 Scal𝑡 𝑑𝑡/ Vol(𝑀, g𝑡 ) is the average of the scalar curvature Scal𝑡 of g𝑡 (Remark 3.46(2)). Remark 6.76. (1) For a Riemannian surface (𝑆, g), let 𝑝 ∈ 𝑆 and let (𝑒 1 , 𝑒 2 ) be an orthonormal basis of 𝑇𝑝 𝑆. The Gaussian curvature of 𝑆 is 𝜅(𝑝) = 𝑅(𝑒 1 , 𝑒 2 , 𝑒 1 , 𝑒 2 ), so the Ricci tensors are Ric(𝑒 1 , 𝑒 1 ) = 𝜅(𝑝), Ric(𝑒 1 , 𝑒 2 ) = 0, and Ric(𝑒 2 , 𝑒 2 ) = 𝜅(𝑝). Thus Ric = 𝜅 g, and (6.35) is the curvature flow in the case of surfaces. (2) For a smooth manifold 𝑀, 𝑝 ∈ 𝑀 and 𝑣 ∈ 𝑇𝑝 𝑀 a unitary vector, we have that Ric(𝑣, 𝑣) is the average of the sectional curvatures of the planes containing 𝑣. Therefore (6.35) means that the metric increases or decreases in the direction of 𝑣 depending on how negative or positive the sectional curvature is (in average) in that direction. Again, this should geometrically serve to spread out the curvature along 𝑀. (3) The evolution of g𝑡 following the Ricci flow is not in a conformal class, except when g𝑡 is Einstein (Remark 6.58(6)). (4) By Remark 6.58(4), the only possible terms in the right hand side of an equation like (6.35) can only involve the Ricci tensor and the scalar curvature. 𝜕g There is certainly a scalar curvature flow given by 𝜕𝑡𝑡 = −(Scal𝑡 −𝑠0 ) g𝑡 , where 𝑠0 is the average of the scalar curvature. This is a conformal flow, i.e., the metric g𝑡 evolves in its conformal class. The normalized scalar curvature flow is the Yamabe flow. Its stationary solutions are the metrics with constant scalar curvature in a given conformal class. So the limit of the flow is a metric of constant scalar curvature, and it can be proved to exist for compact connected manifolds. (5) The equation (6.35) is not parabolic, but only semiparabolic. This means that the linearization of the term Ric𝑡 is only semi-elliptic (the symbol is semidefinite positive). This is due to the fact that the Ricci flow may evolve via diffeomorphisms 𝜑𝑡 , that is g𝑡 = 𝜑∗𝑡 g0 (Exercise 6.26). To avoid this, one has to

6.4. The curvature flow

385

modify the equation (called the DeTurck trick) that transforms it into a parabolic equation by blocking the action of the diffeomorphisms. Note that the sign of (6.35) is the right one to have a parabolic problem. (6) Ric is a symmetric bilinear map in 𝑇𝑝 𝑀. Then there exists an orthonormal basis (𝑒 1 , . . . , 𝑒 𝑛 ) such that Ric(𝑒 𝑖 , 𝑒𝑗 ) = 0 if 𝑖 ≠ 𝑗, and Ric(𝑒 𝑖 , 𝑒 𝑖 ) = 𝜆𝑖 , for 1 ≤ 𝑖, 𝑗 ≤ 𝑛. In dimension 3, if we write 𝐾𝑝 (⟨𝑒 1 , 𝑒 2 ⟩) = 𝜇3 , 𝐾𝑝 (⟨𝑒 1 , 𝑒 3 ⟩) = 𝜇2 and 𝐾𝑝 (⟨𝑒 2 , 𝑒 3 ⟩) = 𝜇1 , we have 𝜇1 + 𝜇2 = 2𝜆3 , 𝜇2 + 𝜇3 = 2𝜆1 , 𝜇1 + 𝜇3 = 2𝜆2 . 1 1 1 Solving, we get 𝜇1 = 2 (𝜆2 + 𝜆3 − 𝜆1 ), 𝜇2 = 2 (𝜆1 + 𝜆3 − 𝜆2 ), 𝜇3 = 2 (𝜆1 + 𝜆2 − 𝜆3 ). Hence in dimension 3 the Ricci tensor is equivalent to the sectional curvature. In this dimension the Ricci flow has been particularly successful. (7) By the previous item, a connected complete Einstein manifold (Remark 6.58(6)) in dimension 3 is automatically a space form. Therefore, not every manifold admits an Einstein metric. On the other hand, by (4) all connected compact manifolds admit metrics of constant scalar curvature (cf. Remark 4.84). (8) The Thurston geometrization conjecture says that a connected compact 3manifold can be cut along 2-spheres and 2-tori into pieces which are of geometrically simple nature: they are open manifolds which admit a locally homogeneous metric, that is, modelled in one of the eight different possible geometries in dimension 3 (Remark 3.110). Hamilton proposed using the Ricci flow to prove this, and Perelman completed the proof in 2004, for which he earned the Fields medal. The idea is to take an initial metric g0 and let it flow. It may develop singularities associated to the cut locus, but the limit in the rest of the manifold, suitably normalized, gives the geometric pieces of the decomposition. This proves (in particular) the Poincaré conjecture: if we start with a simply connected compact 3-manifold, and let a metric flow via the Ricci flow, it ends up with positive constant curvature. (9) The curvature flow and the Ricci flow are intrinsic geometric flows. There are similar interesting extrinsic flows, which depend on the immersion of a smooth manifold into an ambient (Riemannian) space, for instance, the flow by the mean curvature whose critical points are minimal submanifolds (submanifolds of minimal volume or surfaces of minimal area). The easiest case, which is very interesting in itself, is the study of embedded loops in the plane 𝛾 ∶ 𝑆 1 → ℝ2 , with the flow given by the curvature vector of the curve. If we deform the curve in the direction of the curvature vector, then we straighten the bumpy parts. The equation is 𝜕𝛾𝑡 = 𝐤𝛾𝑡 , 𝜕𝑡 for 𝛾 ∶ 𝑆 1 × [0, ∞) → ℝ2 . This flow (again) can be normalized to preserve the condition that 𝛾𝑡 (𝑠) is parametrized by arc length. So if we start with an embedded curve, then the flow by the curvature vector shrinks it to a point. If we normalize the length (equivalent to expanding by a homothety depending

386

6. Global analysis

on 𝑡), we see that the normalized flow makes the curve a round circle (i.e., a curve with constant curvature).

Problems Exercise 6.1. For a 3-dimensional Riemannian manifold (𝑀, g), determine in a local chart the curvature of a conformally equivalent metric ĝ = 𝑒2𝜑 g in terms of the curvature of g. Exercise 6.2. Consider the Mercator projection from 𝑆2 onto the cylinder, given in Exercise 4.10. As it is conformal, describe the complex structure induced on the cylinder, and determine explicitly the biholomorphism. Exercise 6.3. Let 𝑆 be a non-orientable surface. Prove that the following are equivalent: • A structure of a Klein surface on 𝑆. • A conformal structure on 𝑆. • A (smooth) way to measure non-oriented angles ∠(𝑢, 𝑣) for non-zero vectors 𝑢, 𝑣 ∈ 𝑇𝑝 𝑆, 𝑝 ∈ 𝑆. • A complex curve 𝑆 ̂ with an antiholomorphic involution 𝐴 without fixed points such ̂ that 𝑆 ≅ 𝑆/⟨𝐴⟩. Exercise 6.4. Prove that for a vector field 𝑋, div(𝑋) = 0 if and only if the flow 𝜙𝑡 of 𝑋 preserves the volume for all 𝑡 ∈ ℝ. Exercise 6.5. Prove the Cartan formula for the Lie derivative 𝐿𝑋 = 𝑑𝑖𝑋 + 𝑖𝑋 𝑑 on Ω𝑠 (𝑀), for 𝑋 ∈ 𝔛(𝑀). Hint: check that both are derivations that coincide on functions and 1-forms and satisfy the Leibniz rule. Exercise 6.6. Prove that the Hodge star operator commutes with the covariant derivative. Exercise 6.7. Let g and ĝ = 𝜇2 g be two conformal metrics, with 𝜇 a smooth positive function, ̂ = on an 𝑛-manifold 𝑀. Let ⋆ and ⋆̂ be the corresponding Hodge star operators. Prove that ⋆𝛼 𝜇𝑛−2𝑠 ⋆ 𝛼 for an 𝑠-form 𝛼. Conclude that if 𝑛 is even and 𝑠 = 𝑛/2, then ⋆ only depends on the conformal class [g]. Exercise 6.8. Let (𝑀, g) be a Riemannian manifold and (𝑥1 , . . . , 𝑥𝑛 ) local coordinates. Prove that ⟨𝑑𝑥𝑖 , 𝑑𝑥𝑗 ⟩ = 𝑔𝑖𝑗 , where (𝑔𝑖𝑗 ) is the inverse matrix of (𝑔𝑖𝑗 ). Exercise 6.9. Let (𝑀, g) be a Riemannian manifold. We define the Hessian of a smooth function 𝑓 by Hessg 𝑓(𝑋, 𝑌 ) = ⟨∇𝑋 (grad 𝑓), 𝑌 ⟩. Prove that Hessg 𝑓 is a symmetric (2, 0)-tensor. Give its expression on local coordinates, and prove that it is independent on the metric at critical points of 𝑓 (where grad 𝑓(𝑝) = 0). Exercise 6.10. Let (𝑀, g) be a Riemannian manifold, 𝑝 ∈ 𝑀, and let (𝐸1 , . . . , 𝐸𝑛 ) be vector fields in a neighbourhood 𝑈 𝑝 which are an orthonormal basis at every point. Prove that Δg 𝑓 = ∑ ∇𝐸𝑗 ∇𝐸𝑗 𝑓 on 𝑈, for a smooth function 𝑓. Conclude that Δg 𝑓 = trg (↑ Hessg 𝑓), where ↑ is the raising index operator. 𝑑

Exercise 6.11. Let 𝐴(𝑡) be a smooth path in GL(𝑛, ℝ). Prove that 𝑑𝑡 |𝑡=0 det 𝐴(𝑡) = det 𝐴 tr(𝐴−1 𝑉), with 𝐴 = 𝐴(0), 𝑉 = 𝐴′ (0). Exercise 6.12. Let (𝑀, 𝐽, h) be a Kähler manifold. Prove that 𝜔 = Im h is harmonic. Exercise 6.13. For a complex curve (𝑆, 𝐽) and a real harmonic function 𝑢 on 𝑆, is there always a holomorphic function 𝑓 on 𝑆 such that 𝑢 = Re(𝑓)? Prove that this is the case if 𝑆 is simply connected.

Problems

387

Exercise 6.14. If (𝑆, 𝐽) is a complex curve, and 𝑓 is holomorphic, then the following functions are harmonic: • The norm |𝑓|, on the locus {|𝑓| > 0}. • The argument arg 𝑓, on a simply connected set 𝑈 ⊂ 𝑆, where the argument can be defined univocally. Exercise 6.15. Determine the eigenvalues and eigenfunctions of the Laplacian on the square torus 𝑇 2 = 𝔼2 /ℤ2 with the flat metric. Exercise 6.16. Let ℋ be a Hilbert space, and let 𝑊 ⊂ ℋ be a closed affine subspace. Prove that there exists a unique 𝑎 ∈ 𝑊 such that ||𝑎|| = min{||𝑎′ || | 𝑎′ ∈ 𝑊}. Prove that 𝑎 ⟂ 𝑊 and that this characterises the element 𝑎. Exercise 6.17. Compute the symbol of the Laplacian on Ω𝑠 (𝑀). It is useful to use geodesic coordinates. Exercise 6.18. Let (𝑀, g) be a compact oriented Riemannian manifold, and let ∇, ∇′ be two connections. Prove that there are constants 𝐶1 , 𝐶2 > 0 such that 𝐶1 ||𝛼||𝑊 𝑘,2 ≤ ||𝛼||′𝑊 𝑘,2 ≤ 𝐶2 ||𝛼||𝑊 𝑘,2 , for all 𝑠-forms 𝛼. Prove also that the form 𝛼 ∈ Ω𝑠𝑊 𝑘,2 (𝑀) if and only if the coefficients of 𝛼 have (weak) derivatives up to order 𝑘 and all of them are in 𝐿2 . Exercise 6.19. Prove that the cohomology of the cochain complex (𝒱 • , 𝑑), where 𝒱 𝑘 = {𝛼 ∈ Ω𝑘𝐿2 (𝑀)|𝑑𝛼 ∈ Ω𝑘+1 𝐿2 (𝑀)}, is isomorphic to the de Rham cohomology. Exercise 6.20. Prove the Hodge Theorem 6.37 for compact non-orientable manifolds. Exercise 6.21. Let (𝑀, g) be a compact Riemannian manifold. Prove that a cohomology class 𝑠 𝑎 = [𝛼0 ] ∈ 𝐻𝑑𝑅 (𝑀) has a unique harmonic representative 𝛼 ∈ Ω𝑠 (𝑀) by the variational method, minimizing the Dirichlet functional 𝒟(𝛼) = ||𝛼||2𝐿2 on 𝐻𝑎 = 𝛼0 + im 𝑑 ⊂ Ω𝑠𝐿2 (𝑀). Exercise 6.22. Let 𝑆 = ℝ𝑃 2 and consider a metric g0 on it. Prove that there is a conformal metric g ∈ [g0 ] with constant curvature. Exercise 6.23. Prove that given a compact orbisurface 𝑋 with an orbifold metric g0 , there exists a conformal metric g ∈ [g0 ] with constant curvature. Give criteria for uniqueness. Exercise 6.24. Let 𝑆 1 , 𝑆 2 be two distinct smooth exotic spheres of some dimension 𝑛 ≥ 7. Show that there is a map 𝑓 ∶ 𝑆 1 → 𝑆 2 that is a homeomorphism and a diffeomorphism outside one point, but that it cannot be modified to a diffeomorphism (cf. Remark 1.22 and Exercise 1.11). Exercise 6.25. Let 𝑀 be a Riemannian manifold and 𝑝, 𝑞 ∈ 𝑀. Prove that a curve 𝛾 ∈ Ω𝑝𝑞 (𝑀) 1 is a geodesic if and only if it is a critical point of the functional ℰ(𝛾) = ∫0 ||𝛾′ ||2 𝑑𝑡. Prove that 𝛾 1 is a pregeodesic if and only if it is a critical point of the length functional ℓ(𝛾) = ∫0 ||𝛾′ ||𝑑𝑡. Exercise 6.26. Let (𝑀, g0 ) be a (compact, connected, oriented) Riemannian manifold. The metric g0 is a Ricci soliton if there exists a function 𝑓 ∈ 𝐶 ∞ (𝑀) such that Ricg0 + Hessg0 𝑓 = 0. Show that g𝑡 = 𝜑∗𝑡 g0 , where 𝜑𝑡 is the flow of grad 𝑓, is a solution of the Ricci flow. Exercise 6.27. Consider the metrics on the plane ℝ2 given by 4 g𝑡 = 2𝑡 (𝑑𝑢2 + 𝑑𝑣2 ). 𝑒 + 𝑢2 + 𝑣 2 • Check that g𝑡 is a solution of the Ricci flow, and that it is a soliton with 𝑓 = − log(1 + 𝑢2 + 𝑣2 ). This is called the cigar soliton, or a Witten black hole in the physics literature. • Check that the curvature 𝜅0 of g0 is radial, 𝜅0 ≥ 0 and limᵆ2 +𝑣2 →∞ 𝜅0 (𝑢, 𝑣) = 0. • Show that for any (𝑢, 𝑣), we have lim𝑡→∞ 𝜅𝑡 (𝑢, 𝑣) = 1.

388

6. Global analysis

Exercise 6.28. Let g0 be a Riemannian metric on a manifold 𝑀 of dimension 𝑛 > 2. Let g = 𝑒2ᵆ g0 be a conformal metric such that Scalg ≡ 𝑘0 (the solution to the Yamabe problem). Show that 𝑎 Δg0 𝑢 + (Scalg0 )𝑢 = 𝑢1−𝑝 , where 𝑎 = 4(𝑛 − 1)/(𝑛 − 2), 𝑝 = 2𝑛/(𝑛 − 2).

References and extra material Basic reading. Hodge theory appears in the books [War] and [Wel]. The book [War] follows a more elementary approach using Fourier series, and [Wel] develops all functional analysis tools. For functional analysis applied to partial differential equations, we recommend [Eva]. In [Tay] the equation of constant curvature is solved. [Eva] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, 2nd Edition, AMS, 2010. [Tay] M.E. Taylor, Partial Differential Equations III: Nonlinear Equations, Applied Mathematical Sciences, Vol. 117, Springer, 2013. [War] F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics, Vol. 94, 1983. [Wel] R.O. Wells, Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, 3rd Edition, 2008. Further reading. The following topics can be studied to enlarge the contents of this chapter. • Hodge theory. It is good to give a complete treatment of Hodge theorem of harmonic forms on compact manifolds. See [War] and [Wel]. • Kähler geometry. The theory of Kähler manifolds and results on Hodge theory of Kähler manifolds appear in [Wel]. • Conformal geometry. Conformal geometry in higher dimensions (𝑛 > 2) has special features. [Laf] J. Lafontaine, Conformal geometry from the Riemannian viewpoint, Conformal Geometry: A Publication of the Max-Planck-Institut für Mathematik, Bonn (R. S. Kulkarni, U. Pinkall, eds.), Aspects of Mathematics, Vol. 12, Verlag, pp. 65-92, 1988. • Curvature flow. The complete theory can be too involved, but we recommend that the student fully develop some particular cases or particular features. [IMS] J. Isenberg, R. Mazzeo, N. Sesum, Ricci flow in two dimensions, Surveys in geometric analysis and relativity, Adv. Lect. Math. 20, 259-280, 2011. • Curve shortening. A very nice result accessible to students on the flow of a plane curve by the curvature vector (Remark 6.76(9)). This can be applied to the Jordan-Schönflies theorem (Theorem 1.36). [Gag] M.E. Gage, Curve shortening makes convex curves circular, Inventiones Mathematicae 76, 357-364, 1984.

References and extra material

389

References. [BBF] G. Bazzoni, I. Biswas, M. Fernández, V. Muñoz, A. Tralle, Homotopic properties of Kähler orbifolds, In: Special Metrics and Group Actions in Geometry, Springer INdAM Series, vol 23, 23-57, 2017. [Cho] B. Chow, The Ricci flow on the 2-sphere, Jour. Diff. Geom. 33, 325-334, 1991. [Do4] S.K. Donaldson, Lecture notes for TCC course geometric analysis, 2008, wwwf.imperial.ac.uk/~skdona/GEOMETRICANALYSIS.PDF [Ein] A. Einstein, The foundation of the general theory of relativity, Annalen der Physik 354, 769-822, 1916. [Ham] R.S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity, vol. 71, Amer. Math. Soc., Providence, 1988, 237-262. [Jo1] J. Jost, Riemannian Geometry and Geometric Analysis, Universitext, Springer, 2008. [K-M] A. Kriegl, P.W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, Vol. 53, American Mathematical Society, 1997. [Lan] S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics, Springer, 1999. [Mi3] J.W. Milnor, Morse Theory, Annals of Mathematics Studies, Vol. 51, Princeton University Press, 1963. [Mi7] J. Milnor, Towards the Poincaré conjecture and the classification of 3-manifolds, Notices of the AMS. 50, 2003, 1226-1233. [Sm2] S. Smale, What is global analysis? The Collected Papers of Stephen Smale, Vol. 2, pp. 544-549, World Scientific, 2000. [Top] P. Topping, Lectures on the Ricci Flow, University of Warwick, 2006, homepages.warwick.ac.uk/~maseq/topping_RF_mar06.pdf [Voi] C. Voisin, Hodge Theory and Complex Algebraic Geometry, I, Cambridge Studies in Advanced Mathematics, Vol. 76, Cambridge University Press, 2007.

Subject Index

A priori estimate, 368, 382 Abelian Abelianization, 66 Free Abelian group, 67, 85 semigroup, 46 see also category, 89 see also variety, 337 Absolute convergence, 328 Acceleration, 151 Normal acceleration, 152 Tangential acceleration, 152 Action Free action, 72 Free and proper, 191, 221, 247, 338 Non-proper action, 194 on the left, 223 on the right, 223 Proper action, 72, 179 Adjacent cells/polyhedra, 26 Adjoint Adjointness property, 120 Formal adjoint, 354 matrix, 211 of a linear map, 354 see also operator, 359 Alexandroff compactification, 207, 316 Algebraic curve, 301 geometry, 297 map, 10, 334 numbers, 315 variety, 10, 299 Almost complex integrable structure, 278, 280 manifold, 277 structure, 277, 301, 346, 347

Analytic function, 269 manifold, 9, 273 Annulus, 18 Antiholomorphic map, 267 Antisymmetric Antisymmetrization of tensors, 192 bilinear map, 124, 131, 285 multilinear map, 139 tensor, 139 Arc length, 148 Area form of a surface, 161 of 𝕊2 , 204 of ℝ𝑃2 , 214 of a surface, 114 of hyperbolic plane, 238 of hyperbolic triangle, 245, 250 Arrow, 3 Arzelá-Ascoli theorem, 369, 383 Atlas, 1 Complex atlas, 271 Conformal atlas, 344 Maximal atlas, 8 Orbifold atlas, 177 Positive atlas, 36 Smooth atlas, 8 Attaching a cell, 15, 99 Ball, 1 Closed ball, 1 Open ball, 1 Banach fixed point theorem, 382, 383 Belyı̆ theorem, 315 Betti number, 103 of a complex manifold, 292 Bézout theorem, 307, 311

391

392

Bianchi identity, 158 Bieberbach manifold, 231 Biholomorphism, 266, 271, 272, 309, 370 of ℂ, 316 of ℂ, 210 of a complex curve, 322 Bonnet-Myers theorem, 176 Bootstrapping, 369, 383 Boundary 𝑘-boundaries, 85 map, 85, 89, 95 point, 11 see also manifold, 11 Bouquet, 64 Branch of a plane curve, 304 Calabi Calabi-Yau manifold, 369 conjecture, 369 Cartan formula, 351, 386 lemma, 185, 187 Category, 3 Abelian category, 89 Arrow category, 46 Opposite category, 49 Quotient category, 54 Cauchy sequence, 352 theorem, 145, 267, 337 Cauchy-Riemann equations, 266, 269 Cell, 𝑘-cell, 21, 99 Cellular homology, 99, 338 Chain 𝑘-chain, 85 complex, 85, 89 homotopy, 89, 90 morphism, 90 Change of charts, of coordinates, 7 for an orbifold, 177 for complex manifold, 271 rule for derivatives, 274 Chart, 1 Affine chart, 14, 207, 213 Induced orbifold chart, 177 Orbifold chart, 177 Chow theorem, 299, 334 Chow-Hamilton theorem, 383 Christoffel symbols, 142, 163, 344 Circle Generalized circles in ℂ, 209 Geodesic circle, 167 in hyperbolic space, 236 in sphere, 202 Maximal circle of an 𝑛-sphere, 201 Classification

Subject Index

Homotopy, 56, 68 of complex curves, 316 of manifolds, 25 of smooth surfaces, 373 of topological surfaces, 68 of triangulated surfaces, 41 problem, 3, 4 Clockwise, anticlockwise, 31, 40 Co-H-space, 57 Coboundary, 107 Cochain complex, 107 Double cochain complex, 281 Cocycle, 107 Coherent topology, 20, 99 Cohomology of ℂ𝑃𝑛 , 293 Sheaf cohomology, 119, 282 Singular cohomology, 120 see also de Rham, 104 Collapsing, 13 Complete manifold, 155, 183, 184 metric, 118 vector space, 352 see also geodesically, 155 Complex atlas, 271 curve, 271, 300, 315 manifold, 271, 299 projective line, 272 projective space, 292 structure, 272, 276, 347 surface, 271 torus, 317, 337 vector space, 276, 285 Composition, 3 Cone angle, 250 Conformal, 260 class, 344, 363, 384 Conformally flat structure, 345, 347 coordinates, 206, 345, 348, 371 group, 344, 348 Locally conformal, 205, 267, 344 Locally conformally flat, 345 map, 205, 207, 242, 344 metric, 362, 386 orbisurface, 349 structure, 344, 347 structure on orbifold, 349 Conic, 306 𝑛-connected, 87 Connected sum, 16 Smooth connected sum, 19 Twisted connected sum, 20 Connection, 141 compatible with metric, 150 Levi-Civita, 151

Subject Index

Space of connections, 143 Torsion of a connection, 143 Trivial connection, 146, 151, 217, 235 Conservative vector field, 110, 117 Constant curvature Constant scalar curvature, 255, 384 Constant sectional curvature, 183, 185, 187, 191, 254 metrics on 𝑆2 , 322 metrics on orbisurfaces, 370 metrics on surfaces, 254, 321, 362 Continuity method, 368 Contractible space, 55 Locally contractible, 132 Contraction of tensors, 140 operator on forms, 161, 351 Convolution, 377 Coordinates Conformal coordinates, 167, 206, 345, 348, 371 Coordinate functions, 1 Harmonic coordinates, 167, 372 Isothermal coordinates, 167, 372 Orthogonal coordinates, 164, 165, 363 Polar geodesic coordinates, 164 Coproduct, 62 Corner angle, 145, 169 point, 145, 169 Cosmology Cosmological constant, 375 Space form, universe, 183, 190 Cotangent bundle, 138 space, 138 Covariant derivative, 144 Cover, 69, 338 Group of the cover, 70 Orbifold cover, 178, 370 Ramified cover, 80, 308, 325, 334, 339 Section of a cover, 77 Covering, 20 Good covering, 118 Locally finite covering, 20 Cube 𝑘-cube, 85 Degenerate cube, 85 Small cubes, 92 Smooth 𝑘-cubes, 115 Cubic, 306 curve, plane cubic, 323, 333 in standard form, 325 in Weierstrass form, 326, 333 Parametrization of a nodal cubic, 313 Currents, 128 Curvature

393

by holonomy, 170 by length of circle, 168 flow, 375, 379 flow for a curve, 386 Gaussian curvature, 161, 363 Geodesic curvature, 163 Normal curvature, 163 Normalized curvature flow, 379 of Euclidean space, 218 of hyperbolic space, 235 of sphere, 202 Orbifold Gaussian curvature, 180 Riemannian curvature, curvature tensor, 157 Curve 𝐶 1 curve, 144 Elliptic curve, 370 Irreducible, 303 Length of a curve, 147 Piecewise 𝐶 1 curve, 145, 147 Piecewise 𝐶 2 curve, 151, 169 Plane curve, 303 Plane projective curve, 306 Smooth projective curve, 297 Cusp of a plane curve, 304 of a Riemann surface, 249, 320 CW-complex, 99 ℂ𝑃𝑛 as a CW-complex, 293 𝑘-cycles, 85 Cyclic order in 𝑆1 , 242 Cylinder, 15 Complex cylinder, 317 Riemannian cylinder, 222 Darboux frame, 163 theorem, 291 De Rham cohomology, 104, 108, 283 cohomology with compact support, 120 complex, 107 integral cohomology, 296 theorem, 116 Deck transformation, 72, 250, 257, 315 Degree of a cover, number of sheets, 69 of a map, 124 of a plane curve, 306, 308 Degree-genus formula, 308, 324, 339 Dehomogenized polynomial, 298, 302, 303, 324 Diffeomorphism, 8 Differentiable function, 7 manifold, 7 map, 8 see also smooth, 7 Differential equation, 145, 152

394

Elliptic, 358 Non-linear elliptic, 363 Parabolic, 376, 383 see also operator, 353 Differential of a map, 140 Complex differential, 270 Dilation, 205, 344 factor, 205, 344 Dimension, 2, 21, 99 Complex dimension, 271 Generic dimension, 300 of a projective variety, 300 of an orbifold, 176 Dirac delta, 117, 128, 170, 180, 377 Disc, 1 Discriminant, 326, 334 Distribution, 117, 170, 180, 377 Integrable, 279 on a manifold, 279 Divergence, 107, 110, 115, 351, 352, 386 Division algebra, 280 Divisor, 331 Canonical divisor, 332 Dolbeault cohomology, 282, 284, 361 complex, 282 operator, 282 Double coset space, 223 point, 303 ratio, 209 Doubling process from ℝ to ℂ, 267 Duality, 62 Edge, 21 Eigenvalues of a differential operator, 358 Einstein, 183, 232 field equations, 183, 375 manifold, 375 metric, 384, 385 Elliptic curve, 314, 323, 370 differential equation, 363, 365 invariants, 334, 335 regularity, 360, 365 see also differential equation, 358 see also operator, 358 Embedding, 8 Equicontinuous, 369 Euclid Fifth postulate of Euclid, 182, 258 Postulates of Euclid, 258 Euclidean space, 191, 197, 218 Euler identity, 298 Euler-Lagrange equation, 367 Euler-Poincaré characteristic, 33, 76, 103, 124, 172, 308, 369

Subject Index

Orbifold Euler-Poincaré characteristic, 178 Evolution equations, 374, 380 Exact functor, 101 Long exact sequence, 91 sequence, 91 Short exact sequence, 91 Split exact sequence, 92 see also Mayer-Vietoris, 92 Exponential map, 154 of Euclidean space, 218 of hyperbolic space, 235 of sphere, 201 Exterior differential, 106, 276, 353 Antiholomorphic exterior differential, 276 Holomorphic exterior differential, 276 Face, 𝑘-face, 21 Fibered product, 62 Field of meromorphic functions, 334 Finitely generated, finitely presented, 64 Flat connection, 217 Klein bottle, 230 orientable surface, 220 Oriented flat torus, 228 Riemannian manifold, 217 surface, 220 torus, 222, 318 Flow Curvature flow, 375 intrinsic, extrinsic, 385 Normalized Ricci flow, 384 of a vector field, 192 of metrics, 375 Ricci flow, 384 Scalar curvature flow, Yamabe flow, 255, 384, 388 Flux, 115, 161 Form (𝑝, 𝑞)-forms, 276, 280 Compactly supported 𝑘-form, 120 Differential 𝑘-form, 105, 140 with complex coefficients, 109 Fourier series, 371 transform, 358 Frame, 189 Frobenius theorem, 279 Fubini-Study metric, 294, 295 Fuchsian group, 249 Function Bump, 117, 127 Characteristic, 113 Differentiable, 7 Entire function, 328 Simply periodic, 340

Subject Index

Smooth, 7 see also Weierstrass, 329 see also holomorphic, 266 see also meromorphic, 268 Functional, 366 Energy functional, 367, 375 Functor, 5, 88 Abelianization functor, 66 Contravariant, 5, 106 Covariant, 5 Exact functor, 101 Forgetful functor, 9 Homology functor, 119 Quotient functor, 54 Fundamental class, 94, 97, 124 domain, 74, 224, 250 First fundamental form, 155, 161 form of ℂ𝑃𝑛 , 294 form of a Hermitian manifold, 289 group, 2, 5, 57 groupoid, 4 Second fundamental form, 156 solution, 377 theorem of integration, 115 Gauge theory, 158 Gauss egregium theorem, 164 lemma, 165 map, 161 Gauss-Bonnet theorem, 364 global version, 172 local version, 169 orbifold case, 180 General relativity, 183, 190, 232 Generators of a group, 64 Genus, 308 of a compact connected surface, 47 Geodesic, 151, 226, 367 ball, 154 chart, 154 circle, 167 curvature, 151, 163, 168 in ℝ𝑃2 , 214 in Beltrami-Klein model, 238 in Euclidean space, 218 in hyperbolic space, 235 in Poincaré disc model, 240, 243 in Poincaré half-plane model, 245 in sphere, 201 in stereographic chart, 206 loop, 255 polygon, 171 Pregeodesic, 152 triangle, 171, 202 unitary, 152

395

Geodesically complete, 155 convex, 118 Geometric structure, 137, 230, 279, 362, 375 Soft, non-soft, 291 Geometry Cartan geometry, 260 Classical geometries, 258 Elliptic geometry, 259 Euclidean geometry, 259 Hyperbolic geometry, 259 Non-Euclidean, 259 Projective geometry, 259 Spherical geometry, 259 Germ of a function, 274 Graded ring, 105 Gradient, 107, 110, 149, 351 Green operator, 359, 360, 368 theorem, 115, 170 Grothendieck group, 𝐾-theory group, 46 Group Alternating group, 217 Dihedral group, 227 Discrete group, 191, 194, 249, 261 Free group, 63 Fuchsian group, 249 General linear group, 35, 208 Projective linear group, 208, 245 Simple group, 217 Special linear group, 35, 208 see also Lie, 189 see also fundamental, 57 see also homology, 85 see also homotopy, 56 see also orthogonal, 197 see also unitary, 211 Groupoid, 4 Hadamard theorem, 176 Hamilton theorem, 382 Handle, 49 Harmonic conjugate, 356, 372 form, 355, 358, 360, 386 function, 356, 361 Heat equation, 372, 377 backward, 377 Helicoidal movement, 231 Hermitian Almost Hermitian, 286 manifold, 286 metric, 284, 286, 337, 363 metric, indefinite, 241 product, 211 Standard Hermitian metric, 284 Hessian, 323, 339, 361, 386

396

Hilbert basis, 352 Nullstellensatz, 297 space, 352, 387 Hodge diamond, 292 manifold, 299, 337 number, 282, 292 star operator, 350, 362, 386 theorem, 291, 355, 359 theory for orbifolds, 371 Holomorphic function, 266, 272 Locally holomorphic set, 299 map, 269, 272, 334 parametrization, 335 Sheaf of holomorphic functions, 272 Holonomy, 146, 202, 369 group, 146, 338 of 𝔼𝑛 , 217 Homogeneous geometry, 191, 260 metric, 181, 184, 261, 374 polynomial, 297 space, 190, 213, 249, 294 Homological algebra, 89 Homology, 84 class, 85 functor, 119 group, 85 of chain complex, 90 Simplicial homology, 95 Singular homology, 84 Singular homology of smooth chains, 119 Homothetic metrics, 203, 364, 379 Homothety, 172, 205, 259 Classical homothety, 210 in hyperbolic plane, 242 Homotopy, 16, 54 category, 54, 88 equivalence, 55 groups, 56 groups for a cover, 71 groups of spheres, 66 groups of surfaces, 254 Homotopic, homotopic relative to, 54 inverse, 55 Pointed homotopy category, 54 Same homotopy type, 55 Weak homotopy equivalence, 99 Weak homotopy type, 56 Hopf map, 213, 261 for ℂ𝑃𝑛 , 293 Hopf-Rinow theorem, 155 Horocycle, 261 Hurewicz theorem, 87 Hurwitz formula, 308, 325

Subject Index

Hyperbolic plane, Beltrami-Klein model, 236 plane, Poincaré disc model, 239 plane, Poincaré half-plane model, 243 polygon, 251 space, 191, 232, 233, 245 trigonometry, 262 Hyperboloid, 233 Hypersurface, 300 Ideal Homogeneous ideal, 297 in a polynomial ring, 297 point in Beltrami-Klein model, 239 point in hyperbolic plane, 239 point in Poincaré disc model, 243 Radical ideal, 297, 339 triangle, 243 see also rotation, 246 Identity, 3 Immersion, 8 Implicit function theorem, 298 Incidence number, 95 Incident cells/polyhedra, 26 Index of a point of an orbifold, 176 Inflection, 310 point, 323, 333 Injectivity radius, 155 of hyperbolic space, 235 of sphere, 201 Integration, 113 by parts, 381 by quadratures, 314 of chains, 115 using Weierstrass function, 335 Interior point, 11 Intersection form, 130 index, 307, 311 product, 129 Invariance of dimension, 2, 122, 132 Inverse function theorem for Hilbert spaces, 368 of a loop, 60 of morphism, 4 Inversion, 210 Invertible matrix over ℤ, 223 morphism, 4 Irreducible curve, 303 polynomial, 300 projective variety, 300 Iso-areal, 260 Isometry, 149 group, 189 in Beltrami-Klein model, 238

Subject Index

397

in Poincaré disc model, 240, 261 Linear isometry, 149 Local isometry, 149 of ℍ2 , 245 of ℝ𝑃2 , 214 of 𝑆2 , 200, 211 of a flat torus, 226 of Euclidean space, 218 of hyperbolic space, 234 orientation reversing of ℍ2 , 261 Oriented isometry, 228 Isomorphism, isomorphic, 4 Isotopy, 16, 257 Isotropic, 181, 188, 189, 200, 215, 218, 234, 235, 374 Isotropy group, 81, 190, 213

in Beltrami-Klein model, 239 in Poincaré disc model, 243 Line in projective plane, 306 Straight line, 258 Liouville theorem, 211, 315, 328, 337 Locally homogeneous, 181, 185, 260 isotropic, 181, 185, 187, 188 linearizable action, 81, 176 symmetric, 181, 185, 189 Long line, 47, 133 Loop, 60 Lorentz metric, Lorentz manifold, 232 Lowering index, 149 Loxodrome, 260

𝐽-invariant, 327, 335 Jacobian, 36 Jordan curve, Jordan arc, 18, 26 Jordan-Schönflies theorem, 18 Juxtaposition, 4, 60

Manifold, 1 𝐶 𝑟 manifold, 9, 158 𝑛-manifold, 2 Analytic manifold, 9 Complex manifold, 271 Differentiable, smooth manifold, 7 PL manifold, 25 Smooth manifold with boundary, 12 Topological manifold, 1 with boundary, 11 with corners, 110, 133, 228 Mapping class group, 16, 257 torus, 15, 48 Maximum principle, 337 Mayer-Vietoris exact sequence, 92, 112, 122 property, 117 Mean value property, 356 Measurable set, 113 Meromorphic function, 268, 330, 333 of several variables, 269 on a elliptic curve, 327 on a manifold, 273 Metric 𝐿2 metric on 𝑠-forms, 352 on 𝑠-forms, 350 Space of metrics, 149, 193, 374 Standard metric, 149, 151, 217 see also Hermitian, 284 see also Riemannian, 147 Minimal submanifold, 385 Minimizing sequence, 366 Minkowski space, 232 Möbius band, 15, 39, 44 Extended Möbius group, 209 group, 208 map, 208 Moduli

Kähler ℂ𝑃𝑛 is Kähler, 296 form, 289 integral Kähler class, 299 manifold, 289, 362 Klein bottle, 15, 24, 44 Square Klein bottle, 231 surface, 386 Knot, 304 Kodaira embedding theorem, 299, 339 Kondrachov-Rellich lemma, 359, 367 Kronecker delta, 138 Laplacian, 166 for conformal metrics, 356 Geometric Laplacian, 355 holomorphic formula, 357 on forms, Laplace-Beltrami operator, 355 on functions, 355 on the torus, 371 Standard Laplacian, 355, 372 Lattice, 220, 316, 318 Laurent series, 268 Leibniz rule, 106, 144, 274 Lens space, 56, 216 Levi-Civita connection, 150, 151 Lie algebra, 190 bracket, 143, 192 derivative, 143, 192, 351 group, 189, 190, 198 Lift of a path, of a map, 70 Limiting parallel, 239, 259

398

of complex tori, 322, 335 of cubic curves in standard form, 325, 336 of cubic curves in Weierstrass form, 326, 335 of flat Klein bottles, 230 of flat tori, 222, 322 of oriented flat tori, 228 space, 223, 325 Morphism, 3 Morse theory, 375 Multi-index, 105 Multiplicity of a non-regular point, 126 of a tangency, 310 Natural transformation, 117 Newlander-Nirenberg theorem, 278, 301, 347 for compact surface, 371 Nijenhuis tensor, 278, 301, 338 Node, nodal point, 304, 308, 309 Non-degenerate (0, 2)-tensor, 232 2-form, 285 bilinear map, 116, 124 cube, 85 Non-orientable, 15 surface, 230, 254 Norm of vector, 147 Normal, see unitary, 161 Object, 3 Final object, 45 Initial object, 45 Zero object, 89 Octonions, 280 Operator Bounded operator, 353 Compact operator, 359 Differential operator, 353, 375 Domain of definition, 353 Linear, 353 Linear elliptic, 358, 368 Non-linear elliptic, 363 Positive, 368 Regularizing operator, 360 Self-adjoint, 357, 359, 368 Semi-elliptic, 385 Seminegative, 357 Semipositive, 357 Unbounded operator, 353 Orbifold, 82, 176, 349 𝑛-orbifold, 176 almost complex structure, 350 atlas, 177 chart, 177 Differentiable orbifold, 177 Gaussian curvature, 180 Hodge theory for, 362

Subject Index

Integral on orbifold, 180 Orbifold cover, 178 Orbifold Euler-Poincaré characteristic, 178 Orbisurface, 177, 249 partition of unity, 193 Riemannian, 369 Riemannian orbifold metric, 179 tensor, 177 Topological orbifold, 176 Orbit, 72 Order of a pole, 268 of a singularity, 303 of a zero, 267 Orientation, 132 Compatible orientations, 38, 40 Induced orientation, 37, 95 of a 𝑛-polyhedron, 37 of a vector space, 35 Orientable manifold, 35, 189 preserving map, 41, 125, 267 reversing map, 41, 125, 268 Oriented chart, positive chart, 36 cover, 76, 364 manifold, 35 orbisurface, 177 polyhedron, 95 surface, 346 Orthogonal Affine orthogonal group, 218 frame bundle, 189 group, 189, 197 in Beltrami-Klein model, 238 Special orthogonal group, 199 see also coordinates, 165 Orthonormal basis, 198, 284 Outward pointing vector, 37 Parallel, 258 in Beltrami-Klein model, 239 lines, 239 transport, 145, 187 vector field, 145 see also limiting parallel, 239 Parametrization by arc length, arc length parameter, 147 Global parametrization of a conic, 312 of a nodal cubic, 313 Partition of unity, 114, 143, 283 see also orbifold, 193 Path, 60 Path connected components, 56 Path connected space, 64 Perfect group, 216 pairing, 116, 123

Subject Index

Periodic, Λ-periodic, doubly periodic, 328, 330 Pillowcase, 83, 179, 180 PL, Piecewise linear manifold, 25 map, 25 structure, 20, 21, 25 Planar model of a curve, 306, 308 representation of a surface, 29 Poincaré conjecture, 191, 216 dodecahedral space, 216 duality, 123, 361 estimate, 366 lemma, 88, 110, 121, 283 Pointed spaces, 53 union, 15, 57 Polar line respect to a conic, 215 Pole of a meromorphic function, 268, 328 respect to a conic, 215, 238 Polygon, 29, 257 see also geodesic, 171 see also hyperbolic, 251 Polyhedron, 𝑘-polyhedron, 21 Curvilinear polyhedron, 22 Potential, 110 Preissman theorem, 176 Presentation of a group, 63 Presheaf, see sheaf, 6 Principal curvature, 163 direction, 163 Product, 62 Amalgamated, 63 Free product, 63 Projection, 62 Archimedes projection, 260 in projective space, 302, 308 Mercator projection, 260 Stereographic projection, 203, 312 Projective complex linear group, 208 matrix, 212 real linear group, 214 Real projective plane, 39, 213 Real projective space, 14 see also complex, 292 see also variety, 297 Proper homotopy, 121 map, 121 see also action, 72 Pseudoholomorphic function, 280 map, 277

399

Pseudotriangulation, 22 Pullback in a category, 62 of forms, 105 of tensors, 140 Pushforward, 140 Pushout, 63 Quaternions, 280 Quotient category, 54 functor, 54 map, 12 topology, 12 Raising index, 149 Ramification index, 81, 308 point, 81 value, 81 Rank of Abelian group, 100 of lattice, 220 Rational curve, 313 Rationalization, 100 Reflection, 210 Central reflection, 182 Glide reflection, 220, 230 Glide reflection (hyperbolic), 261 Regular cover, 73 curve, 144 point of a map, 125 point, smooth point, 300, 303 triangulation, 22 value, 81, 125, 149 Regularizing property, 360, 377 Relation in a group, 64 Residue, 268 𝑛th residue map, 331 Resolution Acyclic resolution, 283 Injective resolution, 283 Retract, 55 Deformation retract, 55, 89 Ricci flow, 384 Normalized Ricci flow, 384 soliton, 387 tensor, 160, 375, 384, 385 Riemann mapping theorem, 315 sphere, 207 surface, 300, 364 zeta function, 337 Riemann-Roch theorem, 331 Riemannian

400

cover, 149, 188 distance, 148 Inextensible, 155 manifold, 147 metric, 147, 287 open immersion, 149 orbifold metric, 179 submanifold, 155 volume form, 148, 350 see also curvature, 157 see also surface, 164 Rotation, 210 Central rotation (hyperbolic), 245 Ideal rotation, 246 Rotational, 107, 110, 351 Sard theorem, 125 Saturated set, 12 Scalar curvature, 160, 255, 375, 384 Extension of scalars, 99 see also flow, 384 Sectional curvature, 159 Holomorphic sectional curvature, 338 Seifert-van Kampen theorem, 64 Semi-Riemannian, 232, 375 Semigroup, 46 Serre duality, 362 Shape operator, 162 Sheaf, 6 Acyclic sheaf, 283 Canonical sheaf, 292, 332 cohomology, 119, 282 Fine sheaf, 283 Injective sheaf, 282 Morphism of sheaves, 282 of continuous functions, 6 of smooth functions, 7 Presheaf, 6 Restriction of a sheaf, 6 Section of a sheaf, 6 Structure sheaf, 272 Short time existence, 383 Signature of a manifold, 124 of a semi-Riemannian metric, 232 Simplex, 𝑘-simplex, 22, 89 Simplicial chains, 95 complex, 22 homology, 95 map, 25 Simply connected space, 2, 61, 188 Single ratio, 324 Singleton, 5, 55 Singular cohomology, 120

Subject Index

homology, 84 locus, 176 point of an orbifold, 176 point, non-smooth point, 300, 303 𝑘-skeleton, 98, 99 Smooth 𝑘-cubes, 115 atlas, 8 Exotic smooth structure, 10, 20 function, 7 manifold, 7 manifold with boundary, 12 map, 8 point of an orbifold, 176 structure, 7 Sobolev embedding theorem, 353 norm, 𝑊 𝑘,2 norm, 352 Solution Barrier solution, 381, 382 Fundamental solution, 377 Static solution, 378, 379 Supersolution, subsolution, 381, 382 Space form, 190 Simply connected space form of curvature −1, 235 of curvature 1, 203 of vanishing curvature, 218 Spacetime, 232 Spectral theorem, 358 Speed, 151 Sphere, 1, 191 2-sphere, 22 Milnor spheres, 20 Round sphere, 199, 346 Stokes theorem, 114, 337 Subdivision Barycentric subdivision, 24 operator, 92 Submanifold, 8, 155 Support, 120 Surface Non-orientable surface of genus 𝑘/2, 32 Orientable surface of genus 𝑔, 32 Parametrized surface, 160 Riemannian surface, 164, 301 Smooth surface, 373 Subsurface, 39 Topological surface, 26, 68 Triangulated surface, 22, 26, 41 Suspension, 47 Symbol of a differential operator, 358 Symmetric bilinear map, 147 connection, torsion free, 144 multilinear map, 140

Subject Index

space, 181, 184, 190, 261 tensor, 140, 147 Symplectic, 285 form, 290, 338 geometry, 290 manifold, 290 Synge theorem, 176 Tacnode, 304 Tangent bundle, 138, 153 cone, 303 Holomorphic tangent space, 274 Ordinary tangent, 310 space, 35, 138 to a projective variety, 300 Taylor series, 267 Teichmüller space, 257, 262, 321, 374 Tensor (𝑟, 𝑠)-tensor, 138 Cotton tensor, 346 Curvature tensor, 157 product, 99, 137 Ricci tensor, 160 Weyl tensor, 160, 346 Tessellation, 250 hyperbolic, 255 Klein bottle, 75 of the plane, 227 Thom form, 129 Thurston geometries, 260 geometrization program, 191, 385 Torus, 15 see also complex, 317 see also flat, 222 Translation along a geodesic, 246, 247, 256 in Euclidean plane, 210, 220 Transversal intersection, 129, 307 Triangulation, 20, 21 Trisecant, 305 Uniform convergence, 330 Uniformization theorem, 315, 371, 373 Uniformly bounded, 369 Unique factorization domain, 300 Unitary group, 211, 294 normal, 161 Special unitary group, 211 Universal cover, 71, 315 property, 46, 64 Universe, 226, 375 Variational method, 366, 387

401

Variety, 299 Abelian variety, 337 Projective variety, 297 Quasi-projective variety, 299 Secant variety, 302 Smooth projective variety, 297 Tangent variety, 302 see also algebraic, 10 Vector field, 138 Vertex, 21 Volume, 114 form of Hermitian manifold, 288 form of Riemannian manifold, 113 Weak derivative, 352, 360 topology, 99 Wedge of spaces, pointed union, 15, 57 product, 105, 108 Weierstrass ℘ function, 329 see also cubic, 326 Weighted projective line, 326, 340 Weingarten operator, 162 Whitney embedding theorem, 50, 299 Wirtinger operators, 266, 269 Word, 31, 63 Canonical, 32, 253 Empty word, 32 Zariski closure, 302 open subset, 300 topology, 10, 300 Zero measure set, 113, 125 of a holomorphic function, 267

Symbols

𝐵𝜖𝑛 (𝑝), 𝐵 𝑛 , open ball, 𝑆𝜖𝑛−1 (𝑝), 𝑆 𝑛−1 , sphere, (𝑈, 𝜑), chart, 𝒜 = {(𝑈𝛼 , 𝜑𝛼 )}, atlas, dim 𝑀, dimension of a manifold, 𝒞, category, Obj(𝒞), objects of 𝒞, Mor𝒞 (𝑋, 𝑌 ), morphisms of 𝒞, 1𝑋 , Id, identity, 𝑔 ∘ 𝑓, composition, 𝑓

𝑓 ∶ 𝑋 → 𝑌 , 𝑋 ⟶ 𝑌 , 𝑋 → 𝑌 , morphism, 𝐒𝐞𝐭, category of sets, 𝐓𝐨𝐩, 𝐓𝐨𝐩∗ , categories of topological spaces, 𝑛 𝑛 𝐓𝐌𝐚𝐧, 𝐓𝐌𝐚𝐧 , 𝐓𝐌𝐚𝐧𝑐 , 𝐕𝐞𝐜𝐭𝐤 , 𝐆𝐫𝐨𝐮𝐩, 𝐀𝐛𝐞𝐥, Π1 (𝑋), fundamental groupoid, Ω𝑥,𝑦 (𝑋), paths from 𝑥 to 𝑦, 𝑓−1 , inverse of 𝑓, 𝑋 ≅𝒞 𝑌 , 𝑋 ≅ 𝑌 , isomorphism, 𝕃𝒞 , classification list of 𝒞, ⋆, singleton, 𝐹 ∶ 𝒞1 → 𝒞2 , functor, 𝑓∗ = 𝐹(𝑓), covariant functor, 𝑓∗ = 𝐹(𝑓), contravariant functor, 𝐎𝐩𝐞𝐧(𝑋), category of open subsets, 𝑠|𝑉 , restriction, 𝐶 0 , sheaf of continuous functions, 𝐶 0 (𝑋, 𝑌 ), continuous maps, 𝐶 ∞ , smooth functions, (𝑀, S), smooth structure, 𝐶 ∞ (𝑀, 𝑁), smooth maps,

1 1 1 1 2 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 6 6 6 7 7 7 8

𝑛

𝑛

𝐃𝐌𝐚𝐧, 𝐃𝐌𝐚𝐧 , 𝐃𝐌𝐚𝐧𝑐 , 𝐶 𝜔 , real analytic functions, 𝑐𝑜 𝕃𝒞 , list of connected spaces, ℝ𝑛+ = {(𝑥1 , . . . , 𝑥𝑛 ) ∈ ℝ𝑛 |𝑥1 ≥ 0}, 𝐵𝑟+ (0), semiball, Int(𝑀), interior, 𝜕𝑀, boundary, 𝑛 𝑛 𝑛 𝐓𝐌𝐚𝐧𝜕 , 𝐓𝐌𝐚𝐧𝜕,𝑐 , 𝐃𝐌𝐚𝐧𝜕 , 𝑛 𝐃𝐌𝐚𝐧𝜕,𝑐 , ∼, equivalence relation, 𝑋 ⋆ = 𝑋/∼, quotient space, 𝑋/𝑍, collapsing, ℝ𝑃𝑛 , projective space, [𝑥0 , . . . , 𝑥𝑛 ], point of projective space, 𝑋 ∪𝑓 𝑌 , attaching, 𝑋 ∨ 𝑌 , wedge or pointed union, 𝑇𝑓 , mapping torus, Mob, Möbius band, Kl , Klein bottle, 𝑀1 #𝑓 𝑀2 , 𝑀1 #𝑀2 , connected sum, MCS(𝑋, 𝑌 ), MCG(𝑋), mapping class group, |𝐼|, cardinal of 𝐼, 𝑃𝑘 , 𝑘-polyhedron, Int 𝑃 𝑘 , interior of polyhedron, 𝐶𝛼𝑘 , 𝑘-cell, 𝜏 = {(𝑃𝛼𝑘 , 𝑓𝛼 )}, triangulation, ℝ𝑃2 , projective plane, 𝑇 2 , torus, 𝑛 𝑛 𝐏𝐋𝐌𝐚𝐧 , 𝐏𝐋𝐌𝐚𝐧𝑐 , P𝑆 , planar representation of

8 9 9 11 11 11 11 12 12 12 13 14 14 15 15 15 15 16 16 16 20 21 21 21 21 22 23 25

403

404

Symbols

a surface, 𝐩𝑆 , word for a surface, 𝐩1 ≈ 𝐩2 , equivalence of words, Σ𝑔 , orientable surface of genus 𝑔, 𝑋𝑘 , non-orientable surface of genus 𝑘/2, 𝜒(𝑋), Euler-Poincaré characteristic, GL(𝑉), GL(𝑛, ℝ), SL(𝑛, ℝ), Or(𝑉), orientations of a vector space, (𝑀, 𝑜), oriented manifold, 𝒜+ , positive atlas, 𝑜|𝜕𝑀 , induced orientation, ∅, empty word, 2 𝐏𝐋𝐌𝐚𝐧𝜕,𝑐 , 𝐑𝐢𝐧𝐠, category of rings, 𝐾(𝐺), Grothendieck group, S(𝑋), suspension of 𝑋, ± GL (𝑛, ℝ), Mob𝑛 , Kl 𝑛 , Map(𝑋, 𝑌 ) = Mor𝐓𝐨𝐩 (𝑋, 𝑌 ), continuous maps, Map∗ (𝑋, 𝑌 ) = Mor𝐓𝐨𝐩∗ (𝑋, 𝑌 ), pointed maps, 𝐓𝐨𝐩𝐏, category of pairs, (𝑋, 𝐴), pair of spaces, 𝑓0 ∼ 𝑓1 , homotopic maps, 𝒞, quotient category, 𝐇𝐨𝐓𝐨𝐩, 𝐇𝐨𝐓𝐨𝐩∗ , homotopy category, [𝑋, 𝑌 ] = Map(𝑋, 𝑌 )/∼, [𝑋, 𝑌 ]∗ = Map∗ (𝑋, 𝑌 )/∼, 𝑋 ∼ 𝑌 , same homotopy type, 𝑛 𝑛 𝐇𝐨𝐌𝐚𝐧𝑐 , 𝐇𝐨𝐌𝐚𝐧𝑐,∗ , 𝑛 𝜋𝑛 (𝑋, 𝑥0 ) = [𝑆 , 𝑋]∗ , homotopy groups, ℎ𝑍 (𝑌 ) = Mor𝒞 (𝑍, 𝑌 ), 𝜋0 (𝑋), path connected components, 𝜋1 (𝑋, 𝑥0 ), fundamental group, 𝑆 𝑛± = 𝑆 𝑛 ∩ {±𝑥𝑛 ≥ 0}, hemispheres, Ω(𝑋, 𝑥0 ), space of loops, 𝛾1 ∗ 𝛾2 , juxtaposition,

29 31 31 32

𝛾(𝑡) = 𝛾(1 − 𝑡), reverse loop, 𝑋1 ×𝑌 𝑋2 , 𝑋1 × 𝑋2 , product, pullback, 𝑋1 ⋆𝑌 𝑋2 , 𝑋1 ⋆ 𝑋2 , coproduct, pushout, 𝐺 1 ⋆𝐻 𝐺 2 , 𝐺 1 ⋆ 𝐺 2 , amalgamated/free product, 𝐹(𝑎𝑖 |𝑖 ∈ 𝐼), 𝐹𝑟 = 𝐹(𝑎1 , . . . , 𝑎𝑟 ), free group, ⟨⟨𝑅𝑗 |𝑗 ∈ 𝐽⟩⟩, normal subgroup generated by,

60 62 63

←

32 33 35 35 36 36 37 42 45 46 46 48 48 49 53 53 54 54 54 54 55 55 55 55 56 56 56 56 57 58 60 60

63 64 64

⟨𝑎𝑖 , 𝑖 ∈ 𝐼|𝑅𝑗 (𝑎𝑖 ), 𝑗 ∈ 𝐽⟩, presentation of a group, 𝑊𝑟 = 𝑆 1 ∨ 𝑆 1 ∨ ⋯ ∨ 𝑆 1 , bouquet, [𝑔, ℎ] = 𝑔ℎ𝑔−1 ℎ−1 , commutator, [𝐺, 𝐺], commutator subgroup, Ab(𝐺) = 𝐺/[𝐺, 𝐺], Abelianization, 2 𝐇𝐨𝐌𝐚𝐧𝜕,𝑐 , ′ 𝜋 ∶ 𝑋 → 𝑋, cover, 𝐻(𝑋 ′ ,𝜋) < 𝜋1 (𝑋, 𝑝), group of the cover, 𝜋̃ ∶ (𝑋,̃ 𝑝)̃ → (𝑋, 𝑝), universal cover, 𝐂𝐨𝐯(𝑋, 𝑝), 𝐂𝐨𝐯(𝑋), covers of 𝑋, 𝐒𝐮𝐛𝐠𝐫(𝐺), category of subgroups of 𝐺, Homeo(𝑋), group of homeomorphisms, [𝑝] = {𝑔 𝑥|𝑔 ∈ Γ}, orbit of an action, 𝑋/Γ, quotient by an action, Deck(𝜋), deck transformations, |𝐺 ∶ 𝐻|, index of a subgroup, 𝜋̂ ∶ 𝑀̂ → 𝑀, oriented cover, ± Homeo (𝑀), 𝑚𝑞 , ramification index (for ramified cover), 𝜎 ∶ [0, 1]𝑘 → 𝑋, 𝑘-cube, 𝑄𝑘 (𝑋), 𝐷𝑘 (𝑋), 𝐶𝑘 (𝑋), 𝑘-chains, 𝐹𝑖 𝜎, 𝐵𝑖 𝜎, front and back 𝑖-faces, 𝜕 ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘−1 (𝑋), boundary map, (𝐶• (𝑋), 𝜕), 𝐶• (𝑋) = ⨁𝑘 𝐶𝑘 (𝑋), 𝑍𝑘 (𝑋), 𝑘-cycles, 𝐵𝑘 (𝑋), 𝑘-boundaries, 𝐻𝑘 (𝑋) = 𝑍𝑘 (𝑋)/𝐵𝑘 (𝑋), singular homology, (𝐴• , 𝜕), chain complex, 𝑓 ∶ (𝐴• , 𝜕) → (𝐵• , 𝜕), chain morphism, 𝐂𝐡𝒜 , category of chain complexes on 𝒜, 𝐻𝑘 (𝐴• , 𝜕), homology of chain complex, 𝑓

𝑔

𝐴 ⟶ 𝐵 ⟶ 𝐶, exact sequence, 𝐶𝑘 (𝑋; 𝔘), 𝐻𝑘 (𝑋; 𝔘), small cubes, div ∶ 𝐶𝑘 (𝑋) → 𝐶𝑘 (𝑋), subdivision operator, 𝑒𝛼𝑘 = (𝑃𝛼𝑘 , 𝑜𝛼𝑘 ), oriented polyhedron, [𝑒𝛼𝑘 ∶ 𝑒𝑘−1 𝛽 ], incidence number, 𝐶𝑘𝜏 (𝑋), simplicial chains,

64 64 66 67 67 68 69 70 71 71 71 72 72 72 72 73 76 78 81 85 85 85 85 85 85 85 85 90 90 90 90 91 92 92 95 95 95

Symbols

𝑍𝑘𝜏 (𝑋), 𝐵𝑘𝜏 (𝑋), 𝐻𝑘𝜏 (𝑋), simplicial homology, 95 [𝑀], fundamental class, 97 𝑋𝑘 , 𝑘-skeleton, 98 𝐻𝑘𝐶𝑊 (𝑋), cellular homology, 99 𝑀 ⊗𝐴 𝑁, 𝑀 ⊗ 𝑁, tensor product, 99 Mod𝐴 , category of 𝐴-modules, 99 𝑀ℚ = 𝑀 ⊗ ℚ, rationalization, 100 rank 𝑀, rank of Abelian group, 100 𝐻𝑘 (𝑋, 𝑅), homology with coefficients in 𝑅, 101 𝑏𝑘 (𝑋), Betti number, 103 Ω𝑘 (𝑀), 𝑘-forms, 105 𝑑𝑥𝐼 = 𝑑𝑥𝑖1 ∧ ⋯ ∧ 𝑑𝑥𝑖𝑘 , for multi-index 𝐼 = {𝑖1 , . . . , 𝑖𝑘 }, 105 ∧, wedge of forms, 105 GrAlgℝ , category of graded algebras, 106 𝑑 ∶ Ω𝑘 (𝑀) → Ω𝑘+1 (𝑀), exterior differential, 106 𝐻 𝑘 (𝐴• ), cohomology of cochain complex, 107 𝐜𝐨𝐂𝐡𝒜 , category of cochain complexes on 𝒜, 108 (Ω• (𝑀), 𝑑), Ω• (𝑀) = ⨁𝑘 Ω𝑘 (𝑀), 108 𝑘 𝐻𝑑𝑅 (𝑀), de Rham cohomology, 108 𝐇𝐨𝐃𝐌𝐚𝐧, homotopy smooth category, 111 supp(𝜔), support of a form, 113 ∫𝑀 𝜔, integral of an 𝑛-form, 113 {𝜌𝛼 }, partition of unity, 114 Vol(𝑅), area(𝑆), volume and area, 114 𝐻𝑘𝑠𝑚 (𝑀, ℝ), homology of smooth 𝑘-cubes, 116 𝜏 ∶ 𝐹 → 𝐺, natural transformation of functors, 118 Ω𝑘𝑐 (𝑀), compactly supported forms, 120 𝐻𝑐𝑘 (𝑀), cohomology with compact support, 120 deg 𝑓, degree of a map, 124 𝑛−𝑘 Ψ ∶ 𝐻𝑘 (𝑀, ℝ) ≅ 𝐻𝑑𝑅 (𝑀), Poincaré duality, 127 𝜈𝑇 , Thom form, 127 𝑄𝑀 , intersection form, 130 𝑉1 ⊗ ⋯ ⊗ 𝑉 𝑘 , tensor product, 137 𝑣1 ⊗ ⋯ ⊗ 𝑣 𝑘 , 137 𝑇𝑝 𝑀, tangent space, 138

405

𝑇𝑝∗ 𝑀, cotangent space, 138 𝑑𝑝 𝑓 = 𝑓∗ ∶ 𝑇𝑝 𝑀 → 𝑇𝑓(𝑝) 𝑁, differential of 𝑓, 138 𝑇𝑝𝑟,𝑠 𝑀, (𝑟, 𝑠)-tensors at 𝑝 ∈ 𝑀, 138 𝛿𝑖𝑗 , Kronecker delta, 138 𝔛(𝑀), vector fields, 138 𝜕 𝜕𝑥𝑖 = 𝜕𝑥 , tangent vector, 138 𝑖 𝒯 𝑟,𝑠 (𝑀), (𝑟, 𝑠)-tensors, 138 𝑑𝑥𝑗1 ⋅ . . . ⋅ 𝑑𝑥𝑗𝑘 , symmetric tensor, 140 (𝑟,𝑠)

(𝑟−1,𝑠−1)

𝐶𝑎𝑏 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝑀, contraction, 140 𝜑∗ (𝑇), pushforward of tensor, 140 𝜑∗ (𝑇 ′ ), pullback of tensor, 140 ∇, connection, 141 Γ𝑖𝑗𝑘 , Christoffel symbols, 142 𝒞(𝑀), space of connections, 143 𝜏∇ (𝑋, 𝑌 ), torsion of ∇, 143 [𝑋, 𝑌 ], Lie bracket, 143 𝐿𝑋 (𝑇), Lie derivative, 143 𝐷 , covariant derivative, 144 𝑑𝑡 𝔛(𝑐), 𝒯 𝑟,𝑠 (𝑐), vector fields and tensors along a curve, 144 𝔛∥ (𝑐), parallel vector fields, 145 ev𝑡0 (𝑋) = 𝑋(𝑡0 ), evaluation map, 145 𝑡 ,𝑡 𝑃𝑐 0 1 , parallel transport, 145 𝑝 Hol∇ , holonomy group, 146 g, Riemannian metric, 147 (𝑀, g), Riemannian manifold, 147 ⟨𝑋, 𝑌 ⟩ = g(𝑋, 𝑌 ), 147 ∥𝑣∥= √⟨𝑣, 𝑣⟩, norm of vector, 147 ∠(𝑣, 𝑤), angle of two vectors, 147 ℓ(𝑐), length of a curve, 147 𝑑(𝑝, 𝑞), Riemannian distance, 148 𝜈 = √det(𝑔𝑖𝑗 ) 𝑑𝑥1 ∧ ⋯ ∧ 𝑑𝑥𝑛 , Riemannian vol, 𝑋♭ = ∑ 𝑋 𝑖 𝑔𝑖𝑗 𝑑𝑥𝑗 , lower index, 𝛼♯ = ∑ 𝛼𝑗 𝑔𝑖𝑗 𝜕𝑥𝑖 , raise index, ↓ 𝑇, ↑ 𝑇, lowering/raising an index, ℳet(𝑀), space of Riemannian metrics, 𝑛 𝐑𝐢𝐞𝐦 , category of Riemannian 𝑛-manifolds, Isom(𝑀, g), Isom(𝑀), isometry group, g𝑠𝑡𝑑 = 𝑑𝑥12 + ⋯ + 𝑑𝑥𝑛2 , standard metric, 𝐭𝛼 = 𝛼′ / ∥𝛼′ ∥, tangent vector,

148 148 148 149 149 149 149 151 151

406

𝐚 = 𝐚𝑡 + 𝐚𝑛 , acceleration (tangential/normal), 151 𝛾𝑝,𝑣 (𝑡), geodesic, 152 𝑇𝑀 = ⨆𝑝∈𝑀 𝑇𝑝 𝑀, tangent bundle, 153 exp(𝑝, 𝑣), exp𝑝 (𝑣), exponential map, 154 𝐵𝜀𝑑 (𝑝), Riemannian ball, 154 𝐈, first fundamental form, 155 (−)𝑇 ∶ 𝑇𝑝 𝑀 → 𝑇𝑝 𝐻, tangent projection, 155 (−)𝑁 ∶ 𝑇𝑝 𝑀 → (𝑇𝑝 𝐻)⟂ , normal projection, 155 𝐈𝐈, second fundamental form, 156 𝑅(𝑋, 𝑌 )𝑍, Riemannian curvature, 157 ∥𝑢 ∧ 𝑣∥2 =∥𝑢∥2 ∥𝑣∥2 −⟨𝑢, 𝑣⟩2 , 159 𝐾𝑝 (𝜎), sectional curvature, 159 Ric(𝑢, 𝑣), Ric(𝑢), Ricci curvature, 160 Scal, Scalar curvature, 160 𝑊, 𝑊 ± , Weyl tensor, 160 P, parametrization of a surface, 160 𝜕𝑥 𝑥ᵆ = 𝜕ᵆ , derivative, 160 𝐸, 𝐹, 𝐺, first fundamental form of a surface, 161 𝐍𝑝 , unitary normal, 161 𝜅𝑆 (𝑝), Gaussian curvature, 161 𝒮𝑝 = −𝑑𝑝 𝐍, Weingarten or shape operator, 162 𝑒, 𝑓, 𝑔, second fundamental form of a surface, 163 𝐭𝛼 , 𝐧𝛼 , 𝐍, Darboux frame, 163 𝐤𝛼 , 𝐤𝑔 , 𝐤𝑛 , curvature (geodesic/normal), 163 Δ, Laplacian, 166 𝐶𝑟 (𝑝), geodesic circle, 167 𝑘𝑔 , geodesic curvature, 168 Γ𝑝 , group at an orbifold point, 176 𝑆𝑋 , singular locus of an orbifold, 176 𝑠𝑝 =|Γ𝑝 |, index of an orbifold point, 176 Γ𝑝 = {𝑔 ∈ Γ|𝑔 𝑝 ̃ = 𝑝}, ̃ isotropy group, 176 𝜒𝑜𝑟𝑏 (𝑋), orbifold Euler-Poincaré characteristic, 178 𝑂𝑀, orthogonal frame bundle, 189 ± Isom (𝑀, g), 189 𝐺/𝐻, homogeneous space, 190 𝔤, 𝔥, 𝔪, Lie algebras, 190 𝐸𝑘𝑛0 , simply connected

Symbols

space form, 191 1 𝒜(𝑑𝑥𝑖1 ⊗ ⋯ ⊗ 𝑑𝑥𝑖𝑘 ) = 𝑘! 𝑑𝑥𝑖1 ∧ ⋯ ∧ 𝑑𝑥𝑖𝑘 , 192 O(𝑛), orthogonal group, 197 𝐴𝑡 , transpose of a matrix, 197 𝑀𝑛×𝑛 (ℝ), set of 𝑛 × 𝑛 matrices, 197 ± O (𝑛) = {𝐴 ∈ O(𝑛)|± det 𝐴 > 0}, 199 SO(𝑛), special orthogonal group, 199 𝕊𝑛 = (𝑆 𝑛 , 𝑔𝑆𝑛 ), round sphere, 199 ℂ = ℂ ∪ {∞}, Riemann sphere, 207 ℂ𝑃1 , complex projective line, 207 Mob(ℂ), Möbius group, 208 GL(𝑛, ℂ), SL(𝑛, ℂ), PGL(𝑛, ℂ), 208 ˆ Mob(ℂ),extended Möbius group, 209 − Mob (ℂ), 209 ℛ, set of generalized circles, 209 [𝑎 ∶ 𝑏 ∶ 𝑐 ∶ 𝑑], double ratio, 209 𝒯, 𝒫, ℋ, 𝒮, ℐ, translations, rotations, homotheties, reflections, inversions of ℂ, 210 U(𝑛), unitary group, 211 SU(𝑛), special unitary group, 211 𝐴∗ = 𝐴𝑡 , adjoint matrix, 211 PG, projective group, 212 PSL(𝑛, ℂ), PU(𝑛), PSU(𝑛), 212 PGL(𝑛, ℝ), PO(𝑛), 214 𝑟𝑝 , polar line for a conic, 215 𝑝𝑟 , pole of a line for a conic, 215 𝑐𝑜 𝕃𝐑𝐢𝐞𝐦𝑛 , manifolds of 𝐾≡𝑘0

constant curvature, 𝐿(𝑝, 𝑞), lens space, 𝐴𝑛 , alternating group, 𝔼𝑛 = (ℝ𝑛 , g𝑠𝑡𝑑 ), Euclidean space, ± AO(𝑛), AO (𝑛), ASO(𝑛), Λ, lattice, ΛΓ , lattice of a group of translations, Cyl𝑑 = 𝑆 1𝑑/2𝜋 × ℝ, Riemannian cylinder, ℛ, set of rank 2 lattices in ℝ2 , ℳ𝑇 2 , moduli of flat tori, 𝐻\𝑋, quotient by action on the left, GL(𝑛, ℤ), ℳ𝑐𝑦𝑙 , moduli of cylinders, ℳ𝑇𝑜𝑟2 , moduli of oriented flat tori, SL(𝑛, ℤ), ℳ𝑇𝑜𝑟,𝑑 2 , flat tori with fixed

216 216 217 218 218 220 220 222 223 223 223 223 228 228 228

Symbols

407

shortest geodesic, 𝜎ℓ,ᵆ , glide reflection, Mob𝑑 , Kl 𝑑,𝑙 , flat Möbius band, Klein bottle, ℳKl , moduli of Klein bottles, g𝐿 = −𝑑𝑥02 + 𝑑𝑥12 + ⋯ + 𝑑𝑥𝑛2 , Lorentz metric, ℍ𝑛 , hyperbolic space, O(𝑛, 1), orthogonal group for g𝐿 , ± O> (𝑛, 1), 2 ℍ𝐵𝐾 = (𝐵 2 , g𝐵𝐾 ), Beltrami-Klein model, PO(2, 1), ℍ2𝑃𝐷 = (𝐵 2 , g𝑃 ), Poincaré disc model, U(1, 1), SU(1, 1), PU(1, 1), PSU(1, 1), Mob(𝐴)={𝑓∈Mob(ℂ)|𝑓(𝐴)=𝐴}, Bihol(𝐴), biholomorphisms of 𝐴, 𝐻 = {(𝑢, 𝑣) ∈ ℝ2 |𝑣 > 0}, ℍ2𝑃𝐻 = (𝐻, 𝑔𝐻 ), Poincaré half-plane model, + PGL (2, ℝ), PSL(2, ℝ), ℳ𝑇𝑜𝑟,area=1 , oriented flat tori 2 of fixed area, 𝒯𝑔 , Teichmüller space, Hol(𝑈), holomorphic functions, 𝑓′ (𝑧0 ), complex derivative, 𝜕 1 𝜕 𝜕 = 2 ( 𝜕𝑥 − i 𝜕𝑦 ), 𝜕𝑧 𝜕 𝜕𝑧

=

1 2

𝜕

𝜕

( 𝜕𝑥 + i 𝜕𝑦 ),

∫𝛾 𝑓 𝑑𝑧, ord𝑧0 𝑓, order of a zero/pole, 𝚤 ∶ 𝑀𝑚×𝑛 (ℂ) ↪ 𝑀2𝑚×2𝑛 (ℝ), 𝑑𝑧 𝑓, 𝑑𝑧ℂ 𝑓, real/complex differential, 𝒪𝑀 , sheaf of holomorphic functions, Bihol(𝑀), biholomorphisms of 𝑀, 𝑛 𝑛 𝐂𝐌𝐚𝐧 , 𝐂𝐌𝐚𝐧𝑐 , complex manifolds, 𝒯𝑧 𝑀, holomorphic tangent space, 𝑇𝑧 𝑀ℂ = 𝑇𝑧1,0 𝑀 ⊕ 𝑇𝑧0,1 𝑀, Ω1 (𝑀)ℂ = Ω1,0 (𝑀) ⊕ Ω0,1 (𝑀), 𝑑𝑧𝑗 = 𝑑𝑥𝑗 + i𝑑𝑦𝑗 , 𝑑𝑧𝑗 = 𝑑𝑥𝑗 − i𝑑𝑦𝑗 , Ω𝑝,𝑞 (𝑀), (𝑝, 𝑞)-forms, 𝑑𝑧𝐼 = 𝑑𝑧𝑖1 ∧ ⋯ ∧ 𝑑𝑧𝑖𝑝 , 𝑑𝑧𝐽 = 𝑑𝑧𝑗1 ∧ ⋯ ∧ 𝑑𝑧𝑗𝑞 , 𝜕, 𝜕, holomorphic/anti-

229 230 230 230 233 233 234 234 237 238 239 241 242 242 244 244 245 250 257 266 266 266 267 267 270 270 272 272 274 274 275 276 276 276 276

holomorphic differential, 276 (𝑉, 𝐽), (𝑉, 𝑚), complex structure, 276 𝑉ℂ = 𝑉 1,0 ⊕ 𝑉 0,1 , complexification, 277 𝐽𝑠𝑡𝑑 , standard almost complex structure, 277 𝑁𝐽 , Nijenhuis tensor, 278 ℍ, 𝕆, quaternions, octonions, 281 (Ω𝑝,• (𝑀), 𝜕), Dolbeault complex, 282 𝐻 𝑝,𝑞 (𝑀), Dolbeault cohomology, 282 ℎ𝑝,𝑞 = dim 𝐻 𝑝,𝑞 (𝑀), Hodge numbers, 282 Ω 𝑝 , holomorphic 𝑝-forms, 284 h = ∑𝑗,𝑘 ℎ𝑗𝑘 𝑑𝑧𝑗 ⊗ 𝑑𝑧𝑘 , Hermitian metric, 286 𝜔 = −ℑ(h), fundamental form, 289 Ω 1 , canonical sheaf of a curve, 292 ℂ𝑃𝑛 , complex projective space, 292 𝐹 ̂ = 𝐹(1, 𝑤 1 , . . . , 𝑤𝑛 ), dehomogenization, 297 dimℂ 𝑋, complex dimension of variety, 300 𝑋 sing , 𝑋 reg , 300 𝐭𝑥 𝑋 ⊂ ℂ𝑃 𝑛 , tangent for projective variety, 300 Sec (𝐶), secant variety, 302 Tan (𝐶), tangent variety, 302 𝐭𝐜𝑝 𝐶, tangent cone, 303 CopSec (𝐶), secants with coplanar tangents, 305 TriSec (𝐶), trisecant variety, 305 𝐼𝑝 (𝐶1 , 𝐶2 ), intersection index of two curves, 307 ℚ, algebraic closure of ℚ, 315 𝔻 = {𝑧 ∈ ℂ| |𝑧|< 1}, 315 (I) (II) (III) 𝐂𝐌𝐚𝐧 , 𝐂𝐌𝐚𝐧 , 𝐂𝐌𝐚𝐧 , 316 ℳ𝑇ℂ2 , moduli of complex tori, 318 Λ𝜏 = ⟨1, 𝜏⟩, 319 𝒩𝑠𝑡𝑑 , moduli of cubics in standard form, 325 Δ, discriminant of a cubic, 326 𝒩 𝑊 , moduli of cubics in Weierstrass form, 326 ℂ𝑃[𝑑1 0 ,𝑑1 ] , weighted projective line, 326 𝐽𝑊 ∶ 𝒩 𝑊 → ℂ, 𝐽-invariant on Weierstrass form, 327 𝐽𝑠𝑡𝑑 ∶ 𝒩𝑠𝑡𝑑 → ℂ, 𝐽-invariant on standard form, 327 ℘(𝑧), Weierstrass elliptic function, 329

408

𝑆𝑚 =

Symbols

∑ 𝑤∈Λ−{0}

1 , 𝑤𝑚

330

℘′ (𝑧), derivative of ℘, 330 𝐿(𝑛𝑝0 ),meromorphic functions with pole at 𝑝0 , 330 𝑙(𝑛𝑝0 ) = dim 𝐿(𝑛𝑝0 ), 330 res 𝑛𝑝0 ∶ 𝐿(𝑛𝑝0 ) → ℂ, residue map, 331 𝐷 = ∑ 𝑛𝑖 𝑝𝑖 , divisor on a curve, 331 𝐾𝐶 , canonical divisor, 331 𝑔2 = 60 𝑆 4 , 𝑔3 = 140 𝑆 6 , elliptic invariants, 332 ℂ(𝐶), field of meromorphic functions on 𝐶, 334 𝑆𝑟 (Λ), 𝑔𝑟 (Λ), 335 𝑔𝑟 (𝜏) = 𝑔𝑟 (Λ𝜏 ), 335 𝐽(𝜏) = 𝐽(𝐶𝜏 ), 𝐽-invariant for 𝐶𝜏 = ℂ2 /Λ𝜏 , 335 𝜁(𝑧), Riemann zeta function, 337 Conf(𝑀, g), conformal maps, 344 [g], conformal class, 344 𝑛 𝐂𝐨𝐧𝐟 , category of conformal manifolds, 344 (𝑀, [𝑔]), conformal manifold, 344 Conf(𝑀, [g]), conformal group, 344 ℝ∗ ⋅ O(2𝑛), group of dilations, 348 ℳet(𝑋), set of orbifold Riemannian metrics, 349 ⋆ ∶ Ω𝑝𝑠 (𝑀) → Ω𝑝𝑛−𝑠 (𝑀), Hodge star operator, 350 𝑖𝑋 (𝛼), contraction of form with a vector field, 351 ⟨𝛼, 𝛽⟩𝐿2 = ∫𝑀 ⟨𝛼, 𝛽⟩𝜈, 352 2 ∥𝛼∥𝐿2 = √⟨𝛼, 𝛼⟩𝐿2 , 𝐿 norm, 352 ℋ, Hilbert space, 352 𝑘

1/2

∥𝛼∥𝑊 𝑘,2 = (∑𝑖=0 ∥∇𝑖 𝛼∥2𝐿2 ) , Sobolev norm, Ω𝑠𝑊 𝑘,2 (𝑀), Sobolev spaces of 𝑠-forms, 𝑃 ∶ 𝒯(𝑀) → 𝒯 ′ (𝑀), differential operator, 𝑃 ∗ , formal adjoint of operator 𝑃, Δg = 𝑑𝑑 ∗ + 𝑑 ∗ 𝑑, Laplacian, ℋ∆𝑠 g (𝑀), harmonic forms, 𝜎𝑃 (𝑥, 𝜉) = ∑|𝐼|=𝑘 𝐴𝐼 (𝑥)𝜉𝐼 , symbol of 𝑃, 𝐷ᵆ 𝐺, directional derivative of a functional, 𝜕g𝑡 = −2𝜅𝑡 g𝑡 , curvature flow, 𝜕𝑡 𝜅𝑡 , Gaussian curvature of g𝑡 , 𝜕ᵆ 𝜕𝑡

= Δ𝑢, heat equation,

352 352 353 354 355 358 358 367 375 375 377

𝑔 ∗ ℎ, convolution of two functions, 𝜈𝑡 , 𝐴𝑡 , volume form and area of g𝑡 , 𝜕g𝑡 𝜕𝑡

= −2(𝜅𝑡 − 𝑟𝑡 )g𝑡 , normalized curvature flow, 𝑟𝑡 , average of the curvature, 𝑓+ = max{𝑓, 0}, Ric𝑡 , Ricci curvature of g𝑡 ,

377 378 379 379 382 384

This book represents a novel approach to differential topology. Its main focus is to give a comprehensive introduction to the classification of manifolds, with special attention paid to the case of surfaces, for which the book provides a complete classification from many points of view: topological, smooth, constant curvature, complex, and conformal. Each chapter briefly revises basic results usually known to graduate students from an alternative perspective, focusing on surfaces. We provide full proofs of some remarkable results that sometimes are missed in basic courses (e.g., the construction of triangulations on surfaces, the classification of surfaces, the Gauss-Bonnet theorem, the degree-genus formula for complex plane curves, the existence of constant curvature metrics on conformal surfaces), and we give hints to questions about higher dimensional manifolds. Many examples and remarks are scattered through the book. Each chapter ends with an exhaustive collection of problems and a list of topics for further study. The book is primarily addressed to graduate students who did take standard introductory courses on algebraic topology, differential and Riemannian geometry, or algebraic geometry, but have not seen their deep interconnections, which permeate a modern approach to geometry and topology of manifolds.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-208

GSM/208