Geometric Analysis on Real Analytic Manifolds 9783031379123, 9783031379130

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Lecture Notes in Mathematics  2333

Andrew D. Lewis

Geometric Analysis on Real Analytic Manifolds

Lecture Notes in Mathematics Volume 2333

Editors-in-Chief Jean-Michel Morel, Ecole Normale Supérieure Paris-Saclay, Paris, France Bernard Teissier, IMJ-PRG, Paris, France Series Editors Karin Baur, University of Leeds, Leeds, UK Michel Brion, UGA, Grenoble, France Annette Huber, Albert Ludwig University, Freiburg, Germany Davar Khoshnevisan, The University of Utah, Salt Lake City, UT, USA Ioannis Kontoyiannis, University of Cambridge, Cambridge, UK Angela Kunoth, University of Cologne, Cologne, Germany Ariane Mézard, IMJ-PRG, Paris, France Mark Podolskij, University of Luxembourg, Esch-sur-Alzette, Luxembourg Mark Policott, Mathematics Institute, University of Warwick, Coventry, UK Sylvia Serfaty, NYU Courant, New York, NY, USA László Székelyhidi Germany

, Institute of Mathematics, Leipzig University, Leipzig,

Gabriele Vezzosi, UniFI, Florence, Italy Anna Wienhard, Ruprecht Karl University, Heidelberg, Germany

This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes: 1. Research monographs 2. Lectures on a new field or presentations of a new angle in a classical field 3. Summer schools and intensive courses on topics of current research. Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.

Andrew D. Lewis

Geometric Analysis on Real Analytic Manifolds

Andrew D. Lewis Department of Mathematics and Statistics Queens University Kingston, ON, Canada

ISSN 0075-8434 ISSN 1617-9692 (electronic) Lecture Notes in Mathematics ISBN 978-3-031-37912-3 ISBN 978-3-031-37913-0 (eBook) https://doi.org/10.1007/978-3-031-37913-0 Mathematics Subject Classification: 32C05, 46E10, 46T20, 47H30, 53B05, 58A07, 58A20 This work was supported by Natural Sciences and Engineering Research Council of Canada © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Dedicated to the memory of Winston

Preface

This book arose from my own efforts over a number of years to be able to work with things that are real analytic in differential geometry. The difficulties of working with real analytic objects defined on real analytic manifolds are numerous, compared to working with smooth objects. For example, in smooth differential geometry, it is elementary to extend local constructions to global ones using smooth cutoff functions. In the real analytic case, local constructions can typically not be extended to global ones, and one must work with sheaf theoretic constructions or embedding theorems arising from the seminal work of Cartan [13] and Grauert [23]. Similarly, for differential topology, there are various well-understood and classical topologies that one can use in the smooth case, depending on what one wishes to achieve, e.g., [32, 50]. In the real analytic case, the topologies are more rigid as is shown in the work of Martineau [48]. Moreover, usable descriptions of these topologies were not available until fairly recently; as far as we are aware, the first clear and rigorous presentation of seminorms for the real analytic topology on the space of real analytic functions is found in the work of Vogt [66]. It is these seminorms that allow a (mere) user of real analytic topologies to have a chance of proving new theorems, an example being the work of Jafarpour and Lewis [35] on flows of time-varying vector fields. The focus in this book is 1. A careful development of the topology for the space of real analytic functions on real analytic manifolds and, more generally, for the space of real analytic sections of a real analytic vector bundle and real analytic mappings of real analytic manifolds. 2. An illustration of how our characterisations of the real analytic topology can be used to prove basic and useful facts about geometric operations on real analytic manifolds. 3. Do the previous two things in a global, intrinsic, and geometric manner.

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The Approach Let us describe in a little more detail the approach we take to real analytic analysis and at the same time prepare the reader to navigate the rather detailed constructions that we undergo. A barrier to our objectives right at the start is that the appropriate topology for real analytic functions (functions, for simplicity) is not so easily envisaged, in contrast to the smooth case. While a suitable real analytic topology has been around since at least the work of Martineau [48]—who provided two descriptions of such a topology and showed that they agree—there has not been a “user-friendly” description of the real analytic topology, i.e., a description using seminorms, until quite recently. Some useful initial formulae are provided in [52], and seminorms are provided in the lecture notes of [16]. However, as far as we are aware, it is only in the technical note of Vogt [66] that we see a clear proof of the suitability of these seminorms. These were adapted to the geometric setting for sections of a real analytic vector bundle by Jafarpour and Lewis [35]. Part of this development was a decomposition of jet bundles using connections. The initial developments of that monograph are the starting point for our approach here. Another complicating facet of the real analytic theory arises when one considers lifts from the base space to the total space of a real analytic vector bundle .πE : E → M, e.g., vertical lift of a section of .E or horizontal lift of a vector field on .M. The first of these operations requires no additional structure, but the second requires a connection. However, both require (in our approach) connections to study their real analytic continuity, because one needs to provide bounds for the jets on the codomain (i.e., on .E) in terms of jets on the domain (i.e., on .M). To provide seminorms, one also needs Riemannian and vector bundle metrics, and all of this data has to fit together nicely to provide the bounds required. For instance, one has a natural Riemannian metric on the manifold .E arising from (1) a Riemannian metric on .M, (2) a fibre metric on .E, (3) an affine connection on .M, and (4) a linear connection in .E. Using an adaptation of a construction of Sasaki [61], this data gives rise to a Riemannian metric on .E, and so its Levi-Civita affine connection. Moreover, the resulting Riemannian metric makes .πE : E → M a Riemannian submersion, and enables some useful constructions of O’Neill [56]. In order to illustrate the nature of the difficulties one encounters, let us consider a specific and illustrative instance of the sort of argument that one must piece together to prove continuity in the real analytic case. The reader should regard the discussion immediately following as a sort of “thought experiment.” Suppose that we have a real analytic vector bundle .πE : E → M with .∇ πE a real analytic linear connection in .E. Let X be a real analytic vector field on .M which we horizontally lift to a real analytic vector field .Xh on .E. To assess the continuity of the map .X ⎬→ Xh in the real analytic topology, one needs to compute jets of .Xh and relate these to jets of X. Thus, one needs to differentiate .Xh arbitrarily many times. This differentiation must be done on .E, as this is the base on which .Xh is defined. Trying this directly in local coordinates is, in principle, possible, but it is pretty unlikely that one will be able to produce the refined estimates required in this way.

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Thus, in our approach, one needs an affine connection on .E (thinking of .E as just a manifold now). One can now see that there will be a complicated intermingling of the linear connection .∇ πE , an affine connection .∇ M on .M (to compute jets of X), and a fabricated affine connection .∇ E on .E (to compute jets of .Xh ). This is only the beginning of the difficulties one faces. One also needs, not only formulae for the derivatives of .Xh , but also recursive formulae relating how a derivative of .Xh of order, say, k is related to the derivatives of X of orders .0, 1, . . . , k. These recursive formulae are essential for being able to obtain growth estimates for the derivatives needed to relate the seminorms applied to .Xh to those applied to X. Moreover, since the mapping .X ⎬→ Xh is injective, one might hope that the mapping is not just continuous, but is an homeomorphism onto its image. To prove this, one now needs to get estimates for the jets of X from formulae involving the jets of .Xh . Thus, one needs estimates that go “both ways.”1 It is also worth mentioning that the estimates one needs from these recursive formulae are quite unforgiving, and so their form has to be very precisely managed. This requires extensive bookkeeping. This bookkeeping occupies us for a substantial portion of the book. This is contrasted with the smooth case, where very sloppy bookkeeping suffices; we shall say a few words about this contrast at illustrative places in the book. Another difficulty is that the use of connections to compute derivatives for jets forces one to address the matter of whether the topologies defined by the seminorms used for jets, and derived from the use of connections, are actually not dependent on the chosen connection. Thus, one must compare iterated covariant derivatives with respect to different connections and show that these are related to one another in such a way that the resulting real analytic topology is well defined. This, in itself, is a substantial undertaking. It is done in an ad hoc way in [35, Lemma 2.5]; here we do this in a systematic and geometric way that offers many benefits towards the objectives of this book, apart from rendering more attractive the computations of [35]. We mention that the idea of obtaining recursive formulae for derivatives is given in a local setting by Thilliez [64] during the course of the proof of his Proposition 2.5, and can be applied to the mapping .Cω (N) ϶ f ⎬→ Ф∗ f ∈ Cω (M) of pull-back by a real analytic mapping .Ф ∈ Cω (M; N). We are able to extend the ideas in Thilliez’ computations to general classes of geometric operations. For example, as we mention above, a local working out of the estimates for the horizontal lift operation seems like it will be very difficult. However, once one does get these things to work out, it is relatively straightforward to prove the main results of the book, which are the continuity of the fundamental geometric operations mentioned above.

1 Note that this is an open mapping argument, and so raises the question of whether the open mapping theorem holds for the real analytic topology. It does, in fact, using the properties of this topology that we prove in Proposition 2.14. We shall, at times, use the resulting Open Mapping Theorem, but at other times we shall explicitly prove the openness using seminorms since the seminorm approach is the raison d’être of our methodology.

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One of the features of the book is that almost all constructions are done intrinsically. While this may seem to unnecessarily complicate things, this is not, in fact, so. Even were one to work locally, there would still arise two difficult problems that we overcome in our approach, but that still must be overcome in a local approach: (1) the difficulty of lifts as described in detail above; (2) the verification that the topologies do not depend on various choices made (charts in the local calculations, and metrics and connections in the intrinsic calculations). Thus, while the intrinsic calculations are sometimes complicated, they are only a little more complicated than the necessarily already complicated local calculations. And we believe that the intrinsic approach is ultimately easier to use, once one understands how to use it. An objective of this book is to do a lot of tedious hard work required to produce methods and results that are themselves more or less straightforward. As a side-benefit to our approach, we also are able to easily provide proofs in the finitely differentiable and smooth cases. We point out the relevant places where modifications can be made to the real analytic proofs to give the results in the finitely differentiable and smooth cases. An Outline of the Book and Tips for Reading It We open in Chap. 1 by providing an overview of the background required and/or made use of in the book. In Chap. 2, we review real analytic differential geometry and the definition of the real analytic topology as per [48], give some of the properties of this topology, and define in geometric language the seminorms for this topology as constructed in [35]. We give a proof that these seminorms do indeed define the real analytic topology, following [66]. In Chap. 2, we also consider at length the topology of the space of real analytic mappings between real analytic manifolds. We provide a number of descriptions of this topology, which give as a consequence other characterisations of the topology for the set of sections of a real analytic vector bundle. To our knowledge, the topology for the space of real analytic mappings is presented here for the first time. In Chap. 3, we provide a host of geometric constructions whose bearing on the main goal of the book will be difficult to glean on a first reading. Some sketchy motivation for the constructions of this chapter is outlined above in our discussion of the difficulties one will encounter trying to prove continuity of the horizontal lift mapping .X ⎬→ Xh . In Sect. 3.1 we perform constructions with functions, vector fields, and tensors on the total space of a vector bundle. These form the basis for derivative computations done in Sect. 3.2. Particularly, in Sect. 3.2.1, we give .πE : E → M the structure of a Riemannian submersion, following [56]. This allows us to relate, in a natural way, constructions on .E with those on .M. In Sect. 3.3, we provide the crucial recursive formulae that relate derivatives on .E with those on .M. We do this for a few of the standard geometric lifts one has for a vector bundle with a linear connection. Some of these we do because they are intrinsically interesting. Some we do because they are required for our general approach, even if one is not interested in them per se.

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While Chap. 3 is focussed on the geometric constructions we need for our main results, the results of Chap. 4 focus on analytical results, specifically providing norms estimates relating the various operations from Sect. 3.3. In Sect. 4.1, we give fibre norms for various jet bundles that are used to define seminorms corresponding to the geometric constructions of interest. In Sect. 4.2, we put all of our work from Chap. 3 to use to prove Lemma 4.17, the technical lemma that makes everything work. The lemma gives a very precise estimate for the fibre norms of derivatives of coefficients that arise in the recursive constructions of Sect. 3.3. There is no wiggle room in the form of the required estimate, and this is one of the reasons why the computations of Chap. 3 are so laboriously carried out; these computations need to be understood at a high resolution. Once we have these estimates, however, in Sect. 4.3 we show that the fibre norms for jet bundles obtained in Sect. 4.1 behave in the proper way as to make the topologies we construct independent of our choices of connections and metrics. This is stated as Lemma 4.19. The actual proving of the independence of the topologies is carried out by proving in Theorem 4.24 that the topologies are each the same as a topology described using local forms of the seminorms. This device of using a local description carries two benefits. 1. It provides the local description of the seminorms. While our approach is intrinsic as much as this is possible, sometimes in practice one must work locally, and having the explicit local formulae for the fibre norms is beneficial. 2. While we have tried to make our treatment intrinsic, there is a crucial point where a local estimate for the growth of derivatives becomes unavoidable, resting as it does on the Cauchy estimates for holomorphic functions. In our proof of Theorem 4.24 is where this seemingly unavoidable local estimate is not avoided. In Chap. 5, we prove continuity of some representative and some important geometric constructions. There is a long list of these constructions and we only give a few of the more obvious ones; we hope that the tools we develop in the book, and put to use in Chap. 5, will make it easy for researchers down the road to prove some important results in the real analytic setting where continuity is crucial. Acknowledgements The research reported in this book was financially supported by the Natural Sciences and Engineering Research Council of Canada. Kingston, ON, Canada May 2023

Andrew D. Lewis

Contents

1

2

Notation and Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Algebra and Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Real Analysis and Differential Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Riemannian Geometry and Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Jet Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Metric and Uniform Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 Initial and Final Topologies, Inverse and Direct Limits. . . . . . . 1.8 Locally Convex Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Inverse and Direct Limits of Locally Convex Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Metrisable Locally Convex Topological Vector Spaces. . . . . . . 1.8.3 Open Mapping Theorems for Locally Convex Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 Nuclear Locally Convex Topological Spaces. . . . . . . . . . . . . . . . . . 1.8.5 Tensor Products of Locally Convex Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Real Analytic Real Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 5 8 11 12 12 13 14 17

Topology for Spaces of Real Analytic Sections and Mappings . . . . . . . . . . 2.1 Real Analytic Differential Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 The Fundamental Objects of Real Analytic Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Existential Constructions in Real Analytic Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Complexification of Real Analytic Manifolds and Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31

18 21 22 24 24 26

32 32 39

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2.2 Martineau’s Descriptions of the Real Analytic Topology . . . . . . . . . . . . . 2.2.1 Germs of Holomorphic Sections Over Subsets of a Real Analytic Manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 A Natural Direct Limit Topology for the Space of Real Analytic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 The Topology of Holomorphic Germs About a Compact Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 An Inverse Limit Topology for the Space of Real Analytic Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Properties of the Cω -Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Constructions with Jet Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Decompositions for Jet Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Fibre Norms for Jet Bundles of Vector Bundles. . . . . . . . . . . . . . . 2.4 Seminorms for the Cω -Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 A Weighted Direct Limit Topology for Sections of Bundles of Infinite Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Definition of Seminorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Topology for the Space of Real Analytic Mappings . . . . . . . . . . . . . 2.5.1 Motivation for, and Definition of, the Weak-PB Topology . . . 2.5.2 The Topology for the Space of Holomorphic Mappings . . . . . . 2.5.3 Germs of Holomorphic Mappings Over Subsets of a Real Analytic Manifold. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Direct and Inverse Limit Topologies for the Space of Real Analytic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

Geometry: Lifts and Differentiation of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Tensors on the Total Space of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Functions on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Vector Fields on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Linear Mappings on Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Tensors Fields on Vector Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Tensor Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Differentiation of Tensors on the Total Space of a Vector Bundle . . . . 3.2.1 Vector Bundles as Riemannian Submersions . . . . . . . . . . . . . . . . . . 3.2.2 Derivatives of Tensor Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Derivatives of Tensors on the Total Space of a Vector Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Prolongation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Isomorphisms Defined by Lifts and Pull-Backs . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Isomorphisms for Horizontal Lifts of Functions . . . . . . . . . . . . . . 3.3.2 Isomorphisms for Vertical Lifts of Sections . . . . . . . . . . . . . . . . . . . 3.3.3 Isomorphisms for Horizontal Lifts of Vector Fields . . . . . . . . . . 3.3.4 Isomorphisms for Vertical Lifts of Dual Sections . . . . . . . . . . . . . 3.3.5 Isomorphisms for Vertical Lifts of Endomorphisms . . . . . . . . . . 3.3.6 Isomorphisms for Vertical Evaluations of Dual Sections . . . . .

93 94 94 95 97 99 101 105 105 112

41 42 43 47 49 52 53 60 62 62 67 73 74 79 82 87

116 126 137 137 145 152 156 160 164

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3.3.7 3.3.8 3.3.9 4

5

Isomorphisms for Vertical Evaluations of Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 Isomorphisms for Pull-Backs of Functions . . . . . . . . . . . . . . . . . . . . 181 Comparison of Iterated Covariant Derivatives for Different Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Analysis: Norm Estimates for Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fibre Norms for Some Useful Jet Bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Fibre Norms for Horizontal Lifts of Functions . . . . . . . . . . . . . . . . 4.1.2 Fibre Norms for Vertical Lifts of Sections. . . . . . . . . . . . . . . . . . . . . 4.1.3 Fibre Norms for Horizontal Lifts of Vector Fields . . . . . . . . . . . . 4.1.4 Fibre Norms for Vertical Lifts of Dual Sections . . . . . . . . . . . . . . 4.1.5 Fibre Norms for Vertical Lifts of Endomorphisms . . . . . . . . . . . . 4.1.6 Fibre Norms for Vertical Evaluations of Dual Sections . . . . . . . 4.1.7 Fibre Norms for Vertical Evaluations of Endomorphisms . . . . 4.1.8 Fibre Norms for Pull-Backs of Functions . . . . . . . . . . . . . . . . . . . . . 4.2 Estimates Related to Jet Bundle Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Algebraic Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Tensor Field Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Independence of Topologies on Connections and Metrics . . . . . . . . . . . . 4.3.1 Comparison of Metric-Related Notions for Different Connections and Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Local Descriptions of the Real Analytic Topology . . . . . . . . . . .

201 201 202 203 204 206 207 208 209 210 214 215 223 238

Continuity of Some Standard Geometric Operations. . . . . . . . . . . . . . . . . . . . 5.1 Continuity of Algebraic Operations and Constructions . . . . . . . . . . . . . . . 5.1.1 Continuity of Linear Algebraic Operations. . . . . . . . . . . . . . . . . . . . 5.1.2 Direct Sums and Tensor Products of Spaces of Sections . . . . . 5.2 Continuity of Operations Involving Differentiation . . . . . . . . . . . . . . . . . . . 5.3 Continuity of Lifting Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Continuity of Composition Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Real Analytic Composition Operator . . . . . . . . . . . . . . . . . . . . . 5.4.2 The Real Analytic Superposition Operator . . . . . . . . . . . . . . . . . . . .

253 254 254 258 270 275 287 288 292

238 246

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Chapter 1

Notation and Background

In this chapter we review some of the notation and machinery we use in the subsequent chapters. We provide references to aid a reader who may lack some of the prerequisites. While most of the material we overview should be regarded as elementary, it is possible that a reader without a substantial background in functional analysis will benefit from probing more deeply, following the references we give in Sect. 1.8.

1.1 Basic Terminology and Notation When A is a subset of a set X, we write .A ⊆ X. If we wish to exclude the possibility ⎟⎟ that .A = X, we write .A  X. For a⎟⎟family of sets .(Xi )i∈I , we denote by . i∈I Xi the product of these sets. By .prj : i∈I Xi → Xj we denote the projection onto the j th factor. For a family of sets .(Xi )i∈I , their disjoint union is denoted by ◦ ⎟⎟ .

{(x, i) | x ∈ Xi }.

i∈I

The identity map on a set X is denoted by .idX . If .f : X → Y and if .A ⊆ X, .f |A denotes the restriction of f to A. Sometimes we may wish to indicate a mapping without giving it a name, typically in a framework where there is a great deal of existing notation and we do not want to introduce yet another symbol. To do this, we will write a mapping as .(x │ → f (x)), thereby indicating what the mapping does to an element x of its domain. The cardinality of a set X is denoted by .card(X). By .Z we denote the set of integers. We use the notation .Z>0 and .Z≥0 to denote the subsets of positive and nonnegative integers. By .R we denote the set of real

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. D. Lewis, Geometric Analysis on Real Analytic Manifolds, Lecture Notes in Mathematics 2333, https://doi.org/10.1007/978-3-031-37913-0_1

1

2

1 Notation and Background

√ numbers and by .C we denote the set of complex numbers. We denote .i = −1. By .R>0 we denote the subset of positive real numbers. We denote by .Rn the n-fold Cartesian product of .R. A point in .Rn will typically be denoted in a bold font, e.g., .x = (x1 , . . . , xn ). We denote the standard basis for n .R by .(e 1 , . . . , e n ).

1.2 Algebra and Linear Algebra Any intermediate level linear algebra text suffices to provide the concepts we need, e.g., [4]. Tensor algebra at the level we use it is covered in [1, §6.1]. By .Sk we denote the permutation group of .{1, . . . , k}. For .k, l ∈ Z≥0 , we denote by .Sk,l the subset of .Sk+l consisting of permutations .σ satisfying σ (1) < · · · < σ (k),

.

σ (k + 1) < · · · < σ (k + l).

We also denote by .Sk|l the subgroup of .Sk+l consisting of permutations having the form ⎛ ⎞ 1 ··· k k + 1 ··· k + l . σ1 (1) · · · σ1 (k) k + σ2 (1) · · · k + σ2 (l) for .σ1 ∈ Sk and .σ2 ∈ Sl . We note that .Sk+l /Sk|l ≃ Sk,l , so that (1) if .σ ∈ Sk+l , then .σ = σ1 ◦ σ2 for .σ1 ∈ Sk,l and .σ2 ∈ Sk|l and (2) .card(Sk,l ) = (k+l)! k!l! . Note that, similarly, .Sk|l \Sk+1 ≃ Sk,l , and so, if .σ ∈ Sk+l , then .σ = σ1 ◦ σ2 for .σ1 ∈ Sk|l and .σ2 ∈ Sk,l . For .R-vector spaces .U and .V, we denote by .HomR (U; V) the set of .R-linear mappings from .U to .V. We shall also adapt this notation to bilinear mappings. Thus, adding a .R-vector space .W to the conversation, by .HomR (U, V; W) we denote the set of bilinear mappings from .U × V to .W. We denote .EndR (V) = HomR (V; V). We denote by .V∗ = HomR (V; R) the algebraic dual. If .v ∈ V and .α ∈ V∗ , we will denote the evaluation of .α on v at various points by .α(v), .α · v, or ., whichever seems most pleasing to us at the moment. If .A ∈ HomR (U; V), we denote by .A∗ ∈ HomR (V∗ ; U∗ ) the dual of A. If .S ⊆ V, then we denote by ann(S) = {α ∈ V∗ | α(v) = 0, v ∈ S}

.

the annihilator subspace. For a .R-vector space .V, .Tk (V) is the k-fold tensor product of .V with itself. For .r, s ∈ Z>0 , we denote ∗ Trs (V) = V · · ⊗ V ⊗ V · · ⊗ V ∗ .  ⊗ ·  ⊗ ·

.

r times

s times

1.2 Algebra and Linear Algebra

3

By .Sk (V) we denote the k-fold symmetric tensor product Λ of .V with itself, and we think of this as a subset of .Tk (V). In like manner, by . k (V) we denote the k-fold alternating tensors of .V. For .A ∈ Sk (V) and .B ∈ Sl (V), we define the symmetric tensor product of A and B to be Σ

A⊙B =

σ (A ⊗ B).

.

(1.1)

σ ∈Sk,l

We define .Symk : Tk (V) → Sk (V) by .

Symk (v1 ⊗ · · · ⊗ vk ) =

1 Σ vσ (1) ⊗ · · · ⊗ vσ (k) . k! σ ∈Sk

We note that we have the alternative formula A⊙B =

.

(k + l)! Symk+l (A ⊗ B) k!l!

(1.2)

for the product of .A ∈ Sk (V) and .B ∈ Sl (V) [10, Proposition IV.3.3]. We recall that .

dimR (Sk (V)) =

⎞ ⎛ dimR (V) + k − 1 , k

(1.3)

when .V is finite-dimensional. For a .R-vector space .V, let us denote T≤m (V) =

m ⊕

.

Tj (V),

S≤m (V) =

j =0

m ⊕

Sj (V),

j =0

and define Sym≤m : T≤m (V) → S≤m (V) .

(A0 , A1 , . . . , Am ) │ → (A0 , Sym1 (A1 ), . . . , Symm (Am )).

For .R-inner product spaces .(U, GU ) and .(V, GV ), we denote the transpose of L ∈ HomR (U; V) as the linear map .LT ∈ HomR (V; U) defined by

.

GV (L(u), v) = GU (u, LT (v)),

.

u ∈ U, v ∈ V.

By .GV : V → V∗ we denote the isomorphism induced by the inner product .GV on # b .V, and by .G we denote the inverse of .G . V V b

4

1 Notation and Background

1.3 Real Analysis and Differential Calculus A standard reference for real analysis suffices for our purposes, e.g., [62]. Our notation for differential calculus follows [1, Chapter 2]. We shall not use any particular notation for the Euclidean norm for .Rn , and so will just denote this norm by ⎛

⎞1/2 n Σ .‖x‖ = ⎝ |xj |2 ⎠ . j =1

It is sometimes convenient to use other norms for .Rn , particularly the 1- and .∞norms defined, as usual, by ‖x‖1 =

.

n Σ |xj |,

| } { ‖x‖∞ = sup |xj | | j ∈ {1, . . . , n} .

j =1

The following relationships between these norms are useful: ‖x‖ ≤ ‖x‖1 ≤

√ n‖x‖,

‖x‖∞ ≤ ‖x‖ ≤

√

n‖x‖∞ ,

‖x‖∞ ≤ ‖x‖1 ≤ n‖x‖∞ .

.

(1.4)

If we are using a norm whose definition is evident from context, we will simply denote it by .‖·‖, expecting that context will ensure that there is no confusion. For .x ∈ Rn and .r ∈ R>0 , we denote by | } { B(r, x) = y ∈ Rn | ‖y − x‖ < r

.

and | } { B(r, x) = y ∈ Rn | ‖y − x‖ ≤ r

.

the open ball and the closed ball of radius r centred at .x. As with the notation for norms, we shall often use the preceding notation for balls in settings different from n and for .x ∈ Rn , we denote by n .R , and accept the abuse of notation. For .r ∈ R >0 Dn (r, x) =

n ‖

.

(xj − rj , xj + rj )

j =1

and Dn (r, x) =

.

n ‖

[xj − rj , xj + rj ]

j =1

1.4 Differential Geometry

5

the open polydisk and the closed polydisk of radius .r and centre .x. We will use the same notation for the analogous polydisks in .Cn . If .U ⊆ Rn is open and if .⏀ : U → Rm is differentiable at .x ∈ U, we denote its derivative by .D⏀(x) ∈ HomR (Rn ; Rm ). Higher-order derivatives, when they exist, are denoted by .D k ⏀(x), k being the order of differentiation. We recall that, if .⏀ : U → Rm is of class .Ck , .k ∈ Z>0 , then .D k ⏀(x) is symmetric [1, Proposition 2.4.14]. If .U ⊆ Rn1 × Rn2 is open and if .⏀ : U → Rm is differentiable at .(x 01 , x 02 ) ∈ U, then the partial derivatives .D a ⏀(x 01 , x 02 ) ∈ HomR (Rna ; Rm ), .a ∈ {1, 2}, are the derivatives at .x 0a of the mappings x 1 │ → ⏀(x 1 , x 02 ),

.

x 2 │ → ⏀(x 01 , x 2 ).

We shall sometimes find it convenient to use multi-index notation for derivatives. A multi-index with length n is an element of .Zn≥0 , i.e., an n-tuple .I = (i1 , . . . , in ) of nonnegative integers. If .⏀ : U → Rm is a smooth function, then we denote D I ⏀(x) = D i11 · · · D inn ⏀(x),

.

identifying Rn = R × · · · × R

.

n times

and using partial derivative notation as above. We will use the symbol .|I | = i1 + · · · + in to denote the order of the derivative. Another piece of multi-index notation we shall use is a I = a1i1 · · · anin ,

.

for .a ∈ Rn and .I ∈ Zn≥0 . Also, we denote .I ! = i1 ! · · · in !.

1.4 Differential Geometry We shall adopt the notation and conventions of smooth differential geometry of [1]. We shall also make use of real analytic differential geometry, of course. In Sect. 2.1 we shall give an overview of real analytic differential geometry with an emphasis on the ways in which it differs from smooth differential geometry. There are no comprehensive textbook references dedicated to real analytic differential geometry, but the book of [14] contains some of what we shall need. Throughout the book, unless otherwise stated, manifolds are connected, second countable, Hausdorff manifolds. The assumption of connectedness can be dispensed with but is convenient as it allows one to not have to worry about manifolds with

6

1 Notation and Background

components of different dimensions and vector bundles with fibres of different dimensions. We shall work with regularity classes .r ∈ {∞, ω}, “.∞” meaning smooth, “.ω” meaning real analytic. If the reader sees the symbol “.Cr ” without any specific indication of what r is, it is intended that r be either .∞ or .ω. The word “smooth” will always mean “infinitely differentiable.” Sometimes we do not require infinite differentiability, but will hypothesise it anyway. Other times we will precisely specify the regularity needed; but we will be a little sloppy with this as (1) it is not crucial to the purposes of this book and (2) it is typically easy to know when infinite differentiability is hypothesised but not required. We shall make some use of holomorphic regularity, but will introduce our notation for this when we make use of it. The tangent bundle of a manifold .M is denoted by .πTM : TM → M and the cotangent bundle by .πT∗ M : T∗ M → M. We denote by .Cr (M; N) the set of mappings from a manifold .M to a manifold .N of class .Cr . When .N = R, we denote by .Cr (M) = Cr (M; R) the set of scalar-valued functions of class .Cr . For .Ф ∈ C1 (M; N), .T Ф : TM → TN denotes the derivative of .Ф , and .Tx Ф = T Ф |Tx M. A mapping .Ф ∈ C1 (M; N) is a submersion if .Tx Ф is surjective for each .x ∈ M and is an immersion if .Tx Ф is injective for each .x ∈ M. An immersion is an embedding if it is an homeomorphism onto its image. For .f ∈ Cν (M), .ν ∈ Z>0 ∪ {∞, ω}, we denote by .df ∈ ┌ ν−1 (T∗ M) the differential of f , defined by Tx f (vx ) = (f (x), ),

.

vx ∈ Tx M.

We denote by .Tx∗ Ф the dual of .Tx Ф . Let .πE : E → M be a vector bundle of class .Cr . We shall sometimes denote the fibre over .x ∈ M by .Ex , noting that this has the structure of a .R-vector space. If −1 r .A ⊆ M, we denote .E|A = π E (A). By .┌ (E) we denote the set of sections of .E r of class .C . This space has the structure of a .R-vector space with the vector space operations (ξ + η)(x) = ξ(x) + η(x),

.

(aξ )(x) = a(ξ(x)),

x ∈ M,

and of a .Cr (M)-module with the additional operation of multiplication (f ξ )(x) = f (x)ξ(x),

.

x ∈ M,

for .f ∈ Cr (M), .ξ, η ∈ ┌ r (E), and .a ∈ R. By .RkM we denote the trivial bundle k = M × Rk with vector bundle projection being projection onto the first factor. .R M The dual bundle .E∗ of a vector bundle .E is the set of vector bundle mappings from .E to .RM over .idM . We note that there is a natural identification of .┌ r (RM ) with .Cr (M). Given a .Cr -vector bundle .πE : E → M and a mapping .Ф ∈ Cr (N; M), we denote by .Ф ∗ πE : Ф ∗ E → N the pull-back bundle. For .Cr -vector bundles .πE : E → M

1.4 Differential Geometry

7

and .πF : F → M over the same base, we denote by .VBr (E; F) the set of .Cr -vector bundle mappings from .E to .F over .idM . We denote by .Trs (E) the tensor bundle over r k .M whose fibre over x is .Ts (Ex ). We shall also denote by .S (E) the symmetric tensor Λk bundle and by . (E) the alternating tensor bundle. For a vector field X and a differentiable function f , .L X f denotes the Lie derivative of f with respect to X. We might also write .Xf = L X f . For smooth vector fields X and Y , we denote by .[X, Y ] the Lie bracket of these vector fields, this being defined by the formula L [X,Y ] f = L X L Y f − L Y L X f,

.

f ∈ Cr (M).

For .X ∈ ┌ r (TM), the flow of X is denoted by .Ф X t , meaning that, for .x ∈ M, we have .

d X Ф (x) = X ◦ Ф X t (x), dt t

Ф X 0 (x) = x.

Said otherwise, .t │ → Ф X t (x) is the integral curve of X through x at time 0. The Lie derivative for vector fields extends to a derivation of the tensor algebra for a manifold. Specifically, for .X ∈ ┌ ∞ (TM), we denote L X f = ,

.

L X Y = [X, Y ],

f ∈ C∞ (M), X ∈ ┌ ∞ (TM).

For .α ∈ ┌ ∞ (T∗ M), we can then define its Lie derivative with respect to X by = L X − ,

.

Y ∈ ┌ ∞ (TM).

The Lie derivative of a tensor field .A ∈ ┌ ∞ (Trs (TM)) is then defined by L X A(α 1 , . . . , α r , X1 , . . . , Xs ) = L X (A(α 1 , . . . , α r , X1 , . . . , Xs ))

.

−

r Σ

A(α 1 , . . . , L X α j , . . . , α r , X1 , . . . , Xs )

j =1

−

s Σ

A(α 1 , . . . , α r , X1 , . . . , L X Xj , . . . , Xs ).

(1.5)

j =1

Of course, these constructions make sense for tensor fields and vector fields that are less regular than smooth. We shall make occasional use of notation and methods from sheaf theory. A readable introduction to the theory and its uses in differential geometry is the

8

1 Notation and Background

book [60]. By .GEr we denote the sheaf of .Cr -sections of .E. Thus GEr (U) = ┌ r (E|U)

.

when .U ⊆ M is open. By .CMr we denote the sheaf of .Cr -functions on .M. Thus CMr (U) = Cr (U)

.

when .U ⊆ M is open.

1.5 Riemannian Geometry and Connections We shall make use of basic constructions from Riemannian geometry. We also work a great deal with connections, both affine connections and linear connections in vector bundles. We refer to [42] as a standard reference, and [43] is also a useful reference. First suppose that .r ∈ {∞, ω}. A .Cr -fibre metric on a .Cr -vector bundle r 2 ∗ .πE : E → M is .GπE ∈ ┌ (S (E )) such that .GπE (x) is an inner product on .Ex for each .x ∈ M. The associated norm on fibres we denote by .‖·‖G . In case .E is the tangent bundle of .M, then a fibre metric is a Riemannian metric, and we will use the notation .GM in this case. A .Cr -linear connection in a .Cr -vector bundle .πE : E → M will be denoted by π .∇ E ; this defines a mapping π

┌ r (TM) × ┌ r (E) ϶ (X, ξ ) │ → ∇XE ξ ∈ ┌ r (E)

.

that is .R-bilinear, .Cr (M)-linear in X, and that satisfies the following derivation property in .ξ : π

π

∇XE (f ξ ) = f ∇XE ξ + (L X f )ξ.

.

There are many equivalent ways to define a linear connection, all being equivalent to the way we give above. Another way to make the definition, and one we will find useful, is via a connector for .∇ πE . This is a mapping .K∇ πE : TE → E such that the two diagrams

.

(1.6)

1.5 Riemannian Geometry and Connections

9

define vector bundle mappings [43, §11.11]. The relationship between the connection and the connector is π

∇XE ξ = K∇ πE ◦ T ξ ◦ X.

.

(1.7)

Given a linear connection .∇ πE in .π : E → M, one can define the notion of parallel transport along a curve. If .x ∈ M, if .e ∈ Ex , and if .γ : [0, T ] → M is a smooth curve with .γ (0) = x, then we have the initial value problem π

∇γ 'E(t) ξ(t) = 0,

.

ξ(0) = e,

γ

in .E. The solution defines a mapping .τt : Ex → Eγ (t) called parallel transport along .γ . Because .∇ πE is a linear connection, parallel transport is a linear isomorphism of fibres [42, page 114]. In case .E is the tangent bundle of .M, then a linear connection is called an affine connection, and we will denote it by .∇ M . Given a Riemannian metric .GM , there is a distinguished affine connection .∇ M defined by the following requirements: M Y, Z) + G (Y, ∇ M Z), .X, Y, Z ∈ ┌ ∞ (TM); 1. .L X (GM (Y, Z)) = GM (∇X M X M M 2. .∇X Y − ∇Y X = [X, Y ], .X, Y ∈ ┌ ∞ (TM)

[42, Theorem IV.2.2]. The unique affine connection .∇ M with these properties is called the Levi-Civita connection. We will sometimes, but not always, require that the affine connection on .M with which we work be the Levi-Civita affine connection for a Riemannian metric. For an affine connection .∇ M on .M, a .C2 -curve .γ : I → M (.I ⊆ R is an interval) is a geodesic if it satisfies .∇γM' (t) γ ' (t) = 0. A Riemannian manifold, i.e., a manifold .M with a Riemannian metric .GM , has a natural metric space structure as follows. For a piecewise .C1 -curve .γ : [a, b] → M,1 define its length by ⌠ lGM (γ ) =

b

.

√

GM (γ ' (t), γ ' (t)) dt.

a

One can easily show that the length is independent of parameterisation, and so one can then consider curves defined on .[0, 1]. This being the case, the distance between

is continuous and is of class .C1 on a finite number of disjoint open subintervals for which the closure of their union is .[a, b], and the derivative has well-defined limits at the endpoints of the intervals.

1 Thus .γ

10

1 Notation and Background

x1 , x2 ∈ M is

.

| { dGM (x1 , x2 ) = inf lGM (γ )|

.

} γ : [0, 1] → M is piecewise differentiable and γ (0) = x1 and γ (1) = x2 .

This distance function can be verified to have the properties of a metric [1, Proposition 5.5.10]. A linear connection in a vector bundle .πE : E → M induces a splitting of the short exact sequence .

[43, §11.10]. For .e ∈ E, we thus have a splitting of the tangent space .Te E ≃ TπE (e) M⊕EπE (e) . The first component in this splitting we call horizontal and denote by .He E, and the second we call vertical and denote by .Ve E. By .hor and .ver we denote the projections onto the horizontal and vertical subspaces, respectively. We note that covariant differentiation with respect to a vector field X of sections of .E, along with Lie differentiation of functions, gives rise to covariant differentiation of tensors, just as we saw above for .L X . A little more generally, if we have vector bundles .πE : E → M and .πF : F → E, and linear connections .∇ πE and .∇ πF , then we have a connection in .E ⊗ F denoted by .∇ πE ⊗πF and defined by π ⊗πF

∇XE

.

π

π

(ξ ⊗ η) = (∇XE ξ ) ⊗ η + ξ ⊗ (∇XF η).

(1.8)

We will also use the covariant differential for tensor fields. This is defined for an affine connection .∇ M and .A ∈ ┌ ∞ (Tk (T∗ M)) by M ∇ M A(X0 , X1 , . . . , Xk ) = (∇X A)(X1 , . . . , Xk ), 0

.

giving .∇ M A ∈ ┌ ∞ (Tk+1 (T∗ M)). We will especially consider this in the case of tensors .A ∈ ┌ ∞ (Tk (T∗ M) ⊗ E), where the above definition is applied to the first component of the tensor product. If .∇ πE is .Cr -linearΛ connection in a .Cr -vector bundle .πE : E → M, its curvature r tensor is .R∇ πE ∈ ┌ ( 2 (T∗ M) ⊗ E∗ ⊗ E) defined by π

π

π

π

π

E R∇ πE (X, Y )(ξ ) = ∇XE ∇Y E ξ − ∇Y E ∇XE ξ − ∇[X,Y ] ξ,

.

X, Y ∈ ┌ r (TM), ξ ∈ ┌ r (E). For a .Cr -affine connection .∇ M , its torsion tensor is .T∇ M ∈ ┌ r ( defined by M T∇ M (X, Y ) = ∇X Y − ∇YM X − [X, Y ],

.

Λ2

(T∗ M) ⊗ TM)

X, Y ∈ ┌ r (TM).

1.6 Jet Bundles

11

1.6 Jet Bundles We shall make extensive use of jet bundles of various sorts. We can recommend [64] and [43, §12] as useful references. Let .M be a .Cr -manifold and let .m ∈ Z≥0 . In general, an m-jet at .x ∈ M of a function, section, or mapping is an equivalence class of functions, sections, or mappings, defined locally around x, where the equivalence is agreement (say in local coordinates) of all objects to order m. Let us consider the various special cases. For .x ∈ M and .a ∈ R, by .Jm (x,a) (M; R) we denote the m-jets of functions at x taking value a at x. For a .Cr -function f defined in a neighbourhood of x, we denote by .jm f (x) ∈ Jm (x,f (x)) M the m-ket of f . Of particular interest is the set ∗m m .Tx M = J(x,0) (M; R) of jets of functions taking the value 0 at x. This has the structure of a .R-algebra with the algebra structure defined by the three operations jm f (x) + jm g(x) = jm (f + g)(x),

.

(jm f (x))(jm g(x)) = jm (f g)(x), a(jm f (x)) = jm (af )(x), ◦

for functions f and g and for .a ∈ R. We denote .T∗m M = ∪x∈M T∗m x M. For .m, l ∈ Z≥0 with .m ≥ l, we have projections .ρlm : T∗m M → T∗l M. Note that .T∗0 M ≃ M and that .T∗1 M ≃ T∗ M. We abbreviate .ρm  ρ0m : T∗m M → M, which has the structure of a vector bundle. Let .πE : E → M be a .Cr -vector bundle. For .x ∈ M and .m ∈ Z≥0 , .Jm xE denotes the set of m-jets of sections of .E at x. For a .Cr -section .ξ defined in some neighbourhood of x, .jm ξ(x) ∈ Jm x E denotes the m-jet of .ξ . We denote ◦

by .Jm E = ∪x∈M Jm x E the bundle of m-jets. For .m, l ∈ Z≥0 with .m ≥ l, we m m denote by .πE,l : J E → Jl E the projection. Note that .J0 E ≃ E. We abbreviate m m m .πE,m  πE ◦ π E,0 : J E → M, and note that .J E has the structure of a vector bundle over .M, with addition and scalar multiplication defined by jm ξ(x) + jm η(x) = jm (ξ + η)(x),

.

a(jm ξ(x)) = jm (aξ )(x)

12

1 Notation and Background

for sections .ξ and .η and for .a ∈ R. One can show that Jm E ≃ (RM ⊕ T∗m M) ⊗ E.

.

(1.9)

1.7 Topology We shall make extensive use of elementary notions of general topology; the book [73] is a reference.

1.7.1 Basics We shall denote by .int(A) and .cl(A) the interior and closure of a subset A. A subset .A ⊆ X of a topological space .(X, O) is precompact if .cl(A) is compact. Often this property is called “relative compactness.” A base for a topology .O for .X is a collection .B ⊆ O such that every open set is a union of sets from .B. A subbase for a topology is a collection of subsets for which the collection of finite intersections is a base for the topology. A neighbourhood base at .x ∈ X is a collection .B(x) of neighbourhoods of x such that, if .O is a neighbourhood of x, then there exists .B ∈ B(x) such that .B ⊆ O. An Hausdorff topological space .(X, O) is locally compact if every point possesses a base of neighbourhoods that are precompact. We denote by .C0 (X; Y) the set of continuous mappings from a topological space 0 .(X, OX ) to a topological space .(Y, OY ). A mapping .Ф ∈ C (X; Y) is proper if −1 .Ф (K) is compact for every compact .K ⊆ Y. We shall make use of the notion of a compact exhaustion for a topological space .(X, O), by which we mean a countable family .(Kj )j ∈Z>0 of compact subsets of .X with the following properties: 1. .Kj ⊆ int(Kj +1 ), .j ∈ Z>0 ; 2. .∪j ∈Z>0 Kj = X. A second countable, locally compact, Hausdorff topological space always possesses a compact exhaustion [2, Lemma 2.76]. This is a book about function space topologies, particularly for functions that are real analytic. At times we will make reference to the “compact-open” topology. In general, this topology is defined as follows. Let .(X, OX ) and .(Y, OY ) be topological spaces. The compact-open topology for .C0 (X; Y) is the topology with the sets B(K, V) = {Ф ∈ C0 (X; Y) | Ф (K) ⊆ V},

.

K ⊆ X compact, V ∈ OY ,

1.7 Topology

13

as a subbase. Thus open sets in the compact-open topology are unions of finite intersections of subsets of these subbasic sets. We shall provide rather more concrete characterisations of the compact-open topology in cases of interest to us.

1.7.2 Metric and Uniform Spaces We shall, at times, make use of facts about metric spaces, and use [11] as a reference. If .(M, d) is a metric space, if .x0 ∈ M, and if .r ∈ R>0 , then we denote by .Bd (r, x0 ) and .Bd (r, x0 ) the open ball and closed ball, respectively, of radius r and centre .x0 . The metric topology for a metric space .(M, d) is the topology with the set of open balls as a base. We shall also make use of the idea of a uniform space. There are multiple ways of defining the notion of a uniform space, but we choose the most concrete of these, consistent with the theme of the book. Let .X be a set. A semimetric2 for .X is the same as a metric, absent the property that the only pairs of points with distance zero between them are the pairs of identical points. A uniform structure for .X is defined by a family .(di )i∈I of semimetrics. Specifically, one defines a topology for .X by the requirement that the sets k ∩ .

{x ∈ X | dij (x, x0 ) < rj },

r1 , . . . , rk ∈ R>0 , i1 , . . . , ik ∈ I, k ∈ Z>0 ,

j =1

comprise a neighbourhood base at .x0 ∈ X. A topology of this form is special because one has the required structure to define notions such as Cauchy sequences (and so completeness) and uniform continuity normally associated with metric spaces. The semimetric point of view is often undeveloped in modern approaches to uniform spaces. However, [40] gives a good development of the semimetric point of view. We shall also make mention of the notion of Polish spaces and their brethren. A Polish space is a topological space .(X, O) whose topology is the metric topology for some metric whose metric uniformity is complete and whose metric topology is separable. There may be many metrics which give the topology of a Polish space, and one must take care to understand that a “natural” metric on a Polish space may not be the one that makes it a Polish space.3 A Lusin space is an Hausdorff topological space .(X, O) such that there is a Polish space .(X' , O ' ) and a bijective continuous mapping .Ф : X' → X. A Suslin space is an Hausdorff topological space

2 It is not uncommon to call this a “pseudometric.” We shall use “semimetric,” consistent with our use of “seminorm.” 3 For example, .(0, 1) is a Polish space, but the restriction of the standard metric on .R is not the right metric to make this assertion.

14

1 Notation and Background

(X, O) such that there exists a Polish space .(X' , O ' ) and a surjective continuous map .Ф : X' → X. We note that both Lusin and Suslin spaces are separable since the image of a separable space under a continuous map is separable [73, Theorem 16.4(a)]. Another important attribute of topological spaces is that of being sequential [22]. This attribute has a host of equivalent definitions, but perhaps the easiest to understand is the following: a topological space .(X, O) is sequential if a subset .C ⊆ X is closed if and only if every convergent sequence .(xj )j ∈Z>0 in .C has its limit in .C. .

1.7.3 Initial and Final Topologies, Inverse and Direct Limits We shall make frequent use of the notion of initial and final topologies. We let ((Xi , Oi ))i∈I be a family of topological spaces, let .Y be a set, and let .Ф i : Y → Xi and .Ψi : Xi → Y, .i ∈ I , be families of mappings. The initial topology for .Y defined by the mappings .Ф i , .i ∈ I , is the coarsest topology for .Y such that each of the mappings .Ф i , .i ∈ I , is continuous. The subsets −1 .Ф i (Oi ), .Oi ∈ Oi , .i ∈ I , are a base for the initial topology. The initial topology is characterised by the following fact. If .(Z, O) is a topological space, a mapping .Ψ : Z → Y is continuous if and only if the diagram .

.

is a commutative diagram of topological spaces for each .i ∈ I . The final topology for .Y defined by the mappings .Ψi , .i ∈ I , is the finest topology for .Y such that each of the mappings .Ψi , .i ∈ I , is continuous. A subset .U ⊆ Y is open for the final topology if and only if .Ψi−1 (U) ∈ Oi , .i ∈ I . The final topology is characterised by the following fact. If .(Z, O) is a topological space, a mapping .Ф : Y → Z is continuous if and only if the diagram

.

is a commutative diagram of topological spaces for each .i ∈ I . We shall refer to these as the “universal properties” of the initial and final topologies, although they are not strictly universal in the category theoretic sense.

1.7 Topology

15

An important example of an initial topology is the topology induced on a subset of a topological space. Precisely, let .(X, O) be a topological space and let .A ⊆ X with .ιA : A → X the inclusion. Then the induced topology of A from .O is topology OA = {U ∩ A | U ∈ O},

.

and is easily seen to be the initial topology for the family of mappings .(ιA ). An important example of a final topology is the quotient topology. Let .(X, O) be a topological space, let .Y be a set and let .Ф : X → Y be a surjective map. The quotient topology for .Y is the finest topology for which .Ф is continuous. One readily verifies that the quotient topology is the final topology associated with the family of mappings .(Ф ). Other important classes of initial and final topologies arise from inverse and direct limits of topological spaces. Since we routinely use these in not entirely trivial ways, we give the detailed definitions. Definition 1.1 (Inverse Limit of Topological Spaces) Let .(I, ) be a directed set. An inverse system of topological spaces is a pair .(((Xi , Oi ))i∈I , (Ф ii ' )ii ' ) where (i) .(Xi , Oi ), .i ∈ I , is a topological space and (ii) .Ф ii ' ∈ C0 (Xi ' ; Xi ), .i, i ' ∈ I , .i  i ' , are continuous maps satisfying (a) .Ф ii = idXi , .i ∈ I , and (b) .Ф ii '' = Ф ii ' ◦ Ф i ' i '' for all .i, i ' , i '' ∈ I such that .i  i '  i '' . An inverse limit of an inverse system .(((Xi , Oi ))i∈I , (Ф ii ' )ii ' ) is a topological space .(X, OX ) and a family .πi ∈ C0 (X; Xi ), .i ∈ I , of continuous maps such that (iii) the diagram

.

(1.10)

commutes for every .i, i ' ∈ I such that .i  i ' and (iv) if .(Y, OY ) is a topological space and if .Ψi ∈ C0 (Y; Xi ), .i ∈ I , are continuous maps such that the diagram

.

16

1 Notation and Background

commutes for every .i, i ' ∈ I such that .i  i ' , then there exists a unique .Ɵ ∈ C0 (Y; X) such that the diagram

.

commutes for every .i ∈ I . We may sometimes denote the inverse limit by .lim (Xi , Oi ). .◦ ← −i∈I The topology of the inverse limit .(X, O) in the definition is the initial topology defined by the mappings .πi , .i ∈ I . Frequently one sees “projective limit” where we choose to use “inverse limit.” Next we consider direct limits. In the literature, these are typically known as “inductive limits,” but we use the terminology “direct limit” since it agrees with usage in the rest of category theory. Definition 1.2 (Direct Limit of Topological Spaces) Let .(I, ) be a directed set. A directed system of topological spaces is a pair .(((Xi , Oi ))i∈I , (Ф ii ' )ii ' ) where (i) .(Xi , Oi ), .i ∈ I , is a topological space and (ii) .Ф ii ' ∈ C0 (Xi ; Xi ' ), .i, i ' ∈ I , .i  i ' , are continuous maps satisfying (a) .Ф ii = idXi , .i ∈ I ; (b) .Ф ii '' = Ф i ' i '' ◦ Ф ii ' for .i, i ' , i '' ∈ I satisfying .i  i ' and .i '  i '' . A direct limit of a directed system .(((Xi , Oi ))i∈I , (Ф ii ' )ii ' ) of topological spaces is a topological space .(X, OX ) and a family .κi ∈ C0 (Xi ; X), .i ∈ I , of continuous maps such that (iii) the diagram

.

commutes for every .i, i ' ∈ I for which .i  i ' and (iv) if .(Y, OY ) is a topological space and if .Ψi ∈ C0 (Xi ; Y), .i ∈ I , are such that the diagram

.

1.8 Locally Convex Topological Vector Spaces

17

commutes for every .i, i ' ∈ I for which .i  i ' , then there exists a unique 0 .Ɵ ∈ C (X; Y) such that the diagram

.

commutes for every .i ∈ I . We may sometimes denote the direct limit by .lim (Xi , Oi ). .◦ − →i∈I The topology of the direct limit .(X, O) in the definition is the final topology defined by the mappings .κi , .i ∈ I .

1.8 Locally Convex Topological Vector Spaces We shall make occasional use of ideas from the theory of locally convex topological vector spaces. We refer to [37] as a comprehensive reference source for the basic theory. Let .V be a .R-vector space. A seminorm on .V is the same as a norm, absent the property that the only vector with zero “norm” is the zero vector. Unlike for normed vector spaces where the topology is defined by a single norm p, for a locally convex topological vector space the topology is defined by a family of seminorms .(pi )i∈I . By “defined by” we mean that there is a neighbourhood base at .v ∈ V given by k ∩ .

j =1

(v + pi−1 ([0, rj )), j

r1 , . . . , rk ∈ R>0 , i1 , . . . , ik ∈ I, k ∈ Z>0 .

A locally convex topological vector space is evidently a uniform space, and is quite often characterised by means other than by seminorms, just as a uniform space often has a less concrete characterisation than by semimetrics. The uniformity of a locally convex topology means that one has the notion of completeness in a locally convex topological vector space. For linear maps, one has the usual notion of continuity, and we denote by .L(U; V) the set of continuous linear mappings from the locally convex topological vector space .(U, OU ) to the locally convex topological vector space .(V, OV ). One can show that .L ∈ L(U; V) if and only if, for any continuous seminorm q for .(V, OV ), there exists a continuous seminorm p for .(U, OU ) such that .q ◦ L(u) ≤ p(u) for every .u ∈ U. We denote .V' = L(V; R), which is the topological dual of .V. A difference with the theory of normed vector spaces is the absence of the equivalence of the notions of continuity and boundedness. To explain, let .(V, O) be a locally

18

1 Notation and Background

convex topological vector space and let .(pi )i∈I be a family of seminorms defining the topology. A subset .B ⊆ V is bounded if .pi |B is bounded for every .i ∈ I . A linear map .L ∈ HomR (U; V) between locally convex topological vector spaces is bounded if it maps bounded subsets to bounded subsets. One can then show that a continuous linear map is bounded, but the converse may not hold. We shall also have occasion to refer to the notion of a compact continuous linear map, by which we mean one that maps bounded sets to precompact sets. When working with locally convex topological vector spaces, it is advantageous to understand the rôle of specific properties of the topology in ways that do not arise organically in the theory of normed vector spaces. A systematic exploration of the various properties that are possible and what are the benefits of each is precisely the theory of locally convex topological vector spaces, and requires a lengthy book to understand with any level of comprehensiveness. Therefore, we shall limit ourselves here to cursory explanations of those properties of which we shall make use.

1.8.1 Inverse and Direct Limits of Locally Convex Topological Vector Spaces Locally convex limits are an important way of constructing locally convex topologies. We shall give a rapid overview here. We start with inverse limits. In the literature, these are typically known as “projective limits,” but we use the terminology “inverse limit” since it agrees with usage in the rest of category theory. Definition 1.3 (Inverse Limit of Locally Convex Topological Vector Spaces) Let (I, ) be a directed set. An inverse system of locally convex topological vector spaces is a pair .(((Vi , Oi ))i∈I , (φii ' )ii ' ) where

.

(i) .(Vi , Oi ), .i ∈ I , is a locally convex topological vector space and (ii) .φii ' ∈ L(Vi ' ; Vi ), .i, i ' ∈ I , .i  i ' , are continuous linear maps satisfying (a) .φii = idVi , .i ∈ I , and (b) .φii '' = φii ' ◦ φi ' i '' for all .i, i ' , i '' ∈ I such that .i  i '  i '' . An inverse limit of an inverse system .(((Vi , Oi ))i∈I , (φii ' )ii ' ) is a locally convex topological vector space .(V, OV ) and a family .πi ∈ L(V; Vi ), .i ∈ I , of continuous linear maps such that (iii) the diagram

.

commutes for every .i, i ' ∈ I such that .i  i ' and

(1.11)

1.8 Locally Convex Topological Vector Spaces

19

(iv) if .(U, OU ) is a locally convex topological vector space and if .ψi ∈ L(U; Vi ), .i ∈ I , are continuous linear maps such that the diagram

.

commutes for every .i, i ' ∈ I such that .i  i ' , then there exists a unique .θ ∈ L(U; V) such that the diagram

.

commutes for every .i ∈ I . We may sometimes denote the inverse limit by .lim (Vi , Oi ). .◦ ← −i∈I The topology of the inverse limit .(V, O) in the definition is the initial topology defined by the mappings .πi , .i ∈ I . Next we consider direct limits. In the literature, these are typically known as “inductive limits,” but we use the terminology “direct limit.” Definition 1.4 (Direct Limit of Locally Convex Topological Vector Spaces) Let (I, ) be a directed set. A directed system of locally convex topological vector spaces is a pair .(((Vi , Oi ))i∈I , (φii ' )ii ' ) where

.

(i) .(Vi , Oi ), .i ∈ I , is a locally convex topological vector space and (ii) .φii ' ∈ L(Vi ; Vi ' ), .i, i ' ∈ I , .i  i ' , are continuous linear maps satisfying (a) .φii = idVi , .i ∈ I ; (b) .φii '' = φi ' i '' ◦ φii ' for .i, i ' , i '' ∈ I satisfying .i  i ' and .i '  i '' . A locally convex direct limit of a directed system .(((Vi , Oi ))i∈I , (φii ' )ii ' ) of locally convex topological vector spaces is a locally convex topological vector space .(V, OV ) and a family .κi ∈ L(Vi ; V), .i ∈ I , of continuous linear maps such that (iii) the diagram

.

commutes for every .i, i ' ∈ I for which .i  i ' and

20

1 Notation and Background

(iv) if .(U, OU ) is a locally convex topological vector space and if .ψi ∈ L(Vi ; U), .i ∈ I , are such that the diagram

.

commutes for every .i, i ' ∈ I for which .i  i ' , then there exists a unique .θ ∈ L(V; U) such that the diagram

.

commutes for every .i ∈ I . We may sometimes denote the direct limit by .lim (Vi , Oi ). .◦ − →i∈I The topology of the direct limit .(V, O) in the definition is not generally the final topology defined by the mappings .κi , .i ∈ I . It is, instead, the final locally convex topology defined by these mappings. Precisely, this means that, in the definition of final topology above, one requires that the final topology be locally convex, e.g., by taking convex hulls of the open sets from the topological final topology. For direct limits, one typically needs to endow them with additional attributes in order for the direct limit to have useful properties. The following is a list of some such attributes that will be useful for us. Definition 1.5 (Attributes of Direct Limits of Locally Convex Topological Vector Spaces) Let .(I, ) be a directed set, and let .((Vi )i∈I , (φii ' )ii ' ) be a directed system in the category of locally convex topological vector spaces with locally convex direct limit .V and induced mappings .φi ∈ L(Vi ; V), .i ∈ I . The directed system is: (i) compact if the connecting maps .φii ' , .i  i ' , are compact; (ii) regular if, given a bounded set .B ⊆ V, there exists .i ∈ I and a bounded set .Bi ⊆ Vi such that .B ⊆ φi (Bi ); (iii) boundedly retractive if the connecting maps .φii ' , .i  i ' , are injective, and if, given a bounded set .B ⊆ V, there exists .i ∈ I and a bounded set .Bi ⊆ Vi such that .φi |Bi is an homeomorphism onto .B. .◦ As inverse and direct limits, sometimes in combination, are a common means of arriving at locally convex topological vector spaces in practice, it is useful to know what attributes of locally convex topological vector spaces persist under inverse and/or direct limits. For instance, countable inverse limits of Fréchet spaces

1.8 Locally Convex Topological Vector Spaces

21

(see below) are Fréchet spaces, while countable direct limits of Fréchet spaces are generally not. We shall make substantial use of permanence properties of direct limits when we give our construction of the real analytic topology, and we shall reference the appropriate places in the literature at that time.

1.8.2 Metrisable Locally Convex Topological Vector Spaces A locally convex topological vector space .(V, O) is metrisable if it is Hausdorff and if its topology is the metric topology for a translation-invariant metric on .V. One can show that .(V, O) is metrisable if and only if its topology can be defined by a countable or finite family of seminorms. A complete metrisable locally convex topological vector space is a Fréchet space. Typically, Fréchet spaces are to be regarded as somewhat friendly. However, the real analytic topology we define in this book for the space of sections of a real analytic vector bundle is not metrisable. For this reason, one should establish other useful properties for this topology in order for it to be manageable. A useful generalisation of separable Fréchet spaces is to make use of the generalisations of complete separable metric spaces above to Lusin and Suslin spaces. Indeed, we shall be able to make use of the fact that certain Hausdorff countable direct limits of Suslin spaces are Suslin spaces and that all countable inverse limits of Suslin spaces are Suslin spaces. It is difficult to piece together these facts from the literature, so let us illustrate how this can be done. Proposition 1.6 (Direct and Inverse Limits of Suslin Locally Convex Topological Vector Spaces) Let .(Vj , Oj ), .j ∈ Z>0 , be a countable family of Suslin locally convex topological vector spaces. Then the following statements hold: (i) if .Vj ⊆ Vj +1 and if the inclusion of .Vj in .Vj +1 is continuous (i.e., .Oj is finer (V , O ) is Suslin than the induced topology for .Vj from .Oj +1 ), then .lim − →j →∞ j j if it is Hausdorff; (ii) if .φj ∈ L(Vj +1 ; Vj ), .j ∈ Z>0 and if we consider the obvious inverse system associated to this collection of continuous linear maps,4 then .lim (V , O ) ← −j →∞ j j is Suslin. Proof (i) Let .(V, O) = lim (V , O ) be the locally convex direct limit. Let us fix − →j →∞ j j ' .j ∈ Z>0 for a moment. Let .O be the topology for .Vj induced by .O, noting that .Vj j is identified with a linear subspace of .V. Since the definition of direct limit implies that the inclusion of .(Vj , Oj ) in .(V, O) is continuous, we have .Oj' ⊆ Oj . Thus the identity map from .(Vj , Oj ) to .(Vj , Oj' ) is continuous, and so .(Vj , Oj' ) is the is to say, the inverse system where, if .j1 ≤ j2 , then take the linear map in .L(Vj2 ; Vj1 ) to be · · · ◦ φj2 −1 .

4 That .φj1 ◦

22

1 Notation and Background

continuous image of the Suslin space .(Vj , Oj ), and so is itself Suslin (carefully noting that .Oj' is assumed to be Hausdorff). Now we make use of a lemma. Lemma 1 Let .(X, O) be an Hausdorff topological space and let .Sj ⊆ X, .j ∈ Z>0 , be topological subspaces for which (i) the induced topology for .Sj from .(X, O) is Suslin and (ii) .X = ∪j ∈Z>0 Sj . Then .(X, O) is a Suslin space. Proof For each .j ∈ Z>0 , let .(Yj , Oj ) be a Polish space and let .Ф j ∈ C0 (Yj ; Sj ) be a continuous surjective mapping. Since the topology on .Sj is the induced topology, we can regard .Ф j ∈ C0 (Yj ; X). It is easy to verify that the disjoint union ◦

∪j ∈Z>0 (Yj , Oj ) is itself a Polish space.5 Now define

.

Ф : .

◦ ⎟⎟

Yj → X

j ∈Z>0

(yj , j ) │ → Ф j (yj ), which is readily verified to be continuous. Moreover, .image(Ф ) = ∪j ∈Z>0 Sj = X, which gives the result. . This part of the proposition follows immediately from the lemma and what we proved before the lemma. (ii) Here we note some more or less well known facts about Suslin spaces: 1. a countable product of Suslin spaces is a Suslin space [8, Lemma 6.6.5(iii)]; 2. a closed subspace of a Suslin space is a Suslin space [8, Lemma 6.6.5(ii)]. Now, since the inverse limit of an inverse system of Hausdorff locally convex topological vector spaces is a closed subspace of the product of the spaces from the inverse family [37, Proposition 2.6.1], this part of the proposition follows. 

1.8.3 Open Mapping Theorems for Locally Convex Topological Vector Spaces An important theorem in the linear analysis of normed vector spaces is the Open Mapping Theorem which asserts that a surjective continuous linear mapping 5 Countable disjoint unions of separable spaces are pretty clearly themselves separable. Moreover, disjoint unions of metric spaces are also metric spaces if one defines the distance between points in disjoint components of the disjoint union to be distance 1 from one another. If each of the metrics is complete, it is easy to see that the resulting metric for the disjoint union is also complete.

1.8 Locally Convex Topological Vector Spaces

23

between Banach spaces is open. In general, to assert an analogous result for locally convex topological vector spaces requires various sorts of hypotheses on both the domain and codomain. We shall report an approach to this developed by De Wilde [16]. First of all, let us introduce the standard terminology for continuous, open linear mappings between locally convex topological vector spaces. Definition 1.7 (Topological Homomorphism, Topological Monomorphism, Topological Epimorphism) Let .(U, OU ) and .(V, OV ) be locally convex topological vector spaces and let .L ∈ L(U; V). Then L is: (i) a topological homomorphism if it is an open mapping onto .image(L) (with the induced topology); (ii) a topological monomorphism if it is an injective topological homomorphism; (iii) a topological epimorphism if it is a surjective topological homomorphism. .◦ Frequently, one uses the simpler terms “homomorphism,” “monomorphism,” and “epimorphism.” However, this can lead to confusion since, for example, linear monomorphisms are not generally topological monomorphisms, and so it seems best to make the category explicit. For the next piece of terminology, a locally convex topological vector space is ultrabornological if it is an Hausdorff direct limit of Banach spaces. Next, a web in a locally convex topological vector space .(V, O) is a family .Cj1 ,...,jk , .j1 , . . . , jk ∈ Z>0 , .k ∈ Z>0 , of balanced6 convex subsets satisfying the following: 1. .∪j ∈Z>0 Cj = V; 2. .∪j ∈Z>0 Cj1 ,...,jk ,j = Cj1 ,...,jk for all .j1 , . . . , jk ∈ Z>0 , .k ∈ Z>0 ; 3. for each sequence .(jk )k∈Z>0 in .Z>0 , there exists a sequence .(rk )k∈Z>0 in .R>0 such that, Σ for every sequence .(vk )k∈Z>0 for which .vk ∈ Cj1 ,...,jk , .k ∈ Z>0 , the series . ∞ k=1 rk vk converges. A locally convex topological vector space with a web is called webbed. If the notion of a webbed topological vector space seems a little contrived, it is: specifically, it is contrived to give rise to an Open Mapping Theorem. The notion of a web is also useful because it is persistent under countable inverse limits and well-behaved countable direct limits [37, Corollary 5.3.3]. The De Wilde Open Mapping Theorem states that a surjective continuous linear mapping from a webbed locally convex topological vector space onto an ultrabornological locally convex topological vector space is open. A fairly succinct proof can be found in [51, Theorem 24.30].

6A

subset B of a .R-vector space is balanced if .v ∈ B implies that .−v ∈ B.

24

1 Notation and Background

1.8.4 Nuclear Locally Convex Topological Spaces Nuclear locally convex topological vector spaces form a special, yet interesting, class of locally convex topological vector spaces with useful properties resembling finite-dimensional vector spaces. For example, a subset of a nuclear locally convex topological vector space is compact if and only if it is closed and bounded. This implies, as a consequence, that the only normed nuclear spaces are those that are finite-dimensional. To define this class of locally convex topological vector spaces, we first define the notion of a nuclear mapping between Banach spaces. Thus let .(U, pU ) and .(V, pV ) be Banach spaces. A mapping .φ ∈ L(U; V) is nuclear if there exists a sequence ' ' .(λj )j ∈Z>0 in the closed unit ball of .U (giving .U the operator norm), a sequence 1 .(vj )j ∈Z>0 in the closed unit ball of .V, and a sequence .(cj )j ∈Z>0 in .l (Z>0 ; R) for which φ(u) =

∞ Σ

.

cj vj .

j =1

One should think of nuclear linear maps between Banach spaces as having a “small” image. To make use of this Banach space construction to define the notion of a nuclear locally convex topological vector space, we let .(V, O) be a locally convex topological vector space and let p be a continuous seminorm for .V. We can then define the normed vector space .(Vp , p) ˆ by Vp = V/p−1 (0),

.

p(v ˆ + p−1 (0)) = inf{p(v + v ' ) | p(v ' ) = 0},

noting that .p−1 (0) is a subspace. We denote by .(Vp , p) ˆ the Banach space ˆ We then say that .(V, O) is nuclear if, for every continuous completion of .(Vp , p). seminorm p for .V, there exists a continuous seminorm .p' for .V with .p' ≥ p and such that the natural projection .πp,p' : Vp → Vp' is a nuclear mapping between Banach spaces.

1.8.5 Tensor Products of Locally Convex Topological Vector Spaces The subject of tensor products of locally convex topological vector spaces is enormously rich and owes its initial development to the early work of Grothendieck [28]. We require only the most elementary parts of the theory, namely the so-called projective tensor product [37, Chapter 15].

1.8 Locally Convex Topological Vector Spaces

25

Let .(U, OU ) and .(V, OV ) be locally convex topological vector spaces. Note that we have the bilinear and linear mappings U ⊗ V → HomR (U∗ ; V),

U × V → U ⊗ V,

.

respectively, defined by (u, v) │ → u ⊗ v,

.

u ⊗ v │ → (α │ → v).

These algebraic constructions can be used to induce topologies on .U ⊗ V. We shall focus on the first of these mappings, this giving the projective tensor topology. Thus the projective tensor topology can be seen as the topological version of the usual definition of the tensor product in the algebraic setting. The second mapping gives, after some appropriate constructions, the injective tensor topology. The projective tensor topology is the locally convex final topology for .U ⊗ V associated with the bilinear mapping .U × V to .U ⊗ V. One typically denotes by .U ⊗π V the vector space .U ⊗ V equipped with the projective tensor topology. One way to describe the projective tensor topology is to define a family of seminorms that describes the topology. If q is a continuous seminorm for .(U, OU ) and if p is a continuous seminorm for .(V, OV ), then define .q ⊗π p : U ⊗ V → R by ⎧ k ⎨Σ

| | | .q ⊗π p(A) = inf q(uj )p(vj )|| ⎩ | j =1 A=

k Σ j =1

uj ⊗ vj , uj ∈ U, vj ∈ V, j ∈ {1, . . . , k}, k ∈ Z>0

⎫ ⎬ ⎭

.

One can show that .q ⊗π p is indeed a seminorm, and that .q ⊗π p(u⊗v) = q(u)p(v). Moreover, the family of seminorms .q ⊗p, as q and p run over families of seminorms defining the topologies .OU and .OV , gives the projective tensor topology for .U ⊗ V. There is also a characterisation of the projective tensor product that is akin, for locally convex topological vector spaces, to the usual universal property of the algebraic tensor product. To give this characterisation, we add a locally convex topological vector space .(W, OW ) into the discussion. We denote by .L(U, V; W) the set of continuous bilinear mappings from .U × V to .W. Then the projective tensor topology is the unique topology for which the mapping .U × V │ → U ⊗ V is continuous and for which, for any locally convex topological vector .(W, OW ) and

26

1 Notation and Background

β ∈ L(U, V; W), there exists a unique .φβ ∈ L(U ⊗π V; W) such that the diagram

.

.

commutes.

1.9 Real Analytic Real Analysis In order to initiate an uninitiated reader to the subject of real analytic differential geometry, we give a rapid overview of real analytic analysis in Euclidean spaces. A thorough presentation can be found in the book of Krantz and Parks [46]. The notion of real analyticity typically has its origin in its connection to convergent power series. Let us first, therefore, consider convergent power series. Let .n, m ∈ Z>0 . A real power series in n variables with values in .Rm is a formal power series m Σ Σ .

j =1

αIa XI ea ,

I ∈Zn≥0

for coefficients .αIa ∈ R, .a ∈ {1, . . . , m}, .I ∈ Zn≥0 and indeterminants .X =

(X1 , . . . , Xn ). We recall that, if .I = (i1 , . . . , in ) ∈ Zn≥0 , then .XI = X1i1 · · · Xnin . If one wishes to be a little more precise, one can define the series by the mapping {1, . . . , m} × Zn≥0 ϶ (a, I ) │ → αIa ∈ R,

.

which determines the coefficients of the formal series. Writing these coefficients in their rôle of a series with indeterminates connects the precise definition with how one actually thinks about it. We call such a formal power series convergent if there exists some .r ∈ Rn>0 such that the series m Σ Σ .

j =1 I ∈Zn≥0

αIa r I ea

1.9 Real Analytic Real Analysis

27

converges absolutely. In this case, we say that the power series converges for .r. If a formal power series converges for .r, then the series m Σ Σ .

j =1

αIa x I ea

I ∈Zn≥0

converges absolutely for any .x ∈ Dn (r, 0) and uniformly on compact subsets of n .D (r, 0). Since the series obtained by term-by-term differentiation of the original series is itself a power series, it can be subjected to the same arguments. To this end, one can show that the power series obtained by taking any single partial derivative converges for .r if the original series converges for .r. Thus we can see that a formal power series converges to an infinitely differentiable function whose derivatives can be obtained by iterated term-by-term differentiation of the original power series. Additionally, the same arguments as the preceding results also give absolute convergence of the complex power series m Σ Σ .

j =1

αIa zI ea

I ∈Zn≥0

in the complex polydisk {z ∈ Cn | |zj | ≤ rj }

.

and uniform convergence on compact subsets of the corresponding open complex disk. Thus the .Rm -valued functions obtained from real power series are actually convergent complex power series, and so define an holomorphic function from the open complex polydisk to .Cm . Thus the functions obtained from convergent real power series extend to complex power series with an analogous domain of convergence. Thus power series. Let us turn now to functions. For .U ⊆ Rn open, for .f ∈ C∞ (U; Rm ), and for .x 0 ∈ U, the Taylor series for .f at .x 0 is the real power series

.

m Σ Σ D I fa (x 0 ) I ∈Zn≥0 a=1

I!

Xea .

This series will generally not converge. Indeed, it is a theorem of Borel [9] that any real power series—convergent or not—is the Taylor series for some .C∞ -mapping defined in some neighbourhood of .x 0 . This leads to the following definition. Definition 1.8 (Real Analytic Mapping) Let .U ⊆ Rn be open. A mapping m is real analytic if, for each .x ∈ U, the Taylor series for .f at .x .f : U → R 0 0

28

1 Notation and Background

converges and if, in some neighbourhood .V of .x 0 , Σ D I f (x 0 ) (x − x 0 )I . I! n

f (x) =

.

◦

I ∈Z≥0

In brief, a mapping is real analytic when it is equal to its own (convergent) Taylor series in some neighbourhood of every point. Note that, while a function may be real analytic on some domain, its Taylor series at a single point may not converge on the entire domain. To see this, consider the two real analytic functions .

f:R→R

g: R → R

x │ → e , x

x │ →

1 . 1 + x2

For f , we have ∞ Σ D j f (0)

f (x) =

.

j =0

j!

xj =

∞ Σ xj j =0

j!

pointwise on .R, and uniformly on compact subsets of .R. On the other hand, the Taylor series of g about 0 is ∞ Σ D j g(0) .

j!

j =0

∞ Σ = (−1)j x 2j , j =0

and this series converges only on .(−1, 1). This does not contradict real analyticity since, for example, g possesses a (different) Taylor series convergent to g in some neighbourhood of .±1. A question of some importance, and of whose answer we will make essential use, is that of how one can recognise real analytic mappings in the class of .C∞ mappings. Theorem 1.9 (Characterisation of Real Analytic Functions) If .U ⊆ Rn is open and if .f : U → R is infinitely differentiable, then the following statements are equivalent: (i) f is real analytic; (ii) for each .x 0 ∈ U, there exists a neighbourhood .V ⊆ U of .x 0 and .C, r ∈ R>0 such that |D I f (x)| ≤ CI !r −|I |

.

for all .x ∈ V and .I ∈ Zn≥0 .

1.9 Real Analytic Real Analysis

29

Proof It is easy to prove the theorem using holomorphic extensions and Cauchy estimates for holomorphic functions [45, Lemma 2.3.9]. A “real” proof is more difficult, and one such is given in [46, Proposition 2.2.10].  As we have previously indicated, we shall denote by .Cω (U; Rm ) the set of real analytic mappings from .U to .Rm . This set of real analytic mappings is blessed with the algebraic properties one expects. To wit, 1. sums of real analytic mappings are real analytic, 2. compositions of real analytic mappings are real analytic, and 3. for scalar-valued functions, products of real analytic functions are real analytic and the quotient of a real analytic function by a nowhere zero real analytic function is real analytic. If .U, V ⊆ Rn are open, a real analytic diffeomorphism from .U to .V is a real analytic bijection .⏀ : U → V with a real analytic inverse. The Inverse Function Theorem is valid for real analytic functions: if .⏀ : U → Rn is such that .D⏀(x) is invertible, then there is a neighbourhood .U' ⊆ U of .x such that .⏀|U' : U' → ⏀(U' ) is a real analytic diffeomorphism. The real analytic Inverse Function Theorem is essential for real analytic differential geometry, just as the smooth version is essential for smooth differential geometry.

Chapter 2

Topology for Spaces of Real Analytic Sections and Mappings

In this chapter we introduce the main characters of the book, the topologies for the space of real analytic sections of a real analytic vector bundle and for the space of real analytic mappings between real analytic manifolds. We first give an overview of real analytic differential geometry. After this, we will give four descriptions of the real analytic topology for the space of sections of a real analytic vector bundle, two due to Martineau [50] and two given by Jafarpour and Lewis [36], based on the note of Vogt [68]. Finally, in Sect. 2.5, we give a variety of descriptions for topologies for the space of real analytic mappings between real analytic manifolds, and we show that these all agree. In particular, we show how the topology for the space of sections of a vector bundle allows us to introduce a uniform structure for the space of real analytic mappings between real analytic mappings. We note that, as far as we know, these descriptions of the topology of the space of real analytic mappings are being given here for the first time. As can be seen from the above discussion, in this chapter we will arrive at a variety of topologies for spaces of real analytic sections and real analytic mappings. Part of the development will be to show that all of these various topologies agree. This will all contribute to the idea that there is essentially only one meaningful topology for spaces of real analytic sections and mappings. This is a featured shared by the holomorphic analogues, and in contrast to the smooth analogues. We shall have a few words to say about this at the end of the chapter.

2.1 Real Analytic Differential Geometry In this section we consider some of the basic constructions of real analytic differential geometry. We shall also indicate some relationships between real analytic and holomorphic differential geometry. Correspondingly, we also shed some light on some of the existential problems that arise in real analytic differential geometry. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. D. Lewis, Geometric Analysis on Real Analytic Manifolds, Lecture Notes in Mathematics 2333, https://doi.org/10.1007/978-3-031-37913-0_2

31

32

2 Topology for Spaces of Real Analytic Sections and Mappings

2.1.1 The Fundamental Objects of Real Analytic Differential Geometry We assume that the reader is thoroughly familiar with the basic constructions of smooth differential geometry. In particular, we assume that the reader will understand that the basic properties of real analytic mappings as enumerated near the end of Sect. 1.9 allow all of the basic constructions of smooth differential geometry to be applied to the real analytic case with a mere substitution of “.Cω ” for “.C∞ .” Thus we immediately have following constructions: 1. a real analytic manifold has an atlas of charts whose overlap maps are real analytic diffeomorphisms; 2. a real analytic submanifold is one admitting real analytic submanifold charts; 3. a real analytic vector bundle has an atlas of vector bundle charts for which the overlap maps are real analytic local vector bundle isomorphisms; 4. a mapping between real analytic manifolds is real analytic, or class .Cω , if it has real analytic local representatives; 5. a section of a real analytic vector bundle is real analytic if it is a real analytic mapping in the previous sense; 6. the tangent bundle of a real analytic manifold is itself a real analytic manifold; 7. the various tensor bundles associated with a real analytic vector bundle are real analytic vector bundles; 8. the notion of a real analytic linear connection on a vector bundle makes sense. As we mentioned in Sect. 1.4, we shall assume that all real analytic manifolds are Hausdorff, connected, and second countable. We also point out that all of the previous constructions apply with “real analytic” replaced by “holomorphic,” with the initial assumption that charts take values in complex Euclidean spaces. We shall say that functions, mappings, or sections are of class .Chol if they are holomorphic.

2.1.2 Existential Constructions in Real Analytic Differential Geometry Given the breezy way in which we indicate that real analytic and holomorphic differential geometry are “the same as” smooth differential geometry, the uninitiated reader can be forgiven for wondering whether there are any differences between the smooth, and the real analytic and holomorphic cases. The differences arise when one uses constructions in smooth differential geometry that make use of partitions of unity. Indeed, there is no such thing as a real analytic or holomorphic partition of unity. This is a result of the so-called Identity Theorem, a version of which says

2.1 Real Analytic Differential Geometry

33

that, if .M and .N are real analytic or holomorphic manifolds with .M connected,1 if .Ф , Ψ : M → N are real analytic or holomorphic mappings, and if there is an open subset .U ⊆ M such that .Ф |U = Ψ|U, then .Ф = Ψ. In particular, if the Taylor series of .Ф and .Ψ agree at a point, then the two mappings are equal everywhere! We do not know of a statement in the literature of this theorem in the real analytic case (Krantz and Parks [46, §1.2] give a one-dimensional version); this is not necessarily problematic, however, since a proof can be given that follows along the lines of the holomorphic proof. The latter is given in, e.g., [30, Theorem A.3]. What the Identity Theorem indicates is that real analytic and holomorphic differential geometry are far more rigid than smooth differential geometry. This manifests itself in the difficulty of proving basic existential results in the real analytic and holomorphic case that are trivial in the smooth case. There are two principal (and related) tools for solving existential problems in real analytic and holomorphic differential geometry: sheaf theory and embedding theorems. We shall briefly discuss each of these, and give an example of how they can be used. We will work primarily in the real analytic case, but will consider the holomorphic case in a few places. First we consider sheaf theory. We shall not take the time to develop the theory in any substantial way. . . or in any other way, really. . . but rather we illustrate how the theory can be used. We let .r ∈ {∞, ω}, let .M be a .Cr -manifold, and let .πE : E → M be a .Cr -vector bundle. By .CMr we denote the sheaf of .Cr -functions on .M and by .GEr we denote the sheaf of .Cr -sections of .E. A fundamental problem is whether there exist .Cr -sections of .E subject to certain constraints. A tool for ascertaining whether there is a solution to such a problem are the so-called Cartan’s Theorems A and B. These are originally proved by Cartan [12] in the holomorphic case and also by Cartan [13] in the real analytic case. We do not state Cartan’s Theorems, as this is the realm of a different book. However, the following lemma illustrates a use of the real analytic version of Cartan’s Theorem B. We state the result also in the smooth case and we give a sheaf theoretic version of the proof in this case as well. A direct solution using cut-off functions is, of course, elementary. Lemma 2.1 (Extension of Sections with Prescribed Jets) Let .r ∈ {∞, ω}, let πE : E → M be a .Cr -vector bundle, and let .S ⊆ M be a closed .Cr -submanifold. Let r .k ∈ Z≥0 . Let .V be a neighbourhood of .S and let .ξS ∈ Г (E|V). Then there exists r .ξ ∈ Г (E) such that .jk ξ(x) = jk ξS (x). .

Proof We begin with a sublemma. Simpler versions of this result are called Hadamard’s Lemma, but we could not find a reference to the form we require.

1 Our blanket assumption is that all manifolds are connected. But connectedness is fundamental here, so we reiterate it.

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2 Topology for Spaces of Real Analytic Sections and Mappings

Sublemma 1 Let .r ∈ {∞, ω}. Let .U ⊆ Rn be a neighbourhood of .0, let .S ⊆ Rn be the subspace S = {(x 1 , . . . , x n ) ∈ Rn | x 1 = · · · = x s = 0},

.

let .k ∈ Z≥0 , and let .f ∈ Cr (U) satisfy .D j f (x) = 0 for all .j ∈ {0, 1, . . . , k} and n .x ∈ S ∩ U. Let .prS : R → S be the natural projection onto the first s-components. Then there exist a neighbourhood .V ⊆ U of .0 and functions .gI ∈ Cr (V), .I ∈ Zs>0 , .|I | = k + 1, such that Σ

f (x) =

.

x ∈ V.

gI (x) prS (x)I ,

I ∈Zs>0 |I |=k+1

Proof We prove the sublemma by induction on k. For .k = 0, the hypothesis is that f vanishes on .S ∩ U. Let .W ⊆ S be a neighbourhood of .0 and let .∈ ∈ R>0 be such that .B(∈, y) ⊆ U for all .y ∈ W, possibly after shrinking .W. Let V=

| |

B(∈, x).

.

x∈W

Let .x = (x 1 , x 2 ) ∈ V (with .(0, x 2 ) ∈ S) and define γx : [0, 1] → R .

t |→ f (tx 1 , x 2 ).

We calculate f (x) = f (x 1 , x 2 ) = f (x 1 , x 2 ) − f (0, x 2 ) ⌠ 1 γx' (t) dt = γx (1) − γx (0) =

.

⌠ =

0

s 1Σ 0 j =1

Σ ∂f ((tx , x )) dt = x j gj (x), x 1 2 ∂x j s

j

j =1

where ⌠ gj (x) =

.

0

1

∂f (tx 1 , x 2 ) dt, ∂x j

j ∈ {1, . . . , s}.

It remains to show that the functions .g1 , . . . , gs are of class .Cr . By standard theorems on interchanging derivatives and integrals [38, Theorem 16.11], we can conclude that .g1 , . . . , gm are smooth when f is smooth. If the data are holomorphic, swapping integrals and derivatives allows us to conclude that .g1 , . . . , gs are

2.1 Real Analytic Differential Geometry

35

holomorphic when f is holomorphic, by verifying the Cauchy–Riemann equations. In the real analytic case, we can complexify to a complex neighbourhood of .0, and so conclude real analyticity by holomorphicity of the complexification. As a standin for a full proof by induction, let us see how the case .k = 1 follows from the case .k = 0. The general inductive argument is the same, only with more notation. We note that, for .x ∈ V, we have ⌠ Σ ∂g gk (x) + sj =1 x j ∂xjk (x), k ∈ {1, . . . , s}, ∂f (x) = . Σ s j ∂gj ∂x k k ∈ {s + 1, . . . , n}. j =1 x ∂x k (x), Thus .Df (x) = 0 for .x ∈ S ∩ V if and only if .g1 (x) = · · · = gs (x) = 0. Thus one can apply the arguments from the first part of the proof to write gk (x) =

s Σ

.

x j gkj (x)

j =1

on a neighbourhood of .0. Thus f (x) =

s Σ

.

x k x j gkj (x),

j,k=1

giving the desired form of f in this case. .Δ To prove the lemma, let .ZSk be the sheaf of .Cr -sections of .E whose k-jet vanishes on .S. We have the exact sequence .

Note that the stalk of the quotient sheaf at .x ∈ S consists of germs of sections whose k-jets agree on .S. Now, if .x /∈ S, then there is a neighbourhood .U of x such that .U ∩ S = ∅, and so .ZSk (U) = GEr (U). That .ZSk is locally finitely generated at x then follows since r .G is locally finitely generated. If .x ∈ S, choose a submanifold chart .(U, χ ) for .S E about x so that S ∩ U = {y ∈ U | φ(y) = (0, . . . , 0, x s+1 , . . . , x n )}.

.

Then the k-jet of a function f on .U vanishes on .S if and only if it is a .Cr (U)-linear combination of polynomial functions in .x 1 , . . . , x s of degree .k + 1; this follows by the above sublemma. Thus, if .ξ1 , . . . , ξm is a local basis of sections of .E about x, then the (finite) set of products of these sections with the polynomial functions in .x 1 , . . . , x s of degree at least .k + 1 generates .Г r (E|U) as a .Cr (U)-module. This

36

2 Topology for Spaces of Real Analytic Sections and Mappings

shows that .ZSk is locally finitely generated about x. This then shows that .ZSk is coherent in the case .r = ω, by virtue of [25, Consequence A.4.2.2]. Cartan’s Theorem B [13, Proposition 6] shows, in the case .r = ω, that .Ψ is surjective on global sections. The case of .r = ∞ follows in a similar way, using the fact that positive cohomology groups for sheave of modules of smooth functions vanish ([69, Proposition 2.3.11], along with [69, Examples 2.3.4(d,e)] and [69, Proposition 2.3.5]). For the degree 1 cohomology group, this implies that there exists r .ξ ∈ Г (E) such that, for each .x ∈ S, .[ξ ]x = [ξS ]x . This, however, means precisely □ that .jk ξ(x) = jk ξS (x) for each .x ∈ S. Remarks 2.2 (On the Extension of Sections) Let us make two comments on this lemma, as they illustrate the subtle correspondences—and lack of correspondences—between smooth, real analytic, and holomorphic differential geometry. 1. The previous lemma is rather trivial using partitions of unity in the smooth case, and does not require tools like sheaf cohomology. Indeed, one can sometimes regard partitions of unity and sheaf theory as being interchangeable in the smooth case. This is not so in the real analytic case, where sheaf theory provides powerful tools that play the rôle of partitions of unity in the solution of certain kinds of problems. 2. The previous lemma is generally false in the holomorphic case. Indeed, holomorphic differential geometry is far more rigid that real analytic differential geometry. This is most succinctly illustrated by the following fact: the set of sections of an holomorphic vector bundle with a compact base space is finitedimensional.2 ◦ Next we consider the matter of embedding theorems. For smooth manifolds, Whitney’s Embedding Theorem asserts that an Hausdorff, connected, second countable, smooth manifold3 can be properly embedded in a sufficiently highdimensional copy of real Euclidean space [71]. The real analytic version is the following result, which we enunciate clearly for future reference. Theorem 2.3 (Grauert’s Embedding Theorem) If .M is an Hausdorff, connected, second countable, real analytic manifold, then there exist .N ∈ Z>0 and a proper real analytic embedding .ιM ∈ Cω (M; RN ). Proof [24, Theorem 3].

□

We note that there is generally no such embedding theorem for holomorphic manifolds, e.g., by the Maximum Modulus Theorem, there exists no nonconstant holomorphic mapping from a compact connected holomorphic manifold into com-

2 An interesting way to prove this is to note that the space of sections in this case is a normed nuclear space, and so must be finite-dimensional. 3 We recall that the our standing geometric assumptions in Sect. 1.4 are that manifolds are always Hausdorff, connected, and second countable. We reiterate this here since it is essential for the validity of the embedding theorem.

2.1 Real Analytic Differential Geometry

37

plex Euclidean space. Holomorphic manifolds that do admit proper holomorphic embeddings into complex Euclidean space are important, and are called Stein manifolds [26]. The embedding theorem for Stein manifolds was proved by Remmert [61]. As an application of Grauert’s real analytic embedding theorem, we prove the following existential result concerning metrics and connections. Lemma 2.4 (Existence of Real Analytic Connections and Fibre Metrics) If πE : E → M is a real analytic vector bundle, then there exist

.

(i) (ii) (iii) (iv)

a real analytic linear connection on .E, a real analytic affine connection on .M, a real analytic fibre metric on .E, and a real analytic Riemannian metric on .M.

Proof By Theorem 2.3, there exists a proper real analytic embedding .ιE of .E in .RN for some .N ∈ Z>0 . There is then an induced proper real analytic embedding .ιM of N by restricting .ι to the zero section of .E. Let us take the subbundle .E ˆ of .M in .R E N .TR |ιM (M) whose fibre at .ιM (x) ∈ ιM (M) is Eˆ ιM (x) = T0x ιE (V0x E).

.

Now recall that .E ≃ ζ ∗ VE, where .ζ : M → E is the zero section [43, page 55]. Let us abbreviate .ιˆE = T ιE |ζ ∗ VE. We then have the following diagram

.

(2.1)

describing a monomorphism of real analytic vector bundles over the proper ˆ embedding .ιM , with the image of .ιˆE being .E. To prescribe a linear connection in the vector bundle .E, we will take the prescription whereby one defines a connector .K : TE → E such that the diagrams (1.6) are vector bundle mappings. We define K as follows. For .ex ∈ Ex and .Xex ∈ Tex E we have Tex ιˆE (Xex ) ∈ TιˆE (ex ) (RN × RN ) ≃ RN ⊕ RN ,

.

and we define K so that ιˆE ◦ K(Xex ) = pr2 ◦ Tex ιˆE (Xex );

.

this uniquely defines K by injectivity of .ιˆE , and amounts to using on .E the connection induced on .image(ˆιE ) by the trivial connection on .RN × RN . In

38

2 Topology for Spaces of Real Analytic Sections and Mappings

particular, this means that we think of .ιˆE ◦ K(Xex ) as being an element of the fibre of the trivial bundle .RN × RN at .ιM (x). If .vx ∈ TM, if .e, e' ∈ E, and if .X ∈ Te E and .X' ∈ Te' E satisfy .X, X' ∈ T πE−1 (vx ), then note that Te πE (X) = Te' πE (X' ) =⇒ Te (ιM ◦ πE )(X) = Te' (ιM ◦ πE )(X' )

.

=⇒ Te (pr2 ◦ ιˆE )(X) = Te' (pr2 ◦ ιˆE )(X' ) =⇒ TιM (x) pr2 ◦ Te ιˆE (X) = TιM (x) pr2 ◦ Te' ιˆE (X' ). Thus we can write Te ιˆE (X) = (x, e, u, v),

.

Te' ιˆE (X) = (x, e' , u, v ' )

for suitable .x, u, e, e' , v, v ' ∈ RN . Therefore, ιˆE ◦ K(X' ) = (x, v ' ),

ιˆE ◦ K(X) = (x, v),

.

ιˆE ◦ K(X + X' ) = (x, v + v ' ),

from which we immediately conclude that, for addition in the vector bundle T πE : TE → TM, we have

.

ιˆE ◦ K(X + X' ) = ιˆE ◦ K(X) + ιˆE ◦ K(X' ),

.

showing that the diagram on the left in (1.6) makes K a vector bundle mapping. On the other hand, if .ex ∈ E and if .X, X' ∈ Tex E, then we have, using vector bundle addition in .πTE : TE → E, ιˆE ◦ K(X + X' ) = pr2 ◦ Tex ιˆE (X + X' )

.

= pr2 ◦ Tex ιˆE (X) + pr2 ◦ Tex ιˆE (X' ) = ιˆE ◦ K(X) + ιˆE ◦ K(X' ), giving that the diagram on the right in (1.6) makes K a vector bundle mapping. Since K is real analytic, this defines a real analytic linear connection .∇ πE on .E as in [43, §11.11]. The existence of a fibre metric .GπE , a Riemannian metric .GM , and an affine connection .∇ M are straightforward. Indeed, we let .GRN be the Euclidean metric on N .R , and define .GπE and .GM by GπE (ex , ex' ) = GRN (ˆιE (ex ), ιˆE (ex' ))

.

2.1 Real Analytic Differential Geometry

39

and GM (vx , vx' ) = GRN (Tx ιM (vx ), Tx ιM (vx' )).

.

An affine connection .∇ M can be taken to be the Levi-Civita connection of .GM .

□

The upshot of the lemma is that we can always define the following data: 1. 2. 3. 4.

a linear connection .∇ πE in .E; an affine connection .∇ M on .M; a fibre metric .GπE on .E; a Riemannian metric .GM on .M.

We shall frequently assume this data without mention. We note that the constructions of the lemma give the following result of independent interest. We comment that this result is an example of something that is true in real analytic differential geometry, but not generally true in holomorphic differential geometry. Corollary 2.5 (Real Analytic Vector Bundles Are Subbundles of Trivial Bundles) If .π : E → M is a real analytic vector bundle, then .E is a .Cω -subbundle of a N trivial bundle .pr1 : RN M = M × R → M for suitable .N ∈ Z>0 . Proof This follows directly from the constructions that gave rise to the dia□ gram (2.1).

2.1.3 Complexification of Real Analytic Manifolds and Vector Bundles It is not too surprising that, to study real analytic differential geometry, one approach is to “complexify” to holomorphic differential geometry. In this section we describe how one does this. We let .πE : E → M be a real analytic vector bundle. We assume the data required to make the diagram (2.1) giving .πE : E → M as the image of a real analytic vector bundle monomorphism in the trivial vector bundle .RN × RN for some suitable .N ∈ Z>0 . Now √ we complexify. Recall that, if .V is a .C-vector space, then multiplication by . −1 induces a .R-linear map .J ∈ EndR (V).4 A .R-subspace .U of .V is totally real if .U ∩ J (U) = {0}. A submanifold of an holomorphic manifold, thinking of the latter as a smooth manifold, is totally real if its tangent spaces are totally real subspaces. By [72, Proposition 1], for a real analytic manifold .M there exists a complexification .M of .M, i.e., an holomorphic manifold having .M as a totally real

4 Thus

J satisfying .J (v) = iv for .v ∈ V.

40

2 Topology for Spaces of Real Analytic Sections and Mappings

submanifold and where .M has the same .C-dimension as the .R-dimension of .M. As shown by Grauert [24, §3.4], for any neighbourhood .U of .M in .M, there exists a Stein neighbourhood .S of .M contained in .U. By arguments involving extending convergent real power series to convergent complex power series (the conditions on coefficients for convergence are the same for both real and complex power series, as we indicated in Sect. 1.9), one can show that there is an holomorphic extension of .ιM to .ιM : M → CN , possibly after shrinking .M [14, Lemma 5.40]. By applying similar reasoning to the transition maps for the real analytic vector bundle .E, one obtains an holomorphic vector bundle .πE : E → M for which the diagram

.

commutes, where all diagonal arrows are complexification and where the inner diagram is as defined in the proof of Lemma 2.4. One can then define a real analytic Hermitian fibre metric .GπE on .E induced from the standard Hermitian metric on the fibres of the vector bundle .CN × CN and an Hermitian metric .GM on .M induced from the standard Hermitian metric on .CN .

2.2 Martineau’s Descriptions of the Real Analytic Topology We shall give a description of two topologies for the space .Г ω (E) of real analytic sections of a real analytic vector bundle .πE : E → M. The original work of Martineau [50] describes these topologies for the space of real analytic functions, but it is evident that the same considerations apply to sections of a general vector bundle. Each description offers benefits in terms of providing immediately some useful properties of the topology, although showing that they agree is something of an undertaking, and we shall make some comments about this. That all being said, let us turn to Martineau’s characterisations of the topology for .Г ω (E).

2.2 Martineau’s Descriptions of the Real Analytic Topology

41

2.2.1 Germs of Holomorphic Sections Over Subsets of a Real Analytic Manifold In two different places, we will need to consider germs of holomorphic sections. In this section we organise the methodology for doing this to unify the notation. Throughout this section we let .πE : E → M be a real analytic vector bundle and we assume that we have the constructions in place to make use of the complexification constructions of Sect. 2.1.3. Let .A ⊆ M and let .N A be the set of neighbourhoods of A in the complexification hol (E|U) and .η ∈ Г hol (E|V), we say that .ξ is .M. For .U, V ∈ N A , and for .ξ ∈ Г equivalent to .η if there exist .W ∈ N A and .ζ ∈ Г hol (E|W) such that .W ⊆ U ∩ V and such that ξ |W = η|W = ζ .

.

By .G hol we denote the set of equivalence classes, which we call the set of germs A,E

of sections of .E over A. By .[ξ ]A we denote the equivalence class of .ξ ∈ Г hol (E|U) for some .U ∈ N A . For the particular case of functions, i.e., sections of the vector bundle .RM , we denote the set of germs by .C hol . A,M

Now, for .x ∈ M, .Ex is a totally real subspace of .Ex with half the real dimension, and so it follows that Ex = Ex ⊕ J (Ex ),

.

where J is the complex structure on the fibres of .E. For .U ∈ N A , denote by Г hol,R (E|U) those holomorphic sections .ξ of .E|U such that .ξ (x) ∈ Ex for .x ∈ U ∩ M. We think of this as being a locally convex topological .R-vector space with the seminorms .phol , .K ⊆ U compact, defined by

.

K

| { } | hol (ξ ) = sup ‖ξ (x)‖G | x ∈ K , pK

.

E

(2.2)

i.e., we use the locally convex structure induced from the usual compact-open topology on .Г hol (E|U). Remark 2.6 (Closedness of “Real” Sections) We note that .Г hol,R (E|U) is a closed .R-subspace of .Г hol (E) in the compact-open topology, i.e., the restriction of requiring “realness” on .M is a closed condition. This is easily shown, and we often assume it often without mention. ◦

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2 Topology for Spaces of Real Analytic Sections and Mappings

Denote by .G hol,R the set of germs of sections from .Г hol,R (E|U), .U ∈ N A . If A,E

U1 , U2 ∈ N

.

A

satisfy .U1 ⊆ U2 , then we have the restriction mapping

.

rU2 ,U1 : Г hol,R (E|U2 ) → Г hol,R (E|U1 ) ξ |→ ξ |U1 .

This restriction is continuous since, for any compact set .K ⊆ U1 ⊆ U2 and any ξ ∈ Г hol,R (E|U2 ), we have .phol (rU2 ,U1 (ξ )) ≤ phol (ξ ). We also have maps

.

K

K

rU,A : Г hol,R (E|U) → G hol,R A,E

.

ξ |→ [ξ ]A . Note that .N A is a directed set by reverse inclusion; that is, .U2  U1 if .U1 ⊆ U2 . Thus we have the directed system .(Г hol,R (E|U))U∈N A , along with the mappings .r U2 ,U1 , in the category of locally convex topological .R-vector spaces. This then

gives rise to the locally convex direct limit topology for the direct limit .G hol,R , as in A,E Definition 1.4. We shall use this general development in two different ways, one of which we turn our attention to now.

2.2.2 A Natural Direct Limit Topology for the Space of Real Analytic Sections We let .πE : E → M be a real analytic vector bundle. As in Sect. 2.1.3, we shall extend .E to an holomorphic vector bundle .πE : E → M that will serve as an important device for all of our constructions. The following lemma is key. Lemma 2.7 (Real Analytic Sections as Holomorphic Germs) There is a natural R-vector space isomorphism between .Г ω (E) and .G hol,R .

.

M,E

Proof Let .ξ ∈ Г ω (E). As in [14, Lemma 5.40], there is an extension of .ξ to a section .ξ ∈ Г hol,R (E|U) for some .U ∈ N M . We claim that the map .iM : Г ω (E) → G hol,R defined by .iM (ξ ) = [ξ ]M is the desired isomorphism. That .iM is independent M,E

of the choice of extension .ξ is a consequence of the fact that the extension to .ξ is unique inasmuch as any two such extensions agree on some neighbourhood contained in their intersection; this is the uniqueness assertion of [14, Lemma 5.40]. This fact also ensures that .iM is injective. For surjectivity, let .[ξ ]M ∈ G hol,R and let M,E

us define .ξ : M → E by .ξ(x) = ξ (x) for .x ∈ M. Note that the restriction of .ξ to

2.2 Martineau’s Descriptions of the Real Analytic Topology

43

M is real analytic because the values of .ξ |M at points in a neighbourhood of .x ∈ M are given by the restriction of the (necessarily convergent) complex Taylor series of .ξ to .M. Obviously, .iM (ξ ) = [ξ ]M . □ .

Now we use the locally convex direct limit topology on .G hol,R described in M,E Sect. 2.2.1, along with the preceding lemma, to immediately give a locally convex topology for .Г ω (E) that we refer to as the direct limit topology. Let us make an important observation about the direct limit topology for .Г ω (M). Let us denote by .SM the set of all Stein neighbourhoods of .M in .M. As shown by Grauert [24, §3.4], if .U ∈ N M , then there exists .S ∈ SM with .S ⊆ U. Therefore, .SM is cofinal5 in .N M and so the directed systems .(Г hol,R (E|U))U∈N M

and .(Г hol,R (E|S))S∈SM induce the same final topology on .Г ω (E) [29, page 137].

2.2.3 The Topology of Holomorphic Germs About a Compact Set We now turn to the second of Martineau’s topologies for .Г ω (M). This description first makes use of the constructions from Sect. 2.2.1, applied to the case when .K is a compact subset of .M. We continue with the notation from Sect. 2.2.1. For .K ⊆ M compact, we have the locally convex direct limit topology, described above for general subsets .A ⊆ M, on .G hol,R . We seem to have gained nothing, since we have yet another direct limit K,E topology. However, the direct limit can be shown to be of a friendly sort, namely a countable direct limit. Key to doing this is the following construction. Lemma 2.8 (Compact Sets as Countable Intersections of Open Sets) If .(M, d) is a locally compact metric space,6 if .K ⊆ S is compact, and if .NK is the set of neighbourhoods of .K, then there exists a sequence .(Uj )j ∈Z>0 of precompact neighbourhoods of .K such that (i) .cl(Uj +1 ) ⊆ Uj , .j ∈ Z>0 , (ii) .K = ∩j ∈Z>0 Uj , and (iii) if .U ∈ NK , then there exists .j0 ∈ Z>0 such that .Uj0 ⊆ U. Proof We first make use of some metric space constructions. For sets .A, B ⊆ M, denote dist(A, B) = inf{d(x, y) | x ∈ A, y ∈ B}.

.

5A 6A

subset .J ⊆ I is cofinal if, for every .i ∈ I , there exists .j ∈ J with .i  j . manifold, for example.

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2 Topology for Spaces of Real Analytic Sections and Mappings

If .A = {x}, denote .dist({x}, B) = dist(x, B) and so define a function distB : M → R≥0 .

x |→ dist(x, B).

Similarly we denote .dist(A, {y}) = dist(A, y). The following sublemma then records some useful facts. Sublemma 1 If .(M, d) is a metric space, then the following statements hold: (i) if .B ⊆ M, then the function .distB is uniformly continuous in the metric topology; (ii) if .A, B ⊆ M are disjoint closed sets, then .dist(x, B), dist(A, y) > 0 for all .x ∈ A and .y ∈ B. Proof (i) Let .∈ ∈ R>0 and take .δ = 2∈ . Let .y ∈ B be such that .d(x1 , y)−dist(x1 , B) < 2∈ . Then, if .d(x1 , x2 ) < δ, dist(x2 , B) ≤ d(x2 , y) ≤ d(x2 , x1 ) + d(x1 , y) ≤ dist(x1 , B) + ∈.

.

In a symmetric manner one shows that dist(x1 , B) ≤ dist(x2 , B) + ∈,

.

provided that .d(x1 , x2 ) < δ. Therefore, |dist(x1 , B) − dist(x2 , B)| < ∈,

.

provided that .d(x1 , x2 ) < δ, giving uniform continuity, as desired. (ii) Suppose that .dist(x, B) = 0. Then there exists a sequence .(yj )j ∈Z>0 in B such that .d(yj , x) < j1 for each .j ∈ Z>0 . Thus the sequence .(yj )j ∈Z>0 converges to x and so .x ∈ cl(B) = B. Therefore, if .A ∩ B = ∅ we can conclude that, if .dist(x, B) = 0, then .x /∈ A. That is, .dist(x, B) > 0 for every .x ∈ A, and similarly .dist(A, y) > 0 for every .y ∈ B. .Δ Let .x ∈ K and let .rx ∈ R>0 be such that .Bd (rx , x) is compact (by local compactness). Let .Vx ⊆ Bd (rx , x) be a precompact neighbourhood of x. Since k V . Note that .U is .K is compact, let .x1 , . . . , xk ∈ K be such that .K ⊆ U1  ∪ 1 j =1 xj precompact, being a finite union of precompact sets. Let .r1 = min{rx1 , . . . , rxk }. Clear the notation from the preceding paragraph so we can use it again for a different purpose, keeping only .U1 and .r1 . Let .x ∈ K and let .rx ∈ R>0 be such that .rx < r21 and let .Vx ⊆ Bd (rx , x) be a precompact neighbourhood of x. By compactness of .K, let .x1 , . . . , xk ∈ K be such that .K ⊆ U2  ∪kj =1 Vxj . Note that .U2 is precompact and that .cl(U2 ) ⊆ U1 . Let .r2 = min{rx1 , . . . , rxk }.

2.2 Martineau’s Descriptions of the Real Analytic Topology

45

We can continue in this way, each time choosing balls of half the minimum radius of the preceding step, to arrive at a sequence of precompact neighbourhoods having the first property in the statement of the lemma. For the second, we clearly have .K ⊆ ∩j ∈Z>0 Uj . If .x /∈ K, then .x /∈ Uj if we choose j sufficiently large that .rj < dist(K, x), this being possible by part (ii) of the sublemma. We thus conclude that .∩j ∈Z>0 Uj ⊆ K, giving the first part of the lemma. Finally, let .U ∈ NK and note that .distM\U |K is continuous by part (i) of the sublemma and so there exists .r ∈ R>0 such that .distM\U (x) ≥ r for every .x ∈ K. Thus, choosing .j0 ∈ Z>0 such that .rj0 < r, we have .Uj0 ⊆ U. □ By the lemma, there is a countable family .(UK,j )j ∈Z>0 from .N K with the property that .cl(UK,j +1 ) ⊆ UK,j and .K = ∩j ∈Z>0 UK,j . Moreover, the sequence .(UK,j )j ∈Z>0 is cofinal in .N K , i.e., if .U ∈ N K , then there exists .j ∈ Z>0 with .UK,j ⊆ U. Let us fix such a family of neighbourhoods. Let us fix .j ∈ Z>0 for a hol,R moment. Let .Г bdd (E|UK,j ) be the set of bounded sections from .Г hol,R (E|UK,j ), boundedness being taken relative to an Hermitian fibre metric .GπE for .πE : E → M. hol,R If we define a norm on .Г bdd (E|UK,j ) by hol pU

.

K,j

| } { | , ) = sup ‖ξ (x)‖ x ∈ (ξ U | G K,j πE ,∞

hol,R then this makes .Г bdd (UK,j ) into a normed vector space. This gives context for the next purely holomorphic lemma. hol (E)) Let .π : E → M be an holomorphic Lemma 2.9 (The Topology of .Г bdd E vector bundle. Then the following statements hold: hol (E) is a Banach space; (i) .Г bdd hol (E) with the norm topology in .Г hol (E) with the compact(ii) the inclusion of .Г bdd open topology is continuous and compact; (iii) if .U ⊆ M is a precompact open set with .cl(U) ⊆ M, then the restriction map hol (E|U) is continuous. from .Г hol (E) to .Г bdd

Proof hol (E) converges to (i) By [32, Theorem 7.9], a Cauchy sequence .(ξj )j ∈Z>0 in .Г bdd a bounded continuous section .ξ of .E. That .ξ is also holomorphic follows since uniform limits of holomorphic sections are holomorphic [30, page 5]. hol (E) (ii) For continuity, it suffices to show that a sequence .(ξj )j ∈Z>0 in .Г bdd hol converges to .ξ ∈ Г bdd (E) uniformly on compact subsets of .M if it converges in norm. This, however, is obvious. hol (E) be bounded. To show compactness of the inclusion, let .B ⊆ Г bdd hol |B is bounded. This implies that .p hol |B is bounded for any compact Then .pM K hol (E). Since .Г hol (E) is .K ⊆ M. Thus .B is also a bounded subset of .Г nuclear [47, Theorem II.8.2], we conclude that .B is precompact in .Г ν (E|U), giving compactness of the inclusion.

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2 Topology for Spaces of Real Analytic Sections and Mappings

(iii) Since the topology of .Г hol (E) is metrisable [47, Theorem II.8.2], it suffices to show that the restriction of a convergent sequence in .Г hol (E) to .U converges □ uniformly. This, however, follows since .cl(U) is compact. Now, no longer fixing j , the lemma ensures that we have a sequence of inclusions hol,R hol,R Г bdd (E|UK,1 ) ⊆ Г hol,R (E|UK,1 ) ⊆ Г bdd (E|UK,2 ) ⊆

.

hol,R · · · ⊆ Г hol,R (E|UK,j ) ⊆ Г bdd (E|UK,j +1 ) ⊆ · · · . hol,R (UK,j +1 ), .j ∈ Z>0 , is by restriction from The inclusion .Г hol,R (UK,j ) ⊆ Г bdd .UK,j to the smaller .UK,j +1 , keeping in mind that .cl(UK,j +1 ) ⊆ UK,j . Each of the inclusion maps in the preceding sequence is continuous by Lemma 2.9. For .j ∈ Z>0 define hol,R rK,j : Г bdd (E|UK,j ) → G hol,R K,E

.

(2.3)

ξ |→ [ξ ]K . Now one can show that the locally convex direct limit topologies induced on .G hol,R by the directed system .(Г hol,R (E|U))U∈N

K,E

K

of Fréchet spaces and by the directed

hol,R system .(Г bdd (E|UK,j ))j ∈Z>0 of Banach spaces agree [47, Theorem 8.4]. Let us give some properties of this topology.

Proposition 2.10 (Properties of .G hol,R ) Let .πE : E → M be a real analytic vector K,E bundle and let .K ⊆ M be compact and connected. Then the direct limit topology described above for .G hol,R has the following properties: K,E

(i) it is boundedly retractive (meaning that the direct limit is boundedly retractive); (ii) it is Hausdorff; (iii) it is complete; (iv) it is sequential; (v) it is Suslin; (vi) it is nuclear; (vii) it is ultrabornological; (viii) it is webbed. Proof (i) Let .(Uj )j ∈Z>0 be a cofinal sequence in .N K , as in Lemma 2.8. Since .K is connected, the neighbourhoods .Uj of Lemma 2.8 can be chosen to also be

2.2 Martineau’s Descriptions of the Real Analytic Topology

47

connected. Thus the inclusion mapping Г r (E|Uj ) c→ Г r (E|Uj +1 )

.

is injective by the Identity Theorem. By Lemma 2.9, the inclusion map r Г bdd (E|Uj ) c→ Г r (E|Uj )

.

is compact. Thus, by Jarchow [37, Proposition 17.1.1], the inclusion r r Г bdd (E|Uj ) c→ Г bdd (E|Uj +1 )

.

(ii) (iii) (iv) (v) (vi)

(vii)

(viii)

is compact. Thus the direct limit topology is compact, and so is boundedly retractive as follows from [44, Theorem 6’]. By definition, boundedly retractive direct limits are regular. Thus Hausdorffness of the direct limit topology follows from [37, Proposition 4.5.3]. As we saw in the proof of part (i), .G hol,R is a locally convex direct limit with K,E compact connecting maps. Completeness then follows from [44, Theorem 6’]. Fréchet spaces are sequential and Hausdorff direct limits of sequential spaces are sequential [22, Corollary 1.7]. This follows from Proposition 1.6(i). First we note that the space of sections of an holomorphic vector bundle with the compact-open topology is nuclear, essentially by [47, Theorem 8.2]. Thus hol,R .G is a countable Hausdorff locally convex direct limit of nuclear spaces. K,E Its nuclearity then follows from [37, Corollary 21.2.3]. First we note that Fréchet spaces are ultrabornological [37, Corollary 13.1.4]. The space of sections of an holomorphic vector bundle is a Fréchet space by [47, Theorem 8.2], and so is ultrabornological. Thus .G hol,R is ultraK,E bornological by [37, Corollary 13.1.6] as it is a countable Hausdorff locally convex direct limit. The space of sections of an holomorphic vector bundle is a Fréchet space by [47, Theorem 8.2], and so a webbed space by [37, Proposition 5.2.2]. Thus hol,R .G is webbed by [37, Corollary 5.3.3(b)] as it is a countable Hausdorff K,E locally convex direct limit. □

2.2.4 An Inverse Limit Topology for the Space of Real Analytic Sections Now we shall use the constructions from the preceding section to easily arrive at a topology on .Г ω (E) induced by the locally convex topologies on the spaces .G hol,R , K,E .K ⊆ M compact.

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2 Topology for Spaces of Real Analytic Sections and Mappings

For a compact set .K ⊆ M we have an inclusion .iK : Г ω (E) → G hol,R defined K,E

as follows. If .ξ ∈ Г ω (E), then .ξ admits an holomorphic extension .ξ defined on a neighbourhood .U ⊆ M of .M [14, Lemma 5.40]. Since .U ∈ N K we define .iK (ξ ) = [ξ ]K . We can consider the set .KM of compact subsets of .M to be a directed set by inclusion: if .K1 , K2 ∈ KM , we take .K1  K2 if .K1 ⊆ K2 . We then have the inverse system ⎛ .

⎞ (G hol,R )K∈KM , (πK2 ,K1 )K1 ⊆K2 , K,E

(2.4)

where we have πK2 ,K1 : G hol,R → G hol,R K2 ,E

.

K1 ,E

[ξ ]K2 |→ [ξ ]K1 , bearing in mind that .N K2 ⊆ N K1 if .K1 ⊆ K2 . We shall denote by lim G hol,R the corresponding inverse limit in the category of locally convex ← −K∈KM K,E topological vector spaces. The corresponding induced mappings from the direct limit we denote by

.

πK :

.

→ G hol,R . lim G hol,R K,E ← − K' ,E '

K ∈KM

We then have the following result. Lemma 2.11 (Real Analytic Sections as Inverse Limits) There is a natural .Rvector space isomorphism between .Г ω (E) and .lim G hol,R . ← −K∈KM K,E Proof By Lemma 2.7, we have a vector space isomorphism .G hol,R ≃ Г ω (M). Thus M,E it suffices to establish a vector space isomorphism G hol,R ≃ lim G hol,R . M,E ← − K,E

.

K∈KM

We define a linear mapping iK : G hol,R → G hol,R .

M,E

K,E

[ξ ]M |→ [ξ ]K . We claim that this mapping is injective. Indeed, knowledge of a germ .[ξ ]K implies, in particular, knowledge of the germ of .ξ at every .x ∈ K. This, then, uniquely prescribes the germ .[ξ ]M by the Identity Theorem (here we use explicitly our assumption that .M is connected). Now we note that the universal property of inverse

2.2 Martineau’s Descriptions of the Real Analytic Topology

49

limits (used here in the category of .R-vector spaces) gives the unique vertical linear mapping in the following diagram:

.

Since .iK is injective, so too is .πK ◦ ιM . This directly gives injectivity of .ιM . To show that .ιM is surjective, we note that an element of the inverse limit is an element of the product [37, Proposition 2.6.1], and so we can write an element of the inverse limit as .([ξ K ]K )K∈KM , where .[ξ K ]K ∈ G hol,R . We note that the germ .[ξ K ]K specifies K,E the germ at every .x ∈ K. Moreover, the definition of the inverse limit ensures that the germs of .[ξ K1 ]K1 and .[ξ K2 ]K2 agree at each .x ∈ K1 ∩ K2 . Since .M is covered by the union of its compact subsets, this shows that an element of the inverse limit uniquely prescribes a germ of a section at every .x ∈ M; thus this uniquely prescribes the germ of a global section (here we use the fact that the presheaf of real analytic sections is a sheaf). □ Thus the inverse limit topology for .lim G hol,R induces a locally convex ← −K∈KM K,E topology for .Г ω (M) that we call the inverse limit topology, naturally enough. Suppose now that we have a compact exhaustion .(Kj )j ∈Z>0 of .M. Since .N Kj +1 ⊆ N Kj we have a projection πj : G hol,R .

Kj +1 ,E

→ G hol,R Kj ,E

[ξ ]Kj +1 |→ [ξ ]Kj . One can check that, as .R-vector spaces, the inverse limit of the inverse system (G hol,R )j ∈Z>0 is isomorphic to that for the inverse system (2.4). This shows that

.

Kj ,E

the inverse limit topology can be obtained as a countable inverse limit of countable direct limits.

2.2.5 Properties of the Cω -Topology We now have two topologies for .Г ω (E), the direct limit topology of Sect. 2.2.2 and the inverse limit (of direct limit topologies) topology of Sect. 2.2.4. Let us consider the relationship between these two topologies. We first show that the mappings .iK from the proof of Lemma 2.11 are continuous.

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2 Topology for Spaces of Real Analytic Sections and Mappings

Lemma 2.12 (Continuity of Mappings of Germs) For .K ⊆ M compact, the mapping iK : G hol,R → G hol,R .

K,E

M,E

[ξ ]M |→ [ξ ]K is continuous if the domain and codomain have their direct limit topologies. Proof By definition of the direct limit topology for .G hol,R , the lemma will be M,E

proved if we can show that, for any .V ∈ N

M,

in the diagram

.

the diagonal mapping is continuous. Let .(Uj )j ∈Z>0 be a sequence in .N K as in Lemma 2.8. By part (iii) of that lemma, let .j0 ∈ Z>0 be such that .Uj0 ⊆ V. Then we have the diagram

.

which is directly verified to commute in the category of .R-vector spaces. Moreover, the restriction mapping .rV,Uj is continuous in the compact-open topologies, as is 0 directly verified, and the restriction mapping .rUj ,K is continuous by definition of 0

the direct limit topology for .G hol,R . This gives the lemma. K,E

□

Now we can prove the following result. Proposition 2.13 (The Inverse Limit Topology Is Coarser Than the Direct Limit Topology) The identity map on .Г ω (E) is continuous if the domain has the direct limit topology and the codomain has the inverse limit topology.

2.2 Martineau’s Descriptions of the Real Analytic Topology

51

Proof For each compact .K ⊆ M, we have the diagram

.

in the category of .R-vector spaces. If .G hol,R has the direct limit topology, then M,E

the diagonal arrow is continuous, as per the preceding lemma. If .lim G hol,R ← −K∈KM K,E has the inverse limit topology, then this topology is the initial topology associated with the mappings .πK . By the universal property of the inverse limit topology, the vertical arrow is continuous. Bearing in mind Lemmata 2.7 and 2.11, we get the proposition. □ One can now wonder whether the identity map on .Г ω (E) is, in fact, open, which would imply the equality of the direct limit and inverse limit topologies. If there were a suitable Open Mapping Theorem from the direct limit topology to the inverse limit topology, then this would give the result. However, the common Open Mapping Theorems fail to give the result. For example, the De Wilde Open Mapping Theorem would require that the direct limit topology be webbed (something that does not follow easily from the definition of the direct limit topology7 ) and that the inverse limit topology be ultrabornological (something that does not follow easily from the definition of the inverse limit topology8 ). Thus the properties of the domain and codomain are the opposite of what they need to be to apply the De Wilde Open Mapping Theorem. There are various ways in which one can establish the openness of the identity map from Proposition 2.13, and a nice discussion of these ideas can be found in the notes [17]. The original approach in [50, Theorem 1.2(a)] was by ∗ showing that .∪j ∈Z>0 (G hol ) is a dense subspace of the dual of .Г ω (E) equipped Kj ,E

with the direct limit topology, using earlier results in [49] on analytic functionals. A modern approach, using homological methods, equates an inverse limit being ultrabornological with the vanishing of .Proj1 , where .Proj is a functor on inverse systems devised by Palamodov [59]; see also [70]. In all cases, showing equality of the two topologies is not straightforward and we shall just accept this equality as true. In any case, we call the resulting topology, however it is defined, the .Cω topology. The two constructions of the .Cω -topology by Martineau permit some fairly easy conclusions about properties of this topology.

7 The

difficulty is that the direct limit is not countable in this case. difficulty is that subspaces of ultrabornological spaces, even closed subspaces, are not necessarily ultrabornological. 8 The

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2 Topology for Spaces of Real Analytic Sections and Mappings

Proposition 2.14 (Properties of .Г ω (E)) For a real analytic vector bundle ω ω .πE : E → M, the .C -topology for .Г (E) has the following properties: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

it is Hausdorff; it is complete; it is not metrisable; it is sequential; it is Suslin; it is nuclear; it is ultrabornological; it is webbed.

Proof We shall implicitly make use of the conclusions of Proposition 2.10 throughout the proof. We will also make use of both of our equivalent constructions of the real analytic topology, one from Sect. 2.2.2 and one from Sect. 2.2.4. (i) The .Cω -topology is an inverse limit of Hausdorff topologies, and so is Hausdorff by [37, Proposition 2.6.1(a)], noting that a product of Hausdorff topologies is Hausdorff and that a subspace of a Hausdorff space is Hausdorff. (ii) The .Cω -topology is an inverse limit of Hausdorff complete locally convex topological vector spaces, and so is complete by [37, Corollary 3.2.7]. (iii) This is a nontrivial property of the .Cω -topology, and we refer to [20] and [67, Theorem 10] for details. (iv) Fréchet spaces are sequential and Hausdorff direct limits of sequential spaces are sequential [22, Corollary 1.7]. (v) The .Cω -topology is a countable inverse limit of Suslin locally convex topological vector spaces, and so is Suslin by Proposition 1.6(ii). (vi) The .Cω -topology is a countable inverse limit of nuclear locally convex topological vector spaces, and so is nuclear by [37, Corollary 21.2.3]. (vii) The .Cω -topology is an Hausdorff direct limit of ultrabornological spaces, and so is ultrabornological by [37, Corollary 13.1.5]. (viii) The .Cω -topology is a countable inverse limit of webbed spaces, and so is webbed by [37, Corollary 5.3.3(a)]. □

2.3 Constructions with Jet Bundles In this section we provide some constructions that relate jet bundles and connections; these constructions will feature prominently in the book, starting in Sect. 2.4 where we use them to define seminorms for the .Cω -topology. Later, substantial portions of the text will be dedicated to finding certain uniform estimates for jet bundle norms; these estimates will be phrased for tensors arising from constructions with jet bundles and connections. It is fair to say, therefore, that constructions with jet bundles are at the core of our geometric approach.

2.3 Constructions with Jet Bundles

53

2.3.1 Decompositions for Jet Bundles Throughout this section, we will consider .r ∈ {∞, ω}. Let .πE : E → M be a .Cr -vector bundle. As per Lemma 2.4, we suppose that we have a linear connection .∇ πE on the vector bundle .E and an affine connection M on .M, all data being of class .Cr . We then have induced connections, that we .∇ also denote by .∇ πE and .∇ M , in various tensor bundles of .E and .TM, respectively. As in (1.8), the connections .∇ πE and .∇ M extend naturally to connections in various tensor products of .TM and .E, all of these being denoted by .∇ M,πE . Note that, if ∞ .ξ ∈ Г (E), then ∇ M,πE ,m ξ  ∇ M,πE · · · (∇ M,πE (∇ πE ξ )) ∈ Г ∞ (Tm (T∗ M) ⊗ E).

 

.

(2.5)

m−1 times

Now, given .ξ ∈ Г ∞ (E) and .m ∈ Z≥0 , we define D∇mM ,∇ πE (ξ ) = Symm ⊗ idE (∇ M,πE ,m ξ ) ∈ Г ∞ (Sm (T∗ M) ⊗ E),

.

(2.6)

We take the convention that .D∇0 M ,∇ πE (ξ ) = ξ . The following lemma is then key for our presentation. Lemma 2.15 (Decomposition of Jet Bundles) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. The map S∇mM ,∇ πE : Jm E → .

m ⊕

(Sj (T∗ M) ⊗ E)

j =0

jm ξ(x) |→ (ξ(x), D∇1 M ,∇ πE (ξ )(x), . . . , D∇mM ,∇ πE (ξ )(x)) is an isomorphism of .Cr -vector bundles, and, for each .m ∈ Z>0 , the diagram

.

m+1 is the obvious projection, stripping off the last component commutes, where .prm of the direct sum.

Proof We prove the result by induction on m. For .m = 0 the result is a tautology. For .m = 1, as in [43, §17.1], we have a vector bundle mapping .S∇ πE : E → J1 E over .idM that determines the connection .∇ πE by ∇ πE ξ(x) = j1 ξ(x) − S∇ πE (ξ(x)).

.

(2.7)

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2 Topology for Spaces of Real Analytic Sections and Mappings

Let us show that .S∇1 M ,∇ πE is well-defined. Thus let .ξ, η ∈ Г ∞ (E) be such that .j1 ξ(x) = j1 η(x). Then, clearly, .ξ(x) = η(x), and the formula (2.7) shows that .∇ πE ξ(x) = ∇ πE η(x), and so .S∇1 M ,∇ πE is indeed well defined. It is clearly linear on fibres, so it remains to show that it is an isomorphism. This will follow from dimension counting if it is injective. However, if .S∇1 M ,∇ πE (j1 ξ(x)) = 0 then .j1 ξ(x) = 0 by (2.7). For the induction step, we begin with a sublemma. Sublemma 1 Let .F be a field and consider the following commutative diagram of finite-dimensional .F-vector spaces with exact rows and columns:

.

If there exists a mapping .γ2 ∈ HomF (B; C2 ) such that .ψ2 ◦ γ2 = idB (with .p2 ∈ HomF (C2 ; A2 ) the corresponding projection), then there exists a unique mapping .γ1 ∈ HomF (B; C1 ) such that .ψ1 ◦ γ1 = idB and such that .γ2 = ι2 ◦ γ1 . There is also induced a projection .p1 ∈ HomF (C1 ; A1 ). Moreover, if there additionally exists a mapping .σ1 ∈ HomF (A2 ; A1 ) such that .σ1 ◦ ι1 = idA1 , then the projection .p1 is uniquely determined by the condition .p1 = σ1 ◦ p2 ◦ ι2 . Proof We begin by extending the diagram to one of the form

.

2.3 Constructions with Jet Bundles

55

also with exact rows and columns. We claim that there is a natural mapping .φ3 between the cokernels, as indicated by the dashed arrow in the diagram, and that .φ3 is, moreover, an isomorphism. Suppose that .u2 ∈ image(ι1 ) and let .u1 ∈ A1 be such that .ι1 (u1 ) = u2 . By commutativity of the diagram, we have φ2 (u2 ) = φ2 ◦ ι1 (u1 ) = ι2 ◦ φ1 (u1 ),

.

showing that .φ2 (image(ι1 )) ⊆ image(ι2 ). We thus have a well-defined homomorphism φ3 : coker(ι1 ) → coker(ι2 ) .

u2 + image(ι1 ) |→ φ2 (u2 ) + image(ι2 ).

We now claim that .φ3 is injective. Indeed, φ3 (u2 + image(ι1 )) = 0

.

=⇒

φ2 (u2 ) ∈ image(ι2 ).

Thus let .v1 ∈ C1 be such that .φ2 (u2 ) = ι2 (v1 ). Thus 0 = ψ2 ◦ φ2 (u2 ) = ψ2 ◦ ι2 (v1 ) = ψ1 (v1 )

.

=⇒ v1 ∈ ker(ψ1 ) = image(φ1 ). Thus .v1 = φ1 (u'1 ) for some .u'1 ∈ A1 . Therefore, φ2 (u2 ) = ι2 ◦ φ1 (u'1 ) = φ2 ◦ ι1 (u'1 ),

.

and injectivity of .φ2 gives .u2 ∈ image(ι1 ) and so .u2 + image(ι1 ) = 0 + image(ι1 ), giving the desired injectivity of .φ3 . Now note that .

dim(coker(ι1 )) = dim(A2 ) − dim(A1 )

by exactness of the left column. Also, .

dim(coker(ι2 )) = dim(C2 ) − dim(C1 )

by exactness of the middle column. By exactness of the top and middle rows, we have .

dim(B) = dim(C2 ) − dim(A2 ) = dim(C1 ) − dim(A1 ).

This proves that .

dim(coker(ι1 )) = dim(coker(ι2 )).

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2 Topology for Spaces of Real Analytic Sections and Mappings

Thus the homomorphism .φ3 is an isomorphism, as claimed. Now we proceed with the proof, using the extended diagram, and identifying the bottom cokernels with the isomorphism .φ3 . The existence of the stated homomorphism .γ2 means that the middle row in the diagram splits. Therefore, .C2 = image(φ2 ) ⊕ image(γ2 ). Thus there exists a well-defined projection .p2 ∈ HomF (C2 ; A2 ) such that .p2 ◦ φ2 = idA2 [31, Theorem 41.1]. We will now prove that .image(γ2 ) ⊆ image(ι2 ). By commutativity of the diagram and since .ψ1 is surjective, if .w ∈ B then there exists .v1 ∈ C1 such that .ψ2 ◦ ι2 (v1 ) = w. Since .ψ2 ◦ γ2 = idB , we have ψ2 ◦ ι2 (v1 ) = ψ2 ◦ γ2 (w)

.

=⇒

ι2 (v1 ) − γ2 (w) ∈ ker(ψ2 ) = image(φ2 ).

Let .u2 ∈ A2 be such that .φ2 (u2 ) = ι2 (v1 ) − γ2 (w). Since .p2 ◦ φ2 = idA2 we have u2 = p2 ◦ ι2 (v1 ) − p2 ◦ γ2 (w),

.

whence κ1 (u2 ) = κ1 ◦ p2 ◦ ι2 (v1 ) − κ1 ◦ p2 ◦ γ2 (w) = 0,

.

noting that (1) .κ1 ◦ p2 = κ2 (by commutativity), (2) .κ2 ◦ ι2 = 0 (by exactness), and (3) .p2 ◦ γ2 = 0 (by exactness). Thus .u2 ∈ ker(κ1 ) = image(ι1 ). Let .u1 ∈ A1 be such that .ι1 (u1 ) = u2 . We then have ι2 (v1 ) − γ2 (w) = φ2 ◦ ι1 (u1 ) = ι2 ◦ φ1 (u1 ),

.

which gives .γ2 (w) ∈ image(ι2 ), as claimed. Now we define .γ1 ∈ HomF (B; C1 ) by asking that .γ1 (w) ∈ C1 have the property that .ι2 ◦ γ1 (w) = γ2 (w), this making sense since we just showed that .image(γ2 ) ⊆ image(ι2 ). Moreover, since .ι2 is injective, the definition uniquely prescribes .γ1 . Finally we note that ψ1 ◦ γ1 = ψ2 ◦ ι2 ◦ γ1 = ψ2 ◦ γ2 = idB ,

.

as claimed. To prove the final assertion, let us denote .pˆ 1 = σ1 ◦ p2 ◦ ι2 . We then have pˆ 1 ◦ φ1 = σ1 ◦ p2 ◦ ι2 ◦ φ1 = σ1 ◦ p2 ◦ φ2 ◦ ι1 = σ1 ◦ ι1 = idA1 ,

.

using commutativity. We also have pˆ 1 ◦ γ1 = σ1 ◦ p2 ◦ ι2 ◦ γ1 = σ1 ◦ p2 ◦ γ2 = 0.

.

The two preceding conclusions show that .pˆ 1 is the projection defined by the splitting .Δ of the top row of the diagram, i.e., .pˆ 1 = p1 .

2.3 Constructions with Jet Bundles

57

Now suppose that the lemma is true for .m ∈ Z>0 . For any .k ∈ Z>0 , we have a short exact sequence

.

for which we refer to [64, Theorem 6.2.9]. Recall from [64, Definition 6.2.25] that we have an inclusion .ι1,m of .Jm+1 E in .J1 (Jm E) by .jm+1 ξ(x) |→ j1 (jm ξ(x)) (see Sect. 3.2.4). We also have an induced injection ιˆ1,m : Sm+1 (T∗ M) ⊗ E → T∗ M ⊗ Jm E

.

defined by the composition .

Explicitly, the left arrow is defined by α 1 ⊙ · · · ⊙ α m+1 ⊗ ξ |→

m+1 Σ

.

α j ⊗ α 1 ⊙ · · · ⊙ α j −1 ⊙ α j +1 ⊙ · · · ⊙ α m+1 ⊗ ξ,

j =1

⊙ denoting the symmetric tensor product defined in (1.1); see Lemma 3.20. We thus have the following commutative diagram with exact rows and columns:

.

(2.8)

.

We shall define a connection on .(πE,m )1 : J1 (Jm E) → Jm E which gives a splitting .Г 1,m and .P1,m of the lower row in the diagram. By the sublemma, this will give a splitting .Г m+1 and .Pm+1 of the upper row, and so give a projection from .Jm+1 E onto .Sm+1 (T∗ M) ⊗ E, which will allow us to prove the induction step. To compute .Pm+1 from the sublemma, we shall also give a map .λ1,m as in the diagram so that m+1 ∗ .λ1,m ◦ ιˆ1,m is the identity on .S (T M) ⊗ E. We start, under the induction hypothesis, by making the identification Jm E ≃

m ⊕

.

j =0

Sj (T∗ M) ⊗ E,

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2 Topology for Spaces of Real Analytic Sections and Mappings

and consequently writing a section of .Jm E as x |→ (ξ(x), D∇1 M ,∇ πE (ξ(x)), . . . , D∇mM ,∇ πE (ξ(x))).

.

We then have a connection .∇ πE,m on .Jm E given by π

∇XE,m (ξ, D∇1 M ,∇ πE (ξ ), . . . , D∇mM ,∇ πE (ξ ))

.

π

M,πE

= (∇XE ξ, ∇X

M,πE

D∇1 M ,∇ πE (ξ ), . . . , ∇X

D∇mM ,∇ πE (ξ )),

cf. constructions from Sect. 3.2.4. Thus ∇ πE,m (ξ, D∇1 M ,∇ πE (ξ ), . . . , D∇mM ,∇ πE (ξ ))

.

= (∇ πE ξ, ∇ M,πE D∇1 M ,∇ πE (ξ ), . . . , ∇ M,πE D∇mM ,∇ πE (ξ )), which—according to the jet bundle characterisation of connections from [43, §17.1] and which we have already employed in (2.7)—gives the mapping .P1,m in the diagram (2.8) as P1,m (j1 (ξ, D∇1 M ,∇ πE (ξ ), . . . , D∇mM ,∇ πE (ξ )))

.

= (∇ πE ξ, ∇ M,πE D∇1 M ,∇ πE (ξ ), . . . , ∇ M,πE D∇mM ,∇ πE (ξ )). Now we define a mapping .λ1,m for which .λ1,m ◦ ιˆ1,m is the identity on Sm+1 (T∗ M) ⊗ E. We continue to use the induction hypothesis in writing elements of .Jm E, so that we consider elements of .T∗ M ⊗ Jm E of the form

.

(α ⊗ ξ, α ⊗ A1 , . . . , α ⊗ Am ),

.

for .α ∈ T∗ M and .Ak ∈ Sk (T∗ M) ⊗ E, .k ∈ {1, . . . , m}. We then define .λ1,m by 1 m λ1,m (α0 ⊗ ξ, α0 ⊗ α11 ⊗ ξ, . . . , α0 ⊗ αm ⊙ · · · ⊙ αm ⊗ ξ)

.

1 m = Symm+1 ⊗ idE (α0 ⊗ αm ⊙ · · · ⊙ αm ⊗ ξ ).

Note that, with the form of .Jm E from the induction hypothesis, we have ιˆ1,m (α 1 ⊙ · · · ⊙ α m+1 ⊗ ξ ) ⎛ ⎞ m+1 Σ 1 = ⎝0, . . . , 0, α j ⊗ α 1 ⊙ · · · ⊙ α j −1 ⊙ α j +1 ⊙ · · · ⊙ α m+1 ⊗ ξ ⎠ . m+1 .

j =1

2.3 Constructions with Jet Bundles

59

We then directly verify that .λ1,m ◦ ιˆ1,m is indeed the identity. We finally claim that Pm+1 (jm+1 ξ(x)) = D∇m+1 M ,∇ πE (ξ ),

.

(2.9)

which will establish the lemma. To see this, first note that it suffices to define .Pm+1 on .image(∈m+1 ) since 1. .Jm+1 E ≃ (Sm+1 (T∗ M) ⊗ E) ⊕ Jm E, 2. .Pm+1 is zero on .Jm E ⊆ Jm+1 E (thinking of the inclusion arising from the connection-induced isomorphism from the preceding item), and 3. .Pm+1 ◦ ∈m+1 is the identity map on .Sm+1 (T∗ M) ⊗ E. In order to connect the algebra and the geometry, let us write elements of Sm+1 (T∗ M) ⊗ E in a particular way. We let .x ∈ M and let .f 1 , . . . , f m+1 be smooth functions contained in the maximal ideal of .C∞ (M) at x, i.e., .f j (x) = 0, .j ∈ {1, . . . , m+1}. Let .ξ be a smooth section of .E. We then can work with elements of .Sm+1 (T∗ M) ⊗ E of the form .

df 1 (x) ⊙ · · · ⊙ df m+1 (x) ⊗ ξ(x).

.

We then have ∈m+1 (df 1 (x) ⊙ · · · ⊙ df m+1 (x) ⊗ ξ(x)) = jm+1 (f 1 · · · f m+1 ξ )(x);

.

this is easy to see using the Leibniz Rule, cf. [23, Lemma 2.1]. (See [1, Supplement 2.4A] for a description of the higher-order Leibniz Rule.) Now, using the last part of the sublemma, we compute .Pm+1 (jm+1 (f

1

· · · f m+1 ξ )(x))

= λ1,m ◦ P1,m ◦ ι1,m (jm+1 (f 1 · · · f m+1 ξ )(x)) = λ1,m ◦ P1,m (j1 (f 1 · · · f m+1 ξ, D∇1 M ,∇ πE (f 1 · · · f m+1 ξ ), . . . , D∇mM ,∇ πE (f 1 · · · f m+1 ξ ))(x)) = λ1,m (∇ πE (f 1 · · · f m+1 ξ )(x), ∇ M,πE D∇1 M ,∇ πE (f 1 · · · f m+1 ξ )(x), . . . , ∇ M,πE D∇mM ,∇ πE (f 1 · · · f m+1 ξ )(x)) = Symm+1 ⊗ idE (∇ M,πE D∇mM ,∇ πE (f 1 · · · f m+1 ξ )(x)) 1 m+1 ξ )(x), = D∇m+1 M ,∇ πE (f · · · f

which shows that, with .Pm+1 defined as in (2.9), .Pm+1 ◦ ∈m+1 is indeed the identity on .Sm+1 (T∗ M) ⊗ E. The commuting of the diagram in the statement of the lemma follows directly from the recursive nature of the constructions. □

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2 Topology for Spaces of Real Analytic Sections and Mappings

There are a couple of special cases of interest. 1. Jets of functions fit into the framework of the lemma by using the trivial line bundle .RM = M × R. The identification of a function with a section of this bundle is specified by .f |→ ξf , with .ξf (x) = (x, f (x)); we shall make this identification implicitly. In this case, the bundle has a canonical flat connection defined by .∇ πE f = df . Therefore, the decomposition of Lemma 2.15 is determined by an affine connection .∇ M on .M, and so we have a mapping S∇mM : Jm (M; R) → .

m ⊕

Sj (T∗ M)

j =0

(2.10)

f (x) |→ (f (x), df (x), . . . , Symm ◦ ∇ M,m−1 df (x)). This can be restricted to .T∗m M to give the mapping S∇mM : T∗m M → .

m ⊕

Sj (T∗ M)

j =1

(2.11)

f (x) |→ (df (x), . . . , Symm ◦ ∇ M,m−1 df (x)), adopting a mild abuse of notation. We recall that .T∗m x M is an .R-algebra, and the j ∗ induced .R-algebra structure on .⊕m S (T M) is that of polynomial functions x j =1 that vanish at 0 and with degree at most m. 2. Another special case is that of jets of vector fields. In this case, the vector bundle is .πTM : TM → M. We can make use of an affine connection .∇ M on .M to provide everything we need to define the mapping S∇mM : Jm TM → .

m ⊕ (Sj (T∗ M) ⊗ TM) j =0

(2.12)

X(x) |→ (X(x), ∇ M X(x), . . . , Symm ◦ ∇ M,m X(x)). Of course, this applies equally well to jets of one-forms on .M, or any other sections of tensor bundles associated with the tangent bundle. This case of vector fields is the setting of Jafarpour and Lewis [36] in their study of flows of time-varying vector fields.

2.3.2 Fibre Norms for Jet Bundles of Vector Bundles With the decomposition of jet bundles from the preceding section, we now indicate how to provide norms for jet bundles using a Riemannian metric on .M and a fibre

2.3 Constructions with Jet Bundles

61

metric on .E. We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. We shall suppose that we have a .Cr -affine connection .∇ M on .M and a .Cr -vector bundle connection .∇ πE in .E, as in Sect. 2.3.1. This allows us to give the decomposition of .Jm E as in Lemma 2.15. We additionally suppose that we have a .Cr -Riemannian metric .GM on .M and a .Cr -fibre metric .GπE on .E. Note that the existence of the metrics and connections is ensured in the real analytic case by Lemma 2.4; we note that this lemma can be applied equally well in the smooth case. In the smooth case, one can also use standard constructions using partitions of unity give existence of metrics [42, Proposition III.1.4], and the existence of connections is ensured by [42, Theorem II.2.1]. The first step in making the constructions of this section is the following result concerning inner products on tensor products. Lemma 2.16 (Inner Products on Tensor Products) Let .U and .V be finitedimensional .R-vector spaces and let .G and .H be inner products on .U and .V, respectively. Then the element .G ⊗ H of .T2 (U∗ ⊗ V∗ ) defined by G ⊗ H(u1 ⊗ v1 , u2 ⊗ v2 ) = G(u1 , u2 )H(v1 , v2 )

.

is an inner product on .U ⊗ V. Proof Let .(e1 , . . . , em ) and .(f1 , . . . , fn ) be orthonormal bases for .U and .V, respectively. Then {ea ⊗ fj | a ∈ {1, . . . , m}, j ∈ {1, . . . , n}}

.

(2.13)

is a basis for .U ⊗ V. Note that G ⊗ H(ea ⊗ fj , eb ⊗ fk ) = G(ea , eb )H(fj , fk ) = δab δj k ,

.

which shows that .G ⊗ H is indeed an inner product, as (2.13) is an orthonormal basis. □ With .GπE a fibre metric on .E and with .GM a Riemannian metric on .M as above, ∗ let us denote by .G−1 M the associated fibre metric on .T M defined by G−1 M (αx , βx ) = GM (GM (αx ), GM (βx )).

.

#

#

∗ In like manner, one has a fibre metric .G−1 πE on .E . Then, by induction using the preceding lemma, we have a fibre metric in all tensor spaces associated with .TM and .E and their tensor products. We shall denote by .GM,πE any of these various fibre metrics. In particular, we have a fibre metric .GM,πE on .Tj (T∗ M) ⊗ E induced j ∗ by .G−1 M and .GπE . By restriction, this gives a fibre metric on .S (T M) ⊗ E. We can

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2 Topology for Spaces of Real Analytic Sections and Mappings

thus define a fibre metric .GM,πE ,m on .Jm E given by GM,πE ,m (jm ξ(x), jm η(x)) =

m Σ

.

j =0

⎛ GM,πE

⎞ 1 j 1 j D∇ M ,∇ πE (ξ )(x), D∇ M ,∇ πE (η)(x) . j! j! (2.14)

Associated to this inner product on fibres is the norm on fibres, which we denote by .‖·‖GM,π ,m . We shall use these fibre norms continually in our descriptions of our E various topologies for real analytic vector bundles, cf. Sect. 4.1. The appearance of the factorials in the fibre metric (2.14) appears superfluous at this point. However, it is essential in order for the real analytic topology defined by our seminorms to be independent of the choices of .∇ M , .∇ πE , .GM , and .GπE , cf. Theorem 4.24. One may wonder why we have made our constructions of fibre norms for .Jm E so complicated. Since .πE : E → M is a vector bundle, could we not just use this fact to endow it with a fibre metric, and hence a fibre norm? In the smooth case, indeed this is an entirely feasible idea. However, in the real analytic case, one is working, essentially, with infinite jets, and so having fibre norms for .Jm E that vary arbitrarily with .m ∈ Z≥0 will create a theory where the topology is not independent of the choice of these fibre norms. By inducing these fibre norms from data on .M, we ensure that the topologies are not dependent on our choices, although this is not easy to prove, as we shall see in Sect. 4.3.

2.4 Seminorms for the Cω -Topology We now provide seminorms for the .Cω -topology. To do this, we shall first offer another characterisation of the topology on the space .G hol,R of germs of K,E holomorphic sections about a compact set .K.

2.4.1 A Weighted Direct Limit Topology for Sections of Bundles of Infinite Jets Here we provide a direct limit topology for a subspace of the space of continuous sections of the infinite jet bundle of a vector bundle. In Proposition 2.18 we shall connect this direct limit topology to the direct limit topology described above for germs of holomorphic sections about a compact set. The topology we give here has the advantage of providing explicit seminorms for the topology of germs, and subsequently for the space of real analytic sections. For this description, we work with infinite jets, so let us introduce the notation we will use for this, referring to [64, Chapter 7] for details. Let us denote by .J∞ E the bundle of infinite jets of a vector bundle .πE : E → M, this being the inverse

2.4 Seminorms for the Cω -Topology

63

limit (in the category of sets, for the moment) of the inverse system .(Jm E)m∈Z≥0 k , .k, l ∈ Z , .k ≥ l. Precisely, with mappings .πE,l ≥0 ⎧ ⎨

| ⎫ | ⎬ | ∞ k .J E = Jm E || πE,l φ∈ ◦ φ(k) = φ(l), k, l ∈ Z≥0 , k ≥ l . ⎩ ⎭ | m∈Z≥0 | |

∞ : J∞ E → Jm E be the projection defined by .π ∞ (φ) = φ(m); these We let .πE,m E,m mappings serve as the mappings induced by the inverse limit. For .ξ ∈ Г ∞ (E), ∞ ◦ j ξ(x) = j ξ(x). By a theorem of we let .j∞ ξ : M → J∞ E be defined by .πE,m ∞ m ∞ ∞ Borel [9], if .φ ∈ J E, there exist .ξ ∈ Г (E) and .x ∈ M such that .j∞ ξ(x) = φ. We can define sections of .J∞ E in the usual manner: a section is a map . : M → J∞ E ∞ ◦ (x) = x for every .x ∈ M. We shall equip .J∞ E with the initial satisfying .πE,0 ∞ ◦  is continuous for topology so that a section . is continuous if and only if .πE,m every .m ∈ Z≥0 . We denote the space of continuous sections of .J∞ E by .Г 0 (J∞ E). Since we are only dealing with continuous sections, we can talk about sections defined on any subset .A ⊆ M, using the induced topology on A. The continuous sections defined on .A ⊆ M will be denoted by .Г 0 (J∞ E|A). Now let .K ⊆ M be compact and, for .j ∈ Z>0 , denote

| { | Ej (K) =  ∈ Г 0 (J∞ E|K)|

.

∞ sup{j −m ‖πE,m ◦ (x)‖GM,π

E ,m

} | m ∈ Z≥0 , x ∈ K} < ∞ ,

and on .Ej (K) we define a norm .pK,j by { ∞ ◦ (x)‖GM,π pK,j () = sup j −m ‖πE,m

.

E ,m

| } | | m ∈ Z≥0 , x ∈ K .

One readily verifies that, for each .j ∈ Z>0 , .(Ej (K), pK,j ) is a Banach space. Note that .Ej (K) ⊆ Ej +1 (K) and that .pK,j +1 () ≤ pK,j () for . ∈ Ej (K), and so the inclusion of .Ej (K) in .Ej +1 (K) is continuous. We let .E (K) be the direct limit of the directed system .(Ej (K))j ∈Z>0 with the mappings being inclusions. The following attribute of the direct limit topology for .E (K) will be useful. Lemma 2.17 (.E (K) Is A Regular Direct Limit) The direct limit topology for E (K) is regular. As a consequence, .E (K) is Hausdorff.

.

Proof Let .Bj (1, 0) ⊆ Ej (K), .j ∈ Z>0 , be the closed unit ball with respect to the norm topology. We claim that .Bj (1, 0) is closed in the direct limit topology of .E (K). To prove this, we shall prove that .Bj (1, 0) is closed in a topology that is coarser than the direct limit topology. The coarser topology we use is the topology induced by the topology of pointwise convergence in .Г 0 (J∞ E|K). To be precise, let .Eˆ j (K) be the vector space .Ej (K)

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2 Topology for Spaces of Real Analytic Sections and Mappings

with the topology defined by the seminorms { ∞ ◦ (x)‖GM,π px,j () = sup j −m ‖πE,m

.

E ,m

| } | | m ∈ Z≥0 ,

x ∈ K.

Clearly the identity map from .Ej (K) to .Eˆ j (K) is continuous, and so the topology of ˆ j (K) is coarser than the usual topology of .E (K). Now let .Eˆ (K) be the direct limit .E of the directed system .(Eˆ j (K))j ∈Z>0 . Note that, algebraically, .Eˆ (K) = E (K), but the spaces have different topologies, the topology for .Eˆ (K) being coarser than that for .E (K). We will show that .Bj (1, 0) is closed in .Eˆ (K). Let .(I, ) be a directed set and let .(i )i∈I be a convergent net in .Bj (1, 0) in the topology of .Eˆ (K). Thus we have a map . : K → J∞ E|K such that, for each .x ∈ K, .limi∈I i (x) = (x). If . ∈ / Bj (1, 0), then there exists .x ∈ K such that .

{ ∞ ◦ (x)‖GM,π sup j −m ‖πE,m

E ,m

| } | | m ∈ Z≥0 > 1.

Let .∈ ∈ R>0 be such that .

{ ∞ ◦ (x)‖GM,∇ sup j −m ‖πE,m

E ,m

| } | | m ∈ Z≥0 > 1 + ∈

and let .i0 ∈ I be such that .

{ ∞ ∞ ◦ i (x) − πE,m ◦ (x)‖GM,π sup j −m ‖πE,m

E ,m

| } | | m ∈ Z≥0 < ∈

for .i0  i, this by pointwise convergence. We thus have, for all .i0  i, | } { | ∞ ◦ (x)‖GM,π ,m | m ∈ Z≥0 ∈ < sup j −m ‖πE,m E | } { | −m ∞ − sup j ‖πE,m ◦ i (x)‖GM,π ,m | m ∈ Z≥0 E | } { | ∞ ∞ ◦ i (x) − πE,m ◦ (x)‖GM,π ,m | m ∈ Z≥0 < ∈, ≤ sup j −m ‖πE,m

.

E

which contradiction gives the conclusion that . ∈ Bj (1, 0). Since .Bj (1, 0) has been shown to be closed in .E (K), the regularity of the direct limit topology now follows from [7, Corollary 3.7]. Hausdorffness follows from [37, Proposition 4.5.3]. □ The next result explains the relevance of the above constructions.

2.4 Seminorms for the Cω -Topology

65

Proposition 2.18 (.G hol,R Is a Topological Subspace of .E (K)) Let .πE : E → M K,E be a real analytic vector bundle and let .K ⊆ M be compact. Then the mapping LK : G hol,R → E (K) K,E

.

[ξ ]K |→ j∞ ξ |K is a continuous open injection, and so is a topological monomorphism. Proof Throughout the proof, we let .πE : E → M be a complexification of .πE : E → M and we let .(Uj )j ∈Z>0 be a sequence of neighbourhoods of .K as in Lemma 2.8. Let us first prove that .LK is well-defined, i.e., show that, if .[ξ ]K ∈ G hol,R , then K,E

LK ([ξ ]K ) ∈ Ej (K) for some .j ∈ Z>0 . Let .U be a neighbourhood of .K in .M on which the section .ξ is defined, holomorphic, and bounded. Then .ξ |(M ∩ U) is real analytic and so, by Lemma 2.22, there exist .C, r ∈ R>0 such that

.

‖jm ξ(x)‖GM,π

.

E ,m

≤ Cr −m ,

x ∈ K, m ∈ Z≥0 .

If .j > r −1 , it immediately follows that .

{ sup j −m ‖jm ξ(x)‖GM,π

E ,m

| } | | x ∈ K, m ∈ Z≥0 < ∞,

i.e., .LK ([ξ ]K ) ∈ Ej (K). By the universal property of direct limits, to show that .LK is continuous, it hol,R suffices to show that .LK |Г bdd (E|Uj ) is continuous for each .j ∈ Z≥0 . We will prove this by showing that, for each .j ∈ Z>0 , there exists .j ' ∈ Z>0 such hol (E|U )) ⊆ E ' (K) and such that .L that .LK (Г bdd j K is continuous as a map from j hol ' ' .Г bdd (E|Uj ) to .Ej (K). Since .Ej (K) is continuously included in .E (K), this will hol (E|U )) ⊆ E ' (K) for give the continuity of .LK . First let us show that .LK (Г bdd j j ' some .j ∈ Z>0 . By Lemma 2.21, there exist .C, r ∈ R>0 such that ‖jm ξ(x)‖GM,π

.

E ,m

hol (ξ ) ≤ Cr −m pU ,∞ j

hol (E|U ). Taking .j ' ∈ Z ' −1 for every .m ∈ Z≥0 and .ξ ∈ Г bdd j >0 such that .j ≥ r , hol we have .LK (Г bdd (E|Uj )) ⊆ Ej ' (K), as claimed. To show that .LK is continuous hol (E|U ) to .E ' (K), let .(ξ ) hol as a map from .Г bdd j j k k∈Z>0 be a sequence in .Г bdd (E|Uj ) converging to zero. We then have

.

{ lim sup (j ' )−m ‖jm ξk (x)‖GM,π

k→∞

E ,m

| } | | x ∈ K, m ∈ Z≥0 ⎧

≤ lim C sup ‖ξ k (z)‖GM,π k→∞

giving the desired continuity.

E

| ⎫ | | z ∈ Uj = 0, |

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2 Topology for Spaces of Real Analytic Sections and Mappings

Since germs of holomorphic sections are uniquely determined by their infinite jets by the Identity Theorem, injectivity of .LK follows. To show that .LK is open, we cannot use the De Wilde Open Mapping Theorem since .LK is not surjective and since subspaces of ultrabornological spaces are not generally ultrabornological. We will instead use an Open Mapping Theorem from §2 of [5]. This theorem refers to the notion of a DF-space, for which we refer the reader to [37, §12.4]. Accepting this notion, the Open Mapping Theorem we use is the following. Let .(U, OU ) and .(V, OV ) be locally convex topological vector spaces and let .L ∈ L(U; V). Assume the following: (i) (ii) (iii) (iv)

(U, OV ) is Hausdorff ; closed and bounded subsets of .U are compact, e.g., .(U, OU ) is nuclear; .(V, OV ) is a DF-space; −1 (B) is bounded for any bounded subset .B ⊆ V. .L .

Then L is open onto its image. By Proposition 2.10, the direct limit topology of .G hol,R is Hausdorff and nuclear. K,E Note also that .E (K) is a DF-space since Banach spaces are DF-spaces [37, Corollary 12.4.4] and Hausdorff countable direct limits of DF-spaces are DFspaces [37, Theorem 12.4.8]. Therefore, to use the preceding Open Mapping Theorem, we must only show that .L−1 K (B) is bounded when .B is bounded. Thus let .B ⊆ E (K) be bounded. By Lemma 2.17, if .B ⊆ E (K) is bounded, then .B is contained and bounded in .Ej (K) for some .j ∈ Z>0 . Therefore, there exists .C ∈ R>0 such that, if .LK ([ξ ]K ) ∈ B, then ‖jm ξ(x)‖GM,π

.

E ,m

≤ Cj −m ,

x ∈ K, m ∈ Z≥0 .

Let .x ∈ K and let .(Vx , ν x ) be a vector bundle chart for .E about x with corresponding chart .(Ux , χ x ) for .M. Suppose that the fibre dimension of .E over .Ux is k and that n ' .χ x takes values in .R . Let .Ux ⊆ Ux be a precompact neighbourhood of x such that ' ' .cl(Ux ) ⊆ Ux . Denote .Kx = K ∩ cl(Ux ). By Lemma 4.22, there exist .Cx , rx ∈ R>0 such that, if .LK ([ξ ]K ) ∈ B, then |D I ξ a (x)| ≤ Cx I !rx−|I | ,

.

x ∈ χ x (Kx ), I ∈ Zn≥0 , a ∈ {1, . . . , k},

where .ξ is the local representative of .ξ . We can assume, without loss of generality, that .rx < 1. Note that this implies the following for each .[ξ ]K such that .LK ([ξ ]K ) ∈ B and for each .a ∈ {1, . . . , k}: a

1. .ξ admits a convergent power series expansion to an holomorphic function on the polydisk .D(σ x , χ x (x)) for .0 < σx < rx (here .σ x = (σx , . . . , σx ));

2.4 Seminorms for the Cω -Topology

67 a

a

1 )n , recalling that 2. on the polydisk .D(σ x , χ x (x)), .ξ satisfies .|ξ | ≤ Cx ( 1−σ x ∞ Σ .

σxk =

k=0

1 1 − σx

if .0 < σx < 1. It follows that, if .LK ([ξ ]K ) ∈ B and if .x ∈ K, then .ξ has a bounded holomorphic extension in some neighbourhood .Vx ⊆ M whose image under .χ x is a polydisk. Let .x1 , . . . , xk ∈ K be such that .K ⊆ ∪ka=1 Vxa and then let .j ' ∈ Z>0 be such that .Uj ' ⊆ ∪ka=1 Vxa . We then have that, for .[ξ ]K for which .LK ([ξ ]K ) ∈ B, .ξ ∈ hol,R (E|Uj ' ). Thus .L−1 □ Г bdd K (B) is bounded, as claimed.

2.4.2 Definition of Seminorms In this section we provide explicit seminorms for Martineau’s topologies for .Г ω (E). Throughout this section, we will work with a real analytic vector bundle .πE : E → M and the real analytic data .∇ M , .∇ πE , .GM , and .GπE that define the fibre metrics for jet bundles as per Sect. 2.3.2. To define seminorms for .Г ω (E), let .c0 (Z≥0 ; R>0 ) denote the space of sequences in .R>0 , indexed by .Z≥0 , and converging to zero. We shall denote a typical element of .c0 (Z≥0 ; R>0 ) by .a = (aj )j ∈Z≥0 . Now, for .K ⊆ M ω and .a ∈ c0 (Z≥0 ; R>0 ), we define a seminorm .pK,a for .Г ω (E) by { ω pK,a (ξ ) = sup a0 a1 · · · am ‖jm ξ(x)‖GM,π

.

E ,m

| } | | x ∈ K, m ∈ Z≥0 .

ω , .K ⊆ M compact, .a ∈ c (Z ; R ), defines a The family of seminorms .pK,a 0 ≥0 >0 ω locally convex topology on .Г (E). Let us prove that this locally convex topology is actually the .Cω -topology.

Theorem 2.19 (Seminorms for .Г ω (E)) Let .πE : E → M be a real analytic vector bundle. Then the family of seminorms ω pK,a ,

.

K ⊆ M compact, a ∈ c0 (Z≥0 ; R>0 ),

defines a locally convex topology on .Г ω (E) agreeing with the .Cω -topology. Proof First we fix .K and show that the seminorms { ∞ ◦ (x)‖GM,π pK,a = sup a0 a1 · · · am ‖πE,m

.

E ,m

| } | | m ∈ Z≥0 , x ∈ K , a ∈ c0 (Z≥0 ; R>0 ),

define the locally convex direct limit topology for .E (K).

(2.15)

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2 Topology for Spaces of Real Analytic Sections and Mappings

To this end, we first show that the seminorms .pK,a , .a ∈ c0 (Z≥0 ; R>0 ), are continuous on .E (K). By the universal property of final topologies, it suffices to show that .pK,a |Ej (K) is continuous for each .j ∈ Z>0 . Thus, since .Ej (K) is a Banach space, it suffices to show that, if .(k )k∈Z>0 is a sequence in .Ej (K) converging to zero in the norm topology, then .limk→∞ pK,a (k ) = 0. Let .N ∈ Z≥0 be such that .am < j1 for .m ≥ N. Let .C ≥ 1 be such that a0 a1 · · · am ≤ Cj −m ,

m ∈ {0, 1, . . . , N },

.

this being possible since there are only finitely many inequalities to satisfy. Therefore, for any .m ∈ Z≥0 , we have .a0 a1 · · · am ≤ Cj −m . Then, for any 0 ∞ . ∈ Г (J E|K), ∞ ◦ (x)‖GM,π a0 a1 · · · am ‖πE,m

.

E ,m

∞ ◦ (x)‖GM,π ≤ Cj −m ‖πE,m

E ,m

for every .x ∈ K and .m ∈ Z≥0 . From this we immediately have .limk→∞ pK,a (k ) = 0, as desired. This shows that the direct limit topology on .E (K) is finer than the topology defined by the family of seminorms .pK,a , .a ∈ c0 (Z≥0 ; R>0 ). For the converse, we show that every neighbourhood of .0 ∈ E (K) in the direct limit topology contains a neighbourhood of zero in the topology defined by the seminorms .pK,a , .a ∈ c0 (Z≥0 ; R>0 ). Let .Bj (1, 0) denote the open unit ball in .Ej (K). A neighbourhood of 0 in the direct limit topology contains a union of balls .∈j Bj (1, 0) for some .∈j ∈ R>0 , .j ∈ Z>0 , [37, Proposition 6.6.5(a)] and we can assume, without loss of generality, that .∈j ∈ (0, 1) for each .j ∈ Z>0 . We define an increasing sequence .(mj )j ∈Z>0 in .Z≥0 as follows. Let .m1 = 0. Having 1/m

defined .m1 , . . . , mj , define .mj +1 > mj by requiring that .j < ∈j +1 j +1 (j + 1). 1/mj

−1 = ∈ For .m ∈ {mj , . . . , mj +1 − 1}, define .am ∈ R>0 by .am j .m ∈ {mj , . . . , mj +1 − 1}, we have m/mj m

−m am = ∈j

.

j

j . Note that, for

≤ ∈j j m .

Note that .limm→∞ am = 0. If . ∈ Г 0 (J∞ E|K) satisfies .pK,a () ≤ 1, then, for .m ∈ {mj , . . . , mj +1 − 1}, we have ∞ ◦ (x)‖GM,π j −m ‖πE,m

.

E ,m

m ∞ ◦ (x)‖GM,π ≤ am ∈j ‖πE,m

≤

E ,m

∞ ◦ (x)‖GM,π ,m a0 a1 · · · am ∈j ‖πE,m E

≤ ∈j

for .x ∈ K. Thus, if . ∈ Г 0 (J∞ E|K) satisfies .pK,a () ≤ 1, then, for ∞ .m ∈ {mj , . . . , mj +1 − 1}, we have .π E,m ◦  ∈ ∈j Bj (1, 0). Therefore, . ∈

2.4 Seminorms for the Cω -Topology

69

∪j ∈Z>0 ∈j Bj (1, 0), and this shows that, for .a as constructed above, { .

| } | | | ∈j Bj (1, 0),  ∈ Г 0 (J∞ E|K) | pK,a () ≤ 1 ⊆ j ∈Z>0

showing that the seminorms (2.15) give the topology of .E (K). By Proposition 2.18, we conclude that the topology of .G hol,R is defined by the K,E seminorms | { } | m ∈ Z≥0 , x ∈ K , .pK,a ([ξ ]K ) = sup a0 a1 · · · am ‖jm ξ(x)‖G M,π ,m | E

a ∈ c0 (Z≥0 ; R>0 ). Finally, since .Г ω (E) is the inverse limit of the spaces .G hol,R over the directed set K,E of compact subsets .K of .M, we conclude that the seminorms ω pK,a ,

.

K ⊆ M compact, a ∈ c0 (Z≥0 ; R>0 ),

define the .Cω -topology, as asserted in the statement of the theorem.

□

ω . It is sometimes convenient to work with variations of the seminorms .pK,a Specifically, it can be convenient to work with sequences .a with particular properties. To this end, we denote ↓

c0 (Z≥0 ; R>0 ) = {a ∈ c0 (Z≥0 ; R>0 ) | aj ≥ aj +1 , j ∈ Z≥0 },

.

and, for .ρ ∈ R>0 , c0 (Z≥0 ; R>0 ; ρ) = {a ∈ c0 (Z≥0 ; R>0 ) | aj ≤ ρ, j ∈ Z≥0 },

.

and, finally ↓

↓

c0 (Z≥0 ; R>0 ; ρ) = {a ∈ c0 (Z≥0 ; R>0 ) | aj ≤ ρ, j ∈ Z≥0 }.

.

Let us prove that the topology defined by the four flavours of sequences coincide. Lemma 2.20 (Alternative Seminorms for the Real Analytic Topology) Let πE : E → M be a real analytic vector bundle. The following collections of seminorms define the same locally convex topology for .Г ω (E):

.

ω , .K ⊆ M compact, .a ∈ c (Z ; R ); (i) .pK,a 0 ≥0 >0 ↓

ω , .K ⊆ M compact, .a ∈ c (Z ; R ); (ii) .pK,a ≥0 >0 0 ω , .K ⊆ M compact, .a ∈ c (Z ; R ; ρ) for .ρ ∈ R ; (iii) .pK,a 0 ≥0 >0 >0 ↓

ω , .K ⊆ M compact, .a ∈ c (Z ; R ; ρ) for .ρ ∈ R . (iv) .pK,a ≥0 >0 >0 0

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2 Topology for Spaces of Real Analytic Sections and Mappings

Proof Let us denote the topologies defined by the four collections of seminorms by ω O ω , O↓ω , Oρω , O↓,ρ ,

.

respectively. Since the seminorms defining .O↓ω and .Oρω are a subset of those defining ω .O , we immediately have ω O ω ⊆ Oρω ⊆ Oρ,↓ ,

.

ω O ω ⊆ O↓ω ⊆ Oρ,↓ . ρ

For .a ∈ c0 (Z≥0 ; R) and for .ρ ∈ R>0 , we define .a ρ ∈ c0 (Z≥0 ; R>0 ; ρ) by .aj = ↓

max{aj , ρ}, .j ∈ Z≥0 . Given .a ∈ c0 (Z≥0 ; R>0 ), we construct .a ↓ ∈ c0 (Z≥0 ; R>0 ) ↓ as follows. We take .a0 = a0 and let .m0 > 0 be the least integer for which .aj < a0 ↓ for .j > m0 . We then define .aj = a0 , .j ∈ {0, 1, . . . , m0 }. We next let .m1 > m0 be ↓

the least integer for which .aj < am0 +1 for .j > m1 . We then define .aj = am0 +1 , ↓

j ∈ {m0 + 1, . . . , m1 }. We carry on in this way to define .aj , .j ∈ Z≥0 . Then we obviously have

.

ω ω ω ω pK,a (ξ ) ≤ pK,a (ξ ), ρ (ξ ) ≤ pK,a (ξ ), p K,a ↓

K ⊆ M compact, ξ ∈ Г ω (E).

.

From this we conclude that ω Oρ,↓ ⊆ Oρω ⊆ O ω ,

.

ω Oρ,↓ ⊆ O↓ω ⊆ O ω ,

□

and the lemma follows.

While we are primarily interested in the difficult real analytic case in this book, it is useful and illustrative to, at times, make comparisons with the other regularity classes, particularly smooth regularity. With this in mind, let us also define seminorms for the sets .Г ν (E) of sections of class .Cν for .ν ∈ Z≥0 ∪ {∞}. For .ν = m ∈ Z≥0 , we define the seminorms { m pK (ξ ) = sup ‖jm ξ(x)‖GM,π

.

E ,m

| } | | x∈K

(2.16)

for .K ⊆ M compact. For .ν = ∞, the appropriate seminorms are { ∞ pK,m (ξ ) = sup ‖jm ξ(x)‖GM,π

.

E ,m

| } | | x∈K

(2.17)

for .K ⊆ M compact and for .m ∈ Z≥0 . These seminorms give the topology of uniform convergence of derivatives on compact sets. In particular, the .C0 -topology defined in this way is exactly the compact-open topology. A moment’s thought will convince oneself that these seminorms define a Polish topology for .Г ν (E) called the ν .C -topology, .ν ∈ Z≥0 ∪ {∞}. We note that, for the smooth topology, the seminorms are defined for fixed order jets. As we shall indicate as we go along, it is this fact

2.4 Seminorms for the Cω -Topology

71

that leads to simplifications of the results in the book when applied to the smooth case. Let us close this section with seminorm characterisations of holomorphic and real analytic sections. We note that we have already made use of both of the following lemmata at various points in our constructions above, namely in the proof of Proposition 2.18. Moreover, as we see from the proofs of the lemmata, they make use of Lemma 4.22. Additionally, we have made an independent call to Lemma 4.22 in our proof of Proposition 2.18. One might be led to wonder whether this use of the yet-to-be-proved Lemma 4.22 constitutes circular reasoning. It does not; in the diagram in (4.16) we carefully unwind the logical implications that makes everything consistent. First we give geometric analogues of the classical Cauchy estimates for holomorphic mappings. Lemma 2.21 (Cauchy Estimates for Vector Bundles) Let .πE : E → M be an holomorphic vector bundle, let .K ⊆ M be compact, and let .U be a precompact neighbourhood of .K. Then there exist .C, r ∈ R>0 such that ∞ hol pK,m (ξ ) ≤ Cr −m pU,∞ (ξ )

.

hol (E|U). for every .m ∈ Z≥0 and .ξ ∈ Г bdd

Proof Let .z ∈ K and let .(Wz , ν z ) be an holomorphic vector bundle chart about z with .(Uz , χ z ) the associated chart for .M, supposing that .Uz ⊆ U. Let .k ∈ Z>0 be such that .ν z (Wz ) = χ z (Uz ) × Ck . Let .z = χ z (z) and let .ξ : χ z (Uz ) → Ck be the hol (E|U). Note that, when taking real derivatives of .ξ local representative of .ξ ∈ Г bdd with respect to coordinates, we can think of taking derivatives with respect to .

∂ 1 = j ∂z 2

⎛

∂ ∂ −i j j ∂x ∂y

⎞ ,

∂ 1 = j ∂ z¯ 2

⎛

∂ ∂ +i j j ∂x ∂y

⎞ ,

j ∈ {1, . . . , n}.

Since .ξ is holomorphic, the . ∂∂z¯ j derivatives will vanish [45, page 27]. Thus, for the purposes of the multi-index calculations, we consider multi-indices of length n (not 2n). In any case, applying the usual Cauchy estimates [45, Lemma 2.3.9], there exists .r ∈ R>0 such that { } | |D I ξ a (z)| ≤ I !r −|I | sup |ξ a (ζ )| | ζ ∈ D(r, z)

.

hol (E|U). We may choose for every .a ∈ {1, . . . , k}, .I ∈ Zn≥0 , and .ξ ∈ Г bdd .r ∈ (0, 1) such that .D(r, z) is contained in .φ z (Uz ), where .r = (r, . . . , r). Denote −1 ' .Vz = χ z (D(r, z)). By continuity, there exists a neighbourhood .Vz of z such that ' .cl(Vz ) ⊆ Vz and such that

{ } | |D I ξ a (z' )| ≤ 2I !r −|I | sup |ξ a (ζ )| | ζ ∈ D(r, z)

.

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2 Topology for Spaces of Real Analytic Sections and Mappings

hol (E|U), .a ∈ {1, . . . , k}, and .I ∈ Zn . If .|I | ≤ m, for every .z' ∈ χ z (V'z ), .ξ ∈ Г bdd ≥0 then, since we are assuming that .r < 1, we have

.

{ } | 1 I a ' |D ξ (z )| ≤ 2r −m sup |ξ a (ζ )| | ζ ∈ D(r, z) I!

hol (E|U). By Lemma 4.22, it for every .a ∈ {1, . . . , k}, .z' ∈ χ z (V'z ), and .ξ ∈ Г bdd follows that there exist .Cz , rz ∈ R>0 such that

‖jm ξ(z' )‖GM,π

.

E ,m

hol ≤ Cz rz−m pV (ξ ) z ,∞

hol (E|U). Let .z , . . . , z ∈ K be such that for all .z' ∈ V'z , .m ∈ Z≥0 , and .ξ ∈ Г bdd 1 k k ' .K ⊆ ∪ V , and let .C = max{Cz1 , . . . , Czk } and .r = min{rz1 , . . . , rzk }. If .z ∈ K, z j =1 j then .z ∈ V'zj for some .j ∈ {1, . . . , k} and so we have

‖jm ξ(z)‖GM,π

.

hol ≤ Czj rz−m pV j z

j ,∞

E ,m

hol (ξ ) ≤ Cr −m pU,∞ (ξ ),

and taking supremums over .z ∈ K on the left gives the result.

□

The following lemma, providing bounds for real analytic sections, is a global version of the local result stated as Theorem 1.9. Lemma 2.22 (Characterisation of Real Analytic Sections) Let .πE : E → M be a real analytic vector bundle and let .ξ ∈ Г ∞ (E). Then the following statements are equivalent: (i) .ξ ∈ Г ω (E); ∞ (ξ ) ≤ (ii) for every compact set .K ⊆ M, there exist .C, r ∈ R>0 such that .pK,m −m Cr for every .m ∈ Z≥0 . Proof (i). =⇒ (ii) Let .K ⊆ M be compact, let .x ∈ K, and let .(Vx , ν x ) be a vector bundle chart for .E with .(Ux , χ x ) the corresponding chart for .M. Let .ξ : χ x (Ux ) → Rk be the local representative of .ξ . By Theorem 1.9, there exist a neighbourhood ' .Ux ⊆ Ux of x and .Bx , σx ∈ R>0 such that |D I ξ a (x ' )| ≤ Bx I !σx−|I |

.

for every .a ∈ {1, . . . , k}, .x ' ∈ cl(U'x ), and .I ∈ Zn≥0 . We can suppose, without loss of generality, that .σx ∈ (0, 1). In this case, if .|I | ≤ m, .

1 I a ' |D ξ (x )| ≤ Bx σx−m I!

2.5 The Topology for the Space of Real Analytic Mappings

73

for every .a ∈ {1, . . . , k} and .x ' ∈ cl(U'x ). By Lemma 4.22, there exist .Cx , rx ∈ R>0 such that ‖jm ξ(x ' )‖GM,π

.

E ,m

≤ Cx rx−m ,

x ' ∈ cl(U'x ), m ∈ Z≥0 .

Let .x1 , . . . , xk ∈ K be such that .K ⊆ ∪kj =1 U'xj and let .C = max{Cx1 , . . . , Cxk } and ' .r = min{rx1 , . . . , rxk }. Then, if .x ∈ K, we have .x ∈ Ux for some .j ∈ {1, . . . , k} j and so ‖jm ξ(x)‖GM,π

.

E ,m

≤ Cxj rx−m ≤ Cr −m , j

as desired. (ii). =⇒ (ii) Let .x ∈ M and let .(V, ν) be a vector bundle chart for .E such that the associated chart .(U, χ ) for .M is a precompact coordinate chart about x. Let k .ξ : χ (U) → R be the local representative of .ξ . By hypothesis, there exist .C, r ∈ R>0 such that .‖jm ξ(x ' )‖GM,π ,m ≤ Cr −m for every .m ∈ Z≥0 and .x ' ∈ U. Let .U' E be a precompact neighbourhood of x such that .cl(U' ) ⊆ U. By Lemma 4.22, there exist .B, σ ∈ R>0 such that |D I ξ a (x ' )| ≤ BI !σ −|I |

.

for every .a ∈ {1, . . . , k}, .x ' ∈ cl(U' ), and .I ∈ Zn≥0 . We conclude real analyticity of .ξ in a neighbourhood of x by Theorem 1.9. □

2.5 The Topology for the Space of Real Analytic Mappings Next we turn to topologising the space of real analytic mappings between real analytic manifolds .M and .N. We shall provide a few different ways to describe this topology, some of these rather similar to the constructions of Sect. 2.2 for topologising the space of sections of a real analytic vector bundle. Our initial definition of this topology, however, will not be immediately relatable to the vector bundle case. For this initial definition, we shall make a connection to a commonly used topology for finitely differentiable and smooth mappings: the topology of uniform convergence of derivatives on compact sets. After using this as motivation for our definition for the topology for the space of real analytic mappings, we give characterisations of this topology using complexification, in the manner of the two different definitions for a topology for the space of real analytic sections in Sect. 2.2.

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2.5.1 Motivation for, and Definition of, the Weak-PB Topology We first consider a well-known topology for finitely differentiable and smooth mappings. Let us motivate this by recalling the topology for spaces of sections of vector bundles defined by the seminorms (2.16) (in the finitely differentiable case) and the seminorms (2.17) (in the smooth case). We note that these topologies can be regarded as the topology of uniform convergence of finite-order derivatives on compact sets. This idea can be relatively easily adapted to topologies for finitely differentiable and smooth mappings, and we outline this here. We let .M and .N be .C∞ -manifolds and let .ν ∈ Z≥0 ∪ {∞}, with .Cν (M; N), therefore, the space of .Cν -mappings from .M to .N. The topology we consider for this space is called the “weak topology” by [33, §2.1] and the “.COν -topology” by Michor [52, §4.3]. It is the adaptation of the standard compact-open topology for spaces of continuous mappings between topological spaces [73, §43]. Somewhat precisely, it is the topology where a sequence .(Ф j )j ∈Z>0 converges to .Ф if and only if: 1. .ν ∈ Z≥0 : the sequence of all derivatives of .(Ф j )j ∈Z>0 up to and including order .ν converge uniformly on compact subsets; 2. .ν = ∞: for all .k ∈ Z≥0 , the sequence of all derivatives of .(Ф j )j ∈Z>0 up to and including order k converge uniformly on compact subsets. There is another characterisation of this topology, and it is the following. Theorem 2.23 (Characterisation of “Weak Topology” or “.COν -Topology”) For smooth manifolds .M and .N and for .ν ∈ Z≥0 ∪ {∞}, the topology for .Cν (M; N) described above is the same as the initial topology associated with the family of mappings Θf : Cν (M; N) → Cν (M) .

Ф |→ Ф ∗ f,

f ∈ C∞ (N),

where .C∞ (N) has the topology defined by the seminorms (2.17). By the nature of this characterisation of these topologies, the topologies are the uniform topologies associated with the families of semimetrics m dm K,f (Ф 1 , Ф 2 ) = pK (f ◦ Ф 1 − f ◦ Ф 2 )

.

(2.18)

(in the case .ν = m ∈ Z≥0 ) and ∞ d∞ K,m,f (Ф 1 , Ф 2 ) = pK,m (f ◦ Ф 1 − f ◦ Ф 2 ),

.

m ∈ Z≥0 ,

(2.19)

(in the case .ν = ∞) for .f ∈ C∞ (N) and .K ⊆ M compact. While we were not able to pinpoint a proof of the previous theorem anywhere, it will not come as a surprise to those who understand this topology well; its validity

2.5 The Topology for the Space of Real Analytic Mappings

75

boils down to being able to find globally defined coordinate functions about any point in .M and .N, these being furnished, for example, by the Whitney Embedding Theorem. Thanks to the Grauert Embedding Theorem, we are similarly able to find globally defined real analytic coordinate functions about any point on a real analytic manifold. We use this as motivation for the following definition. Definition 2.24 (Weak PB-Topology) Let .M and .N be real analytic manifolds. The weak-PB topology for .Cω (M; N) is the initial topology associated with the family of mappings Θf : Cω (M; N) → Cω (M) .

∗

Ф |→ Ф f,

f ∈ Cω (N), ◦

where .Cω (N) has the .Cω -topology.

The following theorem gives a few alternative means of characterising the weakPB topology, making use of Grauert’s Embedding Theorem. Note that this theorem, applied appropriately in the finitely differentiable and smooth cases, gives a proof of Theorem 2.23. Theorem 2.25 (Characterisation of Weak-PB Topology for Spaces of Real Analytic Mappings) Let .M and .N be .Cω -manifolds and let ι : N → RN .

y |→ (ι1 (y), . . . , ιN (y))

be a proper .Cω -embedding. Then the following topologies for .Cω (M; N) agree: (i) the initial topology associated with the family of mappings Ψf : Cω (M; N) → Cω (M) .

Ф |→ Ф ∗ f,

f ∈ Cω (M);

(ii) the initial topology associated with the family of mappings Ψιj : Cω (M; N) → Cω (M) .

Ф |→ Ф ∗ ιj ,

j ∈ {1, . . . , N };

(iii) the topology induced on .Cω (M; N) ⊆ Cω (M; RN ) by the weak-PB topology for Cω (M; RN ) ≃

N ⊕

.

j =1

Cω (M).

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2 Topology for Spaces of Real Analytic Sections and Mappings

Proof (ii).⊆(i): The topology (ii) is the coarsest topology for which the mappings Ψιj , .j ∈ {1, . . . , N }, are continuous. The topology (i) has this property and so (ii).⊆(i). (iii).⊆(ii): Note that, because the induced topology is the initial topology induced by the inclusion .Cω (M; N) ⊆ Cω (M; RN ), the topology (iii) is the coarsest topology for which that inclusion is continuous. Since the inclusion is given by

.

Ф |→ (ι1 ◦ Ф , . . . , ιN ◦ Ф ) = (Ψι1 (Ф ), . . . , ΨιN (Ф )),

.

by Theorem 5.26 the topology (ii) has the property that this inclusion is continuous, and so (iii).⊆(ii). (i).⊆(iii): The topology (i) is the coarsest topology for which .Ψf is continuous for every .f ∈ Cω (N). We claim that .Ψf is continuous for all .f ∈ Cω (M) for the topology (iii), which will show that (i).⊆(iii). We first state a lemma. Lemma 1 Let .M be a .Cω -manifold and let .S ⊆ M be a .Cω -embedded submanifold with .ιS : S → M the inclusion. Then ι∗S : Cω (M) → Cω (S)

.

is an epimorphism, i.e., continuous, surjective, and open. Proof First note that we can use Lemma 2.1 to show that .ι∗S : Cω (M) → Cω (S) is surjective. It, therefore, remains to show that .ι∗S is continuous and open. Continuity follows from Theorem 5.26. Since .Cω (S) and .Cω (M) are ultrabornological webbed spaces (Proposition 2.14), the De Wilde Open Mapping Theorem implies that .ι∗S is open. .Δ We apply the lemma with .M = RN and .S = N. Let .f ∈ Cω (M) and, by the lemma, let .fˆ ∈ Cω (RN ) be such that .f = fˆ ◦ ι. Now consider the commutative diagram

.

where .Cω (M; N) has the topology (iii) and (as required by the definition of the topology (iii)) .Cω (M; RN ) has the weak-PB topology. Thus the vertical and horizontal arrows are continuous, whence the diagonal arrow is continuous, as desired. Finally, let us prove the assertion made in part (iii) that the weak-PB topology ω for .Cω (M; RN ) is topologically isomorphic to .⊕N j =1 C (M). This is equivalent,

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77

given what we have already proven, to the assertion that the initial topology for Cω (M; RN ) associated with the mappings

.

Ψj : Cω (M; RN ) → Cω (M) .

Ф |→ Ф j ,

ω 1 N j ∈ {1, . . . , N}, is isomorphic to .⊕N j =1 C (M). Here we write .Ф = (Ф , . . . , Ф ). However, this assertion follows from a general assertion regarding initial topologies [73, Theorem 8.12]. Indeed, this general result asserts that the mapping

.

Ψ : C (M; R ) → ω

N

N ⊕

Cω (M)

j =1

.

Ф |→ (Ф , . . . , Ф N ) 1

is an homeomorphism onto its image. Since it is surjective, it is, therefore, an homeomorphism. (Here we also make use of the fact that finite direct sums are topologically isomorphic to products [37, Proposition 4.3.2].) Our proof, in fact, yields a slightly more general conclusion related to part (iii). Corollary 2.26 (Weak-PB Topology for Space of Real Analytic Mappings with Values in a Submanifold) Let .M and .N be .Cω -manifolds and let .S ⊆ N be an embedded submanifold. Then the weak-PB topology of .Cω (M; S) is the topology induced by the weak-PB topology of .Cω (M; N) and the inclusion .Cω (M; S) ⊆ Cω (M; N). The preceding theorem, along with some of our continuity results from Chap. 5, lead to the following result of independent interest. Proposition 2.27 (“Weak” Characterisations for Topology for Spaces of Real Analytic Sections) For a .Cω -vector bundle .πE : E → M, the following topologies for .Г ω (E) agree: (i) the .Cω -topology; (ii) the topology induced by the inclusion .Г ω (E) ⊆ Cω (M; E) and the weak-PB topology for .Cω (M; E); (iii) the initial topology associated with the mappings evα : Г ω (E) → Cω (M) .

ξ |→ (x |→ ),

α ∈ Г ω (E∗ ).

Proof In the proof, we shall make use of Corollary 2.5 which gives .E as a real N ⊥ analytic subbundle of a trivial bundle .pr1 : RN M = M × R → M. We also define .E

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as the subbundle of .RN M whose fibres are the orthogonal complements of the fibres of .E, using the Euclidean fibre metric. Thus .E⊥ is a also a real analytic subbundle. (i).=(ii) By Theorem 2.25(iii) and Theorem 5.4 below, the obvious vector space isomorphism Cω (M; RN ) ≃ Г ω (RN M)

.

is an homeomorphism, if .Cω (M; RN ) has the weak-PB topology and .Г ω (RN M ) has the .Cω -topology. Now note that our recollections from the first paragraph of the proof give the diagram

.

As indicated, the top arrow is a topological isomorphism. The bottom arrow and the vertical arrows are the natural inclusions. The left inclusion is a topological embedding by Corollary 2.26 (noting that a subbundle is necessarily an embedded submanifold). By Theorem 5.4, we have the topological isomorphism ω ω ⊥ Г ω (RN M ) ≃ Г (E) ⊕ Г (E ).

.

Moreover, from Theorem 5.4 it also follows that the right inclusion in the diagram is a topological monomorphism. Now the bottom inclusion is verified to be continuous and open by elementary arguments with open sets, using the facts just given. This shows that the first two topologies agree. (ii).=(iii) With .E as a subbundle of .RN M , define ιj : Cω (M; E) → Cω (M) .

ξ |→ prj ◦ ξ,

where prj : RN M →R .

(x, (v1 , . . . , vN )) |→ vj .

Note that .ιj |Г ω (E) = evαj for .αj ∈ Г ω (E∗ ) given by = prj ◦ ιE (ξ(x)).

.

2.5 The Topology for the Space of Real Analytic Mappings

79

By Theorem 2.25(iii), the initial topology for .Cω (M; E) associated with the mappings .ιj , .j ∈ {1, . . . , N }, is equal to the weak-PB topology. Thus the initial topology for .Г ω (E) associated to the mappings .ιj |Г ω (E) = evαj , .j ∈ {1, . . . , N}, is equal to the topology (ii). □ Note that the weak-PB topology can be characterised as being the uniform topology associated with the family of semimetrics dωK,a,f : Cω (M; N) × Cω (M; N) → R .

ω (Ф 1 , Ф 2 ) |→ pK,a (Ф ∗1 f − Ф ∗2 f ),

(2.20)

for .K ⊆ M compact, .a ∈ c0 (Z≥0 ; R>0 ), and .f ∈ Cω (N). Moreover, the previous theorem ensures that it suffices to restrict attention to the semimetrics of this form for .f ∈ {ι1 , . . . , ιN }, where .ι : N → RN is a proper .Cω -embedding. For the remainder of this section, we shall consider descriptions of the weak-PB topology for mappings that mirror the constructions of Sect. 2.2 for the .Cω -topology for the space of real analytic sections of a real analytic vector bundle.

2.5.2 The Topology for the Space of Holomorphic Mappings Since our constructions will be derived from topologies for spaces of holomorphic mappings, we should first specify and understand this topology. We let .M and .N be holomorphic manifolds. For the space .Chol (M; N) of holomorphic mappings, we use the compact-open topology as per [73, §43]. This is the topology with the subbase B(K, V) = {Ф ∈ Chol (M; N) | Ф (K) ⊆ V},

.

K ⊆ M compact, V ⊆ N open.

The topology for the space of holomorphic sections of an holomorphic vector bundle πE : E → M with the seminorms

.

| } { hol pK (ξ ) = sup ‖ξ(z)‖GE | z ∈ K ,

.

K ⊆ M compact,

(cf. (2.2)) is exactly the compact-open topology, thinking of a section as a mapping from .M to .E. In the following result, we give a few different characterisations of the compactopen topology for the space of holomorphic mappings, some valid only for Stein manifolds.

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Theorem 2.28 (Characterisation of the Compact-Open Topology for Space of Holomorphic Mappings) If .M and .N are holomorphic manifolds, then the following topologies for .Chol (M; N) are the same: (i) the compact-open topology; (ii) the topology of uniform convergence on compact subsets of .M. Additionally, if .M and .N are Stein manifolds and if .ι : N → CN is a proper holomorphic embedding, then the preceding two topologies are the same as the following two: (iii) the initial topology associated with the family of mappings Θf : Chol (M; N) → Chol (M) .

∗

Ф |→ Ф f,

f ∈ Chol (N),

where .Chol (N) has the compact-open topology; (iv) the topology induced on .Chol (M; N) ⊆ Chol (M; CN ) by the compact-open topology for Chol (M; CN ) ≃

N ⊕

.

Chol (M).

j =1

Proof We shall only sketch the proof since the results are essentially well known or follow along the lines of results we have already proved. (i).=(ii) This is a standard general result, e.g., [73, Theorem 43.7]. (ii).=(iii) In the same manner as Theorem 2.23 follows from the existence of smooth coordinate functions in a neighbourhood of any point, the assertion in this case follows since, for a Stein manifold, there are holomorphic coordinate functions in a neighbourhood of any point. (iii).=(iv) This follows in the same manner as the corresponding assertion from Theorem 2.25, after one makes the following observations: 1. for .Ф ∈ Chol (M; N), the mapping .Ф ∗ : Chol (N) → Chol (M) is continuous; 2. if .M is a Stein manifold, then coherent sheaves of .CMhol -modules have vanishing cohomology [12]; 3. as a consequence, the holomorphic analogue of Lemma 2.1 holds when .M is a Stein manifold and .S is a Stein submanifold; 4. as a consequence, the holomorphic analogue of Lemma 1 from the proof of Theorem 2.25 holds. The only one of these observations that we shall prove independently is the first. Let K ⊆ M be compact and let .(gj )j ∈Z>0 be a Cauchy sequence in .Chol (N). Thus, since

.

2.5 The Topology for the Space of Real Analytic Mappings

81

Ф (K) is compact, for every .∈ ∈ R>0 , there exists .N ∈ Z>0 such that

.

|gj (y) − gk (y)| ≤ ∈,

.

j, k ≥ N, y ∈ Ф (K).

Therefore, |Ф ∗ gj (z) − Ф ∗ gk (z)| < ∈,

.

j, k ≥ N, z ∈ K,

showing that .(Ф ∗ gj )j ∈Z>0 is a Cauchy sequence in .Chol (M). This suffices to prove continuity of .Ф ∗ . □ Note that a consequence of the theorem is that the topology of .Chol (M; N) is the restriction of the .C0 -topology to holomorphic mappings. We also have the corresponding analogues of Corollary 2.26 and Proposition 2.27 for Stein manifolds. The proofs follow in exactly the manner as their real analytic counterparts. Corollary 2.29 (Weak-PB Topology for Space of Holomorphic Mappings with Values in a Submanifold) Let .M and .N be Stein manifolds and let .S ⊆ N be an embedded Stein submanifold. Then the compact-open topology of .Chol (M; S) is the topology induced by the compact-open topology of .Chol (M; N) and the inclusion hol .C (M; S) ⊆ Chol (M; N). Proposition 2.30 (“Weak” Characterisations for Topology for Spaces of Holomorphic of Sections) For an holomorphic vector bundle .πE : E → M with .M a Stein manifold, the following topologies agree: (i) the compact-open topology; (ii) the topology induced by the inclusion .Г hol (E) ⊆ Chol (M; E) and the compactopen topology for .Chol (M; E); (iii) the initial topology associated with the mappings evα : Г hol (E) → Chol (M) .

ξ |→ (z |→ ),

α ∈ Г hol (E∗ ).

Proof The only missing ingredient in being able to use the same strategy as in the proof of Proposition 2.27 is to note that the total space of an holomorphic vector bundle over a Stein manifold is a Stein manifold [39, Proposition 54.B.4]. □ Also note that the compact-open topology is the uniform topology defined by the family of semimetrics dhol K (Ф , Ψ) = sup{d(Ф (z), Ψ(z)) | z ∈ K},

.

K ⊆ M compact,

(2.21)

where .d is a metric on .M for which the metric topology is the manifold topology, e.g., the metric associated with a Riemannian metric. Since one can consider the semimetrics associated to a compact exhaustion .(Kj )j ∈Z>0 of .M, this shows that the compact-open topology for .Chol (M; N) is metrisable.

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2.5.3 Germs of Holomorphic Mappings Over Subsets of a Real Analytic Manifold We next adapt our discussion from Sect. 2.2.1 to mappings rather than sections. Let .M and .N be real analytic manifolds with complexifications .M and .N. Let .A ⊆ M and let .N A be the set of neighbourhoods of A in .M. For .U, V ∈ N A , and for .Ф ∈ Chol (U; N) and .Ψ ∈ Chol (V; N), we say that .Ф is equivalent to .Ψ if there exist .W ∈ N A and .Θ ∈ Chol (W; N) such that .W ⊆ U ∩ V and such that Ф |W = Ψ|W = Θ.

.

By .CAhol (M; N) we denote the set of equivalence classes, which we call the set of germs of mappings from .M to .N over A. By .[Ф ]A we denote the equivalence class of .Ф ∈ Chol (U; N) for some .U ∈ N A . For .U ∈ N M , we let Chol,R (U; N) = {Ф ∈ Chol (U; N) | Ф (U ∩ M) ⊆ N}.

.

Note that, if .U ∈ N A and if .Ф ∈ Chol,R (U; N), then, for any mapping .Ψ ∈ Chol (V; N) equivalent to .Ф , we have .Ψ ∈ Chol,R (V; N). By .CAhol,R (M; N) we denote the set of equivalence classes of mappings from .Chol,R (U; N). Now we have mappings

.

rU,A : Chol,R (U; N) → CAhol,R (M; N) Ф |→ [Ф ]A .

If .U1 , U2 ∈ N

A

satisfy .U1 ⊆ U2 , we have the restriction mapping rU2 ,U1 : Chol,R (U2 ; N) → Chol,R (U1 ; N).

.

We claim that this restriction mapping is continuous. Indeed, if .f ∈ Chol (N), consider the diagram

.

2.5 The Topology for the Space of Real Analytic Mappings

83

and this gives continuity of the vertical arrow by the universal property of the initial topology. Thus we have the directed system ⎛ .

(Chol,R (U; N))U∈N A , (rU2 ,U1 )U1 ⊆U2

⎞

in the category of topological spaces. This gives the direct limit topology for the direct limit .CAhol,R (M; N). Let us show that this direct limit topology for the space of germs is independent of complexification. We note that the issue of the independence on complexification also arises in Sect. 2.2, although we did not address it. However, the first part of our (elementary) arguments in the proof of the following lemma apply as well to the case of sections of a vector bundle. Lemma 2.31 (Independence of Direct Limit Topology on Complexification) ' ' Let .M and .N be .Cω -manifolds with complexifications .M and .M , and .N and .N , ' respectively. Let .N A and .N A be the directed sets of neighbourhoods of .A ⊆ M in ' .M and .M , respectively. Then the direct limit topologies for the directed systems ⎛ ⎞ (Chol,R (U; N))U∈N A , (rU2 ,U1 )U1 ⊆U2 , ⎛ ⎞ ' ' (Chol,R (U ; N ))U' ∈N ' , (rU' ,U' )U' ⊆U'

.

2

A

1

1

2

are isomorphic in the category of topological spaces. Proof First we show independence on complexification of .M. By uniqueness of '' complexification, there is a complexification .M which is regarded as an open holo' morphic submanifold of both .M and .M . Since the directed set of neighbourhoods '' ' of .M in .M is cofinal in the directed sets of neighbourhoods of .M in both .M and .M , the direct limits of the directed systems .

⎞ ⎛ (Chol,R (U; N))U∈N A , (rU2 ,U1 )U1 ⊆U2 , ⎛ ⎞ ' (Chol,R (U ; N))U' ∈N ' , (rU' ,U' )U' ⊆U' 2

A

1

1

2

agree in the category of topological spaces. As concerns the independence on the complexification of .N, the reason that there is something to think about is that it is possible that there is a .U ∈ N M and .Ф ∈ ' Chol,R (U; N) such that .Ф (U) /⊆ N . The way one rectifies this is by choosing a ' smaller neighbourhood .U on which to represent the germ; one choice is '

'

U = Ф −1 (N ∩ N ) ∩ U.

.

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2 Topology for Spaces of Real Analytic Sections and Mappings '

In this case, .Ф and .Ф |U are equivalent. Said otherwise, every germ with a ' ' ' representative in .Chol,R (U; N) has a representative in .Chol,R (U ; N ) for some .U . We need to show that this observation gives rise to an homeomorphism between the direct limits associated to the directed systems ⎞ ⎛ (Chol,R (U; N))U∈N A , (rU2 ,U1 )U1 ⊆U2 , ⎛ ⎞ ' (Chol,R (U; N ))U∈N A , (rU2 ,U1 )U1 ⊆U2 .

.

To do this, we need some notation. By .[(Ф , U)]A and .[(Ф ' , U)]'A we denote the ' germs of .Ф ∈ Chol,R (U; N) and .Ф ' ∈ Chol,R (U; N ); note that we bookkeep the open set on which a representative of a germ is defined, as we will need this added resolution. ' We first consider the case that .N ⊆ N, and we denote the inclusion by .ιN' . We define a mapping '

κA :

lim Chol,R (U; N) → lim Chol,R (U; N ) − → − →

U∈N

.

U∈N

A

A

'

'

[(Ф , U)]A |→ [(Ф |Ф −1 (N ) ∩ U, Ф −1 (N ) ∩ U)]'A . First we show that .κA is well-defined in that the definition does not depend on the representative .(Ф , U) of a germ .[(Ф , U)]A . Suppose that .[(Ф 1 , U1 )]A = [(Ф 2 , U2 )]A . Then there exists .V ⊆ U1 ∩ U2 such that .Ф 1 |V = Ф 2 |V. Note that, '

'

−1 Ф −1 1 (N ) ∩ V ⊆ Ф 1 (N ) ∩ U1 ,

.

'

'

−1 Ф −1 2 (N ) ∩ V ⊆ Ф 2 (N ) ∩ U2

and '

'

−1 Ф 1 |Ф −1 1 (N ) ∩ V = Ф 2 |Ф 2 (N ) ∩ V,

.

which gives the well-definedness of .κA . Now we show that .κA is injective. Suppose that .κA ([Ф 1 , U1 ]A ) = κA ([Ф 2 , U2 ]A ). Then there exists '

'

−1 V ⊆ Ф −1 1 (N ) ∩ U1 ∩ Ф 2 (N ) ∩ U2

.

such that .Ф 1 |V = Ф 2 |V, and so .[(Ф 1 , U1 )]A = [(Ф 2 , U2 )]A . To show that .κA is ' surjective, consider a germ .[(Ф ' , U)]'A . Since .N ⊆ N, we immediately have κA ([(Ф , U)]A ) = [(Ф ' , U)]'A ,

.

where .Ф = ιN' ◦ Ф ' .

2.5 The Topology for the Space of Real Analytic Mappings

85

Note that the mapping '

λA :

lim Chol,R (U; N ) → lim Chol,R (U; N) − → − →

U∈N

.

[(Ф

'

U∈N

A

, U)]'A

A

'

|→ [(ιN' ◦ Ф , U)]A

is the inverse of .κA , as is clear from the preceding arguments. For .U ∈ N A , define '

.

λU : Chol,R (U; N ) → Chol,R (U; N) Ф ' |→ ιN' ◦ Ф ' ,

and note that the diagram

(2.22)

.

commutes. To show continuity of .κA , for .U ∈ N A , consider the diagram

.

where the diagonal arrow is defined just so the diagram commutes. By the universal property of the direct limit topology, continuity of .κA will follow if the diagonal arrow is continuous for every .U. Note that the preimage in .Chol,R (U; N) of the diagonal arrow is, by definition of .κA , the set of mappings .Ф ∈ Chol,R (U; N) for ' ' which .image(Ф ) ⊆ N . We claim that, if .O ⊆ lim Chol,R (U; N ) is open, then − →U∈N A −1 .(κA ◦ r U,A ) (O) is open. Indeed, observe that, since −1 −1 −1 (κA ◦ rU,A )−1 (O) = (λ−1 A ◦ rU,A ) (O) = (rU,A ◦ λ ) (O)

.

U

'

and since .r −1 (O) is open in .Chol,R (U; N ) since the direct limit topology is the U,A

final topology induced by the mappings .rU,A , .U ∈ N A , we need only show

86

2 Topology for Spaces of Real Analytic Sections and Mappings '

that .λU (Chol,R (U; N )) is open in .Chol,R (U; N). To see that this is true, let .Ф ∈ ' λU (Chol,R (U; N )) and let .K ⊆ U and .V ⊆ N be such that .Ф (K) ⊆ V. Since ' ' .image(Ф ) ⊆ N , we can choose the open set .V such that .cl(V) ⊆ N (by local compactness of .N). Then let .W ⊆ N be such that .

'

cl(V) ⊆ W ⊆ N .

Then the subbasic open set .B(K, W) for the compact-open topology of Chol,R (U; N) is a neighbourhood of .Ф contained in

.

⎞

⎛ '

λU ⎝ lim Chol,R (U; N )⎠ , − →

.

U∈N

A

which shows that this latter set is open. It thus remains to show that .λA is continuous. This will follow from (2.22) and the universal property of the direct limit topology if we can show that .λU is continuous. ' Let .Ф ' ∈ Chol,R (U; N ) and let .K ⊆ U and .V ⊆ N be open such that .B(K, V) is a subbasic neighbourhood of .λU (Ф ' ). Then '

λ−1 (B(K, V)) = B(K, V ∩ N )

.

U

'

is a subbasic neighbourhood of .Ф ' in .Chol,R (U; N ), giving the desired continuity. ' '' ' The above proves the lemma for .N ⊆ N. In general, we define .N = N ∩ N and note that the arguments above give homeomorphisms .

''

lim Chol,R (U; N) ≃ lim Chol,R (U; N ), − → − →

U∈N

U∈N

A

hol,R

lim C − →

U∈N

A

A

''

'

(U; N ) ≃ lim Chol,R (U; N ), − → U∈N

A

and then the lemma follows by transitivity of the relation “homeomorphic.”

□

We shall next use the preceding constructions in two ways, mirroring what we did for sections in Sect. 2.2.

2.5 The Topology for the Space of Real Analytic Mappings

87

2.5.4 Direct and Inverse Limit Topologies for the Space of Real Analytic Mappings Let .M and .N be real analytic manifolds and let .M and .N be complexifications of .M and .N, respectively. Mirroring what we did in Sect. 2.2, in this section we introduce two topologies for .Cω (M; N) using holomorphic extensions to .M and .N. The first is a direct application of the constructions from the preceding section to the case .A = M, and this gives rise to the directed system .

⎛ ⎞ (Chol,R (U; N))U∈N , (rU2 ,U1 )U1 ⊆U2 M

in the category of topological spaces with direct limit .CMhol,R (M; N). Just as in Lemma 2.7, we have a natural bijection Cω (M; N) ≃ CMhol,R (M; N),

.

and so this induces the direct limit topology for .Cω (M; N). Next, for .K ∈ KM a compact set, we have the directed system .

⎛ ⎞ (Chol,R (U; N))U∈N , (rU2 ,U1 )U1 ⊆U2 K

hol,R in the category of topological spaces with direct limit .CK (M; N). This then gives rise to the inverse system

.

⎞ ⎛ hol,R (CK (M; N))K∈KM , (πK2 ,K1 )K1 ⊆K2 ,

where we have

.

hol,R hol,R (M; N) πK2 ,K1 : CK (M; N) → CK 1 2

[Ф ]K2 |→ [Ф ]K1 . The inverse limit is denoted by .lim C hol,R (M; N). In the same manner as ← −K∈KM K Lemma 2.11, we have a natural bijection hol,R Cω (M; N) ≃ lim CK (M; N), ← −

.

K∈KM

and this gives rise to the inverse limit topology for .Cω (M; N). The following theorem establishes the equality of these two holomorphic extension topologies, and also their equality with the weak-PB topology.

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2 Topology for Spaces of Real Analytic Sections and Mappings

Theorem 2.32 (Characterisations of Weak-PB Topology Using Holomorphic Extension) If .M and .N are .Cω -manifolds with complexifications .M and .N, then the following topologies for .Cω (M; N) are the same: (i) the weak-PB topology; (ii) the direct limit topology; (iii) the inverse limit topology. Proof We begin by making a few observations and constructions that will be useful in the proof. First, as shown in [24, §3.4] and without loss of generality due to Lemma 2.31, we shall assume that the complexification .N is a Stein manifold. By the Remmert Embedding Theorem, we let .ˆι : N → Ck ≃ R2k be a proper holomorphic embedding. This then gives a proper real analytic embedding .ι : N → R2k with holomorphic extension .ι : N → C2k . To make our notation more compatible with notation we have already used, let us denote .N = 2k. We note that we have obvious vector space isomorphisms ω N Г ω (RN M ) ≃ C (M; R ),

.

Г hol (CN ) ≃ Chol (M; CN ). M

Moreover, .CN is a complexification of the vector bundle .RN M , as per Sect. 2.1.3. We M

note that .Cω (M; RN ) has various topologies, all of which agree by Theorem 2.25 and Proposition 2.27. Similarly, .Chol,R (M; CN ) has multiple topologies, all of which agree, by Theorem 2.28 and Proposition 2.30. We shall, therefore, use whichever description of these topologies that suits our instantaneous needs. We shall denote by rU,M : Chol,R (U; N) → CMhol,R (M; N),

.

rˆU,M : Chol,R (U; CN ) → CMhol,R (M; CN ),

U∈N

M,

the mappings induced by the direct limits and by πK :

.

hol,R hol,R lim CK (M; N) → CK (M; N), ' ← − '

K ∈KM

πˆ K :

hol,R hol,R (M; CN ), lim CK (M; CN ) → CK ' ← − '

K ∈ KM ,

K ∈KM

the mappings induced by the inverse limits. The following lemma gives a useful characterisation of the direct limit topology.

2.5 The Topology for the Space of Real Analytic Mappings

89

Lemma 1 The direct limit topology for .CMhol,R (M; N) agrees with the topology induced from .CMhol,R (M; CN ) by the inclusion CMhol,R (M; N) ∈ [Ф ]M |→ [ι ◦ Ф ]M ∈ CMhol,R (M; CN ).

.

Proof First let .O ⊆ CMhol,R (M; N) be open in the direct limit topology. Then, for every .U ∈ N M , .r −1 (O) is open in .Chol,R (U; N). By Theorem 2.28(iv), there exists U,M

an open set .O' ⊆ Chol,R (U; CN ) such that r −1 (O) = O' ∩ Chol,R (U; N).

.

U,M

Next let us consider open sets in the induced topology. These are subsets .P ⊆ CMhol,R (M; N) such that there exists an open subset .P' in .CMhol,R (M; CN ) such that hol,R ' .P = P ∩ C (M; N). Openness of .P' in the direct limit topology (making this M choice of the many possible descriptions of the topology) means exactly that, for every .U ∈ N M , .rˆ −1 (P' ) is open in .Chol,R (U; CN ). Note that U,M

r −1 (P) = rˆ −1 (P' ) ∩ Chol,R (U; N).

.

U,M

U,M

The above discussion shows that the induced topology is finer than the direct limit topology. For the opposite inclusion, we consider the diagram

.

for .U ∈ N M , and with the spaces in the top row having the direct limit topologies. Both horizontal arrows are inclusions. The vertical arrows are continuous by definition of the direct limit topologies. By Theorem 2.28(iv), the bottom horizontal arrow is continuous. Thus the dashed diagonal arrow is continuous. By the universal property of direct limits, the top horizontal arrow is continuous. Thus the direct limit topology for .CMhol,R (M; N) is finer than the induced topology from .CMhol,R (M; CN ) since the induced topology is the coarsest for which the inclusion is continuous. .Δ

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2 Topology for Spaces of Real Analytic Sections and Mappings

(i).⊆(ii) Let .f ∈ Cω (N) and let .V ∈ N N and .f ∈ Chol,R (V) be such that .f = f |N [14, Lemma 5.40]. Note that we have the commutative diagram

.

for every .U ∈ N M . As we have seen, the two upper vertical arrows are homeomorphisms when .Cω (M; N) has the direct limit topology. Thus .Θf is defined so that the upper square commutes. By the universal property of the direct limit topology and the continuity of the diagonal arrow, .Θf is continuous. This shows that .Θf is continuous if .Cω (M; N) has the direct limit topology. By definition of the initial topology, this means that the direct limit topology is finer than the weak-PB topology. (ii).⊆(i) We consider the diagram

.

By the lemma above, the direct limit topology for .CMhol,R (M; N) is the initial topology associated with the top horizontal inclusion, where .CMhol,R (M; CN ) has the direct limit topology. The bottom horizontal arrow is continuous by Theorem 2.25(iii), if both spaces have the weak-PB topology. If the right vertical arrow is continuous with the topologies just indicated, then the dashed diagonal arrow is continuous, and this would establish the continuity of the left vertical arrow by the universal property of the initial topology. It thus remains to show that the vector space isomorphism Cω (M; RN ) → CMhol,R (M; CN )

.

is continuous if .Cω (M; RN ) has the weak-PB topology and .CMhol,R (M; CN ) has the direct limit topology. This, however, follows since both topologies are isomorphic ω to .⊕N j =1 C (M). For the weak-PB topology, this was shown during the proof of Theorem 2.25. For the direct limit topology, this follows from Theorem 5.4 below.

2.5 The Topology for the Space of Real Analytic Mappings

91

(ii).⊆(iii) We consider the diagram

.

for .K ∈ KM and .U ∈ N K . The spaces in the first row have their direct limit topologies, the spaces in the second row have their inverse limit topologies, the spaces in the third row have their direct limit topologies, and the spaces in the fourth row have the compact-open topology. The first pair of vertical arrows are the canonical mappings and the right of these arrows is continuous [50, Theorem 1.2(a)], i.e., the opposite conclusion of Proposition 2.13, the second pair of vertical arrows are those defining the inverse limits, and the third pair of vertical arrows are those coming from the direct limit topologies. Thus all vertical arrows are continuous with the given topologies, except for the top left arrow whose continuity we will establish. All horizontal arrows are inclusions. The fourth horizontal arrow is continuous by Theorem 2.28(iv). Thus the bottom dashed diagonal arrow is continuous. Thus the third horizontal arrow is continuous by the universal property of the direct limit topology. It follows that the middle dashed diagonal arrow is continuous, and so the second horizontal arrow is continuous by the universal property of inverse limit topologies. Thus the top dashed diagonal arrow is continuous, and we conclude that the first left vertical arrow is continuous since (by the lemma) the first left direct limit topology equals the induced topology, which is the initial topology induced by the inclusion. This all proves this part of the theorem. (iii).⊆(ii) For this part of the proof, we consider the diagram

.

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2 Topology for Spaces of Real Analytic Sections and Mappings

for .K ∈ KM and .U ∈ N K . The topologies are the expected ones: direct limit topologies for spaces of germs, inverse limit topologies for inverse limits, and the compact-open topology for holomorphic mappings. The top right vertical arrow is the canonical bijection whose continuity we will demonstrate. Note that the diagram

.

commutes in the category of topological spaces if .K1 , K2 ∈ KM satisfy .K1 ⊆ K2 . Thus, by properties of inverse limits, there is an induced dashed diagonal mapping, and this mapping is continuous. Now, by the universal property of the direct limit topology, the top right vertical arrow is continuous, which gives this part of the theorem. □ Our results in this chapter have produced seven characterisations of the topology for the space of real analytic sections of a real analytic vector bundle and four characterisations of the topology for the space of real analytic mappings between real analytic mappings. That all of these characterisations lead to the same topologies speaks to the “uniqueness” of the real analytic topology. Let us say a few words about this. First of all, for topologies for smooth or finitely differentiable mappings and sections, there are other topologies that one might use than the ones we have introduced; the ones we have introduced are presented mainly for the purpose of comparing them to the real analytic topologies. The topologies we have introduced in the finitely differentiable and smooth cases all have to do with characterising open subsets of mappings by their behaviour on some compact subset of the domain, i.e., by uniform convergence in some way on compact subsets. This seems rather analogous to the inverse limit characterisations of the real analytic topologies. For smooth and finitely differentiable sections and mappings, one might also consider topologies where one wants closeness, not just on compact subsets, but on all of the domain. Such topologies are called “strong” by Hirsch [33], and “Whitney” or “wholly open” by Michor [53]. These topologies seem rather analogous to the direct limit characterisations of the real analytic topologies. In the smooth and finitely differentiable cases, the two sorts of topologies are not the same, but they are in the real analytic case. One might wonder why things are as rigid as this in the real analytic case. The reason, really, is the Identity Theorem, perhaps not surprisingly. We note that germs of holomorphic sections or mappings about compact sets contain germs about points as a special case. The Identity Theorem ensures that, if the domain is connected, then the germ of holomorphic sections or mappings at a point uniquely determine the germs at all points. This, of course, does not prove that the inverse limit and direct limit topologies must agree, but perhaps it is suggestive as to why they might.

Chapter 3

Geometry: Lifts and Differentiation of Tensors

Many of the geometric constructions we undertake in the book, and estimates associated with these geometric constructions, involve tensors of various sorts defined on the total space of a vector bundle. In this chapter we fairly laboriously carry out the following constructions for a vector bundle .πE : E → M. 1. In Sect. 3.1 we consider various classes of tensors associated with a vector bundle, most defined by some sort of lifting operation from .M to .E. Some of these lifting constructions can be carried out just using the vector bundle structure, but others involve using a linear connection .∇ πE in .E. 2. In Sect. 3.2 we differentiate the tensors we built in Sect. 3.1. The differentiation is carried out by combining an affine connection .∇ M on .M and a linear connection π .∇ E in .E, much in the way that we did in our jet bundle constructions of Sect. 2.3.1. 3. In Sect. 3.3 we provide recursive formulae that relate derivatives of objects on .M to their lifts on .E, and vice versa. These derivative constructions are essential to our approach for proving the continuity of the various lifting operations. Many of the constructions we give here will seem, on an initial reading, disconnected from the objectives of the book. This is perhaps especially true of the constructions of Sects. 3.1 and 3.2, at least on a first reading. It is only in the later parts of the book that the relevance of all of these constructions will become apparent. For this reason, perhaps a good strategy would be to skip over these two sections in a first reading, coming back to them when they are subsequently needed. There is nothing particularly real analytic with the material in this chapter, so the smooth and real analytic cases are considered side-by-side.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. D. Lewis, Geometric Analysis on Real Analytic Manifolds, Lecture Notes in Mathematics 2333, https://doi.org/10.1007/978-3-031-37913-0_3

93

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3 Geometry: Lifts and Differentiation of Tensors

3.1 Tensors on the Total Space of a Vector Bundle Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. In this section we define lifts of various kinds of tensors on .M to analogous tensors on .E. Some of these lifts can be carried out just using the structure of the vector bundle, while others will rely on the introduction of a linear connection in .E. As we mention in the preamble to the chapter, the constructions in this section will seem unmotivated in a first reading, so we suggest the possibility of skipping the section until it is needed later.

3.1.1 Functions on Vector Bundles Among the geometric constructions we will consider are those associated to a particular set of functions on a vector bundle. Definition 3.1 (Fibre-Linear Functions) Let .r ∈ {∞, ω} and let .πE : E → M be a vector bundle of class .Cr . A function .F ∈ Cr (E) is fibre-linear if, for each .x ∈ M, r r .F |Ex is a linear function. We denote by .Lin (E) the set of .C -fibre-linear functions on .E. ◦ Let us give some elementary properties of the sets of fibre-linear functions. Lemma 3.2 (Properties of Fibre-Linear Functions) Let .r ∈ {∞, ω} and let πE : E → M be a vector bundle of class .Cr . Then the following statements hold:

.

(i) .Linr (E) is a submodule of the .Cr (M)-module .Cr (E); (ii) for .F ∈ Linr (E), there exists .λF ∈ Г r (E∗ ) such that F (e) = ,

.

e ∈ E,

and, moreover, the map .F ⎬→ λF is an isomorphism of .Cr (M)-modules; Proof (i) Let .F ∈ Linr (E) and .f ∈ Cr (M). Then f · F (e) = (f ◦ πE (e))F (e),

.

and so .f · F is fibre-linear since a scalar multiple of a linear function is a linear function. Also, since the pointwise sum of linear functions is a linear function, we conclude that .Linr (E) is indeed a submodule of .Cr (E). (ii) This merely follows by definition of the dual bundle .E∗ . .□ In a rather related manner, we can consider other classes of functions on vector bundles.

3.1 Tensors on the Total Space of a Vector Bundle

95

Definition 3.3 (Lifts and Evaluations of One-Forms and Functions) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. (i) For .λ ∈ Г r (E∗ ), the vertical evaluation of .λ is .λe ∈ Linr (E) defined by e .λ (ex ) = . (ii) For .f ∈ Cr (M), the horizontal lift of f is the function .f h ∈ Cr (E) defined by ∗ h .f = π f . ◦ E

3.1.2 Vector Fields on Vector Bundles Next we turn to vector fields on the total space of a vector bundle. As with our consideration of functions in the preceding section, we restrict attention to vector fields that interact nicely with the vector bundle structure. We begin with the notion of the vertical lift of a section. Definition 3.4 (Vertical Lift of a Section) Let .r ∈ {∞, ω} and let .πE : E → M be a vector bundle of class .Cr . (i) For .ex , ex' ∈ Ex , we define the vertical lift of .ex' to .ex to be ' . vlft(ex , ex )

| d || (ex + tex' ). = dt |t=0

(ii) Given a section .ξ ∈ Г r (E), we define the vertical lift of .ξ to .E to be the vector field ξ v (ex ) = vlft(ex , ξ(x)).

.

◦

Next we consider another sort of lift, this one requiring a .Cr -connection .∇ πE in the vector bundle .πE : E → B. We let .VE = ker(T πE ) be the vertical subbundle. As mentioned in Sect. 1.5, the connection .∇ πE defines a complement .HE to .VE called the horizontal subbundle. We let .ver, hor : TE → TE be the projections onto .VE and .HE, respectively. Definition 3.5 (Horizontal Lift of a Vector Field) Let .r ∈ {∞, ω}, let .πE : E → M be a vector bundle of class .Cr , and let .∇ πE be a .Cr -connection in .E. (i) For .ex ∈ Ex and .vx ∈ Tx M, the horizontal lift of .vx to .ex is the unique vector .hlft(ex , vx ) ∈ Hex E satisfying Tex πE (hlft(ex , vx )) = vx .

.

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3 Geometry: Lifts and Differentiation of Tensors

(ii) For .X ∈ Г r (TM) on .M, we denote by .Xh the horizontal lift of X to .E, this being the vector field .Xh ∈ Г r (TE) satisfying Xh (ex ) = hlft(ex , X(x)).

.

◦

Next we provide formulae for differentiating various sorts of functions with respect to various sorts of vector fields. Lemma 3.6 (Differentiation of Functions on Vector Bundles) Let .r ∈ {∞, ω} and let .πE : E → M a vector bundle of class .Cr . Let .f ∈ Cr (M), .λ ∈ Г r (E∗ ), r r .X ∈ Г (TM), and .ξ ∈ Г (E). Then the following statements hold: (i) .L Xh f h = (L X f )h ; (ii) .L ξ v f h = 0; (iii) .L ξ v λe = h . Additionally, let .∇ πE be a .Cr -linear connection in .πE : E → M. Then π

(iv) .L Xh λe = (∇XE λ)e . Proof (i) We compute L Xh f h (e) = =

.

= = (L X f )h (e). (ii) Since .f h is constant on fibres of .πE and .ξ v is tangent to fibres, we have f h (e + tξ ◦ πE (e)) = f (e).

.

Differentiating with respect to t at .t = 0 gives the result. (iii) Here we compute | d || .L ξ v λ (e) =

dt |t=0 e

= = h , so completing the proof. (iv) Let .e ∈ E and let .t ⎬→ γ (t) be the integral curve for X satisfying .γ (0) = πE (e) and let .t ⎬→ γ h (t) be the integral curve for .Xh satisfying .γ h (0) = e. Then πE h h .t ⎬→ γ (t) is the parallel translation of e along .γ , and as such we have .∇ ' γ (t) γ (t) =

3.1 Tensors on the Total Space of a Vector Bundle

97

0. Then | d || π .L X h λ (e) = = , dt |t=0 e

□

as claimed.

.

In Sect. 3.2 we shall have a great deal more to say about differentiation of objects on the total space of a vector bundle when one has more structure present than we use in the preceding result.

3.1.3 Linear Mappings on Vector Bundles Now we turn to an examination of linear maps associated to a vector bundle πE : E → M. We shall consider vector bundle mappings of two sorts: (1) with values in the trivial line bundle .RM ; (2) with values in .E. The first sort of mappings are, of course, simply sections of the dual bundle, or linear functions of the sort studied in Sect. 3.1.1. Our interest here is in lifting such objects to the total space. First we work with sections of the dual bundle. If we have a connection .∇ πE in a vector bundle .πE : E → M, then this gives us a splitting .TE = HE ⊕ VE, and hence a splitting .T∗ E = H∗ E ⊕ V∗ E with

.

H∗ E = ann(VE),

.

V∗ E = ann(HE).

Note that .H∗e E = image(Te∗ πE ). Definition 3.7 (Lifts of One-Forms and Dual Sections) Let .r ∈ {∞, ω} and let πE : E → M be a .Cr -vector bundle.

.

(i) For .αx ∈ T∗x M and .ex ∈ Ex , the horizontal lift of .αx to .ex is .hlft(ex , αx ) = Te∗x πE (αx ). (ii) The horizontal lift of .α ∈ Г r (T∗ M) is .α h = πE∗ α ∈ Г r (T∗ E). Additionally, let .∇ πE be a connection in .E. (iii) For .λx ∈ E∗x and .ex ∈ E∗x , the vertical lift of .λx is the unique vector ∗ .vlft(ex , λx ) ∈ Ve E satisfying x =

.

for every .ux ∈ Ex .

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3 Geometry: Lifts and Differentiation of Tensors

(iv) The vertical lift of .λ ∈ Г r (E∗ ) is the one-form .λv ∈ Г r (T∗ E) satisfying λv (ex ) = vlft(ex , λ(x)).

.

◦

We also have natural ways of lifting homomorphisms of vector bundles. Definition 3.8 (Vertical Evaluation and Vertical Lift of an Homomorphism) Let .r ∈ {∞, ω}, and let .πE : E → M and .πF : F → M be .Cr -vector bundles. For .L ∈ Г r (F ⊗ E∗ ), (i) the vertical evaluation of L is the section .Le ∈ Г r (πE∗ F) defined by Le (ex ) = (ex , L(ex )).

.

If, additionally, .∇ πE is a connection in .E, (ii) the vertical lift of L is the vector bundle homomorphism .Lv ∈ Г r (πE∗ F ⊗ T∗ E) defined by Lv (Z) = (e, L ◦ ver(Z))

.

for .Z ∈ Te E, noting that .ver(Z) ∈ Ve E ≃ EπE (e) .

◦

We shall be especially interested in two cases of the vector bundle .F. 1. .F = RM : In this case, .F⊗E∗ ≃ E∗ , .πE∗ F ≃ RE , and .πE∗ F⊗T∗ E ≃ T∗ E. One can easily see that, if .λ ∈ Г r (E∗ ), then the vertical evaluation as per Definition 3.8 agrees with that of Definition 3.3, and the vertical lift as per Definition 3.8 agrees with that of Definition 3.7. 2. .F = E: In this case, .F⊗E∗ ≃ T11 (E), i.e., the bundle of endomorphisms of .E. We also have .πE∗ F ≃ VE [42, §6.11]. Thus, for .L ∈ Г r (T11 (E)), .Le is a .VE-valued vector field. Also, .Lv is a .VE-valued endomorphism of .TE. Let us perform some analysis of the vertical evaluation and vertical lift of an homomorphism. First of all, for .e1 , e2 ∈ Ex , Le (e1 + e2 ) = (e1 , L(e1 )) + (e2 , L(e2 )) = Le (e1 ) + Le (e2 ),

.

where addition is with respect to the vector bundle structure

.

3.1 Tensors on the Total Space of a Vector Bundle

99

where Z is the zero section. Thus .Le is a “linear” section over .E. We define the vector bundle mapping PE,F : πE∗ F ⊗ V∗ E → πE∗ F .

Le ⎬→ Le (e)

(3.1)

over .idE , noting that .e ∈ EπE (e) ≃ Ve E. Then, given .A ∈ Г r (πE∗ F ⊗ V∗ E), .PE,F ◦ A is a section of .πE∗ E. Moreover, .PE,F ◦ Lv = Le for .L ∈ Г r (F ⊗ E∗ ). We shall make use of these observations in Sect. 3.3. Let us recast the preceding observations in a slightly different way. To start, note that, given .λ ∈ Г r (E∗ ) and .η ∈ Г r (F), we have .η ⊗ λ ∈ Г r (F ⊗ E∗ ). The tensor product on the left can be thought of as being of .Cr (M)-modules.1 Moreover, such sections of the bundle of endomorphisms locally generate the sections of the homomorphism bundle. Note that (η ⊗ λ)e = ξ v ⊗ λe ,

.

as is directly verified. In this case, since .Cr (M) is a subring of .Cr (E) (by pull-back), we can regard the tensor product as being of .Cr (E)-modules. Therefore, Le ∈ Г r (Г r (πE∗ F) ⊗ Linr (E)).

.

Since .Linr (E) ⊆ Cr (E), the tensor product is mere multiplication in this case. A similar sort of analysis can be made for the vertical lift of an homomorphism. In this case, given .λ ∈ Г r (E ∗ ) and .η ∈ Г r (F), we have .ξ ⊗ λ ∈ Г r (F ⊗ E∗ ), as in the preceding paragraph. In this case, the vertical lift satisfies (ξ ⊗ λ)v = ξ v ⊗ λv .

.

3.1.4 Tensors Fields on Vector Bundles Next we discuss the extension of our lifts of functions, sections, and vector fields to higher-order tensors. The extension is to tensor powers of the pull-back .πE∗ T∗ M of the cotangent bundle to the total space of the vector bundle. Other sorts of lifts are possible, especially in the presence of a connection in the vector bundle. We restrict

1 This

corresponds to the well-known isomorphism .Г

r

(E) ⊗Cr (M) Г r (F) ≃ Г r (E ⊗ F)

of .Cr (M)-modules that we shall prove as Proposition 5.5 below.

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3 Geometry: Lifts and Differentiation of Tensors

ourselves to the tensor powers of the pull-back of .T∗ M since our interest is in jet bundles, and these tensor powers represent derivatives with respect to the base. We make the following definitions. Definition 3.9 (Lifts of Tensors) Let .r ∈ {∞, ω}, and let .πE : E → M and πF : F → M be a .Cr -vector bundles. Let .k ∈ Z>0 .

.

(i) For .A ∈ Г r (Tk (T∗ M)), the horizontal lift of A is .Ah ∈ Г r (Tk (T∗ E)) defined by Ah (Z1 , . . . , Zk ) = A(Te πE (Z1 ), . . . , Te πE (Zk ))

.

for .Z1 , . . . , Zk ∈ Te E.2 (ii) For .A ∈ Г r (Tk (T∗ M) ⊗ E), the vertical lift of A is .Av ∈ Г r (Tk (T∗ E) ⊗ TE) defined by Av (Z1 , . . . , Zk ) = vlft(e, A(Te πE (Z1 ), . . . , Te πE (Zk ))),

.

for .Z1 , . . . , Zk ∈ Te E. (iii) For .A ∈ Г r (Tk (T∗ M) ⊗ F ⊗ E∗ ), the vertical evaluation of A is .Ae ∈ Г r (Tk (T∗ E) ⊗ πE∗ F) defined by Ae (Z1 , . . . , Zk ) = (e, A(Te πE (Z1 ), . . . , Te πE (Zk ))(e)),

.

for .Z1 , . . . , Zk ∈ Te E. Additionally, let .∇ πE be a connection in .E. (iv) For .A ∈ Г r (Tk (T∗ M) ⊗ TM), the horizontal lift of A is .Ah ∈ Г r (Tk (T∗ E) ⊗ TE) defined by Ah (Z1 , . . . , Zk ) = hlft(e, A(Te πE (Z1 ), . . . , Te πE (Zk )))

.

for .Z1 , . . . , Zk ∈ Te E. (v) For .A ∈ Г r (Tk (T∗ M) ⊗ E∗ ), the vertical lift of A is .Av ∈ Г r (Tk (T∗ E) ⊗ T∗ E) defined by Av (Z1 , . . . , Zk ) = vlft(e, A(Te πE (Z1 ), . . . , Te πE (Zk )))

.

for .Z1 , . . . , Zk ∈ Te E.

2 Of course, this is nothing but the usual definition of pull-back, which we repeat for the sake of symmetry.

3.1 Tensors on the Total Space of a Vector Bundle

101

(vi) For .A ∈ Г r (Tk (T∗ M) ⊗ F ⊗ E∗ ), the vertical lift of A is .Av ∈ Г r (Tk (T∗ E) ⊗ πE∗ F ⊗ T∗ E) defined by Av (Z1 , . . . , Zk )(Z) = (e, A(Te πE (Z1 ), . . . , Te πE (Zk ))(ver(Z))),

.

for .Z1 , . . . , Zk , Z ∈ Te E.

◦

3.1.5 Tensor Contractions In our differentiation results of Sect. 3.2, we shall make use of certain generalisations of the contraction operator on tensors. What we have is a sort of “contraction and insertion” operation. We describe this here in the setting of linear algebra, since this is where it most naturally resides. The constructions can, of course, be extended to vector bundles by performing the vector space constructions on fibres. Let .V be a finite-dimensional .R-vector space, let .k ∈ Z>0 and .l ∈ Z≥0 , and let k ∗ l ∗ .α ∈ T (V ) and .β ∈ T (V ) ⊗ V. For .j ∈ {1, . . . , k}, define the j th insertion of .β k+l−1 ∗ in .α by .Insj (α, β) ∈ T (V ) by

Insj (α, β)(v1 , . . . , vk+l−1 )

.

= α(v1 , . . . , vj −1 , β(vj , vk+1 , . . . , vk+l−1 ), vj +1 , . . . , vk ). To be clear, when .l = 0 we have Insj (α, v)(v1 , . . . , vk−1 ) = α(v1 , . . . , vj −1 , v, vj , . . . , vk−1 ).

.

We will also find it helpful to consider tensor contraction when one of the arguments (the second is the one we care about) is fixed. Thus let .β ∈ Tl (V∗ ) ⊗ V and define .Insj,β (α) = Insj (α, β). We shall also need notation for a specific sort of swapping of arguments of a tensor. Let .α ∈ Tk (V) and let .j1 , j2 ∈ {1, . . . , k}. We define pushj1 ,j2 α(v1 , . . . , vk ) ⌠ α(v1 , . . . , vj1 −1 , vj1 +1 , . . . , vj2 , vj1 , vj2 +1 , . . . , vk ), j1 ≤ j2 , = α(v1 , . . . , vj2 −1 , vj1 , vj2 , . . . , vj1 −1 , vj1 +1 , . . . , vk ), j1 > j2 .

.

The idea is that .pushj1 ,j2 drops .vj1 into the .j2 -slot, and shifts the arguments to make room for this. The “insertion” and “push” mappings can be generalised in the obvious way to give .Insj (A, β) and .pushj1 ,j2 (A) for .A ∈ Tk (V∗ ) ⊗ U and l ∗ k ∗ l ∗ .β ∈ (T (V ) ⊗ V) ⊗ U (resp. .A ∈ U ⊗ T (V ) and .β ∈ U ⊗ (T (V ) ⊗ V)), just by acting on the first (resp. second) component of the tensor product.

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3 Geometry: Lifts and Differentiation of Tensors

The final tensor construction we make is that of a linear tensor derivation. Given A ∈ EndR (V), we define a derivation .DA of the tensor algebra .⊕r,s∈Z≥0 Trs (V) by 0 1 .DA (a) = 0 for .a ∈ T (V) ≃ R, and .DA (v) = A(v) for .v ∈ T (V) ≃ V. It 0 0 ∗ ∗ then follows that .DA (α) = −A (α) for .α ∈ V . More generally, we have the following result which expresses a well-known formula, e.g., [53, §3.4], in terms of our insertion operation. .

Lemma 3.10 (Insertion and Tensor Derivation I) Let .V be a finite-dimensional R-vector space, let .A ∈ EndR (V), let .r, s ∈ Z>0 , and let .T ∈ Trs (V). Then

.

DA (T ) =

r Σ

.

Insj (T , A∗ ) −

j =1

s Σ

Insr+j (T , A).

j =1

Proof We have DA (T )(β 1 , . . . , β r , u1 , . . . , us )

.

=

r Σ

v1 ⊗ · · · ⊗ A(vj ) ⊗ . . . vr ⊗ α 1 ⊗ · · · ⊗ α s (β 1 , . . . , β r , u1 , . . . , us )

j =1

−

s Σ

v1 ⊗ · · · ⊗ vr ⊗ α 1 ⊗ · · · ⊗ A∗ (α j ) ⊗ · · · ⊗ α s

j =1

(β 1 , . . . , β r , u1 , . . . , us ) =

r Σ

β 1 (v1 ) · · · β j (A(vj )) · · · β r (vr )α 1 (u1 ) · · · α s (us )

j =1

−

s Σ

β 1 (v1 ) · · · β r (vr )α 1 (u1 ) · · · A∗ (α j )(uj ) · · · α s (us )

j =1

=

r Σ

β 1 (v1 ) · · · A∗ (β j )(vj ) · · · β r (vr )α 1 (u1 ) · · · α s (us )

j =1

−

s Σ

β 1 (v1 ) · · · β r (vr )α 1 (u1 ) · · · α j (A(uj )) · · · α s (us )

j =1

=

r Σ

T (β 1 , . . . , A∗ (β j ), . . . , β r , u1 , . . . , us )

j =1

−

s Σ

T (β 1 , . . . , β r , u1 , . . . , A(uj ), . . . , us ),

j =1

as claimed.

□

.

3.1 Tensors on the Total Space of a Vector Bundle

103

We shall make a minor extension of the preceding notion of a derivation associated to an endomorphism. Let .k, r, s ∈ Z>0 . Here we let .T ∈ Trs (V) and 1 .S ∈ T (V). For .v1 , . . . , vk−1 ∈ V, we define .S(v1 ,...,vk−1 ) ∈ EndR (V) by k S(v1 ,...,vk−1 ) (v) = S(v, v1 , . . . , vk−1 ).

.

Denote .S ∗ ∈ T1k−1 (V) ⊗ V∗ by =

.

so that ∗ S ∗ (β, v1 , . . . , vk−1 ) = S(v (β). 1 ,...,vk−1 )

.

We then define .DS (T ) ∈ Trs+k−1 (V) by DS (T )(β 1 , . . . , β r , u1 , . . . , us+k−1 )=DS(us+1 ...,us+k−1 ) (T )(β 1 , . . . , β r , u1 , . . . , us ). (3.2)

.

The following elementary lemma gives a simpler formula for the previous constructions. Lemma 3.11 (Insertion and Tensor Derivation II) Let .V be a finite-dimensional R-vector space, let .k ∈ Z>0 , let .S ∈ T1k (V), let .r, s ∈ Z>0 , and let .T ∈ Trs (V). Then

.

DS (T ) =

r Σ

.

Insj (T , S ∗ ) −

j =1

s Σ

Insr+j (T , S).

j =1

Proof We have DS (T )(β 1 , . . . ,β r , u1 , . . . , uk+s−1 )

.

=

r Σ

Insj (T , S(us+1 ,...,us+k−1 ) ∗ )(β 1 , . . . , β r , u1 , . . . , us )

j =1

−

s Σ

Insr+j (T , S(us+1 ,...,us+k−1 ) )(β 1 , . . . , β r , u1 , . . . , us )

j =1

=

r Σ

T (β 1 , . . . , S(us+1 ,...,us+k−1 ) ∗ (βj ), . . . , β r , u1 , . . . , us )

j =1

−

s Σ j =1

T (β 1 , . . . , β r , u1 , . . . , S(us+1 ,...,us+k−1 ) (uj ), . . . , us )

104

3 Geometry: Lifts and Differentiation of Tensors

=

r Σ

Insj (T , S ∗ )(β 1 , . . . , β r , u1 , . . . , us+k−1 )

j =1 r Σ

−

Insr+j (T , S)(β 1 , . . . , β r , u1 , . . . , us+k−1 ),

j =1

□

as claimed.

.

Let us summarise this in the cases of interest. The cases of interest will be two in number. The first is when .S ∈ T12 (V) and .T = T0 ⊗ v for .T0 ∈ Tk (V∗ ) and .v ∈ V. In this case the preceding lemma gives DS (T )(v1 , . . . , vk+1 , β) = Insk+1 (T0 ⊗ v, S ∗ )(v1 , . . . , vk+1 , β) −

k Σ

Insj (T0 ⊗ v, S)(v1 , . . . , vk+1 , β)

j =1 .

= T0 (v1 , . . . , vk ) −

k Σ

Insj (T0 , S)(v1 , . . . , vk+1 )

j =1

= T0 (v1 , . . . , vk ) −

k Σ

Insj (T0 , S)(v1 , . . . , vk+1 ).

j =1

(3.3) The second case we will consider is when .S ∈ T12 (V) and .T = T0 ⊗ α for .T0 ∈ Tk (V∗ ) and .α ∈ V∗ . In this case we have DS (T )(v1 , . . . , vk+2 ) = − Insk+1 (T0 ⊗ α, S)(v1 , . . . , vk+2 ) −

k Σ

Insj (T0 ⊗ α, S)(v1 , . . . , vk+2 )

j =1 .

= − T0 (v1 , . . . , vk )α(S(vk+1 , vk+2 )) −

k Σ

Insj (T0 , S)(v1 , . . . , vk+1 ).

j =1

(3.4)

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

105

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle In this section we establish some technical results for differentiation via connections of various objects—functions, vector fields, tensors—on vector bundles. These results will allow us to intrinsically perform the many calculations required to determine the recursive relations given in Sect. 3.3 between jets on .M and jets on .E for a vector bundle .πE : E → M. As with the constructions of the preceding section, the results in this section might seem non sequitur to the objectives of the book. And, as with the results of the preceding section, perhaps a good strategy is to hurdle over this section until the results are subsequently needed. In this section the constructions and results are made and given in both the smooth and real analytic cases.

3.2.1 Vector Bundles as Riemannian Submersions Let .r ∈ {∞, ω}. Let .πE : E → M be a vector bundle with .πTE : TE → E its tangent bundle. We suppose that .∇ M is an affine connection on .M, that .∇ πE is a linear connection on .E, that .GM is a Riemannian metric on .M, and that .GπE is a fibre metric for .E with all data of class .Cr . We shall construct on .E a Riemannian metric in a more or less natural way. Not all constructions require that the affine connection on .M to be the Levi-Civita connection, but we will only work with the case when it is, since there are useful formulae one can prove in this case. Subsequently we shall show that one can just as well use affine connections other than the Levi-Civita connection. The Riemannian metric we construct on the total space .E is a natural adaptation of the Sasaki metric for tangent bundles [61]. To define the inner product, we use the splitting determined by the connection to give the inner product on .Te E by GE (w1 , w2 ) = GM (hor(w1 ), hor(w2 )) + GπE (ver(w1 ), ver(w2 )),

.

w1 , w2 ∈ TeE . (3.5)

This then turns .E into a Riemannian manifold. We denote by .∇ E the Levi-Civita connection associated with .GE . Since the connection giving the splitting is of class r r π .C if .∇ E is of class .C , the Riemannian metric .GE and its Levi-Civita connection r are of class .C if .GM and .GπE are of class .Cr . We note that the choice of metric .GE ensures that .πE : E → M is a Riemannian submersion if we equip .M with its Riemannian metric .GM used to build .GE . Moreover, the fibres of .πE are totally geodesic submanifolds. There are a few constructions involving Riemannian submersions that will be helpful for us, and we review these here, initially in a general setting.

106

3 Geometry: Lifts and Differentiation of Tensors

We let .(F, GF ) and .(M, GM ) be Riemannian manifolds and suppose that we have a surjective submersion .π ∈ Cr (F; M). By .∇ F and .∇ M we denote the Levi-Civita connections associated with .GF and .GM , respectively. We let .VF = ker(T π ) be the vertical subbundle with .HF its .GF -orthogonal complement, which we call the horizontal subbundle .HF. We let .ver, hor : TF → TF be the projections onto .VF and .HF, just as we have done for vector bundles. The submersion .π is a Riemannian submersion if, for each .y ∈ F, GM (Ty π(u), Ty π(v)) = GF (u, v),

.

u, v ∈ Hy F.

For a vector field X on .M, we denote by .Xh the horizontal lift of X to .F. This is the unique .HF-valued vector field satisfying .Ty π(Xh (y)) = X ◦ π(y) for each .y ∈ F. Thus, for example, GE (Xh , Y h ) = GM (X, Y ),

.

X, Y ∈ Г r (TM).

Following [56], for a .Cr -Riemannian submersion .π : F → N, there are two associated tensors that characterise the submersion. Specifically, we define Aπ , Tπ ∈ Г r (T2 (T∗ F) ⊗ TF)

.

by F F Aπ (ξ, η) = ver(∇hor(ξ ) hor(η)) + hor(∇hor(ξ ) ver(η)),

.

(3.6)

F F Tπ (ξ, η) = hor(∇ver(ξ ) ver(η)) + ver(∇ver(ξ ) hor(η))

for .ξ, η ∈ Г ∞ (TF). One can easily verify that .Aπ and .Tπ are indeed tensors as claimed. Since the fibres of .π are submanifolds, we can define the vertical covariant derivative as the projection of the covariant derivative onto sections: ∇Uver V = ver(∇UF V ),

.

U, V ∈ Г r (VF).

Given a submanifold .S of a Riemannian manifold .(M, GM ), .S inherits the Riemannian metric .GS obtained by pulling back .GM by the inclusion .ιS : S → M. The submanifold .S is totally geodesic if every geodesic for .(S, GS ) is also a geodesic for .(M, GM ). With all this background, we have the following result with tells us how to covariantly differentiate vector fields on the total space of a vector bundle. Lemma 3.12 (Covariant Derivatives for Riemannian Submersions) Let .r ∈ {∞, ω}. Let .(F, GF ) and .(M, GM ) be .Cr -Riemannian manifolds with .∇ F and .∇ M the Levi-Civita connections. Let .π : F → M be a .Cr -Riemannian submersion. Let r r .X, Y ∈ Г (TM) and let .U, V ∈ Г (TF) be vertical vector fields. Then the following statements hold:

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

(i) (ii) (iii) (iv) (v) (vi) (vii)

107

F Y h ) = (∇ M Y )h ; hor(∇X h X 1 h h h h .Aπ (X , Y ) = − ver([X , Y ]); 2 ver F .∇ V = ∇ U V + Tπ (U, V ); U F F h h h .∇ X = hor(∇ X ) + Tπ (V , X ); V V F F h .∇ h V = ver(∇ h V ) + Aπ (X , V ); X X F M h h h h .∇ h Y = (∇ Y ) + Aπ (X , Y ); X X 1 F F h h h h h h .GF (∇ X , Y ) = − GF (ver([X , Y ]), V ) = GF (∇ Y , X ). V V 2 .

Additionally, if the fibres of .π are totally geodesic submanifolds of .F, then the following statements hold: (viii) .Tπ = 0; (ix) .∇ ver |Fx is the Levi-Civita connection for the submanifold Riemannian metric on .Fx ; F V ) = ver([X h , V ]); (x) .ver(∇X h (xi) .∇VF Xh is horizontal and .∇VF Xh = Aπ (Xh , V ). Finally, if .F = E is the total space of a vector bundle and if .GE is the Riemannian metric on .E defined above, then the following additional statements hold for sections r .ξ, η ∈ Г (E): (xii) .∇ξEv ηv = 0; E ξ v ) = (∇ π ξ )v . (xiii) .ver(∇X h X Proof We use the Koszul formula for the Levi-Civita connection: 2GF (∇ξF η, ζ ) = L ξ (GF (η, ζ )) + L η (GF (ξ, ζ )) − L ζ (G(ξ, η))

.

+ GF ([ξ, η], ζ ) − GF ([ξ, ζ ], η) − GF ([η, ζ ], ξ )

(3.7)

for vector fields .ξ , .η, and .ζ on .F [41, Page 160]. We shall also use the formulae L ζ (GF (ξ, η)) = GF (∇ζF ξ, η) + GF (ξ, ∇ζF η)

.

(3.8)

(saying that the Levi-Civita connection is a metric connection) and ∇ξF η − ∇ηF ξ = [ξ, η]

.

(3.9)

(saying that the Levi-Civita connection is torsion-free). Let us make some preliminary computations. First, since .Xh and .Y h are .π -related to X and Y , we have that .[Xh , Y h ] is .π -related to .[X, Y ] [1, Proposition 4.2.25]. Thus .

hor([Xh , Y h ]) = [X, Y ]h .

(3.10)

108

3 Geometry: Lifts and Differentiation of Tensors

In like manner, since V is .π -related to the zero vector field and .Xh is .π -related to X, .[V , Xh ] is .π -related to the zero vector field. That is, .

hor([V , Xh ]) = 0.

(3.11)

Next, if f is a function on .M, then L Xh (π ∗ f ) = = ,

.

from which we deduce L Xh (π ∗ f )(y) = ,

.

y ∈ F.

(3.12)

We trivially have L V (π ∗ f ) = 0.

.

(i) One can use (3.7) with .ξ = Xh , .η = Y h , and .ζ = Z h , and the formulae (3.10) and (3.12) to give F h h ∗ M GF (∇X h Y , Z ) = π GM (∇X Y, Z).

.

This shows that .

F h M h hor(∇X h Y ) = (∇X Y ) .

(3.13)

(ii) Now we use (3.7) with .ξ = Xh , .η = Y h , and .ζ = V . We immediately have that the first three terms on the right in (3.7) are zero. By (3.11), the last two terms on the right in (3.7) are zero. Thus we have F h h 2GF (∇X h Y , V ) = GF ([X, Y ] , V ),

.

and so F h Aπ (Xh , Y h ) = ver(∇X hY ) =

.

1 ver([X, Y ]h ). 2

(iii) We have ∇UF V = ver(∇UF V ) + hor(∇UF V ) = ∇Uver V + Tπ (U, V ),

.

as claimed. (iv) We have ∇VF Xh = hor(∇VF Xh ) + ver(∇VF Xh ) = hor(∇VF Xh ) + Tπ (V , Xh ),

.

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

109

as claimed. (v) We have F F F F h ∇X h V = ver(∇X h V ) + hor(∇X h V ) = ver(∇X h V ) + Aπ (X , V ),

.

as claimed. (vi) We have F h F h F h M h h h ∇X h Y = hor(∇X h Y ) + ver(∇X h Y ) = (∇X Y ) + Aπ (X , Y ),

.

using part (i). (vii) This is a direct computation using (3.9), (3.11), (3.8), and part (i): F h h h GF (∇VF Xh , Y h ) = GF (∇X h V , Y ) + GF ([V , X ], Y )

.

F h = − GF (∇X hY , V )

(3.14)

1 1 = − GF ([Xh , Y h ], V ) = − GF (ver([Xh , Y h ]), V ). 2 2 (viii) and (ix) These are properties of totally geodesic submanifolds, so we first prove the result for the following situation. Sublemma 1 Let .(M, GM ) be a Riemannian manifold and let .S ⊆ M be a submanifold. We let .GS = ι∗S GM be the induced Riemannian metric on .S. We let M and .∇ S be the Levi-Civita connections. Then .S is totally geodesic if and only if .∇ M ∞ .∇ Y is tangent to .S whenever .X, Y ∈ Г (TM) are tangent to .S. X Proof We let .NS ⊆ TM|S be the normal bundle. We define the second fundamental form for .S to be the section .| |S of .T2 (T∗ S) ⊗ NS defined by M | |S (X, Y ) = prNS (∇X Y)

.

for vector fields X and Y on .M that are tangent to .S, where .prNS : TM|S → NS is the orthogonal projection onto .NS. We claim that .| |S is symmetric. Indeed, by (3.9) we have | |S (X, Y ) − | |S (Y, X) = prNS ([X, Y ]) = 0,

.

since .[X, Y ] is tangent to .S if X and Y are tangent to .S. M Y ) = ∇ S Y for vector fields X and Y that are tangent Next we claim that .prTS (∇X X to .S, where .prTS : TM|S → TS is the orthogonal projection. To prove this, we show that M (X, Y ) ⎬→ prTS (∇X Y ),

.

110

3 Geometry: Lifts and Differentiation of Tensors

when restricted to .S, satisfies the defining conditions (3.8) and (3.9) for the LeviCivita connection for .GS . Indeed, because .[X, Y ] is tangent to .S whenever X and Y are tangent to .S, we determine that, when restricted to .S, .

M prTS (∇X Y − ∇YM X) = prTS ([X, Y ]) = [X, Y ]

MY ) for all vector fields X and Y tangent to .S. This shows that .(X, Y ) ⎬→ prTS (∇X satisfies (3.9). Also, when we restrict to .S, we have

L Z (GS (X, Y )) = L Z (GM (X, Y )) = GM (∇ZM X, Y ) + GM (X, ∇ZS Y )

.

= GS (prTS (∇ZM X), Y ) + GS (X, prTS (∇ZM Y )) for all vector fields X, Y , and Z that are tangent to .S. This shows that .(X, Y ) ⎬→ M Y ) satisfies (3.8). prTS (∇X Now we can prove the sublemma. First suppose that .S is totally geodesic. Let S satisfying .γ ' (0) = v . Then .γ is .vx ∈ TS and let .t ⎬→ γ (t) be a geodesic for .∇ x M also a geodesic for .∇ . Thus 0 = ∇γM' (t) γ ' (t) = ∇γS' (t) γ ' (t)

.

= prTS (∇γM' (t) γ ' (t)) = prTS (∇γM' (t) γ ' (t)) + prNS (∇γM' (t) γ ' (t)), from which we conclude, evaluating at .t = 0, that .| |S (vx , vx ) = 0. Since .| |S is symmetric, .| |S = 0. Thus S M M ∇X Y = prTS (∇X Y ) = ∇X Y

.

for vector fields X and Y on .M tangent to .S. M Y = ∇ S Y for all vector fields X The converse, that .S is totally geodesic if .∇X X and Y on .M tangent to .S, is clear. .∆ Given the sublemma, let .x ∈ M and let .S = π −1 (x) be the fibre. As we showed in the proof of the sublemma, if U and V are vertical vector fields (in particular, they are tangent to .S), then ∇UF V = ver(∇UF V ) + Tπ (U, V ) = ∇US V .

.

Matching vertical and horizontal parts on .S gives ∇Uver V = ∇US V ,

.

as claimed.

Tπ (U, V ) = 0,

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

111

(xi) It follows immediately from parts (iv) and (viii) that .∇VF Xh is horizontal. We also have F h ∇VF Xh = hor(∇VF Xh ) = hor(∇X h V ) + hor([V , X ])

.

by (3.9). By part (v), the first term on the right is .Aπ (Xh , V ) and, by (3.11), the second term in the right is zero. (x) By (3.9), we have .

F F h h ver(∇X h V ) = ver(∇V X ) + ver([X , V ]).

By part (xi), the first term on the right is zero. (xii) We note here that the fibres of .πE : E → M are vector spaces and the restriction of .GE to .Ex is just the constant Riemannian metric .GπE (x). Thus covariant derivatives on fibres are just ordinary derivatives. Now, since vertical lifts restricted to fibres are constant, their ordinary derivatives are zero, and this gives the assertion. (xiii) Here, by part (x), we have .

E v h v ver(∇X h ξ ) = ver([X , ξ ]).

By (3.11), .[Xh , ξ v ] is vertical. By [1, Proposition 4.2.34], we have | 1 d2 || .[ξ , X ] = | 2 dt 2 | v

h

h

ξv

X ⏀X −t ◦ ⏀−t ◦ ⏀t

h

ξv ◦ ⏀t (e).

t=0

ξv

h

Using the fact that .⏀t (e) = e + tξ ◦ πE (e) and that .⏀X t (e) is the parallel transport γ .t ⎬→ τt along integral curve .γ for X through .πE (e) cf. [41, Proposition III.1.3], we directly calculate h

ξv

X ⏀X −t ◦ ⏀−t ◦ ⏀t

.

h

ξv ◦ ⏀t (e)

γ

= e − t (τ−t (ξ ◦ γ (t)) − ξ ◦ γ (0)).

We recall the relationship π

∇XE ξ(x) =

.

| d || γ τ (ξ ◦ γ (t)) − ξ ◦ γ (0) dt |t=0 −t

between parallel transport and covariant derivative [41, page 114]. By the Leibniz Rule, we have | | d2 || d || γ γ . | (t (τ−t (ξ ◦ γ (t)) − ξ ◦ γ (0))) = 2 | τ−t (ξ ◦ γ (t)) − ξ ◦ γ (0). dt t=0 dt 2 | t=0

π

Thus we have .[ξ v , Xh ] = −(∇XE ξ )v , as claimed.

□

.

112

3 Geometry: Lifts and Differentiation of Tensors

3.2.2 Derivatives of Tensor Contractions In Sect. 3.1.5 we constructed a tensor contraction/insertion operator. Let us consider the derivative of this operation. Lemma 3.13 (Covariant Differential of Insertion I) Let .r ∈ {∞, ω}. Let πE : E → M a vector bundle of class .Cr , let .∇ πE be a .Cr -vector bundle connection in .E, let .k, l ∈ Z>0 , let .A ∈ Г r (Tk (E∗ )), and let .S ∈ Г r (Tl (E∗ ) ⊗ E). For .j ∈ {1, . . . , k} we have .

∇ πE (Insj (A, S)) = Insj (∇ πE A, S) + Insj (A, ∇ πE S).

.

Proof We let .ξa ∈ Г r (E), .a ∈ {1, . . . , k + l − 1}, and .X ∈ Г r (TM). We calculate L X (Insj (A, S)(ξ1 , . . . , ξk+l−1 ))

.

π

= (∇XE Insj (A, S))(ξ1 , . . . , ξk+l−1 ) +

k+l−1 Σ

π

Insj (A, S)(ξ1 , . . . , ∇XE ξa , . . . , ξk+l−1 )

a=1

=

π (∇XE Insj (A, S))(ξ1 , . . . , ξk+l−1 )

+

j −1 Σ

π

A(ξ1 , . . . , ∇XE ξa , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk )

a=1 π

+ A(ξ1 , . . . , ξj −1 , S(∇XE ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk ) k Σ

+

π

A(ξ1 , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ∇XE ξa , . . . , ξk )

a=j +1 k+l−1 Σ

+

π

A(ξ1 , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ∇XE ξa , . . . , ξk+l−1 ), ξj +1 , . . . , ξk ).

a=k+1

We also calculate L X (Insj (A, S)(ξ1 , . . . , ξk+l−1 ))

.

= L X (A(ξ1 , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk )) π

= (∇XE A)(ξ1 , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk ) +

j −1 Σ

π

A(ξ1 , . . . , ∇XE ξa , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk )

a=1 π

+ A(ξ1 , . . . , ξj −1 , (∇XE S)(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk )

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

113

π

+ A(ξ1 , . . . , ξj −1 , S(∇XE ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk ) +

k Σ

π

A(ξ1 , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ∇XE ξa , . . . , ξk )

a=j +1

+

k+l−1 Σ

π

A(ξ1 , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ∇XE ξa , . . . , ξk+l−1 ), ξj +1 , . . . , ξk ).

a=k+1

Comparing the right-hand sides of the preceding calculations gives (∇ πE Insj (A, S))(ξ1 , . . . , ξk+l−1 , X)

.

= (∇ πE A)(ξ1 , . . . , ξj −1 , S(ξj , ξk+1 , . . . , ξk+l−1 ), ξj +1 , . . . , ξk , ξk+l−1 , X) + A(ξ1 , . . . , ξj −1 , (∇ πE S)(ξj , ξk+1 , . . . , ξk+l−1 , X), ξj +1 , . . . , ξk ) = Insj (∇ πE A, S)(ξ1 , . . . , ξk+l−1 , X) + Insj (A, ∇ πE S)(ξ1 , . . . , ξk+l−1 , X), □

and this gives the result.

.

Using this result, we can easily compute the derivative for tensor insertion with one of the arguments fixed. Lemma 3.14 (Covariant Differential of Tensor Insertion II) Let .r ∈ {∞, ω}. Let πE : E → M a vector bundle of class .Cr , let .∇ πE be a .Cr -vector bundle connection in .E, let .l ∈ Z>0 , and let .S ∈ Г r (Tl (E∗ )⊗E). Then, for .k ∈ Z>0 and .j ∈ {1, . . . , k},

.

(∇ πE InsS,j )(A) = Insj (A, ∇ πE S).

.

Proof We have ∇ πE (InsS,j (A)) = (∇ πE InsS,j )(A) + InsS,j (∇ πE A)

.

and ∇ πE (Insj (A, S)) = Insj (∇ πE A, S) + Insj (A, ∇ πE S).

.

Comparing the equations, noting that .InsS,j (∇ πE A) = Insj (∇ πE A, S), the result follows. .□ Related to tensor contraction is the evaluation of a vector bundle mapping. We shall consider the derivative of this evaluation. In stating the result, we use a bit of tensor notation that we now introduce. Let .V be a finite-dimensional .R-vector space and let .A ∈ T1k+1 (V∗ ) and .B ∈ T1l (V). We then denote by .A(B) ∈ Tk+l (V∗ ) the tensor defined by A(B)(v1 , . . . , vk , vk+1 , . . . , vk+l ) = A(v1 , . . . , vk , B(vk+1 , . . . , vk+l ))

.

(3.15)

114

3 Geometry: Lifts and Differentiation of Tensors

Thus .A(B) is shorthand for .Insk+1 (A, B). With this notation, we have the following result. Lemma 3.15 (Leibniz Rule for Tensor Evaluation) Let .r ∈ {∞, ω}, let .πE : E → M and .πF : F → M be .Cr -vector bundles, and let .∇ πE and .∇ πF be .Cr -vector bundle connections in .E and .F, respectively. Let .∇ M be a .Cr -affine connection on .M. Let r ∗ .L ∈ Г (F ⊗ E ). Then k .D M π (L ◦ ξ ) ∇ ,∇ F

=

k ⎛ ⎞ Σ k l=0

l

⎛ ⎞ Symk D∇l M ,∇ πF ⊗πE (L)(D∇k−l (ξ )) , π M ,∇ E

for .k ∈ Z>0 and .ξ ∈ Г r (E). Proof For .A ∈ Tk (V∗ ) and .σ ∈ Sk , we use the notation σ (A)(v1 , . . . , vk ) = A(vσ (1) , . . . , vσ (k) ).

.

First we claim that ∇ M,πF ,k (L ◦ ξ ) =

k Σ

Σ

.

σ (∇ M,πF ⊗∇F ,l L(∇ M,πE ,k−l ξ )).

(3.16)

l=0 σ ∈Sl,k−l

This clearly holds for .k = 1. So suppose it is true up to k for .k ≥ 1. We then compute π M,πF ,k .∇ F (∇ (L ◦ ξ )(X1 , . . . , Xk ))(Xk+1 )

= ∇ M,πF ,k+1 (L ◦ ξ )(X1 , . . . , Xk , Xk+1 ) +

k Σ j =1

= ∇ M,πF ,k+1 (L ◦ ξ )(X1 , . . . , Xk , Xk+1 ) +

M ∇ M,πF ,k (L ◦ ξ )(X1 , . . . , ∇X X , . . . , Xk ) k+1 j

k k Σ Σ

Σ

∇ M,πF ⊗∇F ,l L(∇ M,πE ,k−l ξ )

j =1 l=0 σ ∈Sl,k−l M (Xσ (1) , . . . , ∇X X , . . . , Xσ (k) ), k+1 σ (j )

using the induction hypothesis. We also compute, still using the induction hypothesis, ∇ πF (∇ M,πF ,k (L ◦ ξ )(X1 , . . . , Xk ))(Xk+1 )

.

=

k Σ

Σ

(∇ M,πF ⊗πE ,l+1 L)

l=0 σ ∈Sl,k−l

(Xσ (1) , . . . , Xσ (l) , Xk+1 )(∇ M,πE ,k−l ξ(Xσ (l+1) , . . . , Xσ (k) ))

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

+

k Σ

Σ

115

l Σ M (∇ πF ⊗πE ,l L(Xσ (l) , . . . , ∇X X , . . . , Xσ (l) )) k+1 σ (j )

l=0 σ ∈Sl,k−l j =1

(∇ M,πE ,k−l ξ(Xσ (l+1) , . . . , Xσ (k) )) +

k Σ

Σ

(∇ M,πF ⊗πE ,l L(Xσ (1) , . . . , Xσ (l) ))

l=0 σ ∈Sl,k−l

(∇ M,πE ,k−l+1 ξ(Xσ (l+1) , . . . , Xσ (k) , Xk+1 )) +

k Σ

Σ

k Σ

(∇ M,πF ⊗πE ,l L(Xσ (1) , . . . , Xσ (l) ))

l=0 σ ∈Sl,k−l j =l+1 M X , . . . , Xσ (k) )). (∇ M,πE ,k−l ξ(Xσ (l+1) , . . . , ∇X k+1 σ (j )

Comparing the preceding two equations gives ∇ M,πF ,k+1 (L ◦ ξ )(X1 , . . . , Xk , Xk+1 )

.

=

k Σ

Σ

(∇ M,πF ⊗πE ,l+1 L)

l=0 σ ∈Sl,k−l

(Xσ (1) , . . . , Xσ (l) , Xk+1 )(∇ M,πE ,k−l ξ(Xσ (l+1) , . . . , Xσ (k) )) +

k Σ

Σ

(∇ M,πF ⊗πE ,l L(Xσ (1) , . . . , Xσ (l) ))

l=0 σ ∈Sl,k−l

(∇ M,πE ,k−l+1 ξ(Xσ (l+1) , . . . , Xσ (k) , Xk+1 )) =

k+1 Σ

Σ

(∇ M,πF ⊗πE ,l L(Xσ (1) , . . . , Xσ (l) ))

l=0 σ ∈Sl,k+1−l

(∇ M,πE ,k+1−l ξ(Xσ (l+1) , . . . , Xσ (k+1) )), giving (3.16). For .σ ∈ Sk , write .σ = σ1 ◦ σ2 for .σ1 ∈ Sk,l and .σ2 ∈ Sk|l . Now we compute k

.D M πF (L ◦ ξ ) ∇ ,∇

=

1 Σ σ (∇ M,πF ,k (L ◦ ξ )) k! σ ∈Sk

=

k 1 Σ Σ k!

l=0 σ ∈Sk

Σ σ ' ∈S

l,k−l

σ ' ◦ σ (∇ M,πF ⊗πE ,l L(∇ M,πE ,k−l ξ ))

116

3 Geometry: Lifts and Differentiation of Tensors

=

k 1 Σ k! l=0

=

k Σ l=0

=

k Σ

Σ σ ' ∈S

l,k−l

Σ σ ' ∈S

l,k−l

Σ

l=0 σ ' ∈Sl,k−l

=

k Σ l=0

Σ

Σ

σ ' ◦ σ1 ◦ σ2 (∇ M,πF ⊗πE ,l L(∇ M,πE ,k−l ξ ))

σ1 ∈Sl,k−l σ2 ∈Sl|k−l

Σ σ1 ∈Sl,k−l

l!(k − l)! ' σ ◦ σ1 (D∇l M ,∇ πF ⊗πE L(D∇k−l M ,∇ πE (ξ ))) k!

⎛ ⎞ l!(k − l)! ' ⎝ Σ k! ⎠ σ ◦ Symk (D∇l M ,∇ πF ⊗πE L(D∇k−l M ,∇ πE (ξ ))) k! l!(k − l)! σ ∈Sk

⎛ ⎞ k! Symk (D∇l M ,∇ πF ⊗πE L(D∇k−l M ,∇ πE (ξ ))) , l!(k − l)!

making reference to (1.1) and (1.2) in the penultimate step, and noting that k! . This is the desired result. .□ card(Sl,k−l ) = l!(k−l)!

.

3.2.3 Derivatives of Tensors on the Total Space of a Vector Bundle In Definition 3.9 we introduced a variety of lifts of tensor fields. Here we give formulae for differentiating these. We shall make ongoing and detailed use of the formulae we develop in this section, and decent notation is an integral part of arriving at useable expressions. Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. We consider a .Cr affine connection .∇ M on .M and a .Cr -linear connection .∇ πE in .E. The connection M induces a covariant derivative for tensor fields .A ∈ Г r (Tk (TM)) on .M, .k, l ∈ .∇ l Z≥0 . This covariant derivative we denote by .∇ M , dropping any reference to the particular k and l. Similarly, the connection .∇ πE induces a covariant derivative for sections .B ∈ Г r (Tkl (E)) of the tensor bundles associated with .E, .k, l ∈ Z≥0 . This covariant derivative we denote by .∇ πE , again dropping reference to the particular k and l. We have already made use of these conventions, e.g., in Lemmata 3.13 and 3.14. We will also consider differentiation of sections of .Tkl11 (TM) ⊗ Tkl22 (E). Here we denote the covariant derivative by .∇ M,πE . If we have another .Cr -vector bundle .πF : F → M with a .Cr -affine connection .∇ πF , then .∇ πE and .∇ πF induce a covariant derivative in .Tkl11 (E) ⊗ Tkl22 (F), and we denote this covariant derivative by π ⊗πF . .∇ E Another construction we need in this section concerns pull-back bundles. Let r r .r ∈ {∞, ω}, let .M and .N be .C -manifolds, let .πF : F → M be a .C -vector bundle, r ∗ and let .⏀ ∈ C (N; M). We then have the pull-back bundle .⏀ πF : ⏀∗ F → N, which is a vector bundle over .N. Given a section .η of .F, we have a section .⏀∗ η of .⏀∗ F defined by .⏀∗ η(y) = (y, η ◦ ⏀(y)). Given a .Cr -vector bundle connection .∇ πF in .F,

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

117

we can define a .Cr -connection .⏀∗ ∇ πF in .⏀∗ F by requiring that ⏀∗ ∇ZF ⏀∗ η(y) = ⏀∗ (∇TyF⏀(Z) η) π

.

π

for a .Cr -section .η and for .Z ∈ Ty N. This is the pull-back of .∇ F by .⏀. Given an affine connection .∇ N on .N, we then have an affine connection on .Tk (T∗ N) ⊗ ⏀∗ F induced by tensor product by .∇ N and .⏀∗ ∇ πF . This connection we denote by N,⏀∗ πF , consistent with our notation above. .∇ If we additionally have an injective vector bundle mapping .ψ : ⏀∗ F → TN, then we have ∇ZN (ψ ◦ ⏀∗ η) = ψ ◦ (⏀∗ ∇ZF ⏀∗ η) + Bψ (⏀∗ η, Z) π

.

for some tensor .Bψ ∈ Г r (⏀∗ F∗ ⊗ T11 (TN)). A special case of this is when .⏀ = πE for a vector bundle .πE : E → M and .F = E. In this case, .πE∗ ξ = ξ v and ∗ ∗ .π F ≃ VE and so we indeed have a natural inclusion of .π F in .TE. Moreover, by E E Lemma 3.12(xiii), πE∗ ∇ZE πE∗ ξ = (∇T EπE (Z) ξ )v ,

.

π

π

and so ∇ZE πE∗ ξ = πE∗ ∇ZE πE∗ ξ + AπE (Z, ξ v ). π

.

(3.17)

With the preceding, we can give formulae for differentiating tensors on vector bundles, rather mirroring what we did in Lemma 3.6 for functions. Lemma 3.16 (Differentiation of Lifted Tensors on Vector Bundles) Let .r ∈ {∞, ω}. Let .πE : E → M and .πF : F → M be vector bundles of class .Cr . Let .GM be a .Cr -Riemannian metric on .M, let .∇ M be the Levi-Civita connection, let .GπE be a r r π π .C -fibre metric on .E, and let .∇ E be a linear connection of class .C in .E. Let .∇ F r r be a .C -vector bundle connection in .F. Let .GE be the associated .C -Riemannian metric on .E from (3.5). Define BπE = push1,2 Ins1 (Ins2 (AπE , hor), hor) + Ins2 (AπE , ver) + push1,2 Ins2 (AπE , ver), (3.18)

.

where .AπE is defined as in (3.6). Then we have the following statements, recalling from (3.2) the derivation .DBπE : (i) for .k ∈ Z>0 and .A ∈ Г r (Tk (T∗ M)), we have ∇ E (Ah ) = (∇ M A)h + DBπE (Ah );

.

118

3 Geometry: Lifts and Differentiation of Tensors

(ii) for .A ∈ Г r (Tk (T∗ M) ⊗ E), we have ∇ E (Av ) = (∇ M,πE A)v + DBπE (Av );

.

(iii) for .A ∈ Г r (Tk (T∗ M) ⊗ TM), we have ∇ E (Ah ) = (∇ M A)h + DBπE (Ah );

.

(iv) for .A ∈ Г r (Tk (T∗ M) ⊗ F ⊗ E∗ ), we have ∇ E,πF (Av ) = (∇ M,πE ⊗πF A)v + DBπE (Av );

.

(v) for .A ∈ Г r (Tk (T∗ M) ⊗ T11 (E)), we have ∇ E (Av ) = (∇ M,πE A)v + DBπE (Av );

.

(vi) for .A ∈ Г r (Tk (T∗ M) ⊗ F ⊗ E∗ ), we have ∇ E,πF (Ae ) = (∇ M,πE ⊗πF A)e + DBπE (Ae ) + Av ;

.

(vii) for .A ∈ Г r (Tk (T∗ M) ⊗ T11 (E)), we have ∇ E (Ae ) = (∇ M,πE A)e + DBπE (Ae ) + Av .

.

Proof Before we begin the proof proper, let us justify a “without loss of generality” argument that we will make at various points in the proof. The arguments all have to do with assuming that it is sufficient, when working with differential operators on spaces of tensor products, to work with pure tensor products. Let us be a little specific about this. Let .πE : E → M, .πF : F → M, and .πG : G → M be .Cr -vector bundles. Suppose that .Δ1 , Δ2 : Jm (E ⊗ F) → G are linear differential operators of order m. We wish to give conditions under which .Δ1 = Δ2 . Of course, this is equivalent to giving conditions under which, for a differential operator .Δ : Jm (E ⊗ F) → G, .Δ = 0. To do so, we claim that, without loss of generality, we can simply prove that .Δ(jm (ξ ⊗ η)) = 0 for all .ξ ∈ Г r (E) and .η ∈ Г r (E). Indeed, suppose that we have proved that .Δ(jm (ξ ⊗ η)) = 0 for all .ξ ∈ Г r (E) and .η ∈ Г r (E). Let .x ∈ M r and let .α ∈ T∗m x M. Let .u ∈ Ex and .v ∈ Fx . By Lemma 2.1, there exists .f ∈ C (M) such that .jm f (x) = α. Then, keeping in mind the identification (1.9), Δ(α ⊗ (u ⊗ v)) = Δ(jm (f (ξ ⊗ η))) = Δ(jm ((f ξ ) ⊗ η)) = 0.

.

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

119

Since every element of .Jm x E is a finite linear combination of terms of the form ∗m .α ⊗ (u ⊗ v) for .α ∈ Tx M, .u ∈ Ex , and .v ∈ Fx , we conclude that .Δ(jm A)(x) = 0 for every .A ∈ Г r (E ⊗ F). Now we proceed with the proof. (i) We have L Zk+1 (Ah (Z1 , . . . , Zk ))

.

= (∇ZEk+1 Ah )(Z1 , . . . , Zk ) +

k Σ j =1

Ah (Z1 , . . . , ∇ZEk+1 Zj , . . . , Zk ).

We consider four cases. 1. .Zj = Xjh , .j ∈ {1, . . . , k + 1}: Here we have L Xh (Ah (X1h , . . . , Xkh )) = (L Xk+1 (A(X1 , . . . , Xk )))h

.

k+1

(by Lemma 3.6(i)) and E h h M h Ah (X1h , . . . , ∇X h Xj , . . . , Xk ) = (A(X1 , . . . , ∇Xk+1 Xj , . . . , Xk ))

.

k+1

(by Lemma 3.12(i)). Thus we conclude that h ∇ E Ah (X1h , . . . , Xk+1 ) = ((∇ M A)(X1 , . . . , Xk+1 ))h .

.

v : Here we calculate 2. .Zj = Xjh , .j ∈ {1, . . . , k}, .Zk+1 = ξk+1 h h h h v (A (X , . . . , X )) = L ξ v (A(X1 , . . . , Xk )) = 0 L ξk+1 k 1 k+1

.

(using the definition of .Ah and Lemma 3.6(ii)) and v Ah (X1h , . . . , ∇ξEv Xjh , . . . , Xkh ) = Ah (X1h , . . . , AπE (Xjh , ξk+1 ), . . . , Xkh )

.

k+1

(using Lemma 3.12(xi)). Thus we conclude that E

∇ A

.

h

v (X1h , . . . , Xkh , ξk+1 )

=−

k Σ

v Ah (X1h , . . . , AπE (Xjh , ξk+1 ), . . . , Xkh ).

j =1 h : We calculate 3. .Zj = ξjv for some .j ∈ {1, . . . , k}, .Zk+1 = Xk+1

L Xh (Ah (Z1 , . . . , ξjv , Zk )) = 0

.

k+1

120

3 Geometry: Lifts and Differentiation of Tensors

(by definition of .Ah ) and E v h h v Ah (Z1 , . . . , ∇X h ξj , . . . , Zk ) = A (Z1 , . . . , AπE (Xk+1 , ξj ), . . . , Zk )

.

k+1

(by Lemma 3.12(v)). Thus h h ∇ E Ah (Z1 , . . . , ξjv , . . . , Zk , Xk+1 ) = −Ah (Z1 , . . . , AπE (Xk+1 , ξjv ), . . . , Zk ).

.

v : We have 4. .Zj = ξjv for some .j ∈ {1, . . . , k}, .Zk+1 = ξk+1 h v v (A (Z1 , . . . , ξ , . . . , Zk )) = 0 L ξk+1 j

.

(by definition of .Ah ) and Ah (Z1 , . . . , ∇ξEv ξjv , . . . , Zk ) = 0

.

k+1

(by Lemma 3.12(iii)). Thus v ∇ E Ah (Z1 , . . . , ξjv , . . . , Zk , ξk+1 ) = 0.

.

Putting this all together, and keeping in mind that .AπE is vertical when both arguments are vertical, we have ∇ E Ah (Z1 , . . . , Zk+1 ) = (∇ M A)h (Z1 , . . . , Zk+1 )

.

−

k Σ

Ah (Z1 , . . . , AπE (hor(Zj ), ver(Zk+1 )), . . . , Zk )

j =1

−

k Σ

Ah (Z1 , . . . , AπE (hor(Zk+1 ), ver(Zj )), . . . , Zk ).

j =1

Now we note that BπE (Zj , Zk+1 )

.

= AπE (hor(Zk+1 ), hor(Zj )) + AπE (Zj , ver(Zk+1 )) + AπE (Zk+1 , ver(Zj )) = AπE (hor(Zj ), ver(Zk+1 )) + AπE (hor(Zk+1 ), ver(Zj )) + something vertical,

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

121

using Lemma 3.12(ii) and the definition of .AπE . Thus ∇ E Ah = (∇ M A)h −

k Σ

.

Insj (Ah , BπE ),

j =1

which gives this part of the lemma by Lemma 3.11. (ii) First we compute, for .Z ∈ Г r (TE), M,π

E E ∇ZE ξ v = ∇hor(Z) ξ v + ∇ver(Z) ξ v = (∇T πEE(Z) ξ )v + AπE (T πE (Z), ξ v )

.

M,π

= (∇T πEE(Z) ξ )v + AπE (Z, ξ v ), using Lemma 3.12(iii), (v), and (xiii), and the definition of .AπE . If we note that BπE (ξ v , Z) = AπE (hor(Z), hor(ξ v )) + AπE (ξ v , ver(Z)) + AπE (Z, ver(ξ v ))

.

= AπE (Z, ξ v ) (using the definition of .AπE ), we have M,π

∇ZE ξ v = (∇T πEE(Z) ξ )v + BπE (ξ v , Z).

.

Now, it suffices to prove this part of the lemma for .A = Ah0 ⊗ ξ v for .A0 ∈ and .ξ ∈ Г r (E). For .Z ∈ Г r (TE), we have

Г r (Tk (T∗ M))

∇ZE (Ah0 ⊗ ξ v ) = (∇ZE Ah0 ) ⊗ ξ v + (Ah0 ) ⊗ ∇ZE ξ v

.

= (∇TMπE (Z) A0 )h ⊗ ξ v − π + Ah0 ⊗ (∇T EπE (Z) ξ )v

k Σ

Insj (Ah0 , BπE ,Z ) ⊗ ξ v

j =1

+ Ah0 ⊗ BπE (ξ v , Z).

We have π

(∇TMπE (Z) A0 )h ⊗ ξ v +Ah0 ⊗ (∇T EπE (Z) ξ )v

.

π

= ((∇TMπE (Z) A0 ) ⊗ ξ )v + (Ah0 ⊗ (∇T EπE (Z) ξ ))v M,π

= (∇T πEE(Z) (A0 ⊗ ξ ))v . Thus, by (3.3) and the first part of the lemma, we have Ah0 ⊗ BπE (ξ v , Z) −

k Σ

.

j =1

Insj (Ah0 , BπE ,Z ) ⊗ ξ v = DBπE ,Z (Ah0 ⊗ ξ v ).

122

3 Geometry: Lifts and Differentiation of Tensors

Assembling the preceding three computations gives this part of the lemma. (iii) First note that E E ∇ZE Xh = ∇hor(Z) Xh + ∇ver(Z) Xh

.

= (∇TMπE (Z) X)h + AπE (hor(Z), Xh ) + AπE (Xh , ver(Z)), using Lemma 3.12(xi). Now we have BπE (Xh , Z)

.

= AπE (hor(Z), Xh ) + AπE (Xh , ver(Z)) + AπE (Z, ver(Xh )) = AπE (hor(Z), Xh ) + AπE (Xh , ver(Z)). Thus we have ∇ZE Xh = (∇TMπE (Z) X)h + BπE (Xh , Z).

.

Now it suffices to prove this part of the lemma for .A = Ah0 ⊗ Xh for .A0 ∈ and .X ∈ Г r (TM). In this case we calculate, for .Z ∈ Г r (TE),

Г r (Tk (T∗ M))

∇ZE (Ah0 ⊗ Xh ) = (∇ZE Ah0 ) ⊗ Xh + Ah0 ⊗ ∇ZE Xh

.

= (∇TMπE (Z) A0 )h ⊗ Xh −

k Σ

Insj (Ah0 , BπE ,Z ) ⊗ Xh

j =1

+ Ah0 ⊗ (∇TMπE (Z) X)h + Ah0 ⊗ BπE (Xh , ver(Z)). We have (∇TMπE (Z) A0 )h ⊗ Xh + Ah0 ⊗ (∇TMπE (Z) X)h = (∇TMπE (Z) (A0 ⊗ X))h .

.

We also have, by (3.3) and the first part of the lemma, Ah0 ⊗ BπE (Xh , ver(Z)) −

k Σ

.

j =1

Insj (Ah0 , BπE ,Z ) ⊗ Xh = D(BπE )Z (Ah0 ⊗ Xh ).

Putting the above computations together gives this part of the lemma. (iv) First we need to compute .∇ E λv . We do this by using the formula L Z1 = +

.

in four cases.

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

123

1. .Z1 = X1h and .Z2 = X2h : Here we have L Xh = 0

.

1

and E h v h h =

.

1

(by Lemma 3.12(vi)), giving E v h v h h v h h = − =

.

1

(by Lemma 3.12(ii)). Thus we have E v h ∗ v h h = .

.

1

2. .Z1 = Xh and .Z2 = ξ v : We compute L Xh = (L X )h

.

(by Lemma 3.6(i)) and π

π

E v v v h E E = =

.

(by Lemma 3.12(xiii)). Thus π

E v v h h E = (L X ) −

.

or π

E v v v v E = .

.

3. .Z1 = ξ v and .Z2 = Xh : In this case we compute L ξ v = 0

.

and = 0

.

(by Lemma 3.12(xi)), giving = 0.

.

124

3 Geometry: Lifts and Differentiation of Tensors

4. .Z1 = ξ1v and .Z2 = ξ2v : We have L ξ1v = L ξ1v h = 0

.

(by Lemma 3.6(ii)) and = 0

.

1

(using Lemma 3.12(xii)). This gives = 0.

.

1

Putting the above together, ∇ZE λv = (∇T EπE (Z) λ)v + hor(A∗πE (λv , hor(Z))). π

.

Now we note that =

.

= + + = − = − , using Lemma 3.12(xi). Thus ∇ZE λv = (∇T EπE (Z) λ)v − Bπ∗E (λv , Z).

.

π

Now, it suffices to prove this part of the lemma for .A = A0 ⊗ λ ⊗ η for .A0 ∈ Г r (Tk (T∗ M)), .λ ∈ Г r (E∗ ), and .η ∈ Г r (F). Here we calculate, for .Z ∈ Г r (TE), E,πF

∇Z

.

(Ah0 ⊗ λv ⊗ πE∗ η) = (∇ZE Ah0 ) ⊗ λv ⊗ πE∗ η + Ah0 ⊗ ∇ZE λv ⊗ η + Ah0 ⊗ λv + πE∗ ∇ZF πE∗ η π

= (∇TMπE (Z) A0 )h ⊗ λv ⊗ πE∗ η − + Ah0

π ⊗ (∇T EπE (Z) λ)v

k Σ j =1

⊗ η − Ah0 ⊗ Bπ∗E (λv , Z)

+ Ah0 ⊗ λv ⊗ πE∗ (∇T FπE (Z) η). π

Insj (Ah0 , BπE ,Z ) ⊗ λv ⊗ πE∗ η

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

125

We have (∇TMπE (Z) A0 )h ⊗ λv + Ah0 ⊗ (∇T EπE (Z) λ)v + Ah0 ⊗ λv ⊗ πE∗ (∇T FπE (Z) η) π

.

π

π

π

= (∇TMπE (Z) A0 ⊗ λ ⊗ η)v + (A0 ⊗ ∇T EπE (Z) λ ⊗ η)v + (A0 ⊗ λv ⊗ ∇T FπE (Z) η)v π ⊗π

F = (∇T EπE (Z) (A0 ⊗ λ ⊗ η))v

and, by (3.4) and the first part of the lemma, k Σ −Ah0 ⊗Bπ∗E (λv , Z)⊗πE∗ η− Insj (Ah0 , BπE ,Z )⊗λv ⊗πE∗ η = DBπE ,Z (Ah0 ⊗λv ⊗πE∗ η).

.

j =1

Assembling the preceding computations gives this part of the lemma. (v) This is a slight modification of the preceding part of the proof, taking the formula (3.17) into account. (vi) By Lemma 3.6(iii) and (iv) we have π

∇ZE λe = L Z λe = (∇T EπE (Z) λ)e + .

.

With the constructions following Definition 3.8 in mind, we work with .A = Ah0 ⊗ λe ⊗ πE∗ η for .A0 ∈ Г r (Tk (T∗ M)), .λ ∈ Г r (E∗ ), and .η ∈ Г r (F). If we keep in mind that .λe is a function, then we can simply write .A = A0 ⊗ (λe πE∗ η). We now calculate E,πF

∇Z

.

(Ah0 ⊗λe ⊗ πE∗ η) = (∇ZE Ah0 ) ⊗ λe ⊗ πE∗ η + Ah0 ⊗ (∇ZE λe ) ⊗ πE∗ η + Ah0 ⊗ λe ⊗ (πE∗ ∇ZF πE∗ η) π

= (∇TMπE (Z) A0 )h ⊗ λe ⊗ πE∗ η − π + Ah0 ⊗ (∇T EπE (Z) λ)e

k Σ

Insj (Ah0 , BπE ) ⊗ λe ⊗ πE∗ η

j =1

⊗ πE∗ η + Ah0 ⊗ (λv (Z)) ⊗ πE∗ η

+ Ah0 ⊗ λe ⊗ πE∗ (∇T FπE (Z) η) + Ah0 ⊗ λe ⊗ BπE (πE∗ η, Z). π

We have (∇TMπE (Z) A0 )h ⊗ λe ⊗ πE∗ η + Ah0 ⊗ (∇T EπE (Z) λ)e ⊗ πE∗ η π

.

+ Ah0 ⊗ λe ⊗ πE∗ (∇T FπE (Z) η) π

126

3 Geometry: Lifts and Differentiation of Tensors π

π

= (∇TMπE (Z) A0 ⊗ λ ⊗ η)e + (A0 ⊗ ∇T EπE (Z) λ ⊗ η)e + (A0 ⊗ λ ⊗ ∇T FπE (Z) η)e M,π ⊗π

= (∇T πEE(Z) F (A0 ⊗ λ ⊗ η))e . Next we note that Ah0 ⊗λe ⊗BπE (πE∗ η, Z)−

k Σ

.

j =1

Insj (Ah0 , BπE )⊗λe ⊗πE∗ η = DBπE ,Z (A0 ⊗(λe πE∗ η)),

keeping in mind that .λe is a function, so the tensor products with .λe are just multiplication. Again making reference to the constructions following Definition 3.8, we have Ah0 ⊗ λv ⊗ πE∗ η = (A0 ⊗ λ ⊗ η)v ,

.

and the lemma follows by combining the preceding three formulae. (vii) This is a slight modification of the preceding part of the proof, taking the formula (3.17) into account. .□ The precise complicated definition (3.18) of the tensor .Bπ is a little immaterial. The point is that it is a tensor associated with the vector bundle .E and the induced connection .∇ E , and that it is of class .Cr .

3.2.4 Prolongation In our geometric setting, differentiation means “prolongation” by taking jets. In this section, we illustrate how our decompositions of Sect. 2.3.1 interact with prolongation. As we shall see, this is one place where our geometric framework makes things a little more complicated, compared to using local coordinates. However, the results are interesting, just for this reason, so we give them in detail and make use of them, e.g., in the proof of Theorem 5.11 below. We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle with .Jm E, .m ∈ Z≥0 , its jet bundles. We suppose that we have a .Cr -affine connection .∇ M on .M and a .Cr -vector bundle connection .∇ πE in .E. Because we have the decomposition Jm E ≃

m ⊕

.

j =0

(Sj (T∗ M) ⊗ E)

(3.19)

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

127

by Lemma 2.15, it follows that the vector bundle .Jm E has a .Cr -connection that we denote by .∇ πE,m . Explicitly, ∇XE,m jm ξ = (S∇mM ,∇ πE )−1 (∇XE ξ, ∇X π

π

.

M,πE

M,πE

D∇1 M ,∇ πE (ξ ), . . . , ∇X

D∇mM ,∇ πE (ξ )).

Therefore, the construction of Lemma 2.15 can be applied to .Jm E in place of .E, and all that remains to sort out is notation. To this end, for .k, m ∈ Z≥0 , let us denote by .∇ M,πE,m the connection in k ∗ m M and .∇ πE,m . .T (T M) ⊕ J E induced, via tensor product, by the connections .∇ r Then, for .ξ ∈ Г (E), denote ∇ M,πE,m ,k jm ξ = ∇ M,πE,m · · · (∇ M,πE,m (∇ πE,m jm ξ )) ∈ Г r (Tk (T∗ M) ⊗ Jm E).   

.

k−1 times

We also denote D∇k M ,∇ πE,m (jm ξ ) = Symk ⊗ idJm E (∇ M,πE,m ,k jm ξ ) ∈ Г r (Sk (T∗ M) ⊗ Jm E).

.

This can be refined further by explicitly decomposing .Jm E, so let us provide the notation for making this refinement. For .m, k ∈ Z≥0 and for .A ∈ Г r (Tm (T∗ M)⊗E), we denote ∇ M,πE ,k A = ∇ M,πE · · · ∇ M,πE A ∈ Г r (Tm+k (T∗ M) ⊗ E)

.

k times

and M,πE ,k m D∇k,m A) ∈ Г r (Sk (T∗ M) ⊗ Tm (T∗ M) ⊗ E). M ,∇ πE (A) = Symk ⊗ idT (T∗ M)⊗E (∇

.

Note that, if .A ∈ Г r (Sm (T∗ M) ⊗ E), then r k ∗ m ∗ D∇k,m M ,∇ πE (A) ∈ Г (S (T M) ⊗ S (T M) ⊗ E).

.

An immediate consequence of Lemma 2.15 is then the following result. Lemma 3.17 (Decompositions of Jet Bundles of Jet Bundles) The maps S∇k M ,∇ πE,m : Jk Jm E →

k ⊕

.

j =0

(Sj (T∗ M) ⊗ Jm E)

128

3 Geometry: Lifts and Differentiation of Tensors

defined by S∇k M ,∇ πE,m (jk (jm ξ(x)))

.

= ((jm ξ(x)), D∇1 M ,∇ πE,m ()(jm ξ(x)), . . . , D∇k M ,∇ πE,m ()(jm ξ(x))) and k,m .S M π ∇ ,∇ E

:J

k

⎛ m ⊕ l=0

⎞ ∗

S (T M) ⊗ E → l

k ⊕

⎛

⎛ ∗

S (T M) ⊗ j

j =0

m ⊕

⎞⎞ ∗

S (T M) ⊗ E l

l=0

defined by jk (A0 , A1 , . . . , Am )(x) ⎬→ ((A0 (x), A1 (x), . . . , Am (x)),

.

(D∇1,0M ,∇ πE (A0 )(x), D∇1,1M ,∇ πE (A1 )(x), . . . , D∇1,m M ,∇ πE (Am )(x)), . . . , (D∇k,0M ,∇ πE (A0 )(x), D∇k,1M ,∇ πE (A1 )(x), . . . , D∇k,m M ,∇ πE (Am )(x))) are isomorphisms of vector bundles, and, for each .k ∈ Z>0 , the diagrams

.

and

.

commute, where .prk+1 are the obvious projections, stripping off the last component k of the direct sum. Of course, in understanding the meaning of the preceding lemma, one should bear in mind the isomorphism (3.19), which infers that we have vector bundle

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

129

isomorphisms J J E ≃ k m

.

k ⊕

∗

(S (T M) ⊗ J E) ≃ j

m

j =0

k ⊕

⎛

⎛ ∗

S (T M) ⊗ j

j =0

m ⊕

⎞⎞ ∗

S (T M) ⊗ E l

,

l=0

and the lemma is just an explicit writing down of these isomorphisms. Now we recall [62, Definition 6.2.25] the injective vector bundle mapping over .idM , for .k, m ∈ Z≥0 , ιE,k,m : Jk+m E → Jk Jm E .

jm+k ξ(x) ⎬→ jk jm ξ(x).

(3.20)

Let us understand this mapping using our decompositions of jet bundles. Doing so is a bit more involved than one might imagine due, essentially, to the fact that iterated covariant differentials are not symmetric. In fact, our constructions will illustrate the rôle of curvature of .∇ πE and torsion of .∇ M in this lack of symmetry. Indeed, the next lemma gives an explicit form for the lack of symmetry for second derivatives. Lemma 3.18 (Decomposition of Second Covariant Differential) Let .r ∈ {∞, ω}, let .πE : E → M be a .Cr -vector bundle, and let .∇ M be a .Cr -affine connection on .M and .∇ πE be a .Cr -linear connection in .E. The following formula holds for .ξ ∈ Г r (E): 1 1 ∇ M,πE ,2 ξ(X, Y ) = D∇2 M ,∇ πE (ξ )(X, Y )+ R∇ πE (X, Y )(ξ )− T∇ M (X, Y )(∇ πE ξ ), 2 2

.

X, Y ∈ Г r (TM), where we denote T∇ M (X, Y )(∇ πE ξ ) = ∇ πE ξ(T∇ M (X, Y )).

.

π

Proof We have .∇ πE ξ(Y ) = ∇Y E ξ , whereupon π

π

∇ M,πE ,2 ξ(X, Y ) = ∇XE ∇Y E ξ − ∇

.

πE M Y ξ. ∇X

Thus ∇ M,πE ξ(X, Y )

.

=

1 M,πE 1 (∇ ξ(X, Y ) + ∇ M,πE ξ(Y, X)) + (∇ M,πE ξ(X, Y ) − ∇ M,πE ξ(Y, X)) 2 2

130

3 Geometry: Lifts and Differentiation of Tensors

1 π π π π πE πE = D∇2 M ,∇ πE (ξ )(X, Y ) + (∇XE ∇Y E ξ − ∇XE ∇Y E ξ − ∇[X,Y ] ξ − ∇T∇ M (X,Y ) ξ ) 2 1 1 = D∇2 M ,∇ πE (ξ )(X, Y ) + R∇ πE (X, Y )(ξ ) − ∇ πE ξ(T∇ M (X, Y )), 2 2 □

as desired.

.

Aside 3.19 (Relationship to Covariant Exterior Derivative) The formula of the preceding lemma reflects, but is not equal to, a standard construction concerning connections. In this aside, we clarify the relationships. We refer to [42, §III.9.34, §III.11.5] for a discussion of the concepts we reference here, but do not fully develop. One of the many ways to define the notion of a connection in a vector bundle .πE : E → M is via the connector which is a vector bundle mapping .K∇ πE : TE → E for which the two diagrams (1.6) commute. In this way of thinking about things, we can think of .K∇ πE as defining a .VE-valued one-form on .E, making use of the vector bundle isomorphism .vlft : E ⊕ E → VE. The covariant derivative is defined using the connector by the formula (1.7). In this formulation, one can give an alternative definition of the curvature tensor: R∇ πE (X(πE (e)), Y (πE (e)))(ξ(e)) = −K∇ πE ([Xh , Y h ](e)).

.

In this formulation, we can see that curvature measures the nonintegrability of the horizontal subbundle. This formulation also admits another differential construction, apart from the covariant derivative.ΛTo set this up, we consider .E-valued differential forms on k ∗ .M, i.e., sections of . (T M) ⊗ E. Just as one defines the exterior derivative for differential forms by requiring that it act on functions in the usual way and extends to general differential forms by certain naturality properties [1, Theorem 6.4.1], one can define the exterior covariant derivative, denoted by .d∇ πE , by asking that π .d∇ πE ξ = ∇ E ξ and that .d∇ πE have the same naturality conditions as specified by those for the usual exterior derivative. A place where the usual exterior derivative differs from the covariant exterior derivative is that, for the latter, it is not necessarily true that .d∇ πE ◦ d∇ πE = 0. Indeed, it holds that d∇ πE ◦ d∇ πE ξ(X, Y ) = R∇ πE (X, Y )(ξ ).

.

Thus, in this formulation, curvature measures the extent to which it is not true that d2∇ πE = 0. With all of this as backdrop, one way to interpret Lemma 3.18 is that curvature plays a rôle in measuring the lack of symmetry of .∇ M,πE ,2 . A difference with the constructions of the preceding paragraph is that .∇ M,πE invokes, as well as the linear connection .∇ πE , an affine connection .∇ M on .M. We see, moreover, that this affine connection on .M contributes to the lack of symmetry of .∇ M,πE ,2 through its torsion. ◦

.

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

131

Note that the first term in the decomposition of .∇ M,πE ,2 ξ from Lemma 3.18 is the symmetric part of this tensor, while the remaining two terms are the skew-symmetric part. To extend this to higher-order covariant differentials, we require a preliminary construction. For a .R-vector space .V and for .r, s ∈ Z≥0 , we have an inclusion, Δk,m : Sk+m (V∗ ) → Sk (V∗ ) ⊗ Sm (V∗ ).

.

(3.21)

Let us give an explicit formula for this inclusion. Lemma 3.20 (Inclusions for Symmetric Tensors) For a finite-dimensional .Rvector space .V and for .r, s ∈ Z≥0 , Δk,m = (Symk ⊗ Symk ) ◦ ιk,m ,

.

where ιk,m : Sk+m (V∗ ) → Tk+m (V∗ ) = Tk (V∗ ) ⊗ Tm (V∗ )

.

is the inclusion. Explicitly, Σ

Δk,m (α 1 ⊙ · · · ⊙ α k+m ) =

.

(α σ (1) ⊙ · · · ⊙ α σ (k) ) ⊗ (α σ (k+1) ⊙ · · · ⊙ α σ (k+m) )

σ ∈Sk,m

for .α 1 , . . . , α k+m ∈ V∗ . Proof We note that .Symk+m ◦ Δk,m = idSk+m (V∗ ) , simply since .Δk,m is the inclusion and .Symk+m is projection onto .Sk+m (V∗ ) ⊆ Tk+m (V∗ ). Thus, for the first part of the lemma, it will suffice to show that .

Symk+m ◦ (Symk ⊗ Symm ) ◦ ιk,m = idSk+m (V∗ ) .

For .A ∈ Sk+m (V∗ ) we have .

Symk ⊗ Symm (A)(v1 , . . . , vk+m ) 1 Σ Σ = A(vσ1 (1) , . . . , vσ1 (k) , vk+σ2 (1) , . . . , vk+σ2 (m) ) k!m! σ1 ∈Sk σ2 ∈Sk

= A(v1 , . . . , vk , vk+1 , . . . , vk+m ), since A is symmetric, and so symmetric on the first k and last m entries. Since Symk+m (A) = A, our claim follows, and so does the first part of the lemma.

.

132

3 Geometry: Lifts and Differentiation of Tensors

For the asserted explicit formula, it is evident that the mapping α 1 ⊙ · · · ⊙ α k+m ⎬→

Σ

.

(α σ (1) ⊙ · · · ⊙ α σ (k) ) ⊗ (α σ (k+1) ⊙ · · · ⊙ α σ (k+m) )

σ ∈Sk,m

takes values in .Sk (V∗ ) ⊗ Sm (V∗ ), and so it suffices to show that ⎛ .

Symk+m ⎝

Σ

⎞ (α σ (1) ⊙ · · · ⊙ α σ (k) ) ⊗ (α σ (k+1) ⊙ · · · ⊙ α σ (k+m) )⎠

σ ∈Sk,m

= α 1 ⊙ · · · ⊙ α k+m .

(3.22)

To this end, we calculate Σ . (α σ (1) ⊙ · · · ⊙ α σ (k) ) ⊗ (α σ (k+1) ⊙ · · · ⊙ α σ (k+m) ) σ ∈Sk,m

= =

1 k!m!

Σ

Σ

σ1 ◦ σ2 ((α 1 ⊙ · · · ⊙ α k ) ⊗ (α k+1 ⊙ · · · ⊙ α k+m ))

σ1 ∈Sk,m σ2 ∈Sk|m

(k + m)! Symk+m ((α 1 ⊙ · · · ⊙ α k ) ⊗ (α k+1 ⊙ · · · ⊙ α k+m )) k!m!

= α 1 ⊙ · · · ⊙ α k+m , using (1.2). Applying .Symk+m to the leftmost and rightmost components of this string of equalities gives (3.22). .□ Using the preceding constructions, we next see how the decomposition of Lemma 3.18 for iterated covariant differentials of order 2 are reflected in higherorder iterated derivatives. To state the result, we use the abbreviation θk,m =

.

1 (Δk−1,1 ⊗ Δ1,m−1 ) ◦ (Symk ⊗ Symm ) 2

for .k, m ∈ Z>0 . Lemma 3.21 (Decomposition of Iterated Derivative) Let .r ∈ {∞, ω}, let πE : E → M be a .Cr -vector bundle, and let .∇ M be a .Cr -affine connection on r π .M and .∇ E be a .C -linear connection in .E. The following formulae hold for r .ξ ∈ Г (E) and for .k, m ∈ Z>0 : .

(i) .D∇k,1M ,∇ πE (D∇1 M ,∇ πE (ξ )) = D∇k+1 M ,∇ πE (ξ )+ ⎛ ⎞ k−1 k−1 M,πE ξ ) ; .θk,1 ⊗ idE D M π R∇ πE (ξ ) − D M π T∇ M (∇ ∇ ,∇ E ∇ ,∇ E

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

133

⎛ k+m m−1 m π D∇k−1 (ii) .D∇k,m (D (ξ ))=D (ξ )+θ ⊗id k,m π π π E M ,∇ E M ,∇ πER∇ E (D∇ M ,∇ πE (ξ )) ∇ M ,∇ E ∇ M ,∇ ⎛E k−1 m .− D M π T∇ M D M π (ξ ) ∇ ,∇ E ⎛ ∇ ,∇ E ⎞⎞⎞ m−2 M,πE ξ ) . +θm−1,1 ⊗ idE D R∇ πE (ξ ) − D∇m−2 M ,∇ πE T∇ M (∇ M,∇ πE ∇

Moreover, in both cases the second term on the right takes values in Sk−1 (T∗ M) ⊗

.

Λ2

(T∗ M) ⊗ Sm−1 (T∗ M) ⊗ E;

in particular, its symmetric part is zero. Proof Write ∇ M,πE ,k (∇ M,πE,m ξ ) = ∇ M,πE ,k+m ξ

.

= ∇ M,πE ,k−1 ∇ M,πE ,2 ∇ M,πE ,m−1 ξ = ∇ M,πE ,k−1 D∇2 M ,∇ πE ∇ M,πE ,m−1 ξ 1 + ∇ M,πE ,k−1 R∇ πE (∇ M,πE ,m−1 ξ ) 2 1 − ∇ M,k−1 T∇ M (∇ M,πE ,m ξ ). 2 (The precise meaning of the final terms will become apparent below when we examine these carefully below.) We first consider the tensor .

Symk ⊗ Symm ⊗ idE (∇ M,πE ,k−1 D∇2 M ,∇ πE ∇ M,πE ,m−1 ξ ).

(3.23)

We note that this tensor is symmetric in its first k entries, its last m entries, and in the “middle” two entries .(k, k + 1). Since permutations of the forms ⎛ .

⎞ ⎛ ⎞ 1 ··· k k + 1 ··· k + m 1 ··· k k + 1 ··· k + m , , σ1 (1) · · · σ1 (k) k + 1 · · · k + m 1 · · · k k + σ2 (1) · · · k + σ2 (m) ⎛ ⎞ 1 ··· k k + 1 ··· k + m , σ1 ∈ Sk , σ2 ∈ Sm , 1 ··· k + 1 k ··· k + m

generate .Sk+m , we conclude that the tensor (3.23) is symmetric. Moreover, we compute, for .v ∈ TM, .

Symk ⊗ Symm ⊗ idE (∇ M,πE ,k−1 D∇2 M ,∇ πE ∇ M,πE ,m−1 ξ )(v, . . . , v) =

1 Σ k!m!

Σ

σ1 ∈Sk σ2 ∈Sm

∇ M,πE ,k−1 D∇2 M ,∇ πE ∇ M,πE ,m−1 ξ(v, . . . , v, v, . . . , v)

134

3 Geometry: Lifts and Differentiation of Tensors

=

Σ 1 2k!m!

Σ

∇ M,πE ,k−1 (∇ M,πE ,2 + pushk,k+1 ◦ ∇ M,πE ,2 )

σ1 ∈Sk σ2 ∈Sm

∇ M,πE ,m−1 ξ(v, . . . , v, v, . . . , v) 1 Σ Σ M,πE ,k+m = ∇ ξ(v, . . . , v) k!m! σ1 ∈Sk σ2 ∈Sm

=∇

M,πE ,k+m

(v, . . . , v) = Symk+m (∇ M,πE ,k+m ξ )(v, . . . , v).

Since the tensors (3.23) and .D∇k+m M ,∇ πE (ξ ) are both symmetric, this is sufficient to conclude that .

Symk ⊗ Symm ⊗ idE (∇ M,πE ,k−1 D∇2 M ,∇ πE ∇ M,πE ,m−1 ξ ) = D∇k+m M ,∇ πE (ξ )

cf. [10, Proposition IV.5.4.3]. Next we consider the tensor .

Symk ⊗ Symm ⊗ idE (∇ M,πE ,k−1 R∇ πE (∇ M,πE ,m−1 ξ ) − ∇ M,k−1 T∇ M (∇ M,πE ,m ξ )). (3.24)

Here we note that .

Symk ⊗ Symm ⊗ idE ⎛ ⎞ ∇ M,πE ,k−1 R∇ πE (∇ M,πE ,m−1 ξ ) − ∇ M,k−1 T∇ M (∇ M,πE ,m ξ ) (v1 , . . . , vk+m ) =

1 k!m! ⎛

Σ

Σ

Σ

Σ

σ1 ∈Sk−1,1 σ1' ∈Sk−1|1 σ2 ∈S1,m−1 σ2' ∈S1|m−1

∇ M,πE ,k−1 R∇ πE (vσ1 ◦ σ1' (k) , vk+σ2 ◦ σ2' (1) )(vσ1 ◦ σ1' (1) , . . . , vσ1 ◦ σ1' (k−1) ) ⎞ (∇ M,πE ,m−1 ξ(vk+σ2 ◦ σ2' (2) , . . . , vk+σ2 ◦ σ2' (m) ))

−

1 k!m! ⎛

Σ

Σ

Σ

Σ

σ1 ∈Sk−1,1 σ1' ∈Sk−1|1 σ2 ∈S1,m−1 σ2' ∈S1|m−1

∇ M,k−1 T∇ M (vσ1 ◦ σ1' (k) , vk+σ2 ◦ σ2' (1) )(vσ1 ◦ σ1' (1) , . . . , vσ1 ◦ σ1' (k−1) ) ⎞ (∇ M,πE ,m−1 ∇ πE ξ )(vk+σ2 ◦ σ2' (2) , . . . , vk+σ2 ◦ σ2' (m) )

3.2 Differentiation of Tensors on the Total Space of a Vector Bundle

1 Σ Σ ⎛ k−1 D∇ M ,∇ πE R∇ πE (vj , vk+l )(v1 , . . . ,ˆ v j , . . . , vk−1 ) km k

=

135

m

j =1 l=1

v k+l , . . . , vm ) (D∇m−1 M ,∇ πE (ξ ))(vk+1 , . . . ,ˆ v j , . . . , vk−1 ) + D∇k−1 M ,∇ πE T∇ M (vj , vk+l )(v1 , . . . ,ˆ

⎞ πE (D∇m−1 (∇ ξ ))(v , . . . ,ˆ v , . . . , v ) k+1 k+l m , M ,∇ πE

where the “hat” over an element of a list means that element is omitted. Combining the preceding two computations, and applying Lemma 3.20 to the second of these gives ⎛ k+m k−1 m−1 m π D∇k,m M ,∇ πE (D∇ M ,∇ πE (ξ ))=D∇ M ,∇ πE (ξ )+θk,m ⊗idE D∇ M ,∇ πE R∇ E (D∇ M ,∇ πE (ξ )) ⎞ m−1 πE − D∇k−1 T (D (∇ ξ )) . (3.25) M π π M ,∇ E ∇ ∇ M ,∇ E

.

πE ξ ) that arises in the Now, for .m ≥ 2, we consider the expression .D∇m−1 M ,∇ πE (∇ last term in the preceding equation. For this, the computations from the preceding two paragraphs simplify to 1 D∇m−1,1 M ,∇ πE (D∇ M ,∇ πE (ξ ))

.

⎛ ⎞ M,πE = D∇mM ,∇ πE (ξ ) + θm−1,1 ⊗ idE D m−2 R∇ πE (ξ ) − D∇m−2 ξ) . M ,∇ πE T∇ M (∇ M,∇ πE ∇

This, incidentally, gives the formula in the first part of the lemma. Additionally, a substitution of this into (3.25) gives the second part of the lemma. .□ We can see from the preceding lemma that, if one works in local coordinates with the canonical flat connections in the resulting trivial bundles, then the lemma simply tells us that “the kth-derivative of the mth-derivative is the .(k + m)th-derivative.” This agrees with calculus, fortunately. The preceding lemma simplifies tremendously when the affine connection .∇ M is torsion-free, and results in an elegant formula simply involving curvature. However, we shall retain torsion since we do not, at this point in our presentation, understand the rôle of differing connections in our characterisations of the seminorms for the real analytic topology. Based on the lemma, we define a vector bundle mapping ιˆE,k,m :

k+m ⊕

.

r=0

∗

S (T M) ⊗ E → r

k ⊕ j =0

∗

S (T M) ⊗ j

⎛ m ⊕ l=0

⎞ ∗

S (T M) ⊗ E l

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3 Geometry: Lifts and Differentiation of Tensors

by ιˆE,k,m (A0 , A1 , . . . , Ak+m ) = ([A0 , A1 , . . . , Am ],

.

[A1 , A2 + θ1,1 ⊗ idE (R∇ πE (A0 ) − T∇ M (A1 )), . . . , Am+1 + θ1,m ⊗ idE (R∇ πE (Am−1 )− m−2 π T∇ M (Am + θm−1,1 ⊗ idE (D∇m−2 M ,∇ πE R∇ E (A0 ) − D∇ M ,∇ πE T∇ M (A1 ))))], . . . , k−1 π [Ak+1 , Ak+2 + θk,1 ⊗ idE (D∇k−1 M ,∇ πE R∇ E (A0 ) − D∇ M T∇ M (A1 )), . . . , π Ak+m + θk,m ⊗ idE (D∇k−1 M ,∇ πE R∇ E (Am−1 )−

m−2 m−2 π D∇k−1 M ,∇ πE T∇ M (Am + θm−1,1 ⊗ idE (D∇ M ,∇ πE R∇ E (A0 ) − D∇ M T∇ M (A1 ))))]).

(Here we break a rule we have about parentheses, using square brackets for some of the groupings for the sake of clarity.) To better understand the meaning of this complicated (though elementary) formula, let us observe that it has been designed so that ιˆE,k,m (ξ(x), D∇1 M ,∇ πE (ξ )(x), . . . , D∇k+m M ,∇ πE (ξ )(x))

.

= ([ξ(x), D∇1 M ,∇ πE (ξ )(x), . . . , D∇mM ,∇ πE (ξ )(x)], m [D∇1,0M ,∇ πE (ξ )(x), D∇1,1M ,∇ πE (D∇1 M ,∇ πE (ξ ))(x), . . . , D∇1,m M ,∇ πE (D∇ M ,∇ πE (ξ ))(x)], . . . , m [D∇k,0M ,∇ πE (ξ )(x), D∇k,1M ,∇ πE (D∇1 M ,∇ πE (ξ ))(x), . . . , D∇k,m M ,∇ πE (D∇ M ,∇ πE (ξ ))(x)]),

(3.26) for .ξ ∈ Г r (E). We now have the following result. Lemma 3.22 (Decomposition of Prolongation of Jet Bundles) Let .r ∈ {∞, ω}, let .πE : E → M be a .Cr -vector bundle, let .∇ M be a .Cr -affine connection on .M, and let .∇ πE be a .Cr -vector bundle connection on .E. Then, for .k, m ∈ Z≥0 , the diagram

.

commutes.

3.3 Isomorphisms Defined by Lifts and Pull-Backs

137

Proof This follows from Lemmata 3.17 and 3.21, taking note of the connection between these lemmata via (3.26). .□

3.3 Isomorphisms Defined by Lifts and Pull-Backs Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. In this section we carefully study isomorphisms that arise from lifts of objects on .M to objects on .E, of the sorts introduced in Sects. 3.1.1, 3.1.2, and 3.1.3. In particular, we shall see that jets of geometric objects can be decomposed (as in Sect. 2.3.1) before or after lifting. We wish here to relate these two sorts of decompositions for all of the lifts we consider in the book. This makes use of our constructions of Sect. 3.2 to give explicit decompositions for jets of certain sections of certain jet bundles on the total space of a vector bundle. Indeed, it is the results in the current section that provide the motivation for the rather intricate constructions of Sect. 3.2. For these constructions, we additionally suppose that we have a .Cr -Riemannian metric .GM on .M and a r M is the Levi-Civita connection for .C -fibre metric .GπE on .E. We suppose that .∇ r π .GM and that we have a .C -linear connection .∇ E in .E. This data gives rise to a Riemannian metric .GE on .E with its Levi-Civita connection .∇ E . We break the discussion into nine cases, the first seven of which correspond to the seven parts of Lemma 3.16. The eighth section provides a construction for pull-backs of functions and the ninth provides constructions involving two different affine connections and two different vector bundle connections. The constructions, statements, and proofs are somewhat repetitive, so we do not provide explicit proofs that are essentially identical to previous proofs. While the results are similar, they are not the same, so we elect to go through all of the cases. There is probably a “meta” result here, but it would take a small journey in itself to setup the framework for this. For our purposes, we stick to a treatment that is concrete at the cost of being dull.

3.3.1 Isomorphisms for Horizontal Lifts of Functions Here we consider the horizontal lift mapping Cr (M) ∋ f ⎬→ πE∗ f ∈ Cr (E).

.

We wish to relate the decomposition associated with the jets of f to those associated with the jets of .πE∗ f . Associated with this, let us denote by .P∗m E the subbundle of ∗m E defined by .RE ⊕ T ∗ m P∗m e E = {jm (πE f )(e) | f ∈ C (M)}.

.

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3 Geometry: Lifts and Differentiation of Tensors

Following Lemma 2.15, our constructions have to do with iterated covariant differentials. The basis of all of our formulae will be a formula for iterated covariant differentials of horizontal lifts of functions on .M. Thus we let .f ∈ C∞ (M) and consider ∇ E,m πE∗ f  ∇ E · · · ∇ E πE∗ f,

.

m ∈ Z>0 .

m times

We state the first two lemmata that we will use. We recall from Lemma 3.16 the definition of .BπE . Lemma 3.23 (Iterated Covariant Differentials of Horizontal Lifts of Functions I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ ∗ m ∗ (Am s , idE ) ∈ VB (T (πE T M); T (T E)),

.

s ∈ {0, 1, . . . , m},

such that E,m ∗ .∇ πE f

=

m Σ

∗ M,s Am f) s (πE ∇

s=0 m m for all .f ∈ Cm (M). Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am satisfy the recursion relations prescribed by

A00 (β0 ) = β0 , A11 (β1 ) = β1 , A10 = 0,

.

and m+1 Am+1 (βm+1 ) = βm+1 ,

.

m Asm+1 (βs ) = (∇ E Am s )(βs ) + As−1 ⊗ idT∗ E (βs )

−

s Σ

Am s ⊗ idT∗ E (Insj (βs , BπE )),

j =1

s ∈ {1, . . . , m}, A0m+1 (β0 )

= (∇ E Am 0 )(β0 ),

where .βs ∈ Ts (πE∗ T∗ M), .s ∈ {0, 1, . . . , m}.

3.3 Isomorphisms Defined by Lifts and Pull-Backs

139

Proof The assertion clearly holds for the initial conditions of the recursion, simply because π ∗ f = π ∗ f,

.

d(π ∗ f ) = π ∗ df + 0f.

So suppose it true for .m ∈ Z>0 . Thus ∇ E,m πE∗ f =

m Σ

.

∗ M,s Am f ), s (πE ∇

s=0

where the vector bundle mappings .Aas , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the recursion relations from the statement of the lemma. Then ∇ E,m+1 πE∗ f =

.

m m Σ Σ ∗ M,s E ∗ M,s (∇ E Am f) + Am f) s )(πE ∇ s ⊗ idT∗ E (∇ πE ∇ s=0

=

s=0

m Σ

m Σ

s=0

s=0

∗ M,s (∇ E Am f) + s )(πE ∇

−

s m Σ Σ

∗ M,s+1 Am f) s ⊗ idT∗ E (πE ∇

∗ M,s Am f, BπE )) s ⊗ idT∗ E (Insj (πE ∇

s=0 j =1

=πE∗ ∇ M,m+1 f +

m Σ

⎛

∗ M,s ∗ M,s ⎝(∇ E Am f ) + Am f) s )(πE ∇ s−1 ⊗ idT∗ E (πE ∇

s=1

−

s Σ

⎞ ∗ M,s ∗ Am f, BπE ))⎠ + (∇ E Am s ⊗ idT∗ E (Insj (πE ∇ 0 )(πE f )

j =1

□

by Lemma 3.16(i). From this, the lemma follows.

.

We shall also need to “invert” the relationship of the preceding lemma. Lemma 3.24 (Iterated Covariant Differentials of Horizontal Lifts of Functions II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings (Bsm , idE ) ∈ VBr (Ts (T∗ E); Tm (πE∗ T∗ M)),

.

s ∈ {0, 1, . . . , m},

140

3 Geometry: Lifts and Differentiation of Tensors

such that m Σ

πE∗ ∇ M,m f =

.

Bsm (∇ E,s πE∗ f )

s=0 m satisfy for all .f ∈ Cm (M). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm the recursion relations prescribed by

B00 (α0 ) = α0 , B11 (α1 ) = α1 , B01 = 0,

.

and m+1 Bm+1 (αm+1 ) = αm+1 ,

.

m Bsm+1 (αs ) = (∇ E Bsm )(αs ) + Bs−1 ⊗ idT∗ E (αs ) +

m Σ

Insj (Bsm (αs ), BπE ),

j =1

s ∈ {1, . . . , m}, B0m+1 (α0 ) = (∇ E B0m )(α0 ) +

m Σ

Insj (B0m (α0 ), BπE ),

j =1

where .αs ∈ Ts (T∗ E), .s ∈ {0, 1, . . . , m}. Proof The assertion clearly holds for the initial conditions for the recursion since π ∗ f = π ∗ f,

.

π ∗ (df ) = d(π ∗ f ) + 0f.

So suppose it true for .m ∈ Z>0 . Thus ∗ M,m .π ∇ f E

=

m Σ

Bsm (∇ E,s πE∗ f ),

(3.27)

s=0

where the vector bundle mappings .Bsa , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the recursion relations from the statement of the lemma. Then, by Lemma 3.16(i), we can work on the left-hand side of (3.27) to give ∇ E πE∗ ∇ M,m f = πE∗ ∇ M,m+1 f −

m Σ

.

Insj (πE∗ ∇ M,m f, BπE )

j =1

= πE∗ ∇ M,m+1 f −

m m Σ Σ s=0 j =1

Insj (Bsm (∇ E,s πE∗ f ), BπE ).

3.3 Isomorphisms Defined by Lifts and Pull-Backs

141

Working on the right-hand side of (3.27) gives ∇ E πE∗ ∇ M,m f =

m Σ

.

∇ E Bsm (∇ E,s πE∗ f ) +

s=0

m Σ

Bsm ⊗ idT∗ E (∇ E,s+1 πE∗ f ).

s=0

Combining the preceding two equations gives πE∗ ∇ M,m+1 f

.

=

m Σ

∇ E Bsm (∇ E,s πE∗ f ) +

s=0

+

m Σ s=0

m m Σ Σ

Insj (Bsm (∇ E,s πE∗ f ), BπE )

s=0 j =1

= ∇ E,m+1 πE∗ f +

m Σ

⎛ m ⎝∇ E Bsm (∇ E,s π ∗ f ) + Bs−1 ⊗ idT∗ E (∇ E,s π ∗ f ) E

s=1

+

Bsm ⊗ idT∗ E (∇ E,s+1 πE∗ f )

m Σ

E

⎞

Insj (Bsm (∇ E,s πE∗ f ), BπE )⎠ +∇ E B0m (πE∗ f )+

j =1

m Σ

Insj (B0m (πE∗ f ), BπE ),

j =1

□

and the lemma follows from this.

.

Next we turn to symmetrised versions of the preceding lemmata. We show that the preceding two lemmata induce corresponding mappings between symmetric tensors. Lemma 3.25 (Iterated Symmetrised Covariant Differentials of Horizontal Lifts of Functions I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ ∗ m ∗ ˆm (A s , idE ) ∈ VB (S (πE T M); S (T E)),

.

s ∈ {0, 1, . . . , m},

such that .

Symm ◦ ∇ E,m πE∗ f =

m Σ s=0

for all .f ∈ Cm (M).

∗ M,s ˆm f) A s (Syms ◦ πE ∇

142

3 Geometry: Lifts and Differentiation of Tensors

Proof We define .Am : T≤m (πE∗ T∗ M) → T≤m (T∗ E) by Am (πE∗ f, πE∗ ∇ M f, . . . , πE∗ ∇ M,m f ) ⎛ ⎞ 1 m Σ Σ 0 ∗ 1 ∗ M,s m ∗ M,s = A0 (πE f ), As (πE ∇ f ), . . . , As (πE ∇ f ) .

.

s=0

s=0

Let us organise the mappings we require into the following diagram:

(3.28)

.

ˆm is defined so that the right square commutes. We shall show that the left Here .A square also commutes. Indeed, ˆm ◦ Sym≤m (π ∗ f, π ∗ ∇ M f, . . . , π ∗ ∇ M,m f ) A E E E

.

= (S∇mE )−1 ◦ (idR ⊕jm πE ) ◦ S∇mM ◦ Sym≤m (πE∗ f, πE∗ ∇ M f, . . . , πE∗ ∇ M,m f ) = Sym≤m (πE∗ f, ∇ E πE∗ f, . . . , ∇ E,m πE∗ f ) = Sym≤m ◦ Am (πE∗ f, πE∗ ∇ M f, . . . , πE∗ ∇ M,m f ). Thus the diagram (3.28) commutes. Now we have ˆm ◦ Sym≤m (π ∗ f, π ∗ ∇ M f, . . . , π ∗ ∇ M,m f ) = A E E E

.

⎛

Sym1 ◦ A00 (πE∗ f ),

1 Σ

Sym2 ◦ A1s (πE∗ ∇ M,s f ), . . . ,

s=0

m Σ

⎞ ∗ M,s Symm ◦ Am f) s (πE ∇

.

s=0

Thus, if we define ∗ M,s ∗ M,s ˆm f ) = Symm ◦ Am f ), A s (Syms ◦ πE ∇ s (πE ∇

.

(3.29)

then we have .

Symm ◦ ∇ E,m πE∗ f =

m Σ

∗ M,s ˆm f ), A s (Syms ◦ πE ∇

s=0

as desired.

□

.

3.3 Isomorphisms Defined by Lifts and Pull-Backs

143

Next we consider the “inverse” of the preceding lemma. Lemma 3.26 (Iterated Symmetrised Covariant Differentials of Horizontal Lifts of Functions II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ m ∗ ∗ ˆm (B s , idE ) ∈ VB (S (T E); S (πE T M)),

.

s ∈ {0, 1, . . . , m},

such that .

Symm ◦ πE∗ ∇ M,m f =

m Σ

E,s ∗ ˆm πE f ) B s (Syms ◦ ∇

s=0

for all .f ∈ Cm (M). Proof We define .B m : T≤m (T∗ E) → T≤m (πE∗ T∗ M) by requiring that B m (πE∗ f, . . . , ∇ E,m πE∗ f ) ⎛ ⎞ 1 m Σ Σ = B00 (πE∗ f ), Bs1 (∇ E,s πE∗ f ), . . . , Bsm (∇ E,m πE∗ f ) ,

.

s=0

(3.30)

s=0

as in Lemma 3.24. Note that the mapping .

idR ⊕jm πE : πE∗ (RM ⊕ T∗m M) → P∗m E

is well-defined and a vector bundle isomorphism. Let us organise the mappings we require into the following diagram:

.

(3.31)

ˆm is defined so that the right square commutes. We shall show that the left Here .B square also commutes. Indeed, ˆm ◦ Sym≤m (π ∗ f, ∇ E π ∗ f, . . . , ∇ E,m π ∗ f ) B E E E

.

= (S∇mM )−1 ◦ (idR ⊕jm πE )−1 ◦ S∇mE ◦ Sym≤m (πE∗ f, ∇ E πE∗ f, . . . , ∇ E,m πE∗ f ) = Sym≤m (πE∗ f, πE∗ ∇ M f, . . . , πE∗ ∇ M,m f ) = Sym≤m ◦ B m (πE∗ f, ∇ E πE∗ f, . . . , ∇ E,m πE∗ f ).

144

3 Geometry: Lifts and Differentiation of Tensors

ˆm Thus the diagram (3.31) commutes. Thus, if we define .B s so as to satisfy E,s ∗ ˆm πE f ) = Symm ◦ Bsm (∇ E,s πE∗ f ), B s (Syms ◦ ∇

.

then we have .

Symm ◦ πE∗ ∇ M,m f =

m Σ

E,s ∗ ˆm πE f ), B s (Syms ◦ ∇

s=0

□

as desired.

.

Remark 3.27 (Nonuniqueness of Inverses) We are being a little sloppy in the preceding lemma, and will be similarly sloppy in subsequent related results. The sloppiness that arises is that the mapping .B m in the lemma is not uniquely defined by the condition (3.30). Indeed, .B m is only uniquely defined on .image(Am ). This can be resolved by giving notation to the vector bundle .image(Am ) and then defining m uniquely on this vector bundle. Alternatively, a vector bundle mapping on .B ≤m ∗ m .image(A ) can be arbitrarily extended to .T (T E) and one can work with this since the conditions defining it only depend on its values on .image(Am ). Thus the sloppiness arises from an unwillingness to introduce even more notation than we already use. This is cleaned up in Lemma 3.28 below. ◦ The following lemma provides two decompositions of .P∗m E, one “downstairs” and one “upstairs,” and the relationship between them. The assertion simply results from an examination of the preceding four lemmata. Lemma 3.28 (Decomposition of Jets of Horizontal Lifts of Functions) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. Then there exist .Cr -vector bundle mappings Am ∈ VBr (P∗m E; S≤m (πE∗ T∗ M)), ∇E

.

B∇mE ∈ VBr (P∗m E; S≤m (T∗ E)),

defined by Am (j (π ∗ f )(e)) = Sym≤m (πE∗ f (e), πE∗ ∇ M f (e), . . . , πE∗ ∇ M,m f (e)), ∇E m E

.

B∇mE (jm (πE∗ f )(e)) = Sym≤m (πE∗ f (e), ∇ E πE∗ f (e), . . . , ∇ E,m πE∗ f (e)).

3.3 Isomorphisms Defined by Lifts and Pull-Backs

145

Moreover, .Am is an isomorphism, .B∇mE is injective, and ∇E B∇mE ◦ (Am )−1 ◦ (Sym≤m (πE∗ f (e), πE∗ ∇ M f (e), . . . , πE∗ ∇ M,m f (e)) ∇E ⎛ 1 Σ ˆ1s (Syms ◦ π ∗ ∇ M,s f (e)), . . . , = A00 (π ∗ f (e)), A

.

E

E

s=0 m Σ

⎞ ∗ M,s ˆm f (e)) A s (Syms ◦ πE ∇

s=0

and m −1 ∗ E ∗ E,m ∗ ◦ Sym≤m (πE f (e), ∇ πE f (e), . . . , ∇ Am ◦ (B E ) πE f (e)) ∇E ∇ ⎛ 1 Σ ˆ1s (Syms ◦ ∇ E,s π ∗ f (e)), . . . , = B00 (π ∗ f (e)), B

.

E

E

s=0 m Σ

⎞ m E,s ∗ ˆ B s (Syms ◦ ∇ πE f (e)) ,

s=0

ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, are as in Lemmata 3.25 and 3.26.

3.3.2 Isomorphisms for Vertical Lifts of Sections Next we consider vertical lifts of sections, i.e., the mapping Г r (E) ∋ ξ ⎬→ ξ v ∈ Г r (TE).

.

We wish to relate the decomposition of the jets of .ξ with those of .ξ v . Associated with this, we denote v m V∗m e E = {jm ξ (e) | ξ ∈ Г (E)}.

.

By (1.9), we have ∗m V∗m e E ≃ Pe E ⊗ Ve E.

.

As with the constructions of the preceding section, we wish to use Lemma 2.15 to provide a decomposition of .V∗m E, and to do so we need to understand the covariant

146

3 Geometry: Lifts and Differentiation of Tensors

derivatives ∇ E,m ξ v  ∇ E · · · ∇ E ξ v ,

.

m ∈ Z≥0 .

m times

In our development, we shall use the notation used in the preceding section in a slightly different, but similar, context. This seems reasonable since we have to do more or less the same thing six times, and using six different pieces of notation will be excessively burdensome. The first result we give is the following. Lemma 3.29 (Iterated Covariant Differentials of Vertical Lifts of Sections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist r .C -vector bundle mappings r s ∗ ∗ m ∗ (Am s , idE ) ∈ VB (T (πE T M) ⊗ VE; T (T E) ⊗ VE),

.

s ∈ {0, 1, . . . , m},

such that ∇

.

E,m v

ξ =

m Σ

M,πE ,s v Am ξ) ) s ((∇

s=0 m m for all .ξ ∈ Г m (E). Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am satisfy 0 the recursion relations prescribed by .A0 (β0 ) = β0 and m+1 Am+1 (βm+1 ) = βm+1 ,

.

m Asm+1 (βs ) = (∇ E Am s )(βs )+As−1 ⊗ idT∗ E (βs )−

s Σ

Am s ⊗ idT∗ E (Insj (βs , BπE ))

j =1

+ Am s

⊗ idT∗ E (Inss+1 (βs , Bπ∗E )),

s ∈ {1, . . . , m},

m ∗ A0m+1 (β0 ) = (∇ E Am 0 )(β0 ) + A0 ⊗ idT∗ E (Ins1 (β0 , BπE )),

where .βs ∈ Ts (πE∗ T∗ M) ⊗ VE, .s ∈ {0, 1, . . . , m + 1}. Proof The assertion clearly holds for .m = 0, so suppose it true for .m ∈ Z>0 . Thus ∇ E,m ξ v =

m Σ

.

s=0

M,πE ,s v Am ξ ) ), s ((∇

3.3 Isomorphisms Defined by Lifts and Pull-Backs

147

where the vector bundle mappings .Aas , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the recursion relations from the statement of the lemma. Then ∇ E,m+1 ξ v =

m Σ

.

M,πE ,s v (∇ E Am ξ) ) + s )((∇

s=0

=

m Σ

E M,πE ,s v Am ξ) ) s ⊗ idT∗ E (∇ (∇

s=0 M,πE ,s v (∇ E Am ξ) ) + s )((∇

s=0

−

m Σ

m Σ

M,πE ,s+1 v Am ξ) ) s ⊗ idT∗ E ((∇

s=0

s m Σ Σ

M,πE ,s v Am ξ ) , BπE )) s ⊗ idT∗ E (Insj ((∇

s=1 j =1

+

m Σ

M,πE ,s v Am ξ ) , Bπ∗E )) s ⊗ idT∗ E (Inss+1 ((∇

s=1 v ∗ + Am 0 ⊗ idT∗ E (Ins1 (ξ , BπE ))

= (∇ M,πE ,m+1 ξ )v ⎛ m Σ M,πE ,s v M,πE ,s v ⎝(∇ E Am + ξ ) ) + Am ξ) ) s )((∇ s−1 ⊗ idT∗ E ((∇ s=1

−

s Σ

M,πE ,s v Am ξ ) , BπE )) s ⊗ idT∗ E (Insj ((∇

j =1

⎞

M,πE ,s v + Am ξ ) , Bπ∗E ))⎠ s ⊗ idT∗ E (Inss+1 ((∇ v m v ∗ + (∇ E Am 0 )(ξ ) + A0 ⊗ idT∗ E (Ins1 (ξ , BπE ))

□

by Lemma 3.16(ii). From this, the lemma follows.

.

Now we “invert” the constructions from the preceding lemma. Lemma 3.30 (Iterated Covariant Differentials of Vertical Lifts of Sections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist r .C -vector bundle mappings (Bsm , idE ) ∈ VBr (Tm (T∗ E) ⊗ VE; Tm (πE∗ T∗ M) ⊗ VE),

.

such that (∇ M,πE ,m ξ )v =

m Σ

.

s=0

Bsm (∇ E,s ξ v )

s ∈ {0, 1, . . . , m},

148

3 Geometry: Lifts and Differentiation of Tensors

m satisfy for all .ξ ∈ Г m (E). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm 0 the recursion relations prescribed by .B0 (α0 ) = α0 and m+1 Bm+1 (αm+1 ) = αm+1 ,

.

Bsm+1 (αs )

= (∇

E

m Bsm )(αs ) + Bs−1

⊗ id

T∗ E

(αs ) +

m Σ

Insj (Bsm (αs ), BπE )

j =1

− Insm+1 (Bsm (αs ), Bπ∗E ), B0m+1 (α0 ) = (∇ E B0m )(α0 ) +

m Σ j =1

s ∈ {1, . . . , m},

Insj (B0m (α0 ), BπE ) − Insm+1 (B0m (α0 ), Bπ∗E ),

where .αs ∈ Ts (T∗ E) ⊗ VE, .s ∈ {0, 1, . . . , m + 1}. Proof The assertion clearly holds for .m = 0, so suppose it true for .m ∈ Z>0 . Thus (∇

.

M,πE ,m

ξ) = v

m Σ

Bsm (∇ E,s ξ v ),

(3.32)

s=0

where the vector bundle mappings .Bsa , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the recursion relations from the statement of the lemma. Then, by Lemma 3.16(ii), we can work on the left-hand side of (3.32) to give ∇ E (∇ M,πE ,m ξ )v = (∇ M,πE ,m+1 ξ )v −

m Σ

.

Insj ((∇ M,πE ,m ξ )v , BπE )

j =1

+ Insm+1 ((∇ M,πE ,m ξ )v , Bπ∗E ) = (∇ M,πE ,m+1 ξ )v −

m m Σ Σ

Insj (Bsm (∇ E,s ξ v ), BπE )

s=0 j =1

+

m Σ

Insm+1 (Bsm (∇ E,s ξ v ), Bπ∗E ).

s=0

Working on the right-hand side of (3.32) gives ∇ E (∇ M,πE ,m ξ )v =

m Σ

.

s=0

∇ E Bsm (∇ E,s ξ v ) +

m Σ s=0

Bsm ⊗ idT∗ E (∇ E,s+1 ξ v ).

3.3 Isomorphisms Defined by Lifts and Pull-Backs

149

Combining the preceding two equations gives ∇ M,πE ,m+1 ξ v =

m Σ

.

∇ E Bsm (∇ E,s ξ v ) +

m Σ

s=0

+

Bsm ⊗ idT∗ E (∇ E,s+1 ξ v )

s=0

m m Σ Σ s=0 j =1

Insj (Bsm (∇ E,s ξ v ), BπE ) − Insm+1 ((∇ M,πE ,m ξ )v , Bπ∗E )

= ∇ E,m+1 ξ v +

m Σ

⎛ m ⎝∇ E Bsm (∇ E,s ξ v ) + Bs−1 ⊗ idT∗ E (∇ E,s ξ v )

s=1

+

m Σ j =1

⎞

Insj (Bsm (∇ E,s ξ v ), BπE ) − Insm+1 (Bsm (∇ E,s ξ v ), Bπ∗E )⎠

+ ∇ E B0m (ξ v ) +

m Σ j =1

Insj (B0m (ξ v ), BπE ) − Insm+1 (B0m (ξ v ), Bπ∗E ), □

and the lemma follows from this.

.

Next we turn to symmetrised versions of the preceding lemmata. We show that the preceding two lemmata induce corresponding mappings between symmetric tensors. Lemma 3.31 (Iterated Symmetrised Covariant Differentials of Vertical Lifts of Sections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ ∗ m ∗ ˆm (A s , idE ) ∈ VB (S (πE T M) ⊗ VE; S (T E) ⊗ VE),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idVE ) ◦ ∇ E,m ξ v =

m Σ

.

M,πE ,s v ˆm ξ) ) A s ((Syms ⊗ idVE ) ◦ (∇

s=0

for all .ξ ∈ Г m (E). Proof The proof follows very similarly to that of Lemma 3.25, but taking the tensor product of everything with .VE. We shall present the complete construction here, but will not repeat it for similar proofs that follow.

150

3 Geometry: Lifts and Differentiation of Tensors

We define .Am : T≤m (πE∗ T∗ M) ⊗ VE → T≤m (T∗ E) ⊗ VE by Am (ξ v , (∇ πE ξ )v , . . . , (∇ M,πE ,m ξ )v ) ⎛ ⎞ 1 m Σ Σ 0 v 1 M,πE ,s v m M,πE ,s v = A0 (ξ ), As ((∇ ξ ) ), . . . , As ((∇ ξ) )

.

s=0

s=0

Let us organise the mappings we require into the following diagram:

.

(3.33) ˆm is defined so that the right square commutes. We shall show that the left Here .A square also commutes. Indeed, ˆm ◦ Sym≤m ⊗ idVE (ξ v , (∇ πE ξ )v , . . . , (∇ M,πE ,m ξ )v ) A

.

= (S∇mE ⊗ idVE )−1 ◦ ((idR ⊗jm πE ) ⊗ idVE ) ◦ (S∇mM ,∇ πE ⊗ idVE ) ◦ (Sym≤m ⊗ idVE )(ξ

v

, (∇ πE ξ )v , . . . , (∇ M,πE ,m ξ )v )

= Sym≤m ⊗ idVE (ξ v , ∇ E ξ v , . . . , ∇ E,m ξ v ) = (Sym≤m ⊗ idVE ) ◦ Am (ξ v , (∇ πE ξ )v , . . . , (∇ M,πE ,m ξ )v ). Thus the diagram (3.33) commutes. Thus, if we define M,πE ,s v M,πE ,s v ˆm ξ ) ) = (Symm ⊗ idVE ) ◦ Am ξ ) ), A s ((Syms ⊗ idVE ) ◦ (∇ s ((∇

.

then we have (Symm ⊗ idVE ) ◦ ∇ E,m ξ v =

m Σ

.

M,πE ,s v ˆm ξ ) ), A s ((Syms ⊗ idVE ) ◦ (∇

s=0

as desired.

□

.

The preceding lemma gives rise to an “inverse,” which we state in the following lemma. Lemma 3.32 (Iterated Symmetrised Covariant Differentials of Vertical Lifts of Sections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings

3.3 Isomorphisms Defined by Lifts and Pull-Backs

151

r s ∗ m ∗ ∗ ˆm (B s , idE ) ∈ VB (S (T E) ⊗ VE; S (πE T M) ⊗ VE),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idVE ) ◦ (∇ M,πE ,m ξ )v =

m Σ

.

E,s v ˆm ξ ) B s ((Syms ⊗ idVE ) ◦ ∇

s=0

for all .ξ ∈ Г m (E). Proof This follows along the lines of Lemma 3.26 in the same manner as .□ Lemma 3.31 follows from Lemma 3.25, by taking tensor products with .VE. We can put together the previous four lemmata into the following decomposition result, which is to be regarded as the main result of this section. Lemma 3.33 (Decomposition of Jets of Vertical Lifts of Sections) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. Then there exist .Cr -vector bundle mappings Am ∈ VBr (P∗m E ⊗ VE; S≤m (πE∗ T∗ M) ⊗ VE), ∇E

.

B∇mE ∈ VBr (P∗m E ⊗ VE; S≤m (T∗ E) ⊗ VE), defined by Am (j (ξ v )(e)) = Sym≤m ⊗ idVE (ξ v (e), (∇ πE ξ )v (e), . . . , (∇ M,πE ,m ξ )v (e)), ∇E m

.

B∇mE (jm (ξ v )(e)) = Sym≤m ⊗ idVE (ξ v (e), ∇ E ξ v (e), . . . , ∇ E,m ξ v (e)). Moreover, .Am is an isomorphism, .B∇mE is injective, and ∇E B∇mE ◦ (Am )−1 ◦ (Sym≤m ⊗ idVE )(ξ v (e), (∇ πE ξ )v (e), . . . , (∇ M,πE ,m ξ )v (e)) ∇E ⎛ 1 Σ v ˆ1s ((Syms ⊗ idVE ) ◦ (∇ M,πE ,s ξ )v (e)), . . . , A = ξ (e),

.

s=0 m Σ s=0

⎞ M,πE ,s v ˆm ξ ) (e)) A s ((Syms ⊗ idVE ) ◦ (∇

152

3 Geometry: Lifts and Differentiation of Tensors

and m −1 v E v E,m v ◦ (Sym≤m ⊗ idVE )(ξ (e), ∇ ξ (e), . . . , ∇ Am ◦ (B E ) ξ (e)) ∇E ∇ ⎛ 1 Σ ˆ1s ((Syms ⊗ idVE ) ◦ ∇ E,s ξ v (e)), . . . , B = ξ v (e),

.

s=0 m Σ

⎞ ˆm B s ((Syms

⊗ idVE ) ◦ ∇

E,s v

ξ (e)) ,

s=0

ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, are as in Lemmata 3.31 and 3.32.

3.3.3 Isomorphisms for Horizontal Lifts of Vector Fields Next we consider horizontal lifts of vector fields via the mapping Г r (TM) ∋ X ⎬→ Xh ∈ Г r (TE).

.

We wish to relate the decomposition of the jets of X with the jets of .Xh . Associated with this, we denote h m H∗m e E = {jm X (e) | X ∈ Г (TM)}.

.

By (1.9), we have ∗m H∗m e E ≃ Pe E ⊗ He E.

.

As with the constructions of the preceding sections, we wish to use Lemma 2.15 to provide a decomposition of .H∗m E, and to do so we need to understand the covariant derivatives ∇ E,m Xh  ∇ E · · · ∇ E Xh ,

.

m ∈ Z≥0 .

m times

In this section we omit proofs, since proofs follow along entirely similar lines to those of the preceding section. The first result we give is the following. Lemma 3.34 (Iterated Covariant Differentials of Horizontal Lifts of Vector Fields I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 ,

3.3 Isomorphisms Defined by Lifts and Pull-Backs

153

there exist .Cr -vector bundle mappings r s ∗ ∗ m ∗ (Am s , idE ) ∈ VB (T (πE T M) ⊗ HE; T (T E) ⊗ HE),

.

s ∈ {0, 1, . . . , m},

such that ∇ E,m Xh =

m Σ

.

M,s Am X)h ) s ((∇

s=0 m m for all .X ∈ Г m (TM). Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am 0 satisfy the recursion relations prescribed by .A0 (β0 ) = β0 and m+1 Am+1 (βm+1 ) = βm+1 ,

.

m Asm+1 (βs ) = (∇ E Am s )(βs ) + As−1 ⊗ idT∗ E (βs )

−

s Σ

Am s ⊗ idT∗ E (Insj (βs , BπE ))

j =1 ∗ + Am s ⊗ idT∗ E (Inss+1 (βs , BπE )), s ∈ {1, . . . , m}, m ∗ A0m+1 (β0 ) = (∇ E Am 0 )(β0 ) + A0 ⊗ idT∗ E (Ins1 (β0 , BπE )),

where .βs ∈ Ts (πE∗ T∗ M) ⊗ HE, .s ∈ {0, 1, . . . , m + 1}. Proof This follows in the same manner as Lemma 3.29, making use of Lemma 3.16(iii). .□ The following lemma “inverts” the relations from the preceding one. Lemma 3.35 (Iterated Covariant Differentials of Horizontal Lifts of Vector Fields II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings (Bsm , idE ) ∈ VBr (Ts (T∗ E) ⊗ HE; Tm (πE∗ T∗ M) ⊗ HE),

.

such that (∇ M,m X)h =

m Σ

.

s=0

Bsm (∇ E,s Xh )

s ∈ {0, 1, . . . , m},

154

3 Geometry: Lifts and Differentiation of Tensors

m for all .X ∈ Г m (TM). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm 0 satisfy the recursion relations prescribed by .B0 (α0 ) = α0 and m+1 Bm+1 (αm+1 ) = αm+1 ,

.

Bsm+1 (αs )

= (∇

E

m Bsm )(αs ) + Bs−1

⊗ id

T∗ E

(αs ) +

m Σ

Insj (Bsm (αs ), BπE )

j =1

− Insm+1 (Bsm (αs ), Bπ∗E ), B0m+1 (α0 ) = (∇ E B0m )(α0 ) +

m Σ j =1

s ∈ {1, . . . , m},

Insj (B0m (α0 ), BπE ) − Insm+1 (B0m (α0 ), Bπ∗E ),

where .αs ∈ Ts (T∗ E) ⊗ HE, .s ∈ {0, 1, . . . , m + 1}. Proof This follows in the same manner as Lemma 3.30, making use of Lemma 3.16(iii). .□ Now we can give the symmetrised versions of the preceding lemmata. Lemma 3.36 (Iterated Symmetrised Covariant Differentials of Horizontal Lifts of Vector Fields I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ ∗ m ∗ ˆm (A s , idE ) ∈ VB (S (πE T M) ⊗ HE; S (T E) ⊗ HE),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idHE ) ◦ ∇ E,m Xh =

m Σ

.

M,s ˆm X)h ) A s ((Syms ⊗ idHE ) ◦ (∇

s=0

for all .X ∈ Г m (TM). Proof This follows along the lines of Lemma 3.25 in the same manner as Lemma 3.31 follows from Lemma 3.25, by taking tensor products with .HE. .□ Lemma 3.37 (Iterated Symmetrised Covariant Differentials of Horizontal Lifts of Vector Fields II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ m ∗ ∗ ˆm (B s , idE ) ∈ VB (S (T E) ⊗ HE; S (πE T M) ⊗ HE),

.

s ∈ {0, 1, . . . , m},

3.3 Isomorphisms Defined by Lifts and Pull-Backs

155

such that (Symm ⊗ idHE ) ◦ (∇ M,m X)h =

m Σ

.

E,s h ˆm X ) B s ((Syms ⊗ idHE ) ◦ ∇

s=0

for all .X ∈ Г m (TM). Proof This follows along the lines of Lemma 3.26 in the same manner as Lemma 3.31 follows from Lemma 3.25, by taking tensor products with .HE. .□ We can put together the previous four lemmata into the following decomposition result, which is to be regarded as the main result of this section. Lemma 3.38 (Decomposition of Jets of Horizontal Lifts of Vector Fields) Let r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. Then there exist .Cr -vector bundle mappings

.

Am ∈ VBr (P∗m E ⊗ HE; S≤m (πE∗ T∗ M) ⊗ HE), ∇E

.

B∇mE ∈ VBr (P∗m E ⊗ HE; S≤m (T∗ E) ⊗ HE), defined by Am (j (Xh )(e)) = Sym≤m ⊗ idHE (Xh (e), (∇ M X)h (e), . . . , (∇ M,m X)h (e)), ∇E m

.

B∇mE (jm (Xh )(e)) = Sym≤m ⊗ idHE (Xh (e), ∇ E Xh (e), . . . , ∇ E,m Xh (e)). Moreover, .Am is an isomorphism, .B∇mE is injective, and ∇E B∇mE ◦ (Am )−1 ◦ (Sym≤m ⊗ idHE )(Xh (e), (∇ M X)h (e), . . . , (∇ M,m X)h (e)) ∇E ⎛ 1 Σ ˆ1s ((Syms ⊗ idHE ) ◦ (∇ M,s X)(e)), . . . , = Xh (e), A

.

s=0 m Σ s=0

⎞ M,s ˆm X)h (e)) A s ((Syms ⊗ idHE ) ◦ (∇

156

3 Geometry: Lifts and Differentiation of Tensors

and m −1 h E h E,m h ◦ (Sym≤m ⊗ idHE )(X (e), ∇ X (e), . . . , ∇ Am ◦ (B E ) X (e)) ∇E ∇ ⎛ 1 Σ ˆ1s ((Syms ⊗ idHE ) ◦ ∇ E,s Xh (e)), . . . , B = Xh (e),

.

s=0 m Σ

⎞ ˆm B s ((Syms

⊗ idHE ) ◦ ∇

E,s

h

X (e)) ,

s=0

ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, are as in Lemmata 3.36 and 3.37.

3.3.4 Isomorphisms for Vertical Lifts of Dual Sections Next we consider vertical lifts of sections of the dual bundle, i.e., the mapping defined by Г r (E∗ ) ∋ λ ⎬→ λv ∈ Г r (T∗ E).

.

Our objective is to relate the decomposition of the jets of .λ with the decomposition of the jets of .λv . To do this, we denote v m ∗ F∗m e E = {jm λ (e) | λ ∈ Г (E )}.

.

By (1.9), we have ∗m ∗ F∗m e E ≃ Pe E ⊗ Ve E.

.

As with the constructions of the preceding sections, we wish to use Lemma 2.15 to provide a decomposition of .F∗m E, and to do so we need to understand the covariant derivatives ∇ E,m λv  ∇ E · · · ∇ E λv ,

.

m ∈ Z≥0 .

m times

In this section we omit proofs, since proofs follow along entirely similar lines to those of preceding sections. The first result we give is the following. Lemma 3.39 (Iterated Covariant Differentials of Vertical Lifts of Dual Sections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data

3.3 Isomorphisms Defined by Lifts and Pull-Backs

157

prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ ∗ ∗ m ∗ ∗ (Am s , idE ) ∈ VB (T (πE T M) ⊗ V E; T (T E) ⊗ V E),

.

s ∈ {0, 1, . . . , m},

such that ∇ E,m λv =

m Σ

.

M,πE ,s v Am λ) ) s ((∇

s=0 m m for all .λ ∈ Г m (E∗ ). Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am satisfy 0 the recursion relations prescribed by .A0 (β0 ) = β0 and m+1 Am+1 (βm+1 ) = βm+1 ,

.

m Asm+1 (βs ) = (∇ E Am s )(βs ) + As−1 ⊗ idT∗ E (βs )

−

s Σ

Am s ⊗ idT∗ E (Insj (βs , BπE )),

j =1

s ∈ {1, . . . , m}, m A0m+1 (β0 ) = (∇ E Am 0 )(β0 ) − A0 ⊗ idT∗ E (Ins1 (β0 , BπE )),

where .βs ∈ Ts (πE∗ T∗ M) ⊗ V∗ E, .s ∈ {0, 1, . . . , m + 1}. Proof This follows in the same manner as Lemma 3.29, making use of .□ Lemma 3.16(iv). The “inverse” of the preceding lemma is as follows. Lemma 3.40 (Iterated Covariant Differentials of Vertical Lifts of Dual Sections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings (Bsm , idE ) ∈ VBr (Ts (T∗ E) ⊗ V∗ E; Tm (πE∗ T∗ M) ⊗ V∗ E),

.

such that (∇ M,πE ,m λ)v =

m Σ

.

s=0

Bsm (∇ E,s λv )

s ∈ {0, 1, . . . , m},

158

3 Geometry: Lifts and Differentiation of Tensors

m satisfy for all .λ ∈ Г m (E∗ ). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm 0 the recursion relations prescribed by .B0 (α0 ) = α0 and m+1 Bm+1 (αm+1 ) = αm+1 ,

.

Bsm+1 (αs )

= (∇

E

m Bsm )(αs ) + Bs−1

⊗ id

T∗ E

(αs ) +

m Σ

Insj (Bsm (αs ), BπE ),

j =1

s ∈ {1, . . . , m}, B0m+1 (α0 )

= (∇

E

B0m )(α0 ) +

m+1 Σ

Insj (B0m (α0 ), BπE ),

j =1

where .αs ∈ Ts (T∗ E) ⊗ V∗ E, .s ∈ {0, 1, . . . , m + 1}. Proof This follows in the same manner as Lemma 3.30, making use of .□ Lemma 3.16(iv). Next we turn to symmetrised versions of the preceding lemmata. We show that the preceding two lemmata induce corresponding mappings between symmetric tensors. Lemma 3.41 (Iterated Symmetrised Covariant Differentials of Vertical Lifts of Dual Sections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ ∗ ∗ m ∗ ∗ ˆm (A s , idE ) ∈ VB (S (πE T M) ⊗ V E; S (T E) ⊗ V E),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idV∗ E ) ◦ ∇ E,m λv =

m Σ

.

M,πE ,s v ˆm λ) ) A s ((Syms ⊗ idV∗ E ) ◦ (∇

s=0

for all .λ ∈ Г m (E∗ ). Proof This follows along the lines of Lemma 3.25 in the same manner as .□ Lemma 3.31 follows from Lemma 3.25, by taking tensor products with .V∗ E. The preceding lemma gives rise to an “inverse,” which we state in the following lemma. Lemma 3.42 (Iterated Symmetrised Covariant Differentials of Vertical Lifts of Dual Sections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E.

3.3 Isomorphisms Defined by Lifts and Pull-Backs

159

For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ ∗ m ∗ ∗ ∗ ˆm (B s , idE ) ∈ VB (S (T E) ⊗ V E; S (πE T M) ⊗ V E),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idV∗ E ) ◦ (∇ M,πE ,m λ)v =

m Σ

.

E,s v ˆm λ ) B s ((Syms ⊗ idV∗ E ) ◦ ∇

s=0

for all .λ ∈ Г m (E∗ ). Proof This follows along the lines of Lemma 3.26 in the same manner as .□ Lemma 3.31 follows from Lemma 3.25, by taking tensor products with .V∗ E. We can put together the previous four lemmata into the following decomposition result, which is to be regarded as the main result of this section. Lemma 3.43 (Decomposition of Jets of Vertical Lifts of Dual Sections) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. Then there exist .Cr -vector bundle mappings Am ∈ VBr (P∗m E ⊗ V∗ E; S≤m (πE∗ T∗ M) ⊗ V∗ E), ∇E

.

B∇mE ∈ VBr (P∗m E ⊗ V∗ E; S≤m (T∗ E) ⊗ V∗ E), defined by Am (j (λv )(e)) = Sym≤m ⊗ idV∗ E (λv (e), (∇ πE λ)v (e), . . . , (∇ M,πE ,m λ)v (e)), ∇E m

.

B∇mE (jm (λv )(e)) = Sym≤m ⊗ idV∗ E (λv (e), ∇ E λv (e), . . . , ∇ E,m λv (e)). Moreover, .Am is an isomorphism, .B∇mE is injective, and ∇E B∇mE ◦ (Am )−1 ◦ (Sym≤m ⊗ idV∗ E )(λv (e), (∇ πE λ)v (e), . . . , (∇ M,πE ,m λ)v (e)) = ∇E ⎛ 1 Σ ˆ1s ((Syms ⊗ idV∗ E ) ◦ (∇ M,πE ,s λ)v (e)), . . . , λv (e), A

.

s=0 m Σ s=0

⎞ M,πE ,s v ˆm λ) (e)) A s ((Syms ⊗ idV∗ E ) ◦ (∇

160

3 Geometry: Lifts and Differentiation of Tensors

and m −1 v E v E,m v ◦ (Sym≤m ⊗ idV∗ E )(λ (e), ∇ λ (e), . . . , ∇ Am ◦ (B E ) λ (e)) ∇E ∇ ⎛ 1 Σ ˆ1s ((Syms ⊗ idV∗ E ) ◦ ∇ E,s λv (e)), . . . , B = λv (e),

.

s=0 m Σ

⎞ ˆm B s ((Syms

⊗ idV∗ E ) ◦ ∇

E,s v

λ (e)) ,

s=0

ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, are as in Lemmata 3.41 and 3.42.

3.3.5 Isomorphisms for Vertical Lifts of Endomorphisms Next we consider vertical lifts of endomorphisms defined by the mapping Г r (T11 (E)) ∋ L ⎬→ Lv ∈ Г r (T11 (TE)).

.

We wish to relate the decomposition of the jets of L with those of .Lv . Associated with this, we denote v m 1 L∗m e E = {jm L (e) | L ∈ Г (T1 (E))}.

.

By (1.9), we have ∗m 1 L∗m e E ≃ Pe E ⊗ T1 (Ve E).

.

As with the constructions of the preceding sections, we wish to use Lemma 2.15 to provide a decomposition of .L∗m E, and to do so we need to understand the covariant derivatives ∇ E,m Lv  ∇ E · · · ∇ E Lv ,

.

m ∈ Z≥0 .

m times

In this section we omit proofs, since proofs follow along entirely similar lines to those of preceding sections. The first result we give is the following. Lemma 3.44 (Iterated Covariant Differentials of Vertical Lifts of Endomorphisms I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 ,

3.3 Isomorphisms Defined by Lifts and Pull-Backs

161

there exist .Cr -vector bundle mappings r s ∗ ∗ 1 m ∗ 1 (Am s , idE ) ∈ VB (T (πE T M)⊗T1 (VE); T (T E)⊗T1 (VE)),

.

s ∈ {0, 1, . . . , m},

such that ∇ E,m Lv =

m Σ

.

M,πE ,s Am L)v ) s ((∇

s=0 m m for all .L ∈ Г m (T11 (E)). Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am satisfy the recursion relations prescribed by .A00 (β0 ) = β0 and m+1 Am+1 (βm+1 ) = βm+1 ,

.

m Asm+1 (βs ) = (∇ E Am s )(βs )+As−1 ⊗ idT∗ E (βs )−

s Σ

Am s ⊗ idT∗ E (Insj (βs , BπE ))

j =1 ∗ + Am s ⊗ idT∗ E (Inss+1 (βs , BπE )), s ∈ {1, . . . , m}, m A0m+1 (β0 ) = (∇ E Am 0 )(β0 ) − A0 ⊗ idT∗ E (Ins1 (β0 , BπE )) ∗ + Am 0 ⊗ idT∗ E (Ins2 (β0 , BπE )),

where .βs ∈ Ts (πE∗ T∗ M) ⊗ T11 (VE), .s ∈ {0, 1, . . . , m + 1}. Proof This follows in the same manner as Lemma 3.29, making use of Lemma 3.16(v). .□ The “inverse” of the preceding lemma is as follows. Lemma 3.45 (Iterated Covariant Differentials of Vertical Lifts of Endomorphisms II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings (Bsm , idE ) ∈ VBr (Ts (T∗ E)⊗T11 (VE); Tm (πE∗ T∗ M)⊗T11 (VE)),

.

such that (∇ M,πE ,m L)v =

m Σ

.

s=0

Bsm (∇ E,s Lv )

s ∈ {0, 1, . . . , m},

162

3 Geometry: Lifts and Differentiation of Tensors

m for all .L ∈ Г m (T11 (E)). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm satisfy the recursion relations prescribed by .B00 (α0 ) = α0 and m+1 Bm+1 (αm+1 ) = αm+1 ,

.

m Bsm+1 (αs ) = (∇ E Bsm )(αs ) + Bs−1 ⊗ idT∗ E (αs ) +

m Σ

Insj (Bsm (αs ), BπE )

j =1

− Insm+1 (Bsm (αs ), Bπ∗E ), s ∈ {1, . . . , m}, B0m+1 (α0 ) = (∇ E B0m )(α0 ) +

m Σ

Insj (B0m (α0 ), BπE ) − Insm+1 (B0m (α0 ), BπE ),

j =1

where .αs ∈ Ts (T∗ E) ⊗ T11 (VE), .s ∈ {0, 1, . . . , m + 1}. Proof This follows in the same manner as Lemma 3.30, making use of .□ Lemma 3.16(v). Next we turn to symmetrised versions of the preceding lemmata. We show that the preceding two lemmata induce corresponding mappings between symmetric tensors. Lemma 3.46 (Iterated Symmetrised Covariant Differentials of Vertical Lifts of Endomorphisms I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ ∗ 1 m ∗ 1 ˆm (A s , idE ) ∈ VB (S (πE T M)⊗T1 (VE); S (T E)⊗T1 (VE)),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idT1 (VE) ) ◦ ∇ E,m Lv =

m Σ

.

1

M,πE ,s ˆm L)v ) A s ((Syms ⊗ idT1 (VE) ) ◦ (∇ 1

s=0

for all .L ∈ Г m (T11 (E)). Proof This follows along the lines of Lemma 3.25 in the same manner as Lemma 3.31 follows from Lemma 3.25, by taking tensor products with .T11 (VE). .□ The preceding lemma gives rise to an “inverse,” which we state in the following lemma. Lemma 3.47 (Iterated Symmetrised Covariant Differentials of Vertical Lifts of Endomorphisms II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For

3.3 Isomorphisms Defined by Lifts and Pull-Backs

163

m ∈ Z≥0 , there exist .Cr -vector bundle mappings

.

r s ∗ 1 m ∗ ∗ 1 ˆm (B s , idE ) ∈ VB (S (T E)⊗T1 (VE); S (πE T M)⊗T1 (VE)),

s ∈ {0, 1, . . . , m},

.

such that (Symm ⊗ idT1 (VE) ) ◦ (∇ M,πE ,m L)v =

m Σ

.

1

E,s v ˆm L ) B s ((Syms ⊗ idT1 (VE) ) ◦ ∇ 1

s=0

for all .L ∈ Г m (T11 (E)). Proof This follows along the lines of Lemma 3.26 in the same manner as Lemma 3.31 follows from Lemma 3.25, by taking tensor products with .T11 (VE). .□ We can put together the previous four lemmata into the following decomposition result, which is to be regarded as the main result of this section. Lemma 3.48 (Decomposition of Jets of Vertical Lifts of Endomorphisms) Let r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. Then there exist .Cr -vector bundle mappings

.

Am ∈ VBr (P∗m E ⊗ T11 (VE); S≤m (πE∗ T∗ M) ⊗ T11 (VE)), ∇E

.

B∇mE ∈ VBr (P∗m E ⊗ T11 (VE); S≤m (T∗ E) ⊗ T11 (VE)), defined by Am (j (Lv )(e)) = Sym≤m ⊗ idT1 (VE) (Lv (e), (∇ πE L)v (e), . . . , (∇ M,πE ,m L)v (e)), ∇E m

.

1

B∇mE (jm (Lv )(e))

= Sym≤m ⊗ idT1 (VE) (Lv (e), ∇ E Lv (e), . . . , ∇ E,m Lv (e)). 1

Moreover, .Am is an isomorphism, .B∇mE is injective, and ∇E B∇mE ◦ (Am )−1 ◦ (Sym≤m ⊗ idT1 (VE) )(Lv (e), (∇ πE L)v (e), . . . , (∇ M,πE ,m L)v (e)) ∇E 1 ⎛ 1 Σ ˆ1s ((Syms ⊗ id 1 = Lv (e), ) ◦ (∇ M,πE ,s L)v (e)), . . . , A

.

T1 (VE)

s=0 m Σ

⎞ M,πE ,s ˆm L)v (e)) A s ((Syms ⊗ idT1 (VE) ) ◦ (∇ 1

s=0

164

3 Geometry: Lifts and Differentiation of Tensors

and m −1 v E v E,m v ◦ (Sym≤m ⊗ idT1 (VE) )(L (e), ∇ L (e), . . . , ∇ Am ◦ (B E ) L (e)) ∇E ∇ 1 ⎛ 1 Σ ˆ1s ((Syms ⊗ id 1 = Lv (e), ) ◦ ∇ E,s Lv (e)), . . . , B

.

T1 (VE)

s=0 m Σ

⎞ E,s v ˆm L (e)) , B s ((Syms ⊗ idT1 (VE) ) ◦ ∇ 1

s=0

ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, are as in Lemmata 3.46 and 3.47.

3.3.6 Isomorphisms for Vertical Evaluations of Dual Sections Next we consider vertical evaluations of endomorphisms given by the mapping Г r (E∗ ) ∋ λ ⎬→ λe ∈ Cr (E).

.

To study the relationship between the decomposition of the jets of .λ with those of the jets of .λe , we denote e m ∗ D∗m e E = {jm λ (e) | λ ∈ Г (E )}.

.

By (1.9), we have ∗m D∗m e E ⊆ Pe E.

.

As we shall see, one can be a little more explicit about the nature of .D∗m e E, and see that ∗m ∗ ∗E ∗ D∗m e E ≃ (Pe E ⊗ V E) ⊕ (Pm−1 ⊗ V E).

.

However, this sort of isomorphism is too cumbersome to make explicit, and so we will just keep the notation .D∗m e E. As with the constructions of the preceding sections, we wish to use Lemma 2.15 to provide a decomposition of .D∗m E, and to do so we need to understand the covariant derivatives ∇ E,m λe  ∇ E · · · ∇ E λe ,

.

m times

m ∈ Z≥0 .

3.3 Isomorphisms Defined by Lifts and Pull-Backs

165

The results in this section have a slightly different character than in the preceding sections. We will not give the complete proofs, but will note that they are similar to the complete proofs given in the next section. Our first result is the following. Lemma 3.49 (Iterated Covariant Differentials of Vertical Evaluations of Dual Sections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ ∗ m ∗ (Am s , idE ) ∈ VB (T (πE T M); T (T E)),

.

s ∈ {0, 1, . . . , m},

and (Csm , idE ) ∈ VBr (Ts (πE∗ T∗ M)⊗V∗ E; Tm−1 (T∗ E)⊗V∗ E),

.

s ∈ {0, 1, . . . , m−1},

such that ∇ E,m λe =

m Σ

.

M,πE ,s e Am λ) ) + s ((∇

s=0

m−1 Σ

Csm ((∇ M,πE ,s λ)v )

s=0

for all .λ ∈ Г m (E∗ ) (Here we regard .V∗ E as a subbundle of .T∗ E). Moreover, m m m m m the vector bundle mappings .Am 0 , A1 , . . . , Am and .C0 , C1 , . . . , Cm−1 satisfy the recursion relations prescribed by A00 (β0 ) = β0 , A11 (β1 ) = β1 , A10 (β0 ) = Ins1 (β0 , BπE ), C01 (γ0 ) = γ0 ,

.

and, for .m ≥ 2, m+1 Am+1 (βm+1 ) = βm+1

.

m+1 Am (βm )

=

Am m−1

⊗ id

T∗ E

(βm ) −

m Σ

Insj (βm , BπE )

j =1 m Asm+1 (βs ) = (∇ E Am s )(βm ) + As−1 ⊗ idT∗ E (βs )

−

s Σ

Am s ⊗ idT∗ E (Insj (βs , BπE )),

j =1

s ∈ {1, . . . , m − 1}, A0m+1 (β0 ) = (∇ E Am 0 )(β0 )

166

3 Geometry: Lifts and Differentiation of Tensors

and m+1 m Cm (γm ) = Cm−1 ⊗ idT∗ E (γm ) + γm

.

E m m Csm+1 (γs ) = Am s ⊗ idT∗ E (γs ) + (∇ Cs )(γs ) + Cs−1 ⊗ idT∗ E (γs )

−

s+1 Σ

Csm ⊗ idT∗ E (Insj (γs , BπE )), s ∈ {1, . . . , m − 1},

j =1 E m m C0m+1 (γ0 ) = Am 0 ⊗ idT∗ E (γ0 ) + (∇ C0 )(γ0 ) − C0 ⊗ idT∗ E (Ins1 (γ0 , BπE )),

where .βs ∈ Ts (πE∗ T∗ M), .s ∈ {0, 1, . . . , m + 1}, and .γs ∈ Ts (πE∗ T∗ M) ⊗ V∗ E, .s ∈ {0, 1, . . . , m − 1}. Proof This follows in the same manner as Lemma 3.54 below, making use of Lemma 3.16(vi). .□ Now we “invert” the constructions from the preceding lemma. Lemma 3.50 (Iterated Covariant Differentials of Vertical Evaluations of Dual Sections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings (Bsm , idE ) ∈ VBr (Ts (T∗ E); Tm (πE∗ T∗ M)),

s ∈ {0, 1, . . . , m},

.

and (Dsm , idE ) ∈ VBr (Ts (T∗ E)⊗V∗ E; Tm−1 (πE∗ T∗ M)⊗V∗ E),

.

s ∈ {0, 1, . . . , m−1},

such that (∇ M,πE ,m λ)e =

m Σ

.

Bsm (∇ E,s λe ) +

s=0

m−1 Σ

Dsm (∇ E,s λv )

s=0

m and for all .λ ∈ Г m (E∗ ). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm m m m 0 .D , D , . . . , D 0 1 m−1 satisfy the recursion relations prescribed by .B0 (α0 ) = α0 , 1 .D (γ0 ) = γ0 , 0 m+1 Bm+1 (αm+1 ) = αm+1

.

m+1 m Bm (αm ) = Bm−1 ⊗ idT∗ E (αm ) +

m Σ j =1

Insj (αm , BπE ) − Insm+1 (αm , Bπ∗E )

3.3 Isomorphisms Defined by Lifts and Pull-Backs

m Bsm+1 = (∇ E Bsm )(αs ) + Bs−1 ⊗ idT∗ E (αs ) +

167 m Σ

Insj (Bsm (αs ), BπE )

j =1

− Insm+1 (Bsm (αs ), Bπ∗E ), s ∈ {1, . . . , m − 1}, B0m+1 (α0 ) = (∇ E B0m )(α0 ) +

m Σ j =1

Insj (B0m (α0 ), BπE ) − Insm+1 (B0m (α0 ), Bπ∗E )

and m+1 m Dm (γm ) = Dm−1 ⊗ idT∗ E (γm ) − γm

.

m Dsm (γs ) = (∇ E Dsm )(γs ) + Ds−1 ⊗ idT∗ E (γs ) − B m s (γs ), s ∈ {1, . . . , m − 1},

D0m+1 = (∇ E D0m )(γ0 ) − B m 0 (γ0 ) for .αs ∈ Ts (T∗ E), .s ∈ {0, 1, . . . , m}, and .γs ∈ Ts (T∗ E) ⊗ V∗ E, .s ∈ {0, 1, . . . , m − 1}, and where r s ∗ ∗ m ∗ ∗ ∗ (B m s , idE ) ∈ VB (T (T E) ⊗ V E; T (πE T M) ⊗ V E),

.

s ∈ {0, 1, . . . , m},

are the vector bundle mappings from Lemma 3.40. Proof This follows in the same manner as Lemma 3.55 below, making use of Lemma 3.16(vi). .□ Next we turn to symmetrised versions of the preceding lemmata. We show that the preceding two lemmata induce corresponding mappings between symmetric tensors. Lemma 3.51 (Iterated Symmetrised Covariant Differentials of Vertical Evaluations of Dual Sections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ ∗ m ∗ ˆm (A s , idE ) ∈ VB (S (πE T M); S (T E)),

.

s ∈ {0, 1, . . . , m},

and r s ∗ ∗ ∗ m ∗ ˆm (C s , idE ) ∈ VB (S (πE T M) ⊗ V E; S (T E)),

.

s ∈ {0, 1, . . . , m − 1},

168

3 Geometry: Lifts and Differentiation of Tensors

such that Symm ◦ ∇

.

E,m e

λ =

m Σ

M,πE ,s e ˆm λ) ) A s (Syms ◦ (∇

s=0

+

m−1 Σ

M,πE ,s v ˆm λ) ) C s ((Syms ⊗ idV∗ E ) ◦ (∇

s=0

for all .λ ∈ Г m (E∗ ). Proof The proof here follows along the lines of Lemma 3.56 below.

□

.

The preceding lemma gives rise to an “inverse,” which we state in the following lemma. Lemma 3.52 (Iterated Symmetrised Covariant Differentials of Vertical Evaluations of Dual Sections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ m ∗ ∗ ˆm (B s , idE ) ∈ VB (S (T E); S (πE T M)),

.

s ∈ {0, 1, . . . , m},

and r s ∗ ∗ m ∗ ∗ ˆm (D s , idE ) ∈ VB (S (T E) ⊗ V E; S (πE T M)),

.

s ∈ {0, 1, . . . , m − 1},

such that

.

Symm ◦ (∇ M,πE ,m λ)e =

m Σ

E,s e ˆm λ) B s (Syms ◦ ∇

s=0

+

m−1 Σ

E,s v ˆm λ ) D s ((Syms ⊗ idV∗ E ) ◦ ∇

s=0

for all .λ ∈ Г m (E∗ ). Proof The proof here follows along the lines of Lemma 3.56 below.

□

.

We can put together the previous four lemmata, along with Lemma 3.48, into the following decomposition result, which is to be regarded as the main result of this section. Lemma 3.53 (Decomposition of Jets of Vertical Evaluations of Dual Sections) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. Then there exist .Cr -vector

3.3 Isomorphisms Defined by Lifts and Pull-Backs

169

bundle mappings Am ∈ VBr (D∗m E; S≤m (πE∗ T∗ M)), ∇E

.

B∇mE ∈ VBr (D∗m E; S≤m (T∗ E))

defined by Am (j (λe )(e)) = Sym≤m (λe (e), (∇ πE λ)e (e), . . . , (∇ M,πE ,m λ)e (e)), ∇E m

.

B∇mE (jm (λe )(e)) = Sym≤m (λe (e), ∇ E λe (e), . . . , ∇ E,m λe (e)). Moreover, .Am and .B∇mE are injective, and ∇E B∇mE ◦ (Am )−1 ◦ Sym≤m (λe (e), (∇ πE λ)e (e), . . . , (∇ M,πE ,m λ)e (e)) ∇E ⎞ ⎛ 1 m Σ Σ M,πE ,s e ˆ1s (Syms ◦ (∇ M,πE ,s λ)e (e)), . . . , ˆm λ) (e)) = λe (e), A A s (Syms ◦ (∇ .

s=0

s=0

⎛

+ 0, λv (e),

1 Σ

ˆ2s ((Syms ⊗ idV∗ E ) ◦ (∇ M,πE ,s λ)v (e)), . . . , C

s=0 m−1 Σ

⎞ ˆm C s ((Syms

⊗ idV∗ E ) ◦ (∇

M,πE ,s

v

λ) (e))

s=0

and m −1 e E e E,m e ◦ Sym≤m (λ (e), ∇ λ (e), . . . , ∇ Am ◦ (B E ) λ (e)) ∇E ∇ ⎞ ⎛ 1 m Σ Σ E,s e ˆ1s (Syms ◦ ∇ E,s λe (e)), . . . , ˆm λ (e)) B B = λe (e), s (Syms ◦ ∇

.

s=0

⎛

+ 0, λv (e),

s=0 1 Σ

ˆ2s ((Syms ⊗ idV∗ E ) ◦ ∇ E,s λv (e)), . . . , D

s=0 m−1 Σ

⎞ ˆm D s ((Syms

⊗ idV∗ E ) ◦ ∇

E,s v

λ (e)) ,

s=0

ˆm ˆm ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, and .C s and .D s , .s ∈ {0, 1, . . . , m − 1}, are as in Lemmata 3.51 and 3.52.

170

3 Geometry: Lifts and Differentiation of Tensors

3.3.7 Isomorphisms for Vertical Evaluations of Endomorphisms Next we consider vertical evaluations of endomorphisms, i.e., the mapping given by Г r (T11 (E)) ∋ L ⎬→ Le ∈ Г r (TE).

.

To study the relationship between the decomposition of jets of L with those of .Le , we denote e m 1 C∗m e E = {jm L (e) | L ∈ Г (T1 (E))}.

.

By (1.9), we have ∗m C∗m e E ⊆ Pe E ⊗ Ve E.

.

As we shall see, one can be a little more explicit about the nature of .C∗m e E, and see that ∗m ∗ ∗E ∗ C∗m e E ≃ (Pe E ⊗ V E ⊗ Ve E) ⊕ (Pm−1 ⊗ V E ⊗ Ve E).

.

However, as in the previous section, this sort of isomorphism is too cumbersome to make explicit. As with the constructions of the preceding sections, we wish to use Lemma 2.15 to provide a decomposition of .C∗m E, and to do so we need to understand the covariant derivatives ∇ E,m Le  ∇ E · · · ∇ E Le ,

.

m ∈ Z≥0 .

m times

The results in this section have a slightly different character than in the preceding sections, so we provide complete proofs. The first result we give is the following. Lemma 3.54 (Iterated Covariant Differentials of Vertical Evaluations of Endomorphisms I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For r .m ∈ Z≥0 , there exist .C -vector bundle mappings r s ∗ ∗ m ∗ (Am s , idE ) ∈ VB (T (πE T M) ⊗ VE; T (T E) ⊗ VE),

.

s ∈ {0, 1, . . . , m},

and (Csm , idE ) ∈ VBr (Ts (πE∗ T∗ M) ⊗ T11 (VE); Tm−1 (T∗ E) ⊗ T11 (VE)),

.

s ∈ {0, 1, . . . , m − 1},

3.3 Isomorphisms Defined by Lifts and Pull-Backs

171

such that m Σ

∇ E,m Le =

.

M,πE ,s Am L)e ) + s ((∇

s=0

m−1 Σ

Csm ((∇ M,πE ,s L)v )

s=0

for all .L ∈ Г m (T11 (E)). (Here we regard .T11 (VE) as a subbundle of .T∗ E ⊗ VE by the mapping T11 (VE) ∋ A ⎬→ A ◦ ver ∈ T∗ E ⊗ VE).

.

m m m m m Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am and .C0 , C1 , . . . , Cm−1 satisfy the recursion relations prescribed by

A00 (β0 ) = β0 , A11 (β1 ) = β1 , A10 (β0 ) = Ins1 (β0 , BπE ), C01 (γ0 ) = γ0 ,

.

and, for .m ≥ 2, m+1 Am+1 (βm+1 ) = βm+1

.

m+1 Am (βm )

=

Am m−1

⊗ id

T∗ E

(βm ) −

m Σ j =1

Asm+1 (βs )

= (∇

E

m Am s )(βm ) + As−1

−

s Σ

Insj (βm , BπE ) + Insm+1 (βm , Bπ∗E )

⊗ idT∗ E (βs )

Am s ⊗ idT∗ E (Insj (βs , BπE ))

j =1 ∗ + Am s ⊗ idT∗ E (Inss+1 (βs , BπE )), s ∈ {1, . . . , m − 1}, m ∗ A0m+1 (β0 ) = (∇ E Am 0 )(β0 ) − A0 ⊗ idT∗ E (Ins1 (β0 , BπE ))

and m+1 m Cm (γm ) = Cm−1 ⊗ idT∗ E (γm ) + γm

.

E m m Csm+1 (γs ) = Am s ⊗ idT∗ E (γs ) + (∇ Cs )(γs ) + Cs−1 ⊗ idT∗ E (γs )

−

s+1 Σ j =1

Csm ⊗ idT∗ E (Insj (γs , BπE )) + Csm ⊗ idT∗ E (Inss+1 (γs , Bπ∗E )),

s ∈ {1, . . . , m − 1}, E m m C0m+1 (γ0 ) = Am 0 ⊗ idT∗ E (γ0 ) + (∇ C0 )(γ0 ) − C0 ⊗ idT∗ E (Ins1 (γ0 , BπE ))

+ C0m ⊗ idT∗ E (Ins2 (γ0 , Bπ∗E )),

172

3 Geometry: Lifts and Differentiation of Tensors

where .βs ∈ Ts (πE∗ T∗ M) ⊗ VE, .s ∈ {0, 1, . . . , m}, and .γs ∈ Ts (πE∗ T∗ M) ⊗ T11 (VE), .s ∈ {0, 1, . . . , m − 1}. Proof The assertion is clearly true for .m = 0 and, for .m = 1, we have ∇ E Le = (∇ πE L)e + Ins1 (L, BπE ) + Lv

.

by Lemma 3.16(vii), which gives the result for .m = 1. Thus suppose the result true for .m ≥ 2 so that ∇ E,m Le =

m Σ

.

M,πE ,s Am L)e ) + s ((∇

s=0

m−1 Σ

Csm ((∇ M,πE ,s L)v )

s=0

m for vector bundle mappings .Am s and .Cs satisfying the stated recursion relations. We then compute

∇ E,m+1 Le =

.

m m Σ Σ M,πE ,s e E M,πE ,s (∇ E Am )((∇ L) ) + Am L)e ) s s ⊗ idT∗ E (∇ (∇ s=0

+

m−1 Σ

s=0

(∇ E Csm )((∇ M,πE ,s L)v ) +

m−1 Σ

s=0

=

Csm ⊗ idT∗ E (∇ E (∇ M,πE ,s L)v )

s=0

m m Σ Σ M,πE ,s M,πE ,s+1 e (∇ E Am L)e ) + Am L) ) s )((∇ s ⊗ idT∗ E ((∇ s=0

−

s=0

s m Σ Σ

M,πE ,s Am L)e , BπE )) s ⊗ idT∗ E (Insj ((∇

s=1 j =1

+

m Σ

M,πE ,s Am L)e , Bπ∗E )) s ⊗ idT∗ E (Inss+1 ((∇

s=0

+

m Σ

M,πE ,s Am L)v ) + s ⊗ idT∗ E ((∇

s=0

+

m−1 Σ

m−1 Σ

(∇ E Csm )((∇ M,πE ,s L)v )

s=0

Csm ⊗ idT∗ E ((∇ M,πE ,s+1 L)v )

s=0

−

s m−1 ΣΣ

Csm ⊗ idT∗ E (Insj ((∇ M,πE ,s L)v , BπE ))

s=1 j =1

+

m−1 Σ s=0

Csm ⊗ idT∗ E (Inss+1 ((∇ M,πE ,s L)v , Bπ∗E ))

3.3 Isomorphisms Defined by Lifts and Pull-Backs

173

⎛ M,πE ,m e = (∇ M,πE ,m+1 L)e + ⎝Am L) ) m−1 ⊗ idT∗ E ((∇

−

m Σ j =1

Insj ((∇ M,πE ,m L)e , BπE ) + Insm+1 ((∇ M,πE ,m L)e , Bπ∗E ) ⎞

m + (∇ M,πE ,m L)v + Cm−1 ⊗ idT∗ E ((∇ M,πE ,m L)v )⎠

+

⎛m−1 Σ

M,πE ,s (∇ E Am L)e ) + s )((∇

s=1

−

s m−1 ΣΣ

m−1 Σ

M,πE ,s Am L)e ) s−1 ⊗ idT∗ E ((∇

s=1 M,πE ,s Am L)e ), BπE ) s ⊗ idT∗ E (Insj ((∇

s=1 j =1

+

m−1 Σ

M,πE ,s Am L)e ), Bπ∗E ) s ⊗ idT∗ E (Inss+1 ((∇

s=1

+

m−1 Σ

M,πE ,s Am L)v ) + s ⊗ idT∗ E ((∇

s=1

+

m−1 Σ

m−1 Σ

(∇ E Csm )((∇ M,πE ,s L)v )

s=1 m Cs−1 ⊗ idT∗ E ((∇ M,πE ,s L)v )

s=1

−

s m−1 ΣΣ

Csm ⊗ idT∗ E (Insj ((∇ M,πE ,s L)v , BπE ))

s=0 j =1

+

m−1 Σ

⎞

Csm ⊗ idT∗ E (Inss+1 ((∇ M,πE ,s L)v , Bπ∗E ))

s=1 e m e ∗ m v + (∇ E Am 0 )(L ) + A0 ⊗ idT∗ E (Ins1 (L , BπE )) + A0 ⊗ idT∗ E (L )

+ (∇ E C0m )(Lv ) − C0m ⊗ idT∗ E (Ins1 (Lv , BπE )) + C0m ⊗ idT∗ E (Ins2 (Lv , Bπ∗E )). From these calculations, the lemma follows.

□

.

Now we “invert” the constructions from the preceding lemma. Lemma 3.55 (Iterated Covariant Differentials of Vertical Evaluations of Endomorphisms II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For

174

3 Geometry: Lifts and Differentiation of Tensors

m ∈ Z≥0 , there exist .Cr -vector bundle mappings

.

(Bsm , idE ) ∈ VBr (Ts (T∗ E) ⊗ VE; Tm (πE∗ T∗ M) ⊗ VE),

.

s ∈ {0, 1, . . . , m},

and (Dsm , idE ) ∈ VBr (Ts (T∗ E) ⊗ T11 (VE); Tm−1 (πE∗ T∗ M) ⊗ T11 (VE)),

.

s ∈ {0, 1, . . . , m − 1}, such that (∇ M,πE ,m L)e =

m Σ

.

Bsm (∇ E,s Le ) +

s=0

m−1 Σ

Dsm (∇ E,s Lv )

s=0

m for all .L ∈ Г m (T11 (E)). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm m m m 0 and .D0 , D1 , . . . , Dm−1 satisfy the recursion relations prescribed by .B0 (α0 ) = α0 , 1 .D (γ0 ) = γ0 , 0 m+1 Bm+1 (αm+1 ) = αm+1

.

m+1 m Bm (αm ) = Bm−1 ⊗ idT∗ E (αm ) +

m Σ j =1

Bsm+1

= (∇

E

m Bsm )(αs ) + Bs−1

⊗ id

Insj (αm , BπE ) − Insm+1 (α, Bπ∗E )

T∗ E

(αs ) +

m Σ

Insj (Bsm (αs ), BπE )

j =1

− Insm+1 (Bsm (αs ), Bπ∗E ), B0m+1 (α0 ) = (∇ E B0m )(α0 ) +

m Σ j =1

s ∈ {1, . . . , m − 1},

Insj (B0m (α0 ), BπE ) − Insm+1 (B0m (α0 ), Bπ∗E )

and m+1 m Dm (γm ) = Dm−1 ⊗ idT∗ E (γm ) − γm

.

m Dsm (γs ) = (∇ E Dsm )(γs ) + Ds−1 ⊗ idT∗ E (γs ) − B m s (γs ), s ∈ {1, . . . , m − 1},

D0m+1 = (∇ E D0m )(γ0 ) − B m 0 (γ0 )

3.3 Isomorphisms Defined by Lifts and Pull-Backs

175

for .αs ∈ Ts (T∗ E ⊗ VE), .s ∈ {0, 1, . . . , m + 1}, and .γs ∈ Ts (T∗ E) ⊗ T11 (VE), .s ∈ {0, 1, . . . , m}, and where r s ∗ 1 m ∗ ∗ 1 (B m s , idE ) ∈ VB (T (T E)⊗T1 (VE); T (πE T M)⊗T1 (VE)),

.

s ∈ {0, 1, . . . , m},

are the vector bundle mappings from Lemma 3.45. Proof The assertion is clearly true for .m = 0, so suppose it true for .m ∈ Z>0 . Thus (∇ M,πE ,m L)e =

m Σ

.

Bsm (∇ E,s Le ) +

m−1 Σ

s=0

Dsm ((∇ E,s L)v ).

(3.34)

s=0

Working on the left-hand side of this equation, using Lemma 3.16(vii), we have E

∇ (∇

.

M,πE ,m

L) = (∇ e

M,πE ,m+1

L) − e

m Σ

Insj ((∇ M,πE ,m L)e , BπE )

j =1

+ Insm+1 ((∇ = (∇

M,πE ,m+1

M,πE ,m

L) − e

L)e , Bπ∗E ) + (∇ M,πE ,m L)v

m m Σ Σ

Insj (Bsm (∇ E,s Le ), BπE )

s=0 j =1

+

m Σ

Insm+1 (Bsm (∇ E,s Le ), Bπ∗E ) +

s=0

m Σ

E,s v Bm L ). s (∇

s=0

Working on the right-hand side of (3.34),

∇ E (∇ M,πE ,m L)e =

.

m m Σ Σ (∇ E Bsm )(∇ E,s Le ) + Bsm ⊗ idT∗ E (∇ E,s+1 Le ) s=0

+

s=0 m−1 Σ

(∇ E Dsm )(∇ E,s Lv ) +

s=0

m−1 Σ

Dsm ⊗ idT∗ E (∇ E,s+1 Lv ).

s=0

Combining the preceding two computations, (∇ M,πE ,m+1 L)e

.

=

m m Σ Σ (∇ E Bsm )(∇ E,s Le ) + Bsm ⊗ idT∗ E (∇ E,s+1 Le ) s=0

+

m−1 Σ s=0

s=0

(∇ E Dsm )(∇ E,s Lv ) +

m−1 Σ s=0

Dsm ⊗ idT∗ E (∇ E,s+1 Lv )

176

3 Geometry: Lifts and Differentiation of Tensors

+

m m Σ Σ

Insj (Bsm (∇ E,s Le ), BπE ) −

s=0 j =1

−

m Σ

m Σ

Insm+1 (Bsm (∇ E,s Le ), Bπ∗E )

s=1

E,s v Bm L ) s (∇

s=0

⎛

m m ⊗ idT∗ E (∇ E,m Le ) + Dm−1 ⊗ idT∗ E (∇ E,m Lv ) = ∇ E,m+1 Le + ⎝Bm−1

+

m Σ j =1

+

⎞ Insj (∇ E,m Le , BπE ) − Insm+1 (∇ E,m Le , Bπ∗E ) − (∇ E,m Lv )⎠

⎛m−1 Σ

(∇ E Bsm )(∇ E,s Le ) +

s=1

+

m−1 Σ

+

m Bs−1 ⊗ idT∗ E (∇ E,s Le )

s=1

(∇ E Dsm )(∇ E,s Lv ) +

s=1

m−1 Σ

m−1 Σ

m Ds−1 ⊗ idT∗ E (∇ E,s Lv )

s=1

m m−1 ΣΣ

Insj (Bsm (∇ E,s Le ), BπE )

s=1 j =1

−

m−1 Σ

Insm+1 (Bsm (∇ E,s Le ), Bπ∗E )

m−1 Σ

−

s=1

⎞ E,s v Bm L ) s (∇

s=1

⎛

+ ⎝(∇ E B0m )(Le ) + (∇ E D0m )(Lv ) +

m Σ

Insj (B0m (Le ), BπE )

j =1

⎞

v ⎠ − Insm+1 (B0m (Le ), Bπ∗E ) − B m 0 (L ) .

□

The lemma follows from these computations.

.

Next we turn to symmetrised versions of the preceding lemmata. We show that the preceding two lemmata induce corresponding mappings between symmetric tensors. Lemma 3.56 (Iterated Symmetrised Covariant Differentials of Vertical Evaluations of Endomorphisms I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ ∗ m ∗ ˆm (A s , idE ) ∈ VB (S (πE T M) ⊗ VE; S (T E) ⊗ VE),

.

s ∈ {0, 1, . . . , m},

3.3 Isomorphisms Defined by Lifts and Pull-Backs

177

and r s ∗ ∗ 1 m ∗ ˆm (C s , idE ) ∈ VB (S (πE T M)⊗T1 (VE); S (T E)⊗VE),

s ∈ {0, 1, . . . , m−1},

.

such that (Symm ⊗ idVE ) ◦ ∇ E,m Le =

m Σ

.

M,πE ,s ˆm L)e A s ((Syms ⊗ idVE ) ◦ (∇

s=0

+

m−1 Σ

M,πE ,s ˆm L)v ) C s ((Syms ⊗ idT1 (VE) ) ◦ (∇ 1

s=0

for all .L ∈ Г m (T11 (E)). ˆm Proof Following along the lines of the proof of Lemma 3.31, we define .A s by requiring that M,πE ,s M,πE ,s ˆm L)v ) = (Symm ⊗ idVE ) ◦ Am L)v ), A s ((Syms ⊗ idVE ) ◦ (∇ s ((∇

.

ˆm and .C s by requiring that M,πE ,s ˆm L)e ) = (Symm ⊗ idVE ) ◦ Csm ((∇ M,πE ,s L)e ). C s ((Syms ⊗ idT1 (VE) ) ◦ (∇

.

1

ˆm That this definition of .A s makes sense follows exactly as in the proof of Lemma 3.31. Let us see how the same arguments also apply to the definition ˆm of .C s . For .m ∈ Z>0 , we define .C m : T≤m−1 (πE∗ T∗ M) ⊗ T11 (VE) → T≤m (T∗ E) ⊗ VE by C m (Lv , (∇ πE L)v , . . . , (∇ M,πE ,m−1 L)v ) ⎛ ⎞ 1 m−1 Σ Σ 1 v 2 M,πE ,s v m M,πE ,s v = C0 (L ), Cs ((∇ L) ), . . . , Cs ((∇ L) ) ,

.

s=0

s=0

making the identification of .T11 (VE) with a subspace of .T∗ E⊗VE as in the statement of Lemma 3.54. Note that we have a natural mapping T∗ E ⊗ T∗m−1 E → T∗m E

.

as in Sect. 3.2.4. This then induces a mapping Pm : (RE ⊕ T∗m−1 E) ⊗ T11 (VE) → (RE ⊗ T∗m E) ⊗ VE.

.

178

3 Geometry: Lifts and Differentiation of Tensors

Now define ˆm : π ∗ (RM ⊕ T∗m−1 M) ⊗ T11 (VE) → (RE ⊕ T∗m E) ⊗ VE P E

.

by ˆm = Pm ◦ ((idR ⊕jm−1 πE ) ⊗ id 1 P T (VE) ),

.

1

noting that .

idR ⊕jm−1 πE : πE∗ (RM ⊕ T∗m−1 M) → RE ⊕ T∗m−1 E

is injective. Also define Qm : S≤m−1 (T∗ E) ⊗ T11 (VE) → Sm (T∗ E) ⊗ VE

.

by Qm (A0 ⊗ α0 ⊗ u0 , . . . , Am−1 ⊗ αm−1 ⊗ um−1 )

.

= (Sym1 (A0 ⊗ α0 ) ⊗ u0 , . . . , Symm (Am−1 ⊗ αm−1 ) ⊗ um−1 ). Note that the diagram

.

commutes. We also define ∗ ˆm = Qm ◦ (πm−1 ⊗ idT1 (VE) ), Q

.

1

where ∗ πm−1 : S≤m−1 (πE∗ T∗ M) → S≤m−1 (T∗ E)

.

is the inclusion. Note that the diagram

3.3 Isomorphisms Defined by Lifts and Pull-Backs

179

.

commutes. Let us organise the mappings we require into the following diagram:

.

(3.35) ˆm is defined so that the right square commutes, which is possible since the Here .C horizontal arrows in the right square are isomorphisms. We shall show that the left square also commutes. Indeed, v πE v M,πE ,m v ˆm ◦ (Sym≤m−1 ⊗ id 1 L) ) C T (VE) )(L , (∇ L) , . . . , (∇

.

1

=

(S∇mE

ˆm ◦ (S m−1 ⊗ idVE )−1 ◦ P ⊗ idT1 (VE) ) ∇ M ,∇ πE 1

◦ (Sym≤m−1 ⊗ idT1 (VE) )(L 1

v

, (∇

πE

L) , . . . , (∇ M,πE ,m Lv ) v

= (Sym≤m−1 ⊗ idVE )(Lv , ∇ E Lv , . . . , ∇ E,m Lv ) = (Sym≤m ⊗ idVE ) ◦ C m (Lv , (∇ πE L)v , . . . , (∇ M,πE ,m L)v ). Thus the diagram (3.35) commutes. Thus, if we define M,πE ,s ˆm L)v ) = (Symm ⊗ idVE ) ◦ Csm ((∇ M,πE ,s L)v ), C s ((Syms ⊗ idT1 (VE) ) ◦ (∇

.

1

then we have (Symm ⊗ idVE ) ◦ ∇ E,m Le =

m Σ

.

M,πE ,s ˆm L)v ), C s ((Syms ⊗ idT1 (VE) ) ◦ (∇ 1

s=0

as desired.

□

.

The preceding lemma gives rise to an “inverse,” which we state in the following lemma. Lemma 3.57 (Iterated Symmetrised Covariant Differentials of Vertical Evaluations of Endomorphisms II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE

180

3 Geometry: Lifts and Differentiation of Tensors

on .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ m ∗ ∗ ˆm (B s , idE ) ∈ VB (S (T E) ⊗ VE; S (πE T M) ⊗ VE),

.

s ∈ {0, 1, . . . , m},

and r s ∗ 1 m ∗ ∗ ˆm (D s , idE ) ∈ VB (S (T E)⊗T1 (VE); S (πE T M)⊗VE),

.

s ∈ {0, 1, . . . , m−1},

such that (Symm ⊗ idVE ) ◦ (∇ M,πE ,m L)e

.

=

m Σ

E,s e ˆm L )+ B s ((Syms ⊗ idVE ) ◦ ∇

s=0

m−1 Σ

E,s v ˆm L ) D s ((Syms ⊗ idT1 (VE) ) ◦ ∇ 1

s=0

for all .L ∈ Г m (T11 (E)). ˆm Proof Following along the lines of the proof of Lemma 3.31, we define .B s by requiring that E,s e ˆm L = (Symm ⊗ idVE ) ◦ Bsm (∇ E,s Le ), B s ((Syms ⊗ idVE ) ◦ ∇

.

ˆm and .C s by requiring that E,s v ˆm L ) = (Symm ⊗ idVE ) ◦ Csm (∇ E,s Lv ). C s ((Syms ⊗ idT1 (VE) ) ◦ ∇

.

1

That these definitions make sense follows along the same lines as the proof of Lemma 3.56. .□ We can put together the previous four lemmata into the following decomposition result, which is to be regarded as the main result of this section. Lemma 3.58 (Decomposition of Jets of Vertical Evaluations of Endomorphisms) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle, with the data prescribed in Sect. 3.2.1 to define the Riemannian metric .GE on .E. Then there exist .Cr -vector bundle mappings Am ∈ VBr (C∗m E; S≤m (πE∗ T∗ M) ⊗ VE), ∇E

.

B∇mE ∈ VBr (C∗m E; S≤m (T∗ E) ⊗ VE)

defined by Am (j (Le )(e)) = Sym≤m ⊗ idVE (Le (e), (∇ πE L)e (e), . . . , (∇ M,πE ,m L)e (e)), ∇E m

.

B∇mE (jm (Le )(e)) = Sym≤m ⊗ idVE (Le (e), ∇ E Le (e), . . . , ∇ E,m Le (e)).

3.3 Isomorphisms Defined by Lifts and Pull-Backs

181

Moreover, .Am and .B∇mE are injective, and ∇E B∇mE ◦ (Am )−1 ◦ (Sym≤m ⊗ idVE )(Le (e), (∇ πE L)e (e), . . . , (∇ M,πE ,m L)e (e)) ∇E ⎛ 1 Σ ˆ1s ((Syms ⊗ idVE ) ◦ (∇ M,πE ,s L)e (e)), . . . , = Le (e), A

.

s=0 m Σ

⎞

ˆm A s ((Syms

⊗ idVE ) ◦ (∇

M,πE ,s

e

L) (e))

s=0

⎛

+ 0, Lv (e),

1 Σ

M,πE ,s ˆ2s ((Syms ⊗ id 1 L)v (e)), . . . , C T (VE) ) ◦ (∇ 1

s=0 m−1 Σ

⎞ M,πE ,s ˆm L)v (e)) C s ((Syms ⊗ idT1 (VE) ) ◦ (∇ 1

s=0

and m −1 e E e E,m e ◦ (Sym≤m ⊗ idVE )(L (e), ∇ L (e), . . . , ∇ Am ◦ (B E ) L (e)) ∇E ∇ ⎛ 1 Σ ˆ1s ((Syms ⊗ idVE ) ◦ ∇ E,s Le (e)), . . . , = Le (e), B

.

s=0 m Σ

⎛

⎞

E,s e ˆm L (e)) B s ((Syms ⊗ idVE ) ◦ ∇

s=0

+ 0, Lv (e),

1 Σ

E,s v ˆ2s ((Syms ⊗ id 1 L (e)), . . . , D T (VE) ) ◦ ∇ 1

s=0 m−1 Σ

⎞ ˆm D s ((Syms

⊗ idT1 (VE) ) ◦ ∇ 1

E,s

v

L (e)) ,

s=0

ˆm ˆm ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, and .C s and .D s , .s ∈ {0, 1, . . . , m − 1}, are as in Lemmata 3.56 and 3.57.

3.3.8 Isomorphisms for Pull-Backs of Functions Next we generalise the presentation of Sect. 3.3.1 from the pull-back of a vector bundle projection to the pull-back by a general mapping. The development here is a

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3 Geometry: Lifts and Differentiation of Tensors

little different from the preceding sections, so we first have a little bit of setting up to do. For .Cr -manifolds .M and .N, and for .⏀ ∈ Cr (M; N), we consider the mapping Cr (N) ∋ f ⎬→ ⏀∗ f ∈ Cr (M).

.

We wish to compare the decomposition of jets of f with those of .⏀∗ f , and to do so ∗m we consider the subbundle .T∗m ⏀ M of .T M defined by ∗ m T∗m ⏀,x M = {jm (⏀ f )(x) | f ∈ C (N)}.

.

Following Lemma 2.15, we shall give a formula for iterated covariant differentials of pull-backs of functions on .N. To do this, we let .∇ M and .∇ N be affine connections on .M and .N, respectively. We note that we have the linear connection .⏀∗ ∇ N in the vector bundle .⏀∗ TN over .M. Explicitly, N ∗ (⏀∗ ∇X ⏀ Y )(x) = (x, ∇TNx ⏀(X(x)) Y ).

.

Following our usual mild notational abuse, we shall also denote by .⏀∗ ∇ N the connection in the dual bundle .(⏀∗ TN)∗ ≃ ⏀∗ T∗ N. We have a natural mapping ˆ : TM → ⏀∗ TN ⏀ .

vx ⎬→ (x, Tx ⏀(vx )).

This mapping induces a mappings on sections which we denote by the same symbol; thus we have the mapping ˆ : Г ∞ (TM) → Г ∞ (⏀∗ TN). ⏀

.

The following lemma gives an important tensor for our analysis. Lemma 3.59 (Tensor for Pull-Back Connection) Let .r ∈ {∞, ω}. Let .M and .N be Cr -manifolds and let .∇ M and .∇ N be .Cr -affine connections on .M and .N, respectively. Let .⏀ ∈ Cr (M; N). Then there exists .A⏀ ∈ Г r (T2 (T∗ M) ⊗ ⏀∗ TN) such that, for .x ∈ M, .

M Nˆ ˆ(∇X ⏀ ⏀(Y )(x) = A⏀ (X(x), Y (x)) Y )(x) − ⏀∗ ∇X

.

for .X, Y ∈ Г ∞ (TM). Proof Let .K M : TTM → TM and .K N : TTN → TN be the connectors for .∇ M and N so that .∇ M ∇X Y = K M ◦ T Y ◦ X,

.

X, Y ∈ Г ∞ (TM),

3.3 Isomorphisms Defined by Lifts and Pull-Backs

183

and ∇UN V = K N ◦ T V ◦ U,

.

U, V ∈ Г ∞ (TN).

We, moreover, have M ˆ(∇X ⏀ Y ) = T ⏀ ◦ K M ◦ T Y ◦ X,

.

X, Y ∈ Г ∞ (TM),

and Nˆ ⏀(Y ) = K N ◦ T (T ⏀ ◦ Y ) ◦ X, ⏀ ∗ ∇X

.

X, Y ∈ Г ∞ (TM)

[51, §10.12]. In preparation to use these formulae, we have the following results. Sublemma 1 If .πE : E → M is a smooth vector bundle, if .ξ ∈ Г ∞ (E), and if ∞ .f ∈ C (M), then Tx (f ξ )(vx ) = f (x)Tx ξ(vx ) + ξ v (x).

.

Proof Let .∇ πE be a linear connection in the vector bundle .E which gives the decomposition .TE = HE ⊕ VE. Let .hor and .ver be the horizontal and vertical projections. Let .vx ∈ Tx M and let .γ : I → M be a smooth curve for which ' .γ (0) = vx . Denote .(t) = (f ◦ γ (t))(ξ ◦ γ (t)) the corresponding curve in .E. Then .

hor(' (t)) = hlft(f ◦ γ (t)ξ ◦ γ (t), γ ' (t)),

ver(' (t)) = vlft(f ◦ γ (t)ξ ◦ γ (t), ∇γ 'E(t) (t)). π

We now have ∇γ 'E(t) (t) = f ◦ γ (t)∇γ 'E(t) ξ ◦ γ (t) + 0 ,

m times

and ∇ N,m f = ∇ N · · · ∇ N f,

.

m ∈ Z>0 ,

m times

for .f ∈ C∞ (N). The following lemma gives the first part of this development, playing the rôle of Lemma 3.16 in this case. Lemma 3.60 (Differentiation of Pull-Backs of Covariant Tensors) Let .r ∈ {∞, ω}. Let .M and .N be .Cr -manifolds with .Cr -affine connections .∇ M and .∇ N , respectively. Define .B⏀ = push1,2 A⏀ with .A⏀ as in Lemma 3.59. Then, for .k ∈ Z>0 and .A ∈ Г r (Tk (T∗ N)), ∇ M ⏀∗ A = ⏀∗ ∇ N A + DB⏀ (⏀∗ A).

.

Proof Let .x ∈ M. Let .X1 , . . . , Xk ∈ Г ∞ (TM). For .Xk+1 ∈ Г ∞ (TM), we have L Xk+1 (⏀∗ A(X1 , . . . , Xk ))

.

=

M ⏀∗ A)(X1 , . . . , Xk ) + (∇X k+1

k Σ j =1

M ⏀∗ A(X1 , . . . , ∇X X , . . . , Xk ) k+1 j

and N ˆ(X1 ), . . . , ⏀ ˆ(Xk ))) = (⏀∗ ∇X ˆ(X1 ), . . . , ⏀ ˆ(Xk )) L Xk+1 (⏀∗ A(⏀ ⏀∗ A)(⏀ k+1

.

+

k Σ j =1

N ˆ(Xj ), . . . , ⏀ ˆ(X1 ), . . . , ⏀∗ ∇X ˆ(Xk )), ⏀ ⏀∗ A(⏀ k+1

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3 Geometry: Lifts and Differentiation of Tensors

using the two interpretations of .⏀∗ A. By (3.36) we have, in the above expressions, ˆ(X1 ), . . . , ⏀ ˆ(Xk )). ⏀∗ A(X1 , . . . , Xk ) = ⏀∗ A(⏀

.

By (3.36) again, we have M M ˆ(X1 ), . . . , ⏀ ˆ(∇X ˆ(Xk )). ⏀∗ A(X1 , . . . , ∇X X , . . . , Xk ) = ⏀∗ A(⏀ X ), . . . , ⏀ k+1 j k+1 j

.

Also note that N ˆ(X1 ), . . . , ⏀ ˆ(Xk ))(x) (⏀∗ ∇X ⏀∗ A)(⏀ k+1

.

N ⏀∗ A)(Tx ⏀(X1 (x)), . . . , Tx ⏀(Xk (x))) = (⏀∗ ∇X k+1

= ∇TNx ⏀(Xk+1 (x)) A(Tx ⏀(X1 (x)), . . . , Tx ⏀(Xk+1 (x))) = ∇ N A(Tx ⏀(X1 (x)), . . . , Tx ⏀(Xk+1 (x))) = ⏀∗ ∇ N A(X1 , . . . , Xk+1 )(x). Combining the above gives ∇ M ⏀∗ A(X1 , . . . , Xk+1 )

.

= ⏀∗ ∇ N A(X1 , . . . , Xk+1 ) +

k Σ j =1

N ˆ(Xj ) ˆ(X1 ), . . . , ⏀∗ ∇X ⏀ ⏀∗ A(⏀ k+1

M ˆ(∇X ˆ(Xk )) −⏀ X ), . . . , ⏀ k+1 j

= ⏀∗ ∇ N A(X1 , . . . , Xk+1 ) −

k Σ

ˆ(X1 ), . . . , A⏀ (Xk+1 , Xj ), . . . , ⏀ ˆ(Xk )). ⏀∗ A(⏀

j =1

Thus ∇ M ⏀∗ A = ⏀∗ ∇ N A −

k Σ

.

Insj (⏀∗ A, B⏀ ),

j =1

giving the result by Lemma 3.11.

□

.

We now have the following lemma, the first of two regarding iterated covariant differentials.

3.3 Isomorphisms Defined by Lifts and Pull-Backs

187

Lemma 3.61 (Iterated Covariant Differentials of Pull-Backs of Functions I) Let .r ∈ {∞, ω} and let .M and .N be .Cr -manifolds with .Cr -affine connections .∇ M and .∇ N , respectively. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ ∗ m ∗ (Am s , idM ) ∈ VB (T (⏀ T N); T (T M)),

.

s ∈ {0, 1, . . . , m},

such that ∇ M,m ⏀∗ f =

m Σ

.

∗ N,s Am f) s (⏀ ∇

s=0 m m for all .f ∈ Cm (N). Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am satisfy the recursion relations prescribed by

A00 (β0 ) = β0 , A11 (β1 ) = β1 , A10 = 0,

.

and m+1 Am+1 (βm+1 ) = βm+1 ,

.

m Asm+1 (βs ) = (∇ M Am s )(βs ) + As−1 ⊗ idT∗ M (βs )

−

s Σ

Am s ⊗ idT∗ M (Insj (βs , B⏀ )),

s ∈ {1, . . . , m},

j =1

A0m+1 (β0 ) = (∇ M Am 0 )(β0 ), where .βs ∈ Ts (⏀∗ T∗ N), .s ∈ {0, 1, . . . , m}. Proof The assertion clearly holds for the initial conditions of the recursion, simply because ⏀∗ f = ⏀∗ f,

.

d(⏀∗ f ) = ⏀∗ df + 0f.

So suppose that it holds for .m ∈ Z>0 . Thus ∇ M,m ⏀∗ f =

m Σ

.

s=0

∗ N,s Am f ), s (⏀ ∇

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3 Geometry: Lifts and Differentiation of Tensors

where the vector bundle mappings .Aas , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the stated recursion relations. Then ∇ M,m+1 ⏀∗ f =

.

m m Σ Σ ∗ N,s M ∗ N,s (∇ M Am )(⏀ ∇ f ) + Am f) s s ⊗ idT∗ M (∇ ⏀ ∇ s=0

=

s=0

m m Σ Σ ∗ N,s ∗ N,s+1 (∇ M Am f) + Am f) s )(⏀ ∇ s ⊗ idT∗ M (⏀ ∇ s=0

−

s=0

s m Σ Σ

∗ N,s Am f, B⏀ ) s ⊗ idT∗ M Insj (⏀ ∇

s=0 j =1

= ⏀∗ ∇ N,m+1 f ⎛ m Σ ∗ N,s ∗ N,s ⎝(∇ M Am + f ) + Am f) s )(⏀ ∇ s−1 ⊗ idT∗ M (⏀ ∇ s=1

−

s Σ

⎞ ∗ N,s ∗ Am f, B⏀ ))⎠ + (∇ M Am s ⊗ idT∗ M (Insj (⏀ ∇ 0 )(⏀ f )

j =1

□

by Lemma 3.60. From this the lemma follows.

.

We shall also need to “invert” the relationship of the preceding lemma. Lemma 3.62 (Iterated Covariant Differentials of Pull-Backs of Functions II) Let .r ∈ {∞, ω} and let .M and .N be .Cr -manifolds with .Cr -affine connections .∇ M and .∇ N , respectively. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings (Bsm , idM ) ∈ VBr (Ts (T∗ M); Tm (⏀∗ T∗ N)),

.

s ∈ {0, 1, . . . , m},

such that ⏀∗ ∇ N,m f =

m Σ

.

Bsm (∇ M,s ⏀∗ f )

s=0 m satisfy for all .f ∈ Cm (N). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm the recursion relations prescribed by

B00 (α0 ) = α0 , B11 (α1 ) = α1 , B01 = 0,

.

3.3 Isomorphisms Defined by Lifts and Pull-Backs

189

and m+1 Bm+1 (αm+1 ) = αm+1 ,

.

m Bsm+1 (αs ) = (∇ M Bsm )(αs ) + Bs−1 ⊗ idT∗ M (αs ) +

m Σ

Insj (Bsm (αs ), B⏀ ),

j =1

s ∈ {1, . . . , m}, B0m+1 (α0 ) = (∇ M B0m )(α0 ) +

m Σ

Insj (B0m (α0 ), B⏀ ),

j =1

where .αs ∈ Ts (T∗ M), .s ∈ {0, 1, . . . , m}. Proof The assertion clearly holds for the initial conditions for the recursion because ⏀∗ f = ⏀∗ f,

.

⏀∗ (df ) = d(⏀∗ f ) + 0f.

So suppose it true for .m ∈ Z>0 . Thus ∗

⏀ ∇

.

N,m

f =

m Σ

Bsm (∇ M,s ⏀∗ f ),

(3.37)

s=0

where the vector bundle mappings .Bsa , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the recursion relations from the statement of the lemma. Then, by Lemma 3.60, we can work on the left-hand side of (3.37) to give ∇ M ⏀∗ ∇ N,m f = ⏀∗ ∇ N,m+1 f −

m Σ

.

Insj (⏀∗ ∇ N,m f, B⏀ )

j =1

= ⏀∗ ∇ N,m+1 f −

m m Σ Σ

Insj (Bsm (∇ M,s ⏀∗ f ), B⏀ ).

s=0 j =1

Working on the right-hand side of (3.27) gives M

∗

∇ ⏀ ∇

.

N,m

f =

m Σ s=0

∇

M

Bsm (∇ M,s ⏀∗ f ) +

m Σ s=0

Bsm ⊗ idT∗ M (∇ M,s+1 ⏀∗ f ).

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3 Geometry: Lifts and Differentiation of Tensors

Combining the preceding two equations gives ⏀∗ ∇ N,m+1 f =

m Σ

.

∇ M Bsm (∇ M,s ⏀∗ f ) +

s=0

+

m Σ

Bsm ⊗ idT∗ M (∇ M,s+1 ⏀∗ f )

s=0

m m Σ Σ

Insj (Bsm (∇ M,s ⏀∗ f ), B⏀ )

s=0 j =1

= ∇ M,m+1 ⏀∗ f ⎛ m Σ m ⎝∇ M Bsm (∇ M,s ⏀∗ f ) + Bs−1 + ⊗ idT∗ M (∇ M,s ⏀∗ f ) s=1

+

m Σ

⎞

Insj (Bsm (∇ M,s ⏀∗ f ), B⏀ )⎠ + ∇ M B0m (⏀∗ f )

j =1

+

m Σ

Insj (B0m (⏀∗ f ), B⏀ ),

j =1

□

and the lemma follows from this.

.

With this data, we have the following result. Lemma 3.63 (Iterated Symmetrised Covariant Differentials of Pull-Backs of Functions I) Let .r ∈ {∞, ω} and let .M and .N be .Cr -manifolds with .Cr -affine connections .∇ M and .∇ N , respectively. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ ∗ m ∗ ˆm (A s , idM ) ∈ VB (S (⏀ T N); S (T M)),

.

s ∈ {0, 1, . . . , m},

such that .

Symm ◦ ∇ M,m ⏀∗ f =

m Σ

∗ N,s ˆm f) A s (Syms ◦ ⏀ ∇

s=0

for all .f ∈ Cm (N). Proof This follows from Lemma 3.61 in the same way as Lemma 3.25 follows from Lemma 3.23. .□ Next we consider the “inverse” of the preceding lemma. Lemma 3.64 (Iterated Symmetrised Covariant Differentials of Horizontal Lifts of Functions II) Let .r ∈ {∞, ω} and let .M and .N be .Cr -manifolds with .Cr -affine connections .∇ M and .∇ N , respectively. For .m ∈ Z≥0 , there exist .Cr -vector bundle

3.3 Isomorphisms Defined by Lifts and Pull-Backs

191

mappings r s ∗ m ∗ ∗ ˆm (B s , idM ) ∈ VB (S (T M); S (⏀ T N)),

.

s ∈ {0, 1, . . . , m},

such that .

Symm ◦ ⏀∗ ∇ N,m f =

m Σ

M,s ∗ ˆm ⏀ f) B s (Syms ◦ ∇

s=0

for all .f ∈ Cm (N). Proof This follows from Lemma 3.62 in the same way as Lemma 3.26 follows from Lemma 3.24. .□ The following lemma provides two decompositions of .T∗m ⏀ M, one “in the domain” and one “in the codomain,” and the relationship between them. The assertion simply results from an examination of the preceding four lemmata. Lemma 3.65 (Decomposition of Jets of Pull-Backs of Functions) Let .r ∈ {∞, ω} and let .M and .N be .Cr -manifolds with .Cr -affine connections .∇ M and .∇ N , respectively. Then there exist .Cr -vector bundle mappings ≤m Am ∈ VBr (T∗m (⏀∗ T∗ N)), ⏀ M; S ∇ M ,∇ N

.

≤m ∗ B∇mM ,∇ N ∈ VBr (T∗m (T M)), ⏀ M; S

defined by Am (j (⏀∗ f )(x)) = Sym≤m (⏀∗ f (x), ⏀∗ ∇ N f (x), . . . , ⏀∗ ∇ N,m f (x)), ∇ M ,∇ N m

.

B∇mM ,∇ N (jm (⏀∗ f )(x)) = Sym≤m (⏀∗ f (x), ∇ M ⏀∗ f (x), . . . , ∇ M,m ⏀∗ f (x)). Moreover, .Am is an isomorphism, .B∇mM ,∇ N is injective, and ∇ M ,∇ N B∇mM ,∇ N ◦ (Am )−1 ◦ (Sym≤m (⏀∗ f (e), ⏀∗ ∇ N f (x), . . . , ⏀∗ ∇ N,m f (x)) ∇ M ,∇ N ⎛ 1 Σ ˆ1s (Syms ◦ ⏀∗ ∇ N,s f (x)), . . . , = A00 (⏀∗ f (x)), A

.

s=0 m Σ s=0

⎞ ∗ N,s ˆm f (x)) A s (Syms ◦ ⏀ ∇

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3 Geometry: Lifts and Differentiation of Tensors

and m −1 ∗ M ∗ M,m ∗ ◦ Sym≤m (⏀ f (x), ∇ ⏀ f (x), . . . , ∇ Am ◦ (B M N ) ⏀ f (x)) ∇ M ,∇ N ∇ ,∇ ⎛ 1 Σ ˆ1s (Syms ◦ ∇ M,s ⏀∗ f (x)), . . . , B = B00 (⏀∗ f (x)),

.

s=0 m Σ

⎞ M,s ∗ ˆm ⏀ f (x)) , B s (Syms ◦ ∇

s=0

ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, are as in Lemmata 3.63 and 3.64.

3.3.9 Comparison of Iterated Covariant Derivatives for Different Connections In Sect. 4.3 we will prove that the seminorms for the .Cω -topology are well-defined, in that they do not depend on choices of metrics and connections. In doing this, it is useful to have at hand formulae that relate arbitrary-order covariant differentials with respect to different connections. This is what we do in this section. We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. We consider r M and .∇ M on .M, and .Cr -linear connections .∇ πE and .∇ πE .C -affine connections .∇ in .E. It then holds that M ∇M X Y = ∇X Y + SM (Y, X),

.

π

π

∇ XE ξ = ∇XE ξ + SπE (ξ, X)

for .SM ∈ Г r (T12 (TM)) an .SπE ∈ Г r (E∗ ⊗ T∗ M ⊗ E). First we relate covariant derivatives of higher-order tensors. Lemma 3.66 (Covariant Derivatives of Higher-Order Tensors with Respect to Different Connections) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr vector bundle. Consider .Cr -affine connections .∇ M and .∇ M on .M, and .Cr -linear connections .∇ πE and .∇ πE in .E. If .k ∈ Z>0 and if .B ∈ Г 1 (Tk (T∗ M) ⊗ E), then ∇ M,πE B = ∇ M,πE B −

k Σ

.

j =1

Insj (B, SM ) − Insk+1 (B, SπE ).

3.3 Isomorphisms Defined by Lifts and Pull-Backs

193

Proof We have M,π

L Xk+1 (B(X1 , . . . , Xk , α)) = (∇ Xk+1E B)(X1 , . . . , Xk , α)

.

+

k Σ j =1

π

E B(X1 , . . . , ∇ M Xk+1 Xj , . . . , Xk , α) + B(X1 , . . . , Xk , ∇ Xk+1 α)

M,π

= (∇ Xk+1E B)(X1 , . . . , Xk , α) +

+

k Σ j =1

k Σ j =1

M B(X1 , . . . , ∇X X , . . . , Xk , α) k+1 j

π

B(X1 , . . . , SM (Xj , Xk+1 ), . . . , Xk , α) + B(X1 , . . . , Xk , ∇XEk+1 α)

+ B(X1 , . . . , Xk , SπE (α, Xk+1 )). This gives ∇ M,πE B = ∇ M,πE B −

k Σ

.

Insj (B, SM ) − Insk+1 (B, SπE ),

j =1

□

as claimed.

.

With this lemma, we can provide the following characterisation of iterated covariant differentials of sections of .E with respect to different connections. Lemma 3.67 (Iterated Covariant Differentials of Sections with Respect to Different Connections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. Consider .Cr -affine connections .∇ M and .∇ M on .M, and .Cr -linear connections .∇ πE and .∇ πE in .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ m ∗ (Am s , idE ) ∈ VB (T (T M ⊗ E); T (T M) ⊗ E),

.

s ∈ {0, 1, . . . , m},

such that ∇ M,πE ,m ξ =

m Σ

.

M,πE ,s Am ξ) s (∇

s=0 m m for all .ξ ∈ Г m (E). Moreover, the vector bundle mappings .Am 0 , A1 , . . . , Am satisfy 0 the recursion relations prescribed by .A0 (β0 ) = β0 and m+1 Am+1 (βm+1 ) = βm+1 ,

.

m Asm+1 (βs ) = (∇ M,πE Am s )(βs ) + As−1 ⊗ idT∗ M (βs )

194

3 Geometry: Lifts and Differentiation of Tensors

−

s Σ

m Am s ⊗ idT∗ M (Insj (βs , SM ))−As ⊗ idT∗ M (Inss+1 (βs , SπE )),

j =1

s ∈ {1, . . . , m}, m A0m+1 (β0 ) = (∇ M,πE Am 0 )(β0 ) − A0 ⊗ idT∗ M (Ins1 (β0 , SπE )),

where .βs ∈ Ts (T∗ M) ⊗ E, .s ∈ {0, 1, . . . , m}. Proof The assertion clearly holds for .m = 0, so suppose it true for .m ∈ Z>0 . Thus ∇

.

M,πE ,m

ξ=

m Σ

M,πE ,s Am ξ ), s (∇

s=0

where the vector bundle mappings .Aas , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the recursion relations from the statement of the lemma. Then m m Σ Σ M,πE ,s M,πE M,πE ,s (∇ M,πE Am )(∇ ξ ) + Am ∇ ξ) s s ⊗ idT∗ M (∇

∇ M,πE ,m+1 ξ =

.

s=0

s=0

m m Σ Σ M,πE ,s M,πE ,s+1 = (∇ M,πE Am )(∇ ξ ) + Am ξ) s s ⊗ idT∗ M (∇ s=0

s=0

−

s m Σ Σ

M,πE ,s Am ξ, SM )) s ⊗ idT∗ M (Insj (∇

s=1 j =1

−

m Σ

M,πE ,s Am ξ, SπE ))−Am s ⊗ idT∗ M (Inss+1 (∇ 0 ⊗ idT∗ M (Ins1 (ξ, SπE ))

s=1

= ∇ M,πE ,m+1 ξ +

m Σ

⎛ M,πE ,s M,πE ,s ξ )+Am ξ) (∇ M,πE Am s )(∇ s−1 ⊗ idT∗ M (∇

s=1

−

s Σ

M,πE ,s Am ξ, SM )) s ⊗ idT∗ M (Insj (∇

j =1

−

Am s

⎞ ⊗ idT∗ M (Inss+1 (∇

M,πE ,s

ξ, SπE )) − (∇ M,πE Am 0 )(ξ )

− Am 0 ⊗ idT∗ M (Ins1 (ξ, SπE )) by Lemma 3.66. From this, the lemma follows. The lemma has an “inverse” which we state next.

□

.

3.3 Isomorphisms Defined by Lifts and Pull-Backs

195

Lemma 3.68 (Iterated Covariant Differentials of Sections with Respect to Different Connections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. Consider .Cr -affine connections .∇ M and .∇ M on .M, and .Cr -linear connections .∇ πE and .∇ πE in .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings (Bsm , idE ) ∈ VBr (Ts (T∗ M ⊗ E); Tm (T∗ M) ⊗ E),

s ∈ {0, 1, . . . , m},

.

such that m Σ

∇ M,πE ,m ξ =

.

Bsm (∇ M,πE ,s ξ )

s=0 m satisfy for all .ξ ∈ Г m (E). Moreover, the vector bundle mappings .B0m , B1m , . . . , Bm 0 the recursion relations prescribed by .B0 (α0 ) = β0 and m+1 Bm+1 (αm+1 ) = αm+1 ,

.

Bsm+1 (αs )

= (∇

M,πE

m Bsm )(αs ) + Bs−1

⊗ id

T∗ M

(αs ) +

m Σ

Insj (Bsm (αs ), SM )

j =1

+ Insm+1 (Bsm (αs ξ ), SπE ), B0m+1 (α0 ) = (∇ M,πE B0m )(α0 )+

m Σ

s ∈ {1, . . . , m},

Insj (B0m (α0 ), SM ) + Insm+1 (B0m (α0 ), SπE ),

j =1

where .αs ∈ Ts (T∗ M) ⊗ E, .s ∈ {0, 1, . . . , m}. Proof The lemma is clearly true for .m = 0, so suppose it true for .m ∈ Z>0 . Thus ∇ M,πE ,m ξ =

m Σ

.

Bsm (∇ M,πE ,s ξ ),

(3.38)

s=0

where the vector bundle mappings .Bsa , .a ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , a}, satisfy the recursion relations given in the lemma. Then, working with the left-hand side of this relation, ∇ M,πE ∇ M,πE ,m ξ = ∇ M,πE ,m+1 ξ

.

−

m Σ j =1

Insj (∇ M,πE ,m ξ, SM ) − Insm+1 (∇ M,πE ,m ξ, SπE )

196

3 Geometry: Lifts and Differentiation of Tensors

= ∇ M,πE ,m+1 ξ −

m m Σ Σ

Insj (Bsm (∇ M,πE ,s ξ ), SM )

s=0 j =1

−

m Σ

Insm+1 (Bsm (∇ M,πE ,s ξ ), SπE ),

s=0

by Lemma 3.66. Now, working with the right-hand side of (3.38), ∇ M,πE ∇ M,πE ,m ξ =

m Σ

.

(∇ M,πE Bsm )(∇ M,πE ,m ξ ) +

s=0

m Σ

Bsm ⊗ idT∗ M (∇ M,πE ,m+1 ξ ).

s=0

Combining the preceding two computations, ∇ M,πE ,m+1 ξ = ∇ M,πE ,m+1 ξ ⎛ m Σ m ⎝(∇ M,πE Bsm )(∇ M,πE ,s ξ ) + Bs−1 + ⊗ idT∗ M (∇ M,πE ,s ξ )

.

s=1

+

m Σ

⎞

Insj (Bsm (∇ M,πE ,s ξ ), SM ) + Insm+1 (Bsm (∇ M,πE ,s ξ ), SπE )⎠

j =1

+ (∇ M,πE B0m )(ξ ) +

m Σ

Insj (B0m (ξ ), SM ) + Insm+1 (B0m (ξ ), SπE ),

j =1

□

and from this the lemma follows.

.

Now we give symmetrised versions of the preceding lemmata, since it is these that are required for computations with jets. Lemma 3.69 (Iterated Symmetrised Covariant Differentials of Sections with Respect to Different Connections I) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. Consider .Cr -affine connections .∇ M and .∇ M on .M, and .Cr linear connections .∇ πE and .∇ πE in .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings r s ∗ m ∗ ˆm (A s , idE ) ∈ VB (T (T M ⊗ E); T (T M) ⊗ E),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idE ) ◦ ∇ M,πE ,m ξ =

m Σ

.

s=0

for all .ξ ∈ Г m (E).

M,πE ,s ˆm ξ) A s ((Syms ⊗ idE ) ◦ ∇

3.3 Isomorphisms Defined by Lifts and Pull-Backs

197

Proof We define .Am : T≤m (T∗ M) ⊗ E → T≤m (T∗ M) ⊗ E by Am (ξ, ∇ πE ξ, . . . , ∇ M,πE ,m ξ ) ⎛ ⎞ 1 m Σ Σ 0 1 M,πE ,s m M,πE ,s = A0 (ξ ), As (∇ ξ ), . . . , As (∇ ξ) .

.

s=0

s=0

Let us organise the mappings we require into the following diagram:

(3.39)

.

ˆm is defined so that the right square commutes. We shall show that the left Here .A square also commutes. Indeed, ˆm ◦ Sym≤m ⊗ idE (ξ, ∇ πE ξ, . . . , ∇ M,πE ,m ξ ) A

.

= (S∇mM ,∇ πE )−1 ◦ S∇mM ,∇ πE ◦ (Sym≤m ⊗ idE )(ξ, ∇ πE ξ, . . . , ∇ M,πE ,m ξ ) = Sym≤m ⊗ idVE (ξ, ∇ πE ξ, . . . , ∇ M,πE ,m ξ ) = (Sym≤m ⊗ idE ) ◦ Am (ξ, ∇ πE ξ, . . . , ∇ M,πE ,m ξ ). Thus the diagram (3.39) commutes. Thus, if we define M,πE ,s M,πE ,s ˆm ξ ) = (Symm ⊗ idE ) ◦ Am ξ ), A s ((Syms ⊗ idE ) ◦ ∇ s (∇

.

(3.40)

then we have (Symm ⊗ idE ) ◦ ∇ M,πE ,m ξ =

m Σ

.

M,πE ,s ˆm ξ ), A s ((Syms ⊗ idE ) ◦ ∇

s=0

as desired.

□

.

The previous lemma has an “inverse” which we state next. Lemma 3.70 (Iterated Symmetrised Covariant Differentials of Sections with Respect to Different Connections II) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. Consider .Cr -affine connections .∇ M and .∇ M on .M, and .Cr linear connections .∇ πE and .∇ πE in .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle

198

3 Geometry: Lifts and Differentiation of Tensors

mappings r s ∗ m ∗ ˆm (B s , idE ) ∈ VB (T (T M ⊗ E); T (T M) ⊗ E),

.

s ∈ {0, 1, . . . , m},

such that (Symm ⊗ idE ) ◦ ∇ M,πE ,m ξ =

m Σ

.

M,πE ,s ˆm ξ) B s ((Syms ⊗ idE ) ◦ ∇

s=0

for all .ξ ∈ Г m (E). Proof The proof here is identical with the proof of Lemma 3.69, making the obvious .□ notational transpositions. The preceding four lemmata combine to give the following result. Lemma 3.71 (Decompositions of Jets of Sections with Respect to Different Connections) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. Consider .Cr -affine connections .∇ M and .∇ M on .M, and .Cr -linear connections .∇ πE and .∇ πE in .E. For .m ∈ Z≥0 , there exist .Cr -vector bundle mappings Am ∈ VBr (Jm E; S≤m (T∗ M) ⊗ E),

.

B m ∈ VBr (Jm E; S≤m (T∗ M) ⊗ E),

defined by Am (jm ξ(x)) = Sym≤m ⊗ idE (ξ(x), ∇ πE ξ(x), . . . , ∇ M,πE ,m ξ(x)),

.

B m (jm ξ(x)) = Sym≤m ⊗ idVE (ξ(x), ∇ πE ξ(x), . . . , ∇ M,πE ,m ξ(x)). Moreover, .Am and .B m are isomorphisms, and B m ◦ (Am )−1 ◦ (Sym≤m ⊗ idE )(ξ(x), ∇ πE ξ(x), . . . , ∇ M,πE ,m ξ(x)) ⎛ 1 Σ ˆ1s ((Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(x)), . . . , = ξ(x), A

.

s=0 m Σ s=0

⎞ ˆm A s ((Syms

⊗ idE ) ◦ ∇

M,πE ,s

ξ(x))

3.3 Isomorphisms Defined by Lifts and Pull-Backs

199

and Am ◦ (B m )−1 ◦ (Sym≤m ⊗ idE )(ξ(x), ∇ πE ξ(x), . . . , ∇ M,πE ,m ξ(x)) ⎛ 1 Σ ˆ1s ((Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(e)), . . . , = ξ(x), B

.

s=0 m Σ

⎞ ˆm B s ((Syms

⊗ idE ) ◦ ∇

M,πE ,s

ξ(x)) ,

s=0

ˆm ˆm where the vector bundle mappings .A s and .B s , .s ∈ {0, 1, . . . , m}, are as in Lemmata 3.69 and 3.70.

Chapter 4

Analysis: Norm Estimates for Derivatives

In this chapter, we carry out the principal analytical constructions that are needed to prove the continuity results of Chap. 5. The main objective is to obtain useful bounds for the fibre norms for the tensors obtained in Sect. 3.3 that relate derivatives on the total space of a vector bundle to derivatives on the base space. This bound is given in Lemma 4.17. When we say “usefulness” of the bounds, the property of usefulness is that it permits one to prove the continuity results of Chap. 5. Another measure of “usefulness” of the norm estimates we obtain is that they permit us to show that the topologies we have defined are independent of the data—namely the metrics and connections—used to define them. This important analysis is made in Sect. 4.3. Our geometric constructions in Chap. 3 were made in both the smooth and real analytic setting. In this chapter, a number of results require real analyticity and/or give conclusions that are really only interesting in the real analytic setting. In most such cases, however, there are analogous (and significantly easier) results in the smooth setting. We will occasionally allude to these analogous results.

4.1 Fibre Norms for Some Useful Jet Bundles In Sect. 3.3 we saw how to make decompositions for jets of sections of vector bundles and jets of various lifts to the total space of a vector bundle .πE : E → M, using the Levi-Civita affine connection induced by a natural Riemannian metric on .E. In this section we consider fibre norms for these jet bundles. The fibre norm for the space of jets of sections of a vector bundle is deduced in a natural way from a Riemannian metric on .M and a fibre metric in .πE : E → M, as we saw in Sect. 2.3.2. For fibre norms of lifted objects, the story is more complicated. Since the objects are lifted from .M, there are two natural fibre norms in each case, one coming from the Riemannian metric on .E (for the jets of the lifted objects), and the other coming from the Riemannian metric on .M and the fibre metric on the vector bundle (for the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. D. Lewis, Geometric Analysis on Real Analytic Manifolds, Lecture Notes in Mathematics 2333, https://doi.org/10.1007/978-3-031-37913-0_4

201

202

4 Analysis: Norm Estimates for Derivatives

jets of the unlifted objects). It is important to understand how these fibre metrics are related, and that is the purpose of this section. The setup is the following. We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr vector bundle. We consider a Riemannian metric .GM on .M, a fibre metric .GπE on .E, the Levi-Civita connection .∇ M on .M, and a vector bundle connection .∇ πE in .E, all being of class .Cr . This gives the Riemannian metric .GE of (3.5) and the associated Levi-Civita connection .∇ E . This data gives the fibre metrics for all sorts of tensors defined on the total space .E. We, however, are interested only in the lifted tensors such as are described in Sect. 3.1. The reader will definitely observe a certain repetitiveness to our constructions in this section, rather similar to that seen in Sect. 3.3. However, the ideas here are important and the notation is confusing, so we do not skip anything. Additionally, a few of the cases we consider require considerations different from the majority of the other cases, so it is worth highlighting these differences. While the results in this section are most important in the real analytic setting, we treat the smooth and real analytic cases simultaneously, as we have done in Chap. 3.

4.1.1 Fibre Norms for Horizontal Lifts of Functions We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. For .f ∈ Cr (M), we have .πE∗ f ∈ Cr (E). We can, therefore, think of the m-jet of .πE∗ f as being characterised by .jm f , as well as by .jm πE∗ f , and of comparing these two characterisations. Thus we have the two fibre norms ‖jm f (x)‖2GM,m =

m Σ

.

j =0

1 ‖∇ M,j f (x)‖2GM (j !)2

and ∗ 2 .‖jm π f (e)‖G E E,m

=

m Σ j =0

1 ‖∇ E,j πE∗ f (e)‖2GE . (j !)2

(4.1)

These fibre norms can be related by virtue of Lemma 3.28. To do so, we make use of the following lemma. Lemma 4.1 (Fibre Norms for Horizontal Lifts of Functions) ‖πE∗ ∇ M,m f (e)‖GE = ‖∇ M,m f (πE (e))‖GM .

.

∗ Proof We have the fibre metric .G−1 E on .T E associated with the Riemannian metric .GE . The subbundles .H∗ E and .V∗ E are .G−1 E -orthogonal. We note that ∗ ∗ ∗ .Te πE : T πE (e) M → He E is an isometry. Thus we have the formula

4.1 Fibre Norms for Some Useful Jet Bundles

203

‖πE∗ B‖GE = ‖B‖GM ,

B ∈ ┌ 0 (Tm (T∗ M)),

.

and the assertion of the lemma is merely a special case of this formula.

□

.

We note that the fibre norm (4.4) makes use of the vector bundle mapping B∇mE ∈ VBr (P∗m E; S≤m (T∗ E))

.

from Lemma 3.28. If instead we use the vector bundle mapping Am ∈ VBr (P∗m E; S≤m (πE∗ T∗ M)) ∇E

.

from Lemma 3.28, then we have the alternative fibre norm ‖jm πE∗ f (e)‖'2 GE,m =

m Σ

.

j =0

Σ 1 1 ∗ M,j 2 ‖π ∇ f (e)‖ = ‖∇ M,j f (πE (e))‖2GM . G E E (j !)2 (j !)2 m

j =0

The relationship between the fibre norms .‖·‖GE,m and .‖·‖'GE,m can be phrased as, “What is the relationship between the norms of the jet of the lift and the lift of the jet?” This is a question we will phrase below for other sorts of lifts, and will address comprehensively when we prove the continuity of the various lifting operations in Sect. 5.3.

4.1.2 Fibre Norms for Vertical Lifts of Sections We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. For .ξ ∈ ┌ r (E), we have .ξ v ∈ ┌ r (TE). We can, therefore, think of the m-jet of .ξ v as being characterised by .jm ξ , as well as by .jm ξ v , and of comparing these two characterisations. Thus we have the two fibre norms ‖jm ξ(x)‖2GM,π

.

E ,m

=

m Σ j =0

1 ‖∇ M,πE ,j ξ(x)‖2GM,π E (j !)2

and ‖jm ξ v (e)‖2GE,m =

m Σ

.

j =0

1 ‖∇ E,j ξ v (e)‖2GE . (j !)2

(4.2)

These fibre norms can be related by virtue of Lemma 3.33. To do so, we make use of the following lemma. Lemma 4.2 (Fibre Norms for Vertical Lifts of Sections)

204

4 Analysis: Norm Estimates for Derivatives

‖(∇ M,πE ,m ξ )v (e)‖GE = ‖∇ M,πE ,m ξ(πE (e))‖GM,π .

.

E

Proof The subbundles .HE and .VE are .GE -orthogonal and the subbundles .H∗ E and .V∗ E are .G−1 E -orthogonal. We note that the identification .Ve E ≃ EπE (e) is an isometry and that .Te∗ πE : T∗πE (e) M → H∗e E is an isometry. Thus we have the formula ‖B v ‖GE = ‖B‖GM,π ,

.

E

B ∈ ┌ 0 (Tm (T∗ M) ⊗ E),

and the assertion of the lemma is merely a special case of this formula.

□

.

We note that the fibre norm (4.2) makes use of the vector bundle mapping B∇mE ∈ VBr (P∗m E ⊗ VE; S≤m (T∗ E) ⊗ VE)

.

from Lemma 3.33. If instead we use the vector bundle mapping Am ∈ VBr (P∗m E ⊗ VE; S≤m (πE∗ T∗ M) ⊗ VE) ∇E

.

from Lemma 3.33, then we have the alternative fibre norm ‖jm ξ v (e)‖'2 GE,m =

m Σ

.

j =0

1 ‖(∇ M,πE ,j ξ )v (e)‖2GE (j !)2 =

m Σ j =0

1 ‖∇ M,πE ,j ξ(πE (e))‖2GM,π . E (j !)2

Again, this points out the matter of the relationship between the norms of the jet of a lift versus the lift of the jet, and this matter will be considered in detail in the continuity results of Sect. 5.3.

4.1.3 Fibre Norms for Horizontal Lifts of Vector Fields We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. For .X ∈ ┌ r (TM), we have .Xh ∈ ┌ r (TE). We can, therefore, think of the m-jet of .Xh as being characterised by .jm X, as well as by .jm Xh , and of comparing these two

4.1 Fibre Norms for Some Useful Jet Bundles

205

characterisations. Thus we have the two fibre norms ‖jm X(x)‖2GM,m =

m Σ

.

j =0

1 ‖∇ M,j X(x)‖2GM (j !)2

and ‖jm X

.

h

(e)‖2GE,m

=

m Σ j =0

1 ‖∇ E,j Xh (e)‖2GE . (j !)2

(4.3)

These fibre norms can be related by virtue of Lemma 3.38. To do so, we make use of the following lemma. Lemma 4.3 (Fibre Norms for Horizontal Lifts of Vector Fields) ‖(∇ M,m X)h (e)‖GE = ‖∇ M,m X(πE (e))‖GM .

.

Proof The subbundles .HE and .VE are .GE -orthogonal. We note that the identification .He E ≃ TπE (e) M is an isometry and that .Te∗ πE : T∗πE (e) M → H∗e E is an isometry. Thus we have the formula ‖B h ‖GE = ‖B‖GM ,

.

B ∈ ┌ 0 (Tm (T∗ M) ⊗ TM),

and the assertion of the lemma is merely a special case of this formula.

□

.

We note that the fibre norm (4.3) makes use of the vector bundle mapping B∇mE ∈ VBr (P∗m E ⊗ HE; S≤m (T∗ E) ⊗ HE)

.

from Lemma 3.38. If instead we use the vector bundle mapping Am ∈ VBr (P∗m E ⊗ HE; S≤m (πE∗ T∗ M) ⊗ HE) ∇E

.

from Lemma 3.38, then we have the alternative fibre norm ‖jm Xh (e)‖'2 GE,m =

m Σ

.

j =0

Σ 1 1 ‖(∇ M,j X)h (e)‖2GE = ‖∇ M,j X(πE (e))‖2GM . 2 (j !) (j !)2 m

j =0

Again, this points out the matter of the relationship between the norms of the jet of a lift versus the lift of the jet, and this matter will be considered in detail in the continuity results of Sect. 5.3.

206

4 Analysis: Norm Estimates for Derivatives

4.1.4 Fibre Norms for Vertical Lifts of Dual Sections We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. For .λ ∈ ┌ r (E∗ ), we have .λv ∈ ┌ r (T∗ E). We can, therefore, think of the m-jet of .λv as being characterised by .jm λ, as well as by .jm λv , and of comparing these two characterisations. Thus we have fibre norms ‖jm λ(x)‖2GM,π

.

E ,m

=

m Σ j =0

1 ‖∇ M,πE ,j λ(x)‖2GM,π E (j !)2

and ‖jm λv (e)‖2GE,m =

m Σ

.

j =0

1 ‖∇ E,j λv (e)‖2GE . (j !)2

(4.4)

These fibre norms can be related by virtue of Lemma 3.43. To do so, we make use of the following lemma. Lemma 4.4 (Fibre Norms for Vertical Lifts of Dual Sections) ‖(∇ M,πE ,m λ)v (e)‖GE = ‖∇ M,πE ,m λ(πE (e))‖GM,π .

.

E

Proof The subbundles .H∗ E and .V∗ E are .G−1 E -orthogonal. We note that the identification .V∗e E ≃ E∗πE (e) is an isometry and that .Te∗ πE : T∗πE (e) M → H∗e E is an isometry. Thus we have the formula ‖B v ‖GE = ‖B‖GM,π ,

.

E

B ∈ ┌ 0 (Tm (T∗ M) ⊗ E∗ ),

and the assertion of the lemma is merely a special case of this formula. We note that the fibre norm (4.4) makes use of the vector bundle mapping B∇mE ∈ VBr (P∗m E ⊗ V∗ E; S≤m (T∗ E) ⊗ V∗ E)

.

from Lemma 3.43. If instead we use the vector bundle mapping Am ∈ VBr (P∗m E ⊗ V∗ E; S≤m (πE∗ T∗ M) ⊗ V∗ E) ∇E

.

from Lemma 3.43, then we have the alternative fibre norm ‖jm λv (e)‖'2 GE,m =

m Σ

.

j =0

1 ‖(∇ M,πE ,j λ)v (e)‖2GE (j !)2 =

m Σ j =0

1 ‖∇ M,πE ,j λ(πE (e))‖2GM,π . E (j !)2

□

.

4.1 Fibre Norms for Some Useful Jet Bundles

207

Again, this points out the matter of the relationship between the norms of the jet of a lift versus the lift of the jet, and this matter will be considered in detail in the continuity results of Sect. 5.3.

4.1.5 Fibre Norms for Vertical Lifts of Endomorphisms We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. For .L ∈ ┌ r (T11 (E)), we have .Lv ∈ ┌ r (T11 (E)). We can, therefore, think of the m-jet of .Lv as being characterised by .jm L, as well as by .jm Lv , and of comparing these two characterisations. Thus we have the two fibre norms ‖jm L(x)‖2GM,π

.

E ,m

=

m Σ j =0

1 ‖∇ M,πE ,j L(x)‖2GM,π E (j !)2

and ‖jm L

.

v

(e)‖2GE,m

=

m Σ j =0

1 ‖∇ E,j Lv (e)‖2GE . (j !)2

(4.5)

These fibre norms can be related by virtue of Lemma 3.48. To do so, we make use of the following lemma. Lemma 4.5 (Fibre Norms for Vertical Lifts of Endomorphisms) ‖(∇ M,πE ,m L)v (e)‖GE = ‖∇ M,πE ,m L(πE (e))‖GM,π .

.

E

Proof The subbundles .H∗ E and .V∗ E are .G−1 E -orthogonal. We note that the identifications .Ve E ≃ EπE (e) and .V∗e E ≃ E∗πE (e) are isometries, and that ∗ ∗ ∗ .Te πE : T πE (e) M → He E is an isometry. Thus we have the formula ‖B v ‖GE = ‖B‖GM,π ,

.

E

B ∈ ┌ 0 (Tm (T∗ M) ⊗ T11 (E)),

and the assertion of the lemma is merely a special case of this formula. We note that the fibre norm (4.5) makes use of the vector bundle mapping B∇mE ∈ VBr (P∗m E ⊗ T11 (VE); S≤m (T∗ E) ⊗ T11 (VE))

.

from Lemma 3.48. If instead we use the vector bundle mapping Am ∈ VBr (P∗m E ⊗ T11 (VE); S≤m (πE∗ T∗ M) ⊗ T11 (VE)) ∇E

.

□

.

208

4 Analysis: Norm Estimates for Derivatives

from Lemma 3.48, then we have the alternative fibre norm ‖jm Lv (e)‖'2 GE,m =

m Σ

.

j =0

1 ‖(∇ M,πE ,j L)v (e)‖2GE (j !)2 =

m Σ j =0

1 ‖∇ M,πE ,j L(πE (e))‖2GM,π . E (j !)2

Again, this points out the matter of the relationship between the norms of the jet of a lift versus the lift of the jet, and this matter will be considered in detail in the continuity results of Sect. 5.3.

4.1.6 Fibre Norms for Vertical Evaluations of Dual Sections We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. For .λ ∈ ┌ r (E∗ ), we have .λe ∈ Cr (E). We can, therefore, think of the m-jet of .λe as being characterised by .jm λ, as well as by .jm λe , and of comparing these two characterisations. Thus we have the two fibre norms ‖jm λ(x)‖2GM,π

.

E ,m

=

m Σ j =0

1 ‖∇ M,πE ,j λ(x)‖2GM,π E (j !)2

and ‖jm λe (e)‖2GE,m =

m Σ

.

j =0

1 ‖∇ E,j λe (e)‖2GE . (j !)2

(4.6)

These fibre norms can be related by virtue of Lemma 3.53. To do so, we make use of the following lemma. Lemma 4.6 (Fibre Norms for Vertical Evaluations of Dual Sections) ‖(∇ M,πE ,m λ)e (e)‖GE = ‖∇ M,πE ,m λ(πE (e))(e)‖GM,π .

.

E

Proof The subbundles .H∗ E and .V∗ E are .G−1 E -orthogonal. We note that the identification .V∗e E ≃ E∗πE (e) is an isometry, and that .Te∗ πE : T∗πE (e) M → H∗e E is an isometry. Thus we have the formula ‖B e (e)‖GE = ‖B(πE (e))(e)‖GM,π ,

.

E

B ∈ ┌ 0 (Tm (T∗ M) ⊗ E∗ ),

and the assertion of the lemma is merely a special case of this formula.

□

.

4.1 Fibre Norms for Some Useful Jet Bundles

209

We note that the fibre norm (4.6) makes use of the vector bundle mapping B∇mE ∈ VBr (P∗m E; S≤m (T∗ E))

.

from Lemma 3.53. If instead we use the vector bundle mapping Am ∈ VBr (P∗m E; S≤m (πE∗ T∗ M)) ∇E

.

from Lemma 3.53, then we have the alternative fibre norm ‖jm λe (e)‖'2 GE,m =

m Σ

.

j =0

1 ‖(∇ M,πE ,j λ)e (e)‖2GE (j !)2 =

m Σ j =0

1 ‖∇ M,πE ,j λ(πE (e))(e)‖2GM,π . E (j !)2

Again, this points out the matter of the relationship between the norms of the jet of a lift versus the lift of the jet, and this matter will be considered in detail in the continuity results of Sect. 5.3.

4.1.7 Fibre Norms for Vertical Evaluations of Endomorphisms We let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. For .L ∈ ┌ r (T11 (E)), we have .Le ∈ ┌ r (TE). We can, therefore, think of the m-jet of .Le as being characterised by .jm L, as well as by .jm Le , and of comparing these two characterisations. Thus we have the two fibre norms ‖jm L(x)‖2GM,π

.

E ,m

=

m Σ j =0

1 ‖∇ M,πE ,j L(x)‖2GM,π E (j !)2

and ‖jm Le (e)‖2GE,m =

m Σ

.

j =0

1 ‖∇ E,j Le (e)‖2GE . (j !)2

(4.7)

These fibre norms can be related by virtue of Lemma 3.58. To do so, we make use of the following lemma. Lemma 4.7 (Fibre Norms for Vertical Evaluations of Endomorphisms) ‖(∇ M,πE ,m L)e (e)‖GE = ‖∇ M,πE ,m L(πE (e))(e)‖GM,π .

.

E

210

4 Analysis: Norm Estimates for Derivatives

Proof The subbundles .H∗ E and .V∗ E are .G−1 E -orthogonal. We note that the identification .V∗e E ≃ E∗πE (e) is an isometry and that .Te∗ πE : T∗πE (e) M → H∗e E is an isometry. Thus we have the formula ‖B e (e)‖GE = ‖B(πE (e))(e)‖GM,π ,

.

E

B ∈ ┌ 0 (Tm (T∗ M) ⊗ T11 (E)),

and the assertion of the lemma is merely a special case of this formula.

□

.

We note that the fibre norm (4.7) makes use of the vector bundle mapping B∇mE ∈ VBr (P∗m E ⊗ VE; S≤m (T∗ E) ⊗ VE)

.

from Lemma 3.58. If instead we use the vector bundle mapping Am ∈ VBr (P∗m E ⊗ VE; S≤m (πE∗ T∗ M) ⊗ VE) ∇E

.

from Lemma 3.58, then we have the alternative fibre norm ‖jm Le (e)‖'2 GE,m =

m Σ

.

j =0

Σ 1 1 M,πE ,j e 2 ‖(∇ L) (e)‖ = ‖∇ M,πE ,j L(e)‖2GM,π . G E E (j !)2 (j !)2 m

j =0

Again, this points out the matter of the relationship between the norms of the jet of a lift versus the lift of the jet, and this matter will be considered in detail in the continuity results of Sect. 5.3.

4.1.8 Fibre Norms for Pull-Backs of Functions We let .r ∈ {∞, ω} and let .M and .N be .Cr -manifolds, and let .Ф ∈ Cr (M; N). For r m ∗ ∗ .f ∈ C (N), we have .Ф f ∈ C (M). We can, therefore, think of the m-jet of .Ф f ∗ as being characterised by .jm f , as well as by .jm (Ф f ), and of comparing these two characterisations. Thus we have the two fibre norms ‖jm f (x)‖2GN,m =

m Σ

.

j =0

1 ‖∇ N,j f (x)‖2GN (j !)2

and ‖jm Ф∗ f (e)‖2GM,m =

m Σ

.

j =0

1 ‖∇ M,j Ф∗ f (e)‖2GM . (j !)2

(4.8)

These fibre norms can be related by virtue of Lemma 3.65. To make use of this relationship, we shall also need to relate the norms of the terms in these expressions.

4.1 Fibre Norms for Some Useful Jet Bundles

211

In the preceding sections, this was easy to do since the Riemannian metric on .E was related in a specific way to the Riemannian metric on .M and the fibre metric in .E. Here, this is not so simple since, if we choose a Riemannian metric .GM on .M and a Riemannian metric .GN on .N, these will be have no useful pointwise relationship. So, rather than getting an equality between certain norms, the best we can achieve (and all that we need) is a useful bound, and this is the content of the next lemma. Lemma 4.8 (Fibre Norms for Pull-Backs of Functions) For a compact set .K ⊆ M: (i) there exists .C ∈ R>0 such that ‖Ф∗ ∇ N,m f (x)‖GM ≤ C m ‖∇ N,m f (Ф(x))‖GN ,

.

x ∈ K, m ∈ Z≥0 ;

(ii) if .Ф is a submersion or an injective immersion, then C from part (i) can be chosen so that it also holds that ‖∇ N,m f (Ф(x))‖GN ≤ C m ‖Ф∗ ∇ N,m f (x)‖GM ,

.

x ∈ K, m ∈ Z≥0 .

Proof The essential part of the proof is the following linear algebraic sublemma. Sublemma 1 Let .(U, GU ) and .(V, GV ) be finite-dimensional .R-inner product spaces and let .Ф ∈ HomR (U; V). Then there exists .C ∈ R>0 such that ‖Ф∗ A‖GU ≤ C k ‖A‖GV

.

for every .A ∈ Tk (V∗ ), .k ∈ Z≥0 . If, additionally, .Ф is a surjective or injective, then C can be chosen so that, additionally, it holds that ‖A‖GV ≤ C k ‖Ф∗ A‖GU

.

for every .A ∈ Tk (V∗ ), .k ∈ Z≥0 . Proof Let .(f1 , . . . , fm ) and .(e1 , . . . , en ) be orthonormal bases for .U and .V with dual bases .(f 1 , . . . , f m ) and .(e1 , . . . , en ). Write n Σ

A=

.

Aj1 ···jk ej1 ⊗ . . . ⊗ ejk

j1 ,...,jk =1

and Ф=

m n Σ Σ

.

j =1 a=1

j

Фa ej ⊗ f a .

212

4 Analysis: Norm Estimates for Derivatives

Then Ф∗ A =

n Σ

m Σ

j

j

Фa11 · · · Фakk Aj1 ···jk f a1 ⊗ . . . ⊗ f ak .

.

j1 ,...,jk =1 a1 ,...,ak =1

Denote } {│ j │ │ ‖Ф‖∞ = max │Фa │ │ a ∈ {1, . . . , m}, j ∈ {1, . . . , n} .

.

We have ‖Ф∗ A‖2GU =

m Σ

.

≤

a1 ,...,ak =1 m Σ a1 ,...,ak =1

≤

m Σ a1 ,...,ak =1

⎛

⎞2

n Σ

⎝

Фa11 · · · Фakk Aj1 ···jk ⎠ j

j

j1 ,...,jk =1

⎛

⎞2 │ │ │ j1 │ j │Фa1 · · · Фakk Aj1 ···jk │⎠

n Σ

⎝

j1 ,...,jk =1

⎛

n Σ

⎝

⎞⎛ │ │2 │ j1 j │ │Фa1 · · · Фakk │ ⎠ ⎝

j1 ,...,jk =1

n Σ

⎞ │ │2 │Aj ···j │ ⎠ k 1

j1 ,...,jk =1

≤ (nm‖Ф‖2∞ )k ‖A‖2GV , using Cauchy–Schwartz. The first part of the result follows by taking .C = √ nm‖Ф‖∞ . If .Ф is surjective, let .Ψ ∈ HomR (V; U) be a right-inverse for .Ф. Then, by the first part of the result, there exists .C ∈ R>0 such that ‖A‖GV = ‖(Ф ◦ Ψ)∗ A‖GV = ‖Ψ ∗ Ф∗ A‖GV ≤ C k ‖Ф∗ A‖GU

.

for every .A ∈ Tk (V∗ ), .k ∈ Z≥0 . If .Ф is injective, we choose the orthonormal basis .(e1 , . . . , en ) so that .(e1 , . . . , em ) is a basis for .image(Ф). In this case we have Ф=

m Σ

.

Фba eb ⊗ f a ,

a,b=1

where the .m × m matrix with components .Фba , .a, b ∈ {1, . . . , m}, is invertible, and Ф∗ A =

m Σ

m Σ

.

b1 ,...,bk =1 a1 ,...,ak =1

Фba11 · · · Фbakk Ab1 ···bk f a1 ⊗ . . . ⊗ f ak .

4.1 Fibre Norms for Some Useful Jet Bundles

213

Letting .Ψab , .a, b ∈ {1, . . . , m}, be defined by ⌠ Ψac Фbc =

1,

a = b,

0,

a /= b,

.

we have A=

m Σ

m Σ

.

b1 ,...,bk =1 a1 ,...,ak =1

Ψab11 · · · Ψabkk (Ф∗ A)b1 ···bk ea1 ⊗ . . . ⊗ eak ,

and the conclusion in this case follows just as in the proof of the first part of the sublemma. . Δ (i) Let .x ∈ K and take .Cx ∈ R>0 as in the sublemma such that ‖Ф∗ ∇ N,m f (x)‖GM ≤ Cxm ‖∇ N,m f (Ф(x))‖GN ,

.

m ∈ Z≥0 .

By continuity, and noting the exact form of the constant C from the sublemma (i.e., depending on the size of the derivative of .Tx Ф), there exists a neighbourhood .Ux of x such that ‖Ф∗ ∇ N,m f (x ' )‖GM ≤ (2Cx )m ‖∇ N,m f (Ф(x ' ))‖GN ,

.

x ' ∈ Ux , m ∈ Z>0 .

Then take .x1 , . . . , xk ∈ K such that .K ⊆ ∪kj =1 Uxj . The first part of the lemma then follows by taking C = max{2Cx1 , . . . , 2Cxk }.

.

(ii) Let .x ∈ K and choose coordinate charts .(Ux , χ x ) and .(Vx , ηx ) about x and Ф(x) so that the local representative of .Ф is linear [1, Theorems 3.5.2 and 3.5.7]. One then has two Riemannian metrics, .GM and the Euclidean metric, on .Ux and two Riemannian metrics, .GN and the Euclidean metric, on .Vx . By the second assertion of sublemma and by Lemma 4.19 below, and by shrinking the chart domains .Ux and .Vx appropriately, one has .

‖∇ N,m f (Ф(x ' ))‖GN ≤ (2Cx )m ‖Ф∗ ∇ N,m f (x ' )‖GM ,

.

x ' ∈ Ux , m ∈ Z>0 .

A compactness argument as in the proof of part (i) gives this part of the result. We note that the fibre norm (4.8) makes use of the vector bundle mapping ≤m ∗ B∇mE ∈ VBr (T∗m (T M)) Ф M; S

.

□

.

214

4 Analysis: Norm Estimates for Derivatives

from Lemma 3.65. If instead we use the vector bundle mapping ≤m Am ∈ VBr (T∗m (Ф∗ T∗ N)) Ф M; S ∇E

.

from Lemma 3.65, then we have the alternative fibre norm ‖jm Ф∗ f (e)‖'2 GM,m =

m Σ

.

j =0

1 ‖Ф∗ ∇ N,j f (e)‖2GM . (j !)2

The relationship between the fibre norms .‖·‖GM,m and .‖·‖'GM,m can be phrased as, “What is the relationship between the norms of the jet of the pull-back and the pullback of the jet?”

4.2 Estimates Related to Jet Bundle Norms In Sect. 3.3 we gave formulae relating derivatives of geometric objects to derivatives of their lifts, and vice versa. In Sect. 4.1 we defined fibre metrics associated with spaces of derivatives of lifted objects. In each of the multitude of constructions, m m m there arose certain vector bundle mappings—denoted by .Am s , .Bs , .Cs , and .Ds , .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m}—that satisfied recursion relations. In Lemma 4.17, we will establish quite specific bounds for these vector bundle mappings. We will use these bounds in two ways. ω , .K ⊆ 1. In Sect. 4.3 we prove that the .Cω -topology defined by the seminorms .pK,a M compact, .a ∈ c0 (Z≥0 ; R>0 ), is independent of the choices of metrics and connections used to define these seminorms. To be clear, this does not follow from Theorem 2.19 since in the proof of that theorem we relied in an essential way on Proposition 2.18, and our proof of that proposition relied in an essential way on the results of Lemma 4.22 and Theorem 4.24, which themselves rely crucially on Lemma 4.17. Thus our work in this section—and hence all of the work that leads up to it—lie at the core of the soundness of the approach of this book. 2. In Chap. 5 we will prove a variety of continuity results for basic operations in differential geometry. As we shall see, the bounds of Lemma 4.17 are used in these proofs in a routine way.

The main results in this section are important, but somewhat elaborate. Moreover, the bounds of Sect. 4.2.2 require, for the first time, real analyticity.

4.2 Estimates Related to Jet Bundle Norms

215

4.2.1 Algebraic Estimates To work with the topologies we presented in Sect. 2.4, we will have to compute and estimate high-order derivatives of various sorts of tensors. In this section we collect the purely linear algebraic estimates that we shall need. All norms on tensor products are those induced by an inner product as in Lemma 2.16. For simplicity, therefore, we shall often omit any particular symbols attached to “.‖·‖” to connote which norm we are talking about; all vector spaces have a unique norm (given the data) that we shall use. We start by giving the norm of the identity mapping. Lemma 4.9 (Norm of the Identity Map)√If .V is a finite-dimensional .R-vector space with inner product .G, then .‖idV ‖ = dimR (V). Proof Let .(e1 , . . . , en ) be an orthonormal basis for .V with dual basis .(e1 , . . . , en ) the dual basis. Write .

idV =

n n Σ Σ

δjk ek ⊗ ej .

j =1 k=1

We have ‖A‖2 =

n n Σ Σ

.

(δjk )2 = n,

j =1 k=1

□

as claimed.

.

Next we consider the norm of the tensor product of linear maps. Lemma 4.10 (Norms of Tensor Products) Let .U, .V, .W, and .X be finitedimensional .R-vector spaces with inner products. Then, for .A ∈ HomR (U; V) and .B ∈ HomR (W; X), ‖A ⊗ B‖ = ‖A‖‖B‖.

.

Proof Let .(e1 , . . . , en ), .(f1 , . . . , fm ), .(g1 , . . . , gk ), and .(h1 , . . . , hl ) be orthonormal bases for .U, .V, .W, and .X, respectively. Let .(e1 , . . . , en ), .(f 1 , . . . , f m ), 1 k 1 l .(g , . . . , g ), and .(h , . . . , h ) be the dual bases. Write A=

m n Σ Σ

.

j =1 a=1

Aaj fa ⊗ ej ,

B=

l k Σ Σ i=1 b=1

Bib hb ⊗ g i

216

4 Analysis: Norm Estimates for Derivatives

so that A⊗B =

k Σ m Σ l n Σ Σ

.

Aaj Bib (fa ⊗ hb ) ⊗ (ej ⊗ g i ).

j =1 i=1 a=1 b=1

Then ‖A ⊗ B‖2 =

.

k Σ m Σ l n Σ Σ │ a b │2 │A B │ j

i

j =1 i=1 a=1 b=1

⎞⎛ ⎛ ⎞ m l n Σ k Σ Σ Σ │ a │2 │ b │2 │ │ │ │ ⎠ ⎝ Aj Bi = = ‖A‖2 ‖B‖2 , j =1 a=1

i=1 b=1

□

as claimed.

.

Our next estimate concerns the relationship between norms of tensors evaluated on arguments. Lemma 4.11 (Norm of Tensor Evaluation) Let .U and .V be finite-dimensional .Rvector spaces with inner products .G and .H, respectively. Then ‖L(u)‖ ≤ ‖L‖ ‖u‖

.

for all linear mappings .L ∈ HomR (U; V) and for all .u ∈ U. Proof Let .(f1 , . . . , fm ) and .(e1 , . . . , en ) be orthonormal bases for .U and .V with (f 1 , . . . , f m ) and .(e1 , . . . , en ) their dual bases. For .L ∈ HomR (U; V), write

.

L=

n m Σ Σ

.

j

La ej ⊗ f a .

a=1 j =1

Then we compute, using Cauchy–Schwarz, ‖L(u)‖2 =

⎛ m n Σ Σ

.

j =1

≤

n Σ j =1

a=1

⎞2 j La ua

⎞2 ⎛ m n Σ Σ│ j │ │La ua │ ≤ j =1

a=1

⎛ m ⎞⎛ m ⎞ Σ │ j │2 Σ│ │2 a │La │ │u │ = ‖L‖2 ‖u‖2 , a=1

a=1

giving the lemma. We shall also make use of a sort of “reverse inequality” related to the above.

□

.

4.2 Estimates Related to Jet Bundle Norms

217

Lemma 4.12 (Upper Bound for Norm of Linear Map) Let .U and .V be finite-dimensional .R-vector spaces with inner products .GU and .GV . For .L ∈ HomR (U; V), √ .‖L‖ ≤ dimR (U) sup{‖L(u)‖ | ‖u‖ = 1}. Proof The result is true with equality and without the constant if one uses the induced norm for .HomR (U; V), rather than the tensor norm as we do here. So the statement of the lemma is really about relating the induced norm with the tensor norm. The tensor norm, in the case of linear mappings as we have here, is really the 2 Frobenius norm, and as such √ it is computed as the .l -norm of the vector of the set ∞ T of .dimR (U) eigenvalues of . L ◦ L. On the other √ hand, the induced norm is the .l norm of this same vector of eigenvalues of . LT ◦ L. These interpretations can be found in [6, page 7]. For this reason, an application of (1.4) gives the result. .□ Another tensor estimate we shall find useful concerns symmetrisation. Lemma 4.13 (Norms of Symmetrised Tensors) Let .V be a finite-dimensional .Rvector space and let .G be an inner product on .V. Then ‖Symk (A)‖ ≤ ‖A‖

.

for every .A ∈ Tk (V∗ ) and .k ∈ Z>0 . Proof The result follows from the following sublemma. Sublemma 1 The map .Symk : Tk (V∗ ) → Sk (V∗ ) is the orthogonal projection. Proof Let us simply denote by .G the inner product on .Tk (V∗ ), defined as in Lemma 2.16. It suffices to show that .G(A, S) = G(Symk (A), S) for every .A ∈ Tk (V∗ ) and .S ∈ Sk (V∗ ). It also suffices to show that this is true as A runs over a set of generators for .Tk (V∗ ) and S runs over a set of generators for .Sk (V∗ ). Thus we let .(e1 , . . . , en ) be an orthonormal basis for .V with dual basis 1 n .(e , . . . , e ). Then we have generators e a1 ⊗ . . . ⊗ e ak ,

.

a1 , . . . , ak ∈ {1, . . . , n},

for .Tk (V∗ ) and .

Symk (eb1 ⊗ . . . ⊗ ebk ),

b1 , . . . , bk ∈ {1, . . . , n},

218

4 Analysis: Norm Estimates for Derivatives

for .Sk (V∗ ). For .a1 , . . . , ak , b1 , . . . , bk ∈ {1, . . . , n}, we wish to show that the inner product G(ea1 ⊗ . . . ⊗ eak , Symk (eb1 ⊗ . . . ⊗ ebk )) 1 Σ = G(ea1 ⊗ . . . ⊗ eak , ebσ (1) ⊗ . . . ⊗ ebσ (k) ) k!

.

σ ∈Sk

=

1 Σ G(ea1 , ebσ (1) ) · · · G(eak , ebσ (k) ) k! σ ∈Sk

is equal to G(Symk (ea1 ⊗ . . . ⊗ eak ), Symk (eb1 ⊗ . . . ⊗ ebk )).

.

Unless .{a1 , . . . , ak } and .{b1 , . . . , bk } agree as multisets (i.e., they agree as sets, and also multiplicities of members of the sets agree), we have 0 = G(ea1 ⊗ . . . ⊗ eak , Symk (eb1 ⊗ . . . ⊗ ebk ))

.

= G(Symk (ea1 ⊗ . . . ⊗ eak ), Symk (eb1 ⊗ . . . ⊗ ebk )). Thus we can suppose that .{a1 , . . . , ak } and .{b1 , . . . , bk } agree as multisets. In this case, since .

Symk (ea1 ⊗ . . . ⊗ eak ) = Symk (eb1 ⊗ . . . ⊗ ebk ),

we can assume, without loss of generality, that .aj = bj , .j ∈ {1, . . . , k}. For .l ∈ {1, . . . , n}, let .kla ∈ Z≥0 be the number of occurrences of l in the list .(a1 , . . . , ak ). Let .Sak ⊆ Sk be those permutations .σ for which .aj = aσ (j ) , .j ∈ {1, . . . , k}. Note that .card(Sak ) = k1a ! · · · kna ! since .Sak consists of compositions of permutations that permute all the 1’s, all the 2’s, etc., in the list .(a1 , . . . , ak ). With these bits of notation, we have ea1 ⊗ . . . ⊗ eak = eaσ (1) ⊗ . . . ⊗ eaσ (k)

.

⇐⇒

σ ∈ Sak .

Therefore, ⌠ G(e

.

a1

⊗ ... ⊗ e ,e ak

aσ (1)

⊗ ... ⊗ e

aσ (k)

)=

1,

σ ∈ Sak ,

0,

otherwise.

4.2 Estimates Related to Jet Bundle Norms

219

We then have G(ea1 ⊗ . . . ⊗ eak , Symk (ea1 ⊗ . . . ⊗ eak ))

.

=

k a ! · · · kna ! k1a ! · · · kna ! G(ea1 ⊗ . . . ⊗ eak , ea1 ⊗ . . . ⊗ eak ) = 1 . k! k!

Next we calculate G(Symk (ea1 ⊗ . . . ⊗ eak ), Symk (ea1 ⊗ . . . ⊗ eak )).

.

σ (a)

Let .σ ∈ Sk and, for .l ∈ {1, . . . , n}, let .kl ∈ Z≥0 be the number of occurrences of σ (a) ⊆ Sk be those permutations .σ ' for which l in the list .(aσ (1) , . . . , aσ (k) ). Let .Sk σ (a) σ (a) σ (a) .aσ (j ) = aσ ' (j ) , .j ∈ {1, . . . , k}. As above, .card(S ) = k1 ! · · · kn !. Also as k above, we then have σ (a)

G(eσ (1) ⊗ . . . ⊗ eσ (k) , Symk (ea1 ⊗ . . . ⊗ eak )) =

.

k1

σ (a)

! · · · kn k!

!

=

k1a ! · · · kna ! , k!

if .k1a , . . . , kna are as in the preceding paragraph. Therefore, G(Symk (ea1 ⊗ . . . ⊗eak ), Symk (ea1 ⊗ . . . ⊗ eak )) 1 Σ = G(eσ (1) ⊗ . . . ⊗ eσ (k) , Symk (ea1 ⊗ . . . ⊗ eak )) k!

.

σ ∈Sk

=

k a ! · · · kna ! 1 Σ k1a ! · · · kna ! = 1 , k! k! k! σ ∈Sk

and so we have G(ea1 ⊗ . . . ⊗ eak , Symk (eb1 ⊗ . . . ⊗ ebk ))

.

= G(Symk (ea1 ⊗ . . . ⊗ eak ), Symk (ea1 ⊗ . . . ⊗ eak )), and the sublemma follows.

Δ

.

Now, given .A ∈ Tk (V∗ ), we write .A = Symk (A) + A1 where .A1 is orthogonal to .Sk (V∗ ). We then have .‖A‖2 = ‖Symk (A)‖2 + ‖A1 ‖2 , from which the lemma follows. .□ The sublemma from the preceding lemma is proved, differently, in [56, page 124]. We shall also require the norm of the inclusions .Δr,s defined in (3.21).

220

4 Analysis: Norm Estimates for Derivatives

Lemma 4.14 (Norm of Inclusion of Symmetric Tensors) Let .V be a finitedimensional .R-vector space and let .G be an inner product on .V. Then ‖Δr,s (A)‖ ≤

.

(r + s)! ‖A‖ r!s!

for every .A ∈ Sr+s (V∗ ) and .r, s ∈ Z≥0 . Proof Let .(e1 , . . . , en ) be an orthonormal basis for .V with .(e1 , . . . , en ) the dual basis. Write .A ∈ Sr+s (V∗ ) as n Σ

A=

Aj1 ···jn ej1 ⊗ . . . ⊗ ejr+s ,

.

j1 ,...,jr+s =1

for coefficients satisfying Ajσ (1) ···jσ (r+s) = Aj1 ···jr+s ,

.

σ ∈ Sr+s .

Then, by Lemma 3.20, Δr,s (A)(ej1 , · · · , ejr+s ) =

Σ

.

A(ejσ (1) , . . . , ejσ (r+s) )

σ ∈Sr,s

=

(r + s)! A(ej1 , . . . , ejr+s ), r!s! □

which gives the result.

.

Let us also determine the norm of various insertion operators that we shall use. We shall use notation that is specific to the manner in which we use these estimates, and this will seem unmotivated out of context. Let .U, .V, and .W be finite-dimensional 1 .R-vector spaces, let .m, s, r ∈ Z>0 and .a ∈ {0, 1, . . . , r}, let .S ∈ T r−a+2 (U), and let s+r−a+1 A ∈ Tm+a+1 (U) ⊗ W ⊗ V∗ .

.

We then have the mapping 1 IA,S,j : Ts (U∗ ) ⊗ V → Tm+r+1 (U∗ ) ⊗ W

.

defined by 1 IA,S,j (β) = A(Insj (β, S)).

.

4.2 Estimates Related to Jet Bundle Norms

221

Here we implicitly use the isomorphism κ : Tsm (U) → HomR (Ts (U∗ ); Tm (U∗ )),

.

for a finite-dimensional .R-vector space .U and for .m, s ∈ Z≥0 , via κ(v1 ⊗. . .⊗vs ⊗α 1 ⊗. . .⊗α m )(β 1 ⊗. . .⊗β s ) = · · · α 1 ⊗. . .⊗α m ,

.

for .va ∈ U, .a ∈ {1, . . . , s}, and .α j , β b ∈ U∗ , .b ∈ {1, . . . , s}, .j ∈ {1, . . . , m}. Thus, for additional finite-dimensional .R-vector spaces .V and .W, we have the identification ∗ s ∗ m ∗ Tm s (U) ⊗ W ⊗ V ≃ HomR (T (U ) ⊗ V; T (U ) ⊗ W).

.

We now have the following result. Lemma 4.15 (Norm of Composition with Tensor Insertion I) With the preceding notation, 1 ‖IA,S,j ‖ ≤ ‖A‖‖S‖.

.

Proof Let .(f1 , . . . , fm ) be an orthonormal basis for .U with dual basis (f 1 , . . . , f m ). Let .(e1 , . . . , en ) be an orthonormal basis for .V with .(e1 , . . . , en ) the dual basis. Let .(g1 , . . . , gk ) be an orthonormal basis for .W with .(g 1 , . . . , g k ) the dual basis. Let us write

.

S=

m Σ

m Σ

.

a=1 a1 ,...,ar−a+2 =1

Saa1 ···ar−a+2 fa ⊗ f a1 ⊗ . . . ⊗ f ar−a+2

and A=

m Σ

m Σ

k Σ n Σ

.

a1 ,...,as+r−a+1 =1 b1 ,...,bm+a+1 =1 α=1 l=1

a ···a

α

s+r−a+1 b1 Aba1 ···bm+a+1 ⊗ . . . ⊗ f bm+a+1 l f

⊗ fa1 ⊗ . . . ⊗ fas+1 ⊗ gα ⊗ el . We then have, for .a1 , . . . , as ∈ {1, . . . , m}, .α ∈ {1, . . . , k}, and .l ∈ {1, . . . , n}, InsS,j (f a1 ⊗ . . . ⊗ f as ⊗ gα ⊗ el )

.

= Insj (f a1 ⊗ . . . ⊗ f aj ⊗ . . . ⊗ f as ⊗ gα ⊗ el , S) =

m Σ b1 ,...,br−a+2 =1

a

Sb1j ···br−a+2 f a1 ⊗ . . . ⊗ f aj −1 ⊗ f b1 ⊗ . . . ⊗ f br−a+2

222

4 Analysis: Norm Estimates for Derivatives

⊗ f aj +1 ⊗ . . . ⊗ f as ⊗ gα ⊗ el m Σ

=

m Σ

n Σ

k Σ n Σ

c1 ,...,cj −1 =1 cj +1 ,...,cs =1 b1 ,...,br−a+2 =1 β=1 p=1 a

a

Sb1j ···br−a+2

a

−1 +1 × δca11 · · · δcjj−1 δcjj+1 · · · δcass δαβ δpl f c1 ⊗ . . . ⊗ f cj −1

⊗ f b1 ⊗ . . . ⊗ f br−a+2 ⊗ f cj +1 ⊗ . . . ⊗ f cs ⊗ gβ ⊗ ep . Thus 1 IA,S,j (f a1 ⊗ . . . ⊗ f as ⊗ gα ⊗ el )

.

m Σ

=

m Σ

k Σ n Σ

b1 ,...,br−a+2 =1 d1 ,...,dm+a+1 =1 α=1 l=1

a ···a

b ···br−a+2 aj +1 ···as α

−1 1 Ad11 ···djm+a+1 l

a

× Sb1j ···br−a+2 f d1 ⊗ . . . ⊗ f dm+a+1 ⊗ gα ⊗ el . Then we calculate, using Cauchy–Schwarz, m Σ

1 ‖IA,S,j ‖2 =

m Σ

k Σ n Σ

.

a1 ,...,as =1 d1 ,...,dm+a+1 =1 α=1 l=1

⎛

⎞2

m Σ

⎝

a ···a

b1 ,...,br−a+2 =1 m Σ

≤

b ···br−a+2 aj +1 ···as α aj Sb1 ···br−a+2 ⎠

−1 1 Ad11 ···djm+a+1 l

m Σ

k Σ n Σ

a1 ,...,as =1 d1 ,...,dm+a+1 =1 α=1 l=1

⎛

m Σ

⎝

b1 ,...,br−a+2 =1 m Σ

≤

⎞2 │ │ │ a1 ···aj −1 b1 ···br−a+2 aj +1 ···as α aj │⎠ Sb1 ···br−a+2 │ │Ad1 ···dm+a+1 l

m Σ

k Σ n Σ

a1 ,...,as =1 d1 ,...,dm+a+1 =1 α=1 l=1

⎛

m Σ

⎝

b1 ,...,br−a+2 =1

≤

⎞⎛ │ │ │ a1 ···aj −1 b1 ···br−a+2 aj +1 ···as │2 ⎠ ⎝ │Ad1 ···dm+a+1 l │

m Σ b1 ,...,br−a+2 =1

⎞ │ │2 │ aj │ ⎠ │Sb1 ···br−a+2 │

‖A‖2 ‖S‖2 ,

as claimed.

□

.

4.2 Estimates Related to Jet Bundle Norms

223

Now we perform the same sort of estimate for a similar construction. We take .U, V, and .W as above, and m, s, r, and a as above. We also still take .S ∈ T1r−a+2 (U), but here we take

.

B ∈ Tsm+a (U) ⊗ W ⊗ V∗ .

.

We then have the mapping 2 IB,S,j : Ts (U∗ ) ⊗ V → Tm+r+1 (U∗ ) ⊗ W

.

defined by 2 IB,S,j (β) = Insj (B(β), S)

.

We now have the following result, whose proof follows from direct computation, just as does Lemma 4.15. Lemma 4.16 (Norm of Composition with Tensor Insertion II) With the preceding notation, 2 ‖IB,S,j ‖ ≤ ‖B‖‖S‖.

.

4.2.2 Tensor Field Estimates m m m We next turn to providing estimates for the tensors .Am s , .Bs , .Cs , and .Ds , .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m}, that appear in the lemmata from Sect. 3.3. In this section is where all of our seemingly pointless computations from Sects. 3.1 and 3.2, and our only slightly less seemingly pointless constructions from Sects. 3.3 and 4.1, bear fruit. We first develop a general estimate, and then show how this estimate can be made to apply to all of the specific tensors from Sect. 3.3. We work with real analytic vector bundles .πE : E → M and .πF : F → M. The rôle of .πE : E → M in this discussion and that in Sect. 3.3 is different. One should think of .E in Sect. 3.3 as being played by .M here. This is because the lifted tensors in Sect. 3.3 are defined as having .E as their base space. So here we rename this base space as .M. As a consequence of this, one should think of (1) the rôle of .M in the lemma below as being played by .E in the lemmata of Sect. 3.3 (as we just said), (2) the rôle of .∇ M in the lemma below as being played by .∇ E in the lemmata of Sect. 3.3, and (3) the rôles of .∇ πE and .∇ πF , and consequently .∇ πE ⊗πF , in the lemma below as being played by the induced connection in an appropriate tensor bundle in the lemmata of Sect. 3.3. In our development here, we use the symbol M,πE , etc., to denote the connection induced in any of the myriad bundles formed .∇ by taking tensor products of .TM, .T∗ M, .E, and .E∗ , etc., cf. the constructions at the beginning of Sects. 2.3.1 and 3.2.3.

224

4 Analysis: Norm Estimates for Derivatives

With this notational discussion out of the way, the main technical result we have is the following. Lemma 4.17 (Bound for Families of Real Analytic Tensors Defined by Recursion) Let .πE : E → M and .πF : F → M be real analytic vector bundles, let .∇ M be a real analytic affine connection on .M, let .∇ πE and .∇ πF be real analytic linear connections in .E and .F, respectively. Let .GM be a real analytic Riemannian metric on .M, and let .GπE and .GπF be real analytic fibre metrics for .E and .F, respectively. Suppose that we are given the following data: ∗ (i) .φm ∈ ┌ ω (Tm m (TM) ⊗ F ⊗ E ), .m ∈ Z≥0 ; s s ω (ii) .Фm ∈ ┌ (End(Tm+1 (TM) ⊗ F ⊗ F∗ )), .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m}; (iii) .Ψjs m ∈ ┌ ω (Hom(Tsm (TM) ⊗ F ⊗ E∗ ; Tsm+1 (TM) ⊗ F ⊗ E∗ )), .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m}, .j ∈ {0, 1, . . . , m}; s ∗ ∗ (iv) .^sm ∈ ┌ ω (Hom(Ts−1 m (TM) ⊗ F ⊗ E ; Tm+1 (TM) ⊗ F ⊗ E )), .m ∈ Z≥0 , .s ∈ {1, . . . , m}; ω s ∗ (v) .Am s ∈ ┌ (Tm (TM) ⊗ F ⊗ E ), .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m},

and that the data satisfies the recursion relations prescribed by .A00 = φ0 and m+1 m+1 Am+1 = Фm ◦ φm+1 ,

m ∈ Z≥0

.

Asm+1 = Фsm ◦ ∇ M,πE ⊗πF Am s +

m Σ

s m Ψjs m ◦ Am s + ^m ◦ As−1 ,

m ∈ Z>0 , s ∈ {1, . . . , m},

j =0

A0m+1

=

Ф0m ◦ ∇ M,πE ⊗πF Am 0

+

m Σ

Ψj0m ◦ Am 0 , m ∈ Z≥0 .

j =0

Suppose that the data are such that, for each compact .K ⊆ M, there exist .C1 , σ1 ∈ R>0 satisfying (i) .‖D∇r M ,∇ πE ⊗πF φm (x)‖GM,π

E ⊗πF

≤ C1 σ1−r r!, .m, r ∈ Z≥0 ;

(ii) .‖D∇r M ,∇ πF Фsm (x) ◦ A‖GM,π ⊗π ≤ C1 σ1−r r!‖A‖GM,π ⊗π , E F E F s ∗ .A ∈ T m+1 (Tx M ⊗ Fx ⊗ Ex ), .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m + 1}; (iii) .‖D∇r M ,∇ πE ⊗πF Ψjs m (x) ◦ A‖GM,π ⊗π ≤ C1 σ1−r r!‖A‖GM,π ⊗π , E F E F s ∗ .A ∈ Tm (Tx M ⊗ Fx ⊗ Ex ), .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}, .j ∈ {0, 1, . . . , m}; (iv) .‖D∇r M ,∇ πE ⊗πF ^sm (x) ◦ A‖GM,π ⊗π ≤ C1 σ1−r r!‖A‖GM,π ⊗π , E

F

E

F

∗ A ∈ Ts−1 m (Tx M ⊗ Fx ⊗ Ex ), .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}.

.

for .x ∈ K.

4.2 Estimates Related to Jet Bundle Norms

225

Then, for .K ⊆ M compact, there exist .C, σ, ρ ∈ R>0 such that ‖D∇r M ,∇ πE ⊗πF Am s (x)‖GM,π

.

E ⊗πF

≤ Cσ −m ρ −(m+r−s) (m + r − s)!

for .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}, and .x ∈ K. Proof We prove the lemma with a sort of meandering induction, covering various special cases of m and s, before giving a proof for the general case. Before we embark on the proof, we organise some data that will arise in the estimate that we prove. 1. We take .K ⊆ M compact and define .C1 , σ1 ∈ R>0 as in the statement of the lemma. We shall assume, without loss of generality, that .C1 > 1 and .σ1 < 1. We shall also make use of Lemma 2.22 to further assume, without loss of generality, that ‖D∇r M ,∇ πE R∇ πE (x)‖GM,π ≤ σ1−r r!,

.

E

‖D∇r M T∇ M (x)‖GM ≤ σ1−r r!,

x ∈ K, r ∈ Z≥0 .

2. Choose .β ≥ 2 so that ∞ Σ .

β −k < ∞,

k=0 β and let .α = β−1 > 1 denote the value of this sum. Let .γ = 6α. 3. We note that, for any .a, b, c ∈ Z>0 with .b < c, we have

.

(a + c)! (a + b)! < . b! c!

This is a direct computation: .

(a + c)! (a + b)! = (1 + b) · · · (a + b) < (1 + c) · · · (a + c) = . b! c!

4. For .m ∈ Z≥0 and .s ∈ {0, 1, . . . , m}, we denote ⌠

1, Cm,s = (m−1) s−1 ,

.

We note that (a) (b) (c) (d)

Cm,m = 1, that Cm,s ≤ Cm+1,s , that .Cm,s ≤ Cm+1,s+1 , and that .mCm,s ≤ (m + 1 − s)Cm+1,s . . .

m = 0 or s = 0, otherwise.

(4.9)

226

4 Analysis: Norm Estimates for Derivatives

The first and second of these assertions is obvious. For the third, for .m, s ∈ Z>0 with .s ≤ m, we compute Cm,s =

.

m (m − 1)! (m − 1)! ≤ = Cm+1,s+1 . (s − 1)!(m − s)! s (s − 1)!(m − s)!

For the fourth, for .m ∈ Z>0 and .s ∈ Z>0 satisfying .s ≤ m, we compute mCm,s = m

.

(m − 1)! (s − 1)!(m − s)! ≤ (m − s + 1)

m! = (m − s + 1)Cm+1,s . (s − 1)!(m + 1 − s)!

5. We shall have occasion below, and also elsewhere, to use a standard multinomial estimate. First let .α1 , . . . , αn ∈ R>0 and note that Σ

(α1 + . . . + αn )m =

.

m1 +...+mn

m! α1m1 · · · αnmn , m ! · · · m ! 1 n =m

which is well-known, but can easily be proved by induction on m. Taking .α1 = . . . = αn = 1, we see that .

m! ≤ nm m1 ! · · · mn !

(4.10)

whenever .m1 , . . . , mn ∈ Z≥0 sum to m. Given all of this, we shall prove that ‖D∇r M ,∇ πE ⊗πF Am s (‖GM,π

.

E ⊗πF

≤ C1 (2C1 σ1−1 γ )m Cm,s

⎛

β σ1

⎞m+r−s (m + r − s)! (4.11)

for .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}, and .x ∈ K. Case .m = s = 0: Directly using the hypotheses, we have ‖D∇r M ,∇ πE ⊗πF A00 (x)‖GM,π

.

E ⊗πF

=‖D∇r M ,∇ πE ⊗πF φ0 (x)‖GM,π ⊗π E F ≤ C1 σ1−r r! ≤ C1 (2C1 σ1−1 γ )0 C0,0 for .r ∈ Z≥0 and .x ∈ K. This gives (4.11) in this case.

⎛

β σ1

⎞0+r−0 (0 + r − 0)!

4.2 Estimates Related to Jet Bundle Norms

227

Case .m ∈ Z>0 and .s = m: By Lemma 3.15, we have r m D∇r M ,∇ πE ⊗πF Am m = D∇ M ,∇ πE ⊗πF (Фm−1 ◦ φm )

.

r ⎛ ⎞ Σ r r−a D∇a M ,∇ πE Фm = m−1 (D∇ M ,∇ πE ⊗πF φm ) a a=0

for .m, r ∈ Z≥0 . Therefore, by Lemma 4.11, using the hypotheses, and by (4.9) above, ‖D∇r M ,∇ πE ⊗πF Am m (x)‖GM,π

.

≤

r Σ a=0

≤

E ⊗πF

r! −(m+r−a) (C1 σ1−a a!)(C1 σ1 (r − a)!) a!(r − a)!

C1 C1 σ1−m r!

r Σ

σ1−a

⎛

a=0

≤ C1 (2C1 σ1−1 γ )m Cm,m

β σ1

⎛

⎞r−a

β σ1

≤

C1 C1 σ1−m

⎛

β σ1

⎞r

⎞m+r−m

r!

r Σ

β −a

a=0

(m + r − m)!.

As this holds for every .m ∈ Z>0 , .r ∈ Z≥0 , and .x ∈ K, this gives (4.11) in this case. Case .m = 1 and .s = 0: By Lemma 3.15 we have r 1 .D M π ⊗π A0 ∇ ,∇ E F

=

r ⎛ ⎞ Σ r a=0



a

M,πE ⊗πF 0 D∇a M ,∇ πE ⊗πF Ф00 (D∇r−a A0 ) M ,∇ πE ⊗πF ∇





term 1 r ⎛ ⎞ Σ r 0 0 D∇a M ,∇ πE ⊗πF Ψ00 + (D∇r−a M ,∇ πE ⊗πF A0 ) . a a=0    term 2

If we specialise the computations below from the last case in the proof, we obtain r 1 .‖D M π ⊗π A0 (x)‖G M,πE ⊗πF ∇ ,∇ E F

and this gives (4.11) in this case. Case .m ∈ Z>0 and .s = 0:

≤

C1 (2C1 σ1−1 γ )1 C1,0

⎛

β σ1

⎞1+r−0 (1 + r − 0)!

228

4 Analysis: Norm Estimates for Derivatives

We use induction on m, the desired estimate having been shown to be true for m = 1. By Lemma 3.15 we have

.

D∇r M ,∇ πE ⊗πF A0m+1 =

.

r ⎛ ⎞ Σ r M,πE ⊗πF m D∇a M ,∇ πE Ф0m (D∇r−a A0 ) M ,∇ πE ⊗πF ∇ a a=0    term 1

+

⎞ r 0 m D∇a M ,∇ πE ⊗πF Ψ0m (D∇r−a M ,∇ πE ⊗πF A0 ) a  

r ⎛ Σ a=0



term 2(a) r ⎛ ⎞ m Σ Σ r m D∇a M ,∇ πE ⊗πF Ψj0m (D∇r−a + M ,∇ πE ⊗πF A0 ) . a j =1 a=0    term 2(b)

As in the previous case, we can specialise the computations from the last case in the proof to give ‖D∇r M ,∇ πE ⊗πF A0m+1 (x)‖GM,π

.

E ⊗πF

≤ C1 (2C1 σ1−1 γ )m+1 Cm+1,0

⎛

β σ1

⎞m+1+r−0 (m + 1 + r − 0)!.

This proves (4.11) by induction in this case. Case .m ∈ Z>0 and .s ∈ {1, . . . , m − 1}: We use induction first on m (the result having been proved for the case .m = 0) and, for fixed m, by induction on s (the result having been proved for the case .s = 0). By Lemma 3.15 we have r m+1 .D M π ⊗π As ∇ ,∇ E F

r ⎛ ⎞ Σ r M,πE ⊗πF m D∇a M ,∇ πE Фsm (D∇r−a = As ) M ,∇ πE ⊗πF ∇ a a=0    term 1

r ⎛ ⎞ Σ r s m D∇a M ,∇ πE ⊗πF Ψ0m + (D∇r−a M ,∇ πE ⊗πF As ) a a=0    term 2(a)

4.2 Estimates Related to Jet Bundle Norms

+

229

r ⎛ ⎞ m Σ Σ r m D∇a M ,∇ πE ⊗πF Ψjs m (D∇r−a M ,∇ πE ⊗πF As ) a j =1 a=0    term 2(b)

+

⎞ r m D∇a M ,∇ πE ⊗πF ^sm (D∇r−a M ,∇ πE ⊗πF As−1 ) . a  

r ⎛ Σ a=0



term 3

We shall evaluate the components of this expression one-by-one. For term 1, by Lemma 3.21, we have M,πE ⊗πF m D∇r−a As ) M ,∇ πE ⊗πF (∇

.

r−a−1 m m π = D r−a+1 M,πE ⊗πF As + θr−a,1 ⊗ id(D∇ M ,∇ πE ⊗πF R∇ E (As ))  ∇      (a)

(b)

− θr−a,1 ⊗ id(D∇r−a−1 T∇ M (∇ M,πE ⊗πF Am M s )) .    (c)

We examine each of the three terms on the right separately. For term (a), by the induction hypothesis, m ‖D r−a+1 M,πE ⊗πF As (x)‖GM,π

.

∇

E ⊗πF

≤ C1 (2C1 σ1−1 γ )m Cm,s

⎛

β σ1

⎞m+r−a−s+1 (m + r − a + 1 − s)!.

Therefore, using Lemmata 3.15 and 4.11 and observation (4.9), we estimate ‖term 1(a)(x)‖GM,π

.

≤

r Σ

r! (C1 σ1−a a!) a!(r − a)! a=0 ⎛ ⎛ ×

≤

E ⊗πF

C1 (2C1 σ1−1 γ )m Cm,s

β σ1

⎞

⎞m+r−a+1−s

C1 (2C1 σ1−1 )m+1 γ m Cm,s (m + r

(m + r − a + 1 − s)! + 1 − s)!

r Σ a=0

σ1−a

⎛

β σ1

⎞m+r−a+1−s

230

4 Analysis: Norm Estimates for Derivatives

⎛

≤ C1 (2C1 σ1−1 )m+1 γ m Cm,s

≤ C1 (2C1 σ1−1 )m+1 γ m αCm,s

β σ1

⎛

⎞m+r+1−s

β σ1

(m + r + 1 − s)!

r Σ

β −a

a=0

⎞m+r+1−s

(m + r + 1 − s)!

for .x ∈ K. Now we consider terms (b) and (c). As in observation 1 at the beginning of the proof, we have π ‖D∇r−a−1 M ,∇ πE ⊗πF R∇ E (x)‖GM,π

.

E ⊗πF

−(r−a−1)

≤ σ1

(r − a − 1)!, −(r−a−1)

T∇ E (x)‖GM ≤ σ1 ‖D∇r−a−1 M

(r − a − 1)!

for .x ∈ K. By the induction hypothesis, E ⊗πF

≤ C1 (2C1 σ1−1 γ )m Cm,s

Am s (x)‖GM,πE ⊗πF

C1 (2C1 σ1−1 γ )m Cm,s

‖Am s (x)‖GM,π

.

⎛

β σ1

⎞m−s (m − s)!

and ‖∇

.

M,πE

≤

⎛

β σ1

⎞m+1−s (m + 1 − s)!

for .x ∈ K. By Lemmata 4.9, 4.13, and 4.14, by (4.10), and since .β ≥ 2, ‖θr−a,1 ⊗ id‖GM,π

.

E ⊗πF

1 ‖Δr−a−1,1 ‖GM,π ⊗π ‖Δ1,0 ‖GM,π ⊗π ‖id‖GM,π ⊗π E F E F E F 2 √ 1 (r − a)! m + s ≤ β r−a m. ≤ 2 (r − a − 1)!1!

=

Putting the preceding estimates together and using Lemma 4.11 and observation 4 gives ‖term (b)(x)‖GM,π

.

E ⊗πF

⎛

⎞ β m+r−a−s (r − a − 1)!(m − s)! σ1 ⎛ ⎞m+r−a−s β ≤ C1 2m+1 (C1 σ1−1 γ )m Cm+1,s (m + r − a − s)! σ1

≤ 2C1 (2C1 σ1−1 γ )m mCm,s

4.2 Estimates Related to Jet Bundle Norms

231

and, similarly, ‖term (c)(x)‖GM,π

.

E ⊗πF

≤ C1 2m+1 (C1 σ1−1 γ )m Cm+1,s

⎛

β σ1

⎞m+r−a+1−s (m + r − a + 1 − s)!.

Now a computation like that for term 1(a) gives ‖term 1(b)(x)‖GM,π

.

≤

E ⊗πF

, ‖term 1(c)(x)‖GM,π

E ⊗πF

C1 (2C1 σ1−1 )m+1 γ m αCm+1,s

⎛

β σ1

⎞m+r+1−s (m + r + 1 − s)!.

Let us now consider term 2(a). Here a similar analysis to the above gives ‖term 2(a)(x)‖GM,π

.

E ⊗πF

≤

C1 (2C1 σ1−1 )m+1 γ m αCm,s

⎛

β σ1

⎞m+r−s (m + r − s)!.

For term 2(b) we compute, along similar lines and using observation 4, ‖term 2(b)(x)‖GM,π

.

E ⊗πF

≤ C1 (2C1 σ1−1 )m+1 γ m αmCm,s

⎛

β σ1

⎞m+r−s (m + r − s)!

≤ C1 (2C1 σ1−1 )m+1 γ m α(m − s + 1)Cm+1,s ≤ C1 (2C1 σ1−1 )m+1 γ m αCm+1,s

⎛

β σ1

⎛

⎞m+r−s

β σ1

⎞m+r−s (m + r − s)!

(m + r + 1 − s)!.

An entirely similar analysis can be applied to term 3 to give ‖term 3(x)‖GM,π

.

E ⊗πF

≤ C1 (2C1 σ1−1 )m+1 γ m Cm,s−1 α

⎛

β σ1

⎞m+r−s+1 (m + r − s + 1)!.

Adding these as in the previous case and using our observation 4 above, we have ‖D∇r M ,∇ πE ⊗πF Asm+1 (x)‖GM,π

.

≤

E ⊗πF

C1 (2C1 σ1−1 γ )m+1 Cm+1,s

proving (4.11) by induction in this case.

⎛

β σ1

⎞m+1+r−s (m + 1 + r − s)!,

232

4 Analysis: Norm Estimates for Derivatives

We now note that a standard binomial estimate via (4.10) gives .Cm,s ≤ 2m . The lemma now follows from (4.11) by taking C = C1 , σ = 4C1 σ1−1 γ , ρ =

.

β . σ1

□

.

We now apply the lemma to the recursion relations that we proved in Lemmata 3.23, 3.24, 3.29, 3.30, 3.34, 3.35, 3.39, 3.40, 3.44, 3.45, 3.49, 3.50, 3.54, 3.55, 3.61, and 3.62. We first provide the correspondence between the data from the preceding lemmata with the data of Lemma 4.17. 1. Lemma 3.23: We have (a) .M = E, .E = F = RE , (b) .φm (βm ) = βm , .βm ∈ Tm (T∗ M), .m ∈ Z≥0 , (c) .Фsm (αsm+1 ) = αsm+1 , .αsm+1 ∈ Tsm+1 (T∗ M) ⊗ F ⊗ E∗ ), .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m + 1}, (d) .Ψjs m (αsm )(βs ) = −αsm ⊗ idT∗ M (Insj (βs , BπE )), s ∗ m ∗ m .αs ∈ Hom(T (T M) ⊗ E; T (T M) ⊗ F), s ∗ .βs ∈ T (T M) ⊗ E, .m ∈ Z>0 , .s ∈ {1, . . . , m}, .j ∈ {1, . . . , s}, m ) = α m ⊗ id ∗ , .α m ∈ Hom(Ts−1 (T∗ M) ⊗ E; Tm (T∗ M) ⊗ F), (e) .^sm (αs−1 T M s−1 s−1 .m ∈ Z>0 , .s ∈ {1, . . . , m}, (f) .Ψj0m = 0, .m ∈ Z≥0 , and (g) .^0m = 0, .m ∈ Z≥0 . 2. Lemma 3.24: We have (a) .M = E, .E = F = RE , (b) .φm (βm ) = βm , .βm ∈ Tm (T∗ M), .m ∈ Z≥0 , (c) .Фsm (αsm+1 ) = αsm+1 , .αsm+1 ∈ Tsm+1 (T∗ M) ⊗ F ⊗ E∗ , .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m + 1}, (d) .Ψjs m (αsm )(βs ) = Insj (αsm (βs ), BπE ), s ∗ m ∗ s ∗ m .αs ∈ Hom(T (T M) ⊗ E; T (T M) ⊗ F), .βs ∈ T (T M) ⊗ E, .m ∈ Z>0 , .s ∈ {1, . . . , m}, .j ∈ {1, . . . , m}, and m ) = α m ⊗ id ∗ , .α m ∈ Hom(Ts−1 (T∗ M) ⊗ E; Tm (T∗ M) ⊗ F), (e) .^sm (αs−1 T M s−1 s−1 .m ∈ Z>0 , .s ∈ {0, . . . , m}. 3. Lemma 3.29: We have M = E, E = F = VE,

.

and all other data derived from Lemma 3.29, similarly to the case of Lemma 3.23.

4.2 Estimates Related to Jet Bundle Norms

233

4. Lemma 3.30: We have M = E, E = F = VE,

.

and all other data derived from Lemma 3.30, similarly to the case of Lemma 3.24. 5. Lemma 3.34: We have M = E, E = F = HE,

.

and all other data derived from Lemma 3.34, similarly to the case of Lemma 3.23. 6. Lemma 3.35: We have M = E, E = F = HE,

.

and all other data derived from Lemma 3.35, similarly to the case of Lemma 3.24. 7. Lemma 3.39: We have M = E, E = F = V∗ E,

.

and all other data derived from Lemma 3.39, similarly to the case of Lemma 3.23. 8. Lemma 3.40: We have M = E, E = F = V∗ E,

.

and all other data derived from Lemma 3.39, similarly to the case of Lemma 3.24. 9. Lemma 3.44: We have M = E, E = F = T11 (VE),

.

and all other data derived from Lemma 3.44, similarly to the case of Lemma 3.23. 10. Lemma 3.45: We have M = E, E = F = T11 (VE),

.

and all other data derived from Lemma 3.45, similarly to the case of Lemma 3.24.

234

4 Analysis: Norm Estimates for Derivatives

11. Lemma 3.49: We have (a) .M = E, .E = RE ⊕ V∗ E, .F = RE ⊕ RE , (b) .φm (βm , δm ) = βm , .(βm , δm ) ∈ Tm (T∗ M) ⊗ E, .m ∈ Z≥0 , m+1 , γ m+1 ), (c) .Фsm (αsm+1 , γsm+1 ) = (αm s s m+1 m+1 .(αs , γs ) ∈ Tm+1 (T∗ M) ⊗ F ⊗ E∗ , .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m − 1}, m+1 m+1 ) = (0, 0), .(α m+1 , γ m+1 ) ∈ Tm (T∗ M) ⊗ F ⊗ E∗ , .m ∈ (d) .Фm m (αm , γm m m m+1 Z≥0 , (e) .Ψjs m (αsm , γsm )(βs , δs ) = (−αsm ⊗ idT∗ M (Insj (βs , BπE )), Σs+1 m s m m ∗ ∗ .− j =1 γs ⊗ idT∗ M (Insj (δs , BπE ))), .(αs , γs ) ∈ Tm (T M) ⊗ F ⊗ E , s ∗ .(βs , δs ) ∈ T (T M) ⊗ E, .m ≥ 2, .s ∈ {1, . . . , m − 1}, .j ∈ {1, . . . , s}, m , γ m )(β , δ ) = (−Ins (β , B ), δ ), (f) .Ψjmm (αm m m j m πE m m m ∗ m ∗ m m ∗ .(αm , γm ) ∈ Tm (T M) ⊗ F ⊗ E , .(βm , δm ) ∈ T (T M) ⊗ E, .m ≥ 2, .j ∈ {1, . . . , m}, m (α m , γ m )(β , δ ) = (0, −γ m ⊗ id ∗ (Ins (δ , B ))), (g) .Ψ00 0 0 1 0 πE T M 0 0 0 m ∗ m m ∗ .(α , γ ) ∈ T (T M) ⊗ F ⊗ E , .(β0 , δ0 ) ∈ E, .m ≥ 2, 0 0 0 m ⊗ id ∗ , γ m ⊗ id ∗ ), .m ≥ 2, .s ∈ {1, . . . , m}. (h) .^sm (αsm , γsm ) = (αs−1 T M s−1 T M 12. Lemma 3.50: We have (a) .M = E, .E = RE ⊕ V∗ E, .F = RE ⊕ RE , (b) .φm (βm , δm ) = βm , .(βm , δm ) ∈ Tm (T∗ M) ⊗ E, .m ∈ Z≥0 , m+1 , γ m+1 ), .(α m+1 , γ m+1 ) ∈ Ts ∗ (c) .Фsm (αsm+1 , γsm+1 ) = (αm s s s m+1 (T M) ⊗ F ⊗ ∗ E , .m ∈ Z≥0 , .s ∈ {0, 1, . . . , m − 1}, m+1 m+1 ) = (0, 0), .(α m+1 , γ m+1 ) ∈ Tm (T∗ M) ⊗ F ⊗ E∗ , .m ∈ (d) .Фm m (αm , γm m m m+1 Z≥0 , (e) .Ψjs m (αsm , γsm )(βs , δs ) = (Insj (αsm (βs ), BπE ) ∗ m m .−Insm+1 (αs (βs ), BπE ), −B s ), s s ∗ m m ∗ ∗ .(αs , γs ) ∈ Tm (T M) ⊗ F ⊗ E , .(βs , δs ) ∈ T (T M) ⊗ E, .m ≥ 2, .s ∈ {1, . . . , m − 1}, .j ∈ {1, . . . , m}, ∗ m , γ m )(β , δ ) = (Ins (β , B ) − Ins (f) .Ψjmm (αm m m j m πE m+1 (βm , BπE ), −δm ), m m ∗ m ∗ m m ∗ .(αm , γm ) ∈ Tm (T M) ⊗ F ⊗ E , .(βm , δm ) ∈ T (T M) ⊗ E, .m ≥ 2, .j ∈ {1, . . . , m}, m (α m , γ m )(β , δ ) = (0, −γ m ⊗ id ∗ (Ins (δ , B ))), (g) .Ψ00 0 0 1 0 πE T M 0 0 0 m ∗ m m ∗ .(α , γ ) ∈ T (T M) ⊗ F ⊗ E , .(β0 , δ0 ) ∈ E, .m ≥ 2, 0 0 0 m ⊗ id ∗ , γ m ⊗ id ∗ ), .m ≥ 2, .s ∈ {1, . . . , m}. (h) .^sm (αsm , γsm ) = (αs−1 T M s−1 T M 13. Lemma 3.54: We have M = E, E = VE ⊕ T11 (VE), F = VE ⊕ VE,

.

and all other data derived from Lemma 3.54, similarly to the case of Lemma 3.49.

4.2 Estimates Related to Jet Bundle Norms

235

14. Lemma 3.55: We have M = E, E = VE ⊕ T11 (VE), F = VE ⊕ VE,

.

and all other data derived from Lemma 3.55, similarly to the case of Lemma 3.50. 15. Lemma 3.61: We have M = M, E = Ф∗ T∗ N, F = T∗ M,

.

and all other data derived from Lemma 3.61, similarly to the case of Lemma 3.23. 16. Lemma 3.62: We have M = M, E = T∗ M, F = Ф∗ T∗ N,

.

and all other data derived from Lemma 3.62, similarly to the case of Lemma 3.24. 17. Lemma 3.67: We have M = M, E = E, F = E,

.

and all other data derived from Lemma 3.67, similarly to the case of Lemma 3.23. 18. Lemma 3.68: We have M = M, E = E, F = E,

.

and all other data derived from Lemma 3.68, similarly to the case of Lemma 3.24. Having now translated the lemmata of Sect. 3.3 to the general Lemma 4.17, we now need to show that the data of the lemmata of Sect. 3.3 satisfy the hypotheses of Lemma 4.17 in the real analytic case. As is easily seen, there are a few sorts of expressions that appear repeatedly, and we shall simply give estimates for these terms and leave to the reader the putting together of the pieces. The following lemma gives the required bounds. Lemma 4.18 (Specific Bounds for Terms Coming from Recursion) Let πE : E → M be a real analytic vector bundle, let .∇ M be a real analytic affine connection on .M, and let .∇ πE be a real analytic vector bundle connection in .E. Let .GM be a real analytic Riemannian metric on .M and let .GπE be a real analytic fibre metric for .E. Let .S ∈ ┌ ω (T12 (TM)). Let .K ⊆ M be compact and let n be the larger of the dimension of .M and the fibre dimension of .E and let .σ0 = n−1 . Then we have the following bounds: .

236

4 Analysis: Norm Estimates for Derivatives −(m+r+1)

(i) .‖D∇r M ,∇ πE idTm (T∗ M)⊗E (x)‖GM,π ≤ σ0 , E .x ∈ K, .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}; −(2m+r+1) (ii) .‖D∇r M ,∇ πE idTms (T∗ M)⊗E (x)‖GM,π ≤ σ0 , E .x ∈ K, .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}; (iii) if .Фsm (αsm+1 ) = αsm+1 , .αsm+1 ∈ Tsm+1 (T∗ M) ⊗ E, then ‖D∇r M ,∇ πE Фsm ◦ D∇a M ,∇ πE Asm+1 (x)‖GM,π ≤ ‖D∇a M ,∇ πE Asm+1 (x)‖GM,π ,

.

E

E

for .x ∈ K, .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}; (iv) if Ψjs m (αsm )(βs ) = (αsm ⊗ idT∗ M )(Insj (βs , S)),

.

αsm ∈ Hom(Ts (T∗ M) ⊗ E; Tm (T∗ M) ⊗ E), βs ∈ Ts (T∗ M) ⊗ E, then there exist .C1 , σ1 ∈ R>0 such that −r a m ‖D∇r M ,∇ πE Ψjs m ◦ D∇a M ,∇ πE Am s (x)‖GM,π ≤ C1 σ1 r!‖D∇ M ,∇ πE As (x)‖GM,π ,

.

E

E

for .x ∈ K, .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}; (v) if Ψjs m (αsm )(βs ) = Insj (αsm (βs ), S),

.

αsm ∈ Hom(Ts (T∗ M) ⊗ E; Tm (T∗ M) ⊗ E), βs ∈ Ts (T∗ M) ⊗ E, then there exist .C1 , σ1 ∈ R>0 such that −r a m ‖D∇r M ,∇ πE Ψjs m ◦ D∇a M ,∇ πE Am s (x)‖GM,π ≤ C1 σ1 r!‖D∇ M ,∇ πE As (x)‖GM,π ,

.

E

E

for .x ∈ K, .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}; (vi) if m m ^sm (αs−1 ) = αs−1 ⊗idT∗ M ,

.

m αs−1 ∈ Hom(Ts−1 (T∗ M)⊗E; Tm (T∗ M)⊗E),

then a m ‖D∇r M ,∇ πE ^sm ◦ D∇a M ,∇ πE Am s−1 (x)‖GM,π ≤ ‖D∇ M ,∇ πE As−1 (x)‖GM,π ,

.

E

E

for .x ∈ K, .m, r ∈ Z≥0 , .s ∈ {0, 1, . . . , m}. Proof Parts (i) and (ii) follow from Lemma 4.9 along with the fact that the covariant derivative of the identity tensor is zero. Part (iii) is a tautology, but one that arises in the lemmata of Sect. 3.3.

4.2 Estimates Related to Jet Bundle Norms

237

For the next two parts of the proof, let .C1 , σ1 ∈ R>0 be such that ‖D∇r M S(x)‖GM ≤ C1 σ1−r r!,

x ∈ K,

.

(4.12)

this being possible by Lemma 2.22, and recalling the rôle of the factorials in the definition (2.14) of the fibre norms. (iv) Let us define ˆ sj m (βsm )(αs ) = (βsm )(Insj (αs , S)), Ψ

.

βsm ∈ Hom(Ts (T∗ M) ⊗ E; Tm (T∗ M) ⊗ E), αs ∈ Ts (T∗ M) ⊗ E and τms (αsm ) = αsm ⊗ idT∗ M ,

.

αsm ∈ Hom(Ts (T∗ M) ⊗ E; Tm (T∗ M) ⊗ E)

ˆ s = InsS,j so that, by Lemma 3.14, ˆ sm ◦ τms . Note that .Ψ so that .Ψjs m = Ψ jm ˆ sj m (D a M π Bsm ) = InsD r D∇r M ,∇ πE Ψ ∇ ,∇ E M

.

∇ ,∇ πE

a m S,j (D∇ M ,∇ πE Bs ).

Since the covariant derivative of the identity tensor is zero, a m D∇a M ,∇ πE (Am s ⊗ idT∗ M ) = (D∇ M ,∇ πE As ) ⊗ idT∗ M ),

.

s from which we deduce that .D∇a M ,∇ πE τms = τm+a . Thus r s a m ˆs D∇r M ,∇ πE Ψjs m ◦ D∇a M ,∇ πE Am s = D∇ M ,∇ πE (Ψ j m ◦ τm ) ◦ D∇ M ,∇ πE As

.

= InsD r M

∇ ,∇ πE

a m S,j (D∇ M ,∇ πE As

⊗ idT∗ M ).

By Lemmata 4.10 and 4.15, this part of the lemma follows immediately. (v) Here we have .Ψjs m (αsm ) = InsS,j ◦ αsm and, following the arguments from the preceding part of the proof, s a m ◦ D M πE As = InsD r D∇r M ,∇ πE Ψm ∇ ,∇ M

.

∇ ,∇ πE

a m S,j (D∇ M ,∇ πE As ),

and so this part of the lemma follows from Lemma 4.16. (vi) This follows from Lemma 4.10 and the fact that the covariant derivative of the identity tensor is zero. .□

238

4 Analysis: Norm Estimates for Derivatives

4.3 Independence of Topologies on Connections and Metrics The seminorms introduced in Sect. 2.4 for defining topologies for the space of real analytic sections of a vector bundle .πE : E → M are made upon a choice of various objects, namely (1) an affine connection .∇ M on .M, (2) a linear connection .∇ πE in .E, (3) a Riemannian metric .GM on .M, and (4) a fibre metric .GπE for .E. In order for these topologies to be useful, they should be independent of all of these choices. This is made more urgent by our very specific choice in Sect. 3.2.1 of a Riemannian metric .GE on the total space .E and its Levi-Civita connection. These choices were made because they made available to us the formulae of Sect. 3.2.3 for differentiation of lifts of tensors, which lifts were used in an essential way in Sects. 3.3 and 4.2, and use of which will be made in Chap. 5, as well as in the present chapter. However, as we pointed out in the preamble to Sect. 4.2, it does remain to ω , be shown that the seminorm topology defined by the real analytic seminorms .pK,a .K ⊆ M compact, .a ∈ c0 (Z≥0 ; R>0 ), is independent of the choices of metrics and connections used to define these seminorms. We have at various points indicated that many of our constructions can be carried out in the smooth case. However, in the smooth case, it is relatively easy to see that the topology defined by the seminorms (2.17) is independent of metrics and connections. Indeed, these seminorms can just as well be defined by any fibre metrics on the jet bundles of .πE : E → M, and the resulting topology will be independent of these choices. This is not the case for the topology defined by the real analytic seminorms, and this is where the technical developments of Chap. 3 and the preceding sections of this chapter are important.

4.3.1 Comparison of Metric-Related Notions for Different Connections and Metrics We first consider how various constructions involving Riemannian metrics and fibre metrics vary when one varies these metrics. The first result concerns fibre norms for tensor products induced by a fibre metric. Lemma 4.19 (Comparison of Fibre Norms for Different Fibre Metrics) Let πE : E → M be a smooth vector bundle and let .G1 and .G2 be smooth fibre metrics on .E. Let .K ⊆ M be compact. Then there exist .C, σ ∈ R>0 such that

.

.

σ r+s C ‖A(x)‖G2 ≤ ‖A(x)‖G1 ≤ r+s ‖A(x)‖G2 C σ

for all .A ∈ ┌ 0 (Trs (E)), .r, s ∈ Z≥0 , and .x ∈ K. Proof We begin by proving a linear algebra result.

□

.

4.3 Independence of Topologies on Connections and Metrics

239

Sublemma 1 If .G1 and .G2 are inner products on a finite-dimensional .R-vector space .V, then there exists .C ∈ R>0 such that C −1 G1 (v, v) ≤ G2 (v, v) ≤ CG1 (v, v)

.

for all .v ∈ V. Proof Let .Gj ∈ HomR (V; V∗ ) and .Gj ∈ HomR (V∗ ; V), .j ∈ {1, 2}, be the induced linear maps. Note that b

#

#

b

#

b

G1 (G1 ◦ G2 (v1 ), v2 ) = G2 (v1 , v2 ) = G2 (v2 , v1 ) = G1 (G1 ◦ G2 (v2 ), v1 ),

.

#

b

showing that .G1 ◦ G2 is .G1 -symmetric. Let .(e1 , . . . , en ) be a .G1 -orthonormal basis # b for .V that is also a basis of eigenvectors for .G1 ◦ G2 . The matrix representatives of .G1 and .G2 are then ⎡

10 ⎢0 1 ⎢ .[G1 ] = ⎢ . . ⎣ .. ..

··· ··· .. .

⎤ 0 0⎥ ⎥ .. ⎥ , .⎦

⎡

λ1 ⎢0 ⎢ [G2 ] = ⎢ . ⎣ ..

0 0 ··· 1

0 λ2 .. .

··· ··· .. .

⎤ 0 0⎥ ⎥ .. ⎥ , .⎦

0 0 · · · λn

where .λ1 , . . . , λn ∈ R>0 . Let us assume without loss of generality that λ1 ≤ · · · ≤ λn .

.

Then taking .C = max{λn , λ−1 1 } gives the result, as one can verify directly.

Δ

.

Next we use the preceding sublemma to give the linear algebraic version of the lemma. Sublemma 2 Let .V be a finite-dimensional .R-vector space and let .G1 and .G2 be inner products on .V. Then there exist .C, σ ∈ R>0 such that .

σ r+s C ‖A‖G2 ≤ ‖A‖G1 ≤ r+s ‖A‖G2 C σ

for all .A ∈ Trs (V), .r, s ∈ Z≥0 . Proof As in the proof of Sublemma 1, let .(e1 , . . . , en ) be a .G1 -orthonormal # b basis for .V consisting of eigenvectors for .G1 ◦ G2 . Let .λ1 , . . . , λn ∈ R>0 be the corresponding eigenvalues, supposing that λ1 ≤ · · · ≤ λn .

.

240

4 Analysis: Norm Estimates for Derivatives

Note that .G2 (ej , ek ) = δj k λj , .j ∈ {1, . . . , n}, (.δj k being the Kronecker delta −1 symbol) so that .(eˆ1  λ−1 1 e1 , . . . , eˆn  λn en ) is a .G2 -orthonormal basis. Denote by .(e1 , . . . , en ) and .(eˆ1 , . . . , eˆn ) be the dual bases. Note that .eˆj = λj ej , .j ∈ {1, . . . , n}. Now let .A ∈ Trs (V) and write A=

n Σ

n Σ

j ···j

.

j1 ,...,jr =1 k1 ,...,ks =1

Ak11 ···krs ej1 ⊗ . . . ⊗ ejr ⊗ ek1 ⊗ . . . ⊗ eks

and A=

n Σ

n Σ

.

j1 ,...,jr =1 k1 ,...,ks =1

ˆj1 ···jr eˆj1 ⊗ . . . ⊗ eˆjr ⊗ eˆk1 ⊗ . . . ⊗ eˆks . A k1 ···ks

We necessarily have ˆj1 ···jr = λj1 · · · λjr λ−1 · · · λ−1 Aj1 ···jr , A k1 ks k1 ···ks k1 ···ks

j1 , . . . , jr , k1 , . . . , ks ∈ {1, . . . , n}.

.

We have ⎛

n Σ

‖A‖G1 = ⎝

n Σ

.

j1 ,...,jr =1 k1 ,...,ks =1

⎛ ‖A‖G2 = ⎝

n Σ

n Σ

j1 ,...,jr =1 k1 ,...,ks =1

⎞1/2 │ │ │ j1 ···jr │2 ⎠ , │Ak1 ···ks │ ⎞1/2 │ │2 │ ˆj1 ···jr │ ⎠ . │Ak1 ···ks │

Therefore, if we let .σ = min{λ1 , λ−1 n }, we have ‖A‖G2 ≤ σ −(r+s) ‖A‖G1 .

.

This gives one half of the estimate in the sublemma, and the other is established . Δ analogously. The lemma follows from the preceding sublemma since C and .σ depend only # b on .G1 and .G2 through the largest and smallest eigenvalues of .G1 ◦ G2 , which are uniformly bounded above and below on .K. .□ Remark 4.20 (Generalisation to Tensor Products of Vector Bundles) For simplicity we have stated the preceding result for a single vector bundle .πE : E → M with tensors coming from the tensor algebra of this vector bundle. A moment’s consideration of the proof of the lemma will convince the reader that the result will hold for tensors that are tensor products of elements of the tensor algebra of any

4.3 Independence of Topologies on Connections and Metrics

241

finite number of vector bundles, each equipped with two fibre metrics. We shall use this generalisation without mention. ◦ The following comparison result for the distance function associated to two Riemannian metrics is often useful, although we do not make use of it for our developments here. This is certainly a known result, although we could not locate a proof. Lemma 4.21 (Comparison of Distance Functions for Riemannian Metrics) If G1 and .G2 are .C∞ -Riemannian metrics on a .C∞ -manifold .M with metrics .d1 and .d2 , respectively, and if .K ⊆ M is compact, then there exists .C ∈ R>0 such that .

C −1 d1 (x1 , x2 ) ≤ d2 (x1 , x2 ) ≤ Cd1 (x1 , x2 )

.

for every .x1 , x2 ∈ K. Proof First we give a local version of the result. Let .x ∈ M. Let .N1 and .N2 be geodesically convex neighbourhoods of x with respect to the Riemannian metrics .G1 and .G2 , respectively [42, Proposition IV.3.4]. Thus every pair of points in .N1 can be connected by a unique distance-minimising geodesic for .G1 that remains in .N1 , and similarly with .N2 and .G2 . By Sublemma 1 from the proof of Lemma 4.19, let .Cx ∈ R>0 be such that Cx−2 G1 (vx , vx ) < G2 (vx , vx ) < Cx2 G1 (vx , vx ),

.

vx ∈ Tx M.

By continuity of .G1 and .G2 , we can choose .N1 and .N2 sufficiently small that Cx−2 G1 (vy , vy ) < G2 (vy , vy ) < Cx2 G1 (vy , vy ),

.

y ∈ N1 ∪ N 2 .

Now define .Ux = N1 ∩ N2 . Then every pair of points in .Ux can be connected with a unique distance-minimising geodesic of both .G1 and .G2 that remains in .N1 ∪ N2 . Now let .x1 , x2 ∈ Ux . Let .γ : [0, 1] → M be the unique distance-minimising .G1 geodesic connecting .x1 and .x2 . Then  d2 (x1 , x2 ) ≤ lG2 (γ ) =

1√

.

 ≤ Cx

G2 (γ ' (t), γ ' (t)) dt

0

1√

G1 (γ ' (t), γ ' (t)) ]dt

0

= Cx lG1 (γ ) = Cx d1 (x1 , x2 ). One similarly shows that .d1 (x1 , x2 ) ≤ Cx d2 (x1 , x2 ). We now prove the assertion of the lemma. Let .K ⊆ M be compact and, for each .x ∈ K, let .Ux be a neighbourhood of x and let .Cx ∈ R>0 be as in the preceding

242

4 Analysis: Norm Estimates for Derivatives

paragraph. Let .x1 , . . . , xk ∈ K be such that .K ⊆ ∪kj =1 Uxj . Let Da = sup{da (x, y) | x, y ∈ K},

.

a ∈ {1, 2}.

By the Lebesgue Number Lemma [11, Theorem 1.6.11], let .ra ∈ R>0 be such that, if .x1 , x2 ∈ K satisfy .da (x1 , x2 ) < ra , .a ∈ {1, 2}, then there exists .j ∈ {1, . . . , k} such that .x1 , x2 ∈ Uxj . Let us denote ⌠ D1 D2 . C = max Cx1 , . . . , Cxk , , r2 r1

.

Now let .x1 , x2 ∈ K. If .d1 (x1 , x2 ) < r1 , then let .j ∈ {1, . . . , k} be such that x1 , x2 ∈ Uj . Then

.

d2 (x1 , x2 ) ≤ Cd1 (x1 , x2 ).

.

If .d1 (x1 , x2 ) ≥ r1 , then .

d2 (x1 , x2 )r1 d2 (x1 , x2 )r1 ≤ d1 (x1 , x2 ) ≤ D2 d2 (x1 , x2 )

This gives .d2 (x1 , x2 ) ≤ Cd1 (x1 , x2 ). Swapping the rôles of .G1 and .G2 gives d1 (x1 , x2 ) ≤ Cd2 (x1 , x2 ) for an appropriate C, giving the lemma. .□

.

Now we can compare fibre norms for jet bundles associated with different metrics and connections. Lemma 4.22 (Comparison of Fibre Norms for Jet Bundles for Different Metrics and Connections) Let .r ∈ {∞, ω} and let .πE : E → M be a .Cr -vector bundle. Consider .Cr -affine connections .∇ M and .∇ M on .M, and .Cr -vector bundle connections .∇ πE and .∇ πE in .E. Consider .Cr -Riemannian metrics .GM and .GM for r .M, and .C -fibre metrics .GπE and .GπE for .E. Let .K ⊆ M be compact. Then there exist .C, σ ∈ R>0 such that .

σm C ‖jm ξ(x)‖GM,π ,m ≤ ‖jm ξ(x)‖GM,π ,m ≤ m ‖jm ξ(x)‖GM,π ,m E E E C σ

for all .ξ ∈ ┌ m (E), .m ∈ Z≥0 , and .x ∈ K. Proof We first make some preliminary constructions that will be useful.

4.3 Independence of Topologies on Connections and Metrics

243

By Lemma 3.69, we have ˆ00 ξ(x), ξ(x) = A ˆ11 (∇ πE ξ(x)) + A ˆ10 (ξ(x)), (Sym1 ⊗ idE )∇ πE ξ(x) = A .. .

.

(Symm ⊗ idE ) ◦ ∇ M,πE ,m ξ(x) =

(4.13) m Σ

M,πE ,s ˆm ξ(x)). A s ((Syms ⊗ idE ) ◦ ∇

s=0

In like manner, by Lemma 3.70, we have ˆ00 ξ(x), ξ(x) = B ˆ11 (∇ πE ξ(x)) + B ˆ10 (ξ(x)), (Sym1 ⊗ idE )∇ πE ξ(x) = B .. .

.

(Symm ⊗ idE ) ◦ ∇ M,πE ,m ξ(x) =

(4.14) m Σ

M,πE ,s ˆm ξ(x)). B s ((Syms ⊗ idE ) ◦ ∇

s=0

By Lemma 4.11, we have m ‖Am s (βs )‖GM,π ≤ ‖As ‖GM,π ‖βs ‖GM,π

.

E

E

E

for .βs ∈ Ts (T∗ M) ⊗ E, .m ∈ Z>0 , .s ∈ {0, 1, . . . , m}. By Lemma 4.13, ‖Syms (A)‖GM,π ≤ ‖A‖GM,π

.

E

E

for .A ∈ Ts (T∗ E) and .s ∈ Z>0 . Thus, recalling (3.40), m m ˆm ‖A s (Syms (βs ))‖GM,π = ‖Symm ◦ As (βs )‖GM,π ≤ ‖As ‖GE ‖βs ‖GM,π ,

.

E

E

E

for .βs ∈ Ts (πE∗ T∗ M) ⊗ E, .m ∈ Z>0 , .s ∈ {1, . . . , m}. By Lemmata 4.17 and 4.18 with .r = 0, there exist .σ1 , ρ1 ∈ R>0 such that ‖Aks (x)‖GM,π ≤ σ1−k ρ1−(k−s) (k − s)!,

.

E

k ∈ Z≥0 , s ∈ {0, 1, . . . , k}, x ∈ K.

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4 Analysis: Norm Estimates for Derivatives

Without loss of generality, we assume that .σ1 , ρ1 ≤ 1. Thus, redefining .σ1 = σ1 ρ1 , we have ˆks ((Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(x)‖ ‖A GM,π

.

≤

C1 σ1−k (k

E

− s)!‖(Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(x)‖GM,π

E

for .m ∈ Z≥0 , .k ∈ {0, 1, . . . , m}, .s ∈ {0, 1, . . . , k}, .x ∈ K. By Lemma 4.19 (along with Remark 4.20), let .C2 , σ2 ∈ R>0 be such that .

σ2m+1 C2 ‖A‖GM,π ,m ≤ ‖A‖GM,π ,m ≤ m+1 ‖A‖GM,π ,m E E E C2 σ2

for .A ∈ ┌ ∞ (Sk (T∗ M)⊗E). We shall assume, without loss of generality, that .σ2 ≤ 1. Thus, by (1.4) and (4.13),

.

σ2m+1 ‖jm ξ(x)‖GM,π ,m E C2 m Σ 1 ‖(Symk ⊗ idE ) ◦ ∇ M,πE ,k ξ(x)‖GM,π E k! k=0 ‖ k ‖ m ‖ Σ 1 ‖ ‖Σ ˆk ‖ M,πE ,s = ξ(x))‖ As ((Syms ⊗ idE ) ◦ ∇ ‖ ‖ k! ‖

≤

k=0

≤

k m Σ Σ

GM,πE

s=0

C1 σ1−k

k=0 s=0

s!(k − s)! 1 ‖(Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(x)‖GM,π E k! s!

for .x ∈ K and .m ∈ Z≥0 . Now note that .

s!(k − s)! ≤ 1, k!

C1 σ1−k ≤ C1 σ1−m ,

for .s ∈ {0, 1, . . . , m}, .k ∈ {0, 1, . . . , s}, since .σ1 ≤ 1. Then ΣΣ 1 σ2m+1 ‖(Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(x)‖GM,π . ‖jm ξ(x)‖GM,π ,m ≤ C1 σ1−m E E C2 s! m

k

k=0 s=0

≤ C1 σ1−m

m m Σ Σ 1 ‖(Syms ⊗ idE ) ◦ ∇ M,s ξ(x)‖GM,π E s! k=0 s=0

4.3 Independence of Topologies on Connections and Metrics

245

= (m + 1)C1 σ1−m ×

m Σ 1 ‖(Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(x)‖GM,π , E s! s=0

which gives ‖jm ξ(x)‖GM,π

.

E ,m

≤ (m + 1)C1 C2 σ2 (σ1 σ2 )−m ×

m Σ 1 ‖(Syms ⊗ idE ) ◦ ∇ M,πE ,s ξ(x)‖GM,π E s! s=0

√ ≤ m + 1(m + 1)C1 C2 σ2 (σ1 σ2 )−m ‖jm ξ(x)‖GM,π

E ,m

,

making use of (1.4). Now let .σ < σ1 σ2 and note that .

lim (m + 1)3/2

m→∞

(σ1 σ2 )−m = 0. σ −m

Thus there exists .N ∈ Z>0 such that (m + 1)3/2 C1 C2 σ2 (σ1 σ2 )−m ≤ C1 C2 σ2 σ −m ,

m ≥ N.

.

Let ⌠ C = max C1 C2 σ2 , 2

3/2

.

σ , 33/2 C1 C2 σ2 C1 C2 σ2 σ1 σ2

⎛

σ σ1 σ2

⎞2 ,..., ⎛

(N + 1)

3/2

C1 C2 σ2

σ σ1 σ2

⎞N " .

We then immediately have .(m + 1)3/2 C1 C2 σ2 σ1−m ≤ Cσ −m for all .m ∈ Z≥0 . We then have ‖jm ξ(x)‖GM,π

.

E ,m

≤ Cσ −m ‖jm ξ(x)‖GM,π

E ,m

.

This gives one half of the desired pair of estimates. For the other half of the estimate, we use (4.14), and Lemmata 4.17 and 4.18 in the computations above to arrive at the estimate ‖jm ξ(x)‖GM,π

.

which gives the result.

E ,m

≤ Cσ −m ‖jm ξ(x)‖GM,π

E ,m

, □

.

246

4 Analysis: Norm Estimates for Derivatives

Remark 4.23 (Adaptation to the Smooth Case) An analogous result to the preceding result holds in the smooth case, and with a much easier proof. In the result, one can replace “.Cσ −m ” with a fixed constant “.Cm ” for each m; i.e., one does not need any uniformity of the estimates in m. For this reason, the proof is also far simpler, e.g., one need not keep track of all the factorial terms that give rise to the exponential component in the estimates. ◦

4.3.2 Local Descriptions of the Real Analytic Topology We endeavour to make our presentation as unencumbered of coordinates as possible. While the intrinsic jet bundle characterisations of the seminorms are useful for general definitions and elegant proofs, concrete proofs often require local descriptions of the topologies. In this section we provide these local descriptions of the topologies. By proving that these local descriptions are equivalent to the intrinsic descriptions above, we also prove that these intrinsic descriptions of topologies do not depend on the choice of metrics or connections. Let us develop the notation for working with local descriptions of topologies. Let ω n k .U ⊆ R be an open set. We define local seminorms for .C (U; R ) as follows. Let ω k .Ф ∈ C (U; R ). For .K ⊆ U compact and for .a ∈ c0 (Z≥0 ; R>0 ), denote 'ω pK,a (Ф) = sup

.

│ {a a · · · a 0 1 m │ |D I Фa (x)|│ I!

} x ∈ K, a ∈ {1, . . . , k}, I ∈ Zn≥0 , |I | ≤ m, m ∈ Z≥0 .

These seminorms, defined for all compact .K ⊆ U and .a ∈ c0 (Z≥0 ; R>0 ), define the local .Cω -topology for .Cω (U; Rk ). There are many possible variations of the seminorms that one can use, and these variations are equivalent to the seminorms above. For example, rather than using the .∞-vector norm, one might use the 2-vector norm. In doing so, one uses (1.4) to give

.

sup{|D I Фa (x)| | I ∈ Zn≥0 , |I | = m, a ∈ {1, . . . , k}} ≤ ‖D m Ф(x)‖ √ ≤ knm sup{|D I Фa (x)| | I ∈ Zn≥0 , |I | = m, a ∈ {1, . . . , k}}.

If we define √ √ b0 = 2 ka0 , bj = 2 naj ,

.

j ∈ Z>0 ,

4.3 Independence of Topologies on Connections and Metrics

247

then, noting that .nj ≤ nm for .j ∈ {0, 1, . . . , m} and that .m + 1 ≤ 2m for .m ∈ Z≥0 , we have │ {a a · · · a 0 1 m │ 'ω ‖D |I | Ф(x)‖│ .p K,a (Ф) ≤ sup I! } 'ω x ∈ K, I ∈ Zn≥0 , |I | ≤ m, m ∈ Z≥0 ≤ pK,b (Ф), and this gives equivalence of the topologies using the .∞- and 2-norms. Another variation in the seminorms is that one might scale the derivatives by . |I1|! rather than .

1 I! .

In this case, we use the standard multinomial estimate (4.10) to give .

|I |! ≤ nm . I!

Thus, if we take b0 = a0 , bj = naj ,

.

j ∈ Z>0 ,

we have 'ω .p K,b (Ф)

│ │ a0 a1 · · · am I a ≤ sup |D Ф (x)|││ |I |! ⌠

'ω x ∈ K, a ∈ {1, . . . , k}, I ∈ Zn≥0 , |I | ≤ m, m ∈ Z≥0 ≤ pK,a (Ф).

This gives the equivalence of the topologies defined using the scaling factor . |I1|! for

derivatives in place of . I1! . One can also combine the previous modifications. Indeed, if we use the 2-norm and the scaling factor . |I1!| , then one readily sees that we recover the intrinsic seminorms on the trivial vector bundle .RkU of Sect. 2.4 using (1) the Euclidean inner product for the Riemannian metric on .U and for the fibre metric on .Rk and (2) standard differentiation as covariant differentiation. We shall use this observation in the proof of Theorem 4.24 below. We wish to show that these local topologies can be used to define a topology for .┌ ω (E) that is equivalent to the intrinsic topologies defined in Sect. 2.4 using jet bundles, connections, and metrics. To state the result, let us indicate some notation. Let .(V, ν) be a vector bundle chart for .πE : E → M with .(U, χ ) the induced chart for k .M. Suppose that .ν(V) = χ (U) × R . Given a section .ξ , we define .ν ∗ (ξ ) : χ (U) → k R by requiring that ν ◦ ξ ◦ χ −1 (x) = (x, ν ∗ (ξ )(x)).

.

With this notation, we have the following result.

248

4 Analysis: Norm Estimates for Derivatives

Theorem 4.24 (Agreement of Intrinsic and Local Topologies) Let .πE : E → M be a .Cω -vector bundle. Let .GM be a Riemannian metric on .M, let .GπE be a fibre metric on .E, let .∇ M be an affine connection on .M, and let .∇ πE be a vector bundle connection on .E, with all of these being of class .Cω . Then the following two collections of seminorms for .┌ ω (E) define the same topology: ω , .a ∈ c (Z ; R ), .K ⊆ M compact; (i) .pK,a 0 ≥0 >0 'ω ◦ ν , .a ∈ c (Z ; R ), .(V, ν) is a vector bundle chart for .E with .(U, χ ) (ii) .pK,a ∗ 0 ≥0 >0 the induced chart for .M, .K ⊆ χ (U) compact.

Proof As alluded to in the discussion above, it suffices to use the norm ⎞1/2

⎛

⎟ ⎜ ⎟ ⎜ Σ Σ k ⎟ ⎜ m I a 2⎟ ⎜ .‖D Ф(x)‖2 = |D Ф (x)| ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ I ∈ Zn≥0 a=1 |I | = m for derivatives of .Rk -valued functions on .U ⊆ Rn . If we denote jm Ф(x) = (Ф(x), DФ(x), . . . , D m Ф(x)),

.

then we define ‖jm Ф(x)‖22,m =

m Σ

.

j =0

1 ‖D j Ф(x)‖22 , (j !)2

this norm agreeing with the fibre norms used in Sect. 2.3.2 with the flat connections and with the Euclidean inner products. We use these norms to define seminorms that we denote by .q ' in place of the local seminorms .p' as above. We might like to use Lemma 4.22 in this proof. However, we cannot do so without a moment’s thought. The reason for this is that the proof of Lemma 4.22 makes reference to Lemma 4.17, which itself can be applied only by virtue of Lemma 4.18. The proof of this latter lemma relies on the bound (4.12), which is deduced from Lemma 2.22. The proof of Lemma 2.22, we note, makes use of Lemma 4.22. To intrude on the potential circular logic, we must give a proof of this part of the theorem that does not rely on Lemma 4.22 as it is stated. In fact, the only part of the chain of results that we need to prove independently is the bound (4.12). In particular, if we can show that Lemma 4.22 holds in the current situation where 1. .M = U ⊆ Rn and .E = RkU , 2. .GU and .GπE are the Euclidean inner products, and 3. .∇ M and .∇ πE are the flat connections, this will be enough to make use of Lemma 4.22.

4.3 Independence of Topologies on Connections and Metrics

249

To this end, let .(V, ν) be a vector bundle chart for .E with .(U, χ ) the chart for .M. Theorem 1.9 gives .C1 , σ1 ∈ R>0 such that ‖D∇r U ,∇ πE SU (x)‖2 , ‖D∇r U ,∇ πE SπE (x)‖2 ≤ C1 σ1−r r!,

.

x ∈ K.

(4.15)

This gives the bound (4.12) in this case, and so we can use Lemma 4.18, and then Lemma 4.17, and then the computation of Lemma 4.22 (all in our current local setting) to give .

σm C ‖jm ξ ‖GM,π ,m ≤ ‖jm (ν ∗ (ξ ))(χ(x))‖2,m ≤ m ‖jm ξ ‖GM,π ,m . E E C σ

Now, having established Lemma 4.22 in the case of interest, we proceed with the proof, making use of this fact. Let .K ⊆ χ (U) be compact and let .a ∈ c0 (Z≥0 ; R>0 ). As per our appropriate version of Lemma 4.22, there exist .C, σ ∈ R>0 such that ‖jm (ν ∗ (ξ ))(χ(x))‖2,m ≤

.

C ‖jm ξ(x)‖GM,π ,m E σm

for every .ξ ∈ ┌ ω (E), .x ∈ χ −1 (K), and .m ∈ Z≥0 . Then a0 a1 · · · am ‖jm (ν ∗ (ξ ))(χ(x))‖2,m ≤

.

Ca0 a1 · · · am ‖jm ξ(x)‖GM,π ,m E σm

for every .ξ ∈ ┌ ω (E), .x ∈ χ −1 (K), and .m ∈ Z≥0 . Define .b ∈ c0 (Z≥0 ; R>0 ) by b0 = Ca0 , bj =

.

aj , σ

j ∈ Z>0 .

Then, taking supremums of the preceding inequality gives 'ω ω ◦ ν ∗ (ξ ) ≤ p −1 qK,a (ξ ) χ (K),b

.

for .ξ ∈ ┌ ω (E). Now let .K ⊆ M be compact and let .a ∈ c0 (Z≥0 ; R>0 ). Let .x ∈ K and let .(Vx , ν x ) be a vector bundle chart for .E with .(Ux , χ x ) the chart for .M with .x ∈ Ux . We suppose that .Ux is precompact and that, by our appropriate version of Lemma 4.22, there exist .Cx , σx ∈ R>0 such that ‖jm ξ(y)‖GM,π

.

E ,m

≤

Cx ‖jm (ν ∗ ξ )(y)‖m,2 σxm

250

4 Analysis: Norm Estimates for Derivatives

for .ξ ∈ ┌ ω (E), .y ∈ cl(Ux ), .m ∈ Z≥0 . Therefore, a0 a1 · · · am ‖jm ξ(y)‖GπE ,m ≤

.

Cx a0 a1 · · · am ‖jm (ν ∗ ξ )(y)‖m,2 σxm

for .ξ ∈ ┌ ω (E), .y ∈ cl(Ux ), .m ∈ Z≥0 . Compactness of .K gives .x1 , . . . , xs ∈ K such that .K ⊆ ∪sj =1 Uxj and we then take C = max{Cx1 , . . . , Cxs },

.

σ = min{σx1 , . . . , σxs }.

We define .b ∈ c0 (Z≥0 ; R>0 ) by b0 = Ca0 , bj =

.

aj , σ

j ∈ Z>0 .

We then arrive at the inequality ω 'ω pK,a (ξ ) ≤ qcl(U x

.

1 ),b

'ω ◦ ν x1 ∗ (ξ ) + . . . + qcl(U ),b ◦ ν xs ∗ (ξ ) xs

which is valid for .ξ ∈ ┌ ω (E).

□

.

Since there is some slightly tormented logic necessitated by our invocation of local estimates, it is perhaps worth making it clear how the pertinent results are logically interconnected. This we do in the following diagram:

.

(4.16) Remark 4.25 (Adaptation to the Smooth Case) An analogous version of the preceding theorem holds in the smooth case. The proof is somewhat simpler in the smooth case, unlike in the proof of Lemma 4.22 where the smooth case is significantly simpler than the real analytic case. Note also that, in the smooth case, one does not need the local estimates for derivatives of real analytic functions, so this also significantly simplifies the logic. ◦ An immediate consequence of the theorem is the following.

4.3 Independence of Topologies on Connections and Metrics

251

Corollary 4.26 (Independence of Topologies on Connections and Metrics) Let πE : E → M be a real analytic vector bundle. Let .GM and .GM be .Cω -Riemannian metrics on .M, let .GπE and .GπE be .Cω -fibre metrics on .E, let .∇ M and .∇ M be .Cω affine connections on .M, and let .∇ πE and .∇ πE be .Cω -linear connections on .E, ω giving rise to seminorms .pK,a and .pωK,a , .K ⊆ M compact, .a ∈ c0 (Z≥0 ; R>0 ). Then the topologies defined by the two families of seminorms

.

ω pK,a , p ωK,a ,

.

agree.

K ⊆ M compact, a ∈ c0 (Z≥0 ; R>0 ),

Chapter 5

Continuity of Some Standard Geometric Operations

In this chapter we put to use the developments of the preceding two chapters to prove the continuity of a number of standard algebraic and differential operations on real analytic manifolds. The reader will notice as they go through the proofs that there are definite themes that emerge from the various proofs of continuity. Let us highlight these here, since our belief is that these themes are important in and of themselves, and justify the rather complex developments of Chaps. 3 and 4. We particularly draw attention to the following recurrent ideas. 1. We take full advantage of the algebraic bounds from Sect. 4.2.1 that were nominally developed to prove the bounds of Sect. 4.2.2. This gives independent interest to these algebraic bounds. 2. We make repeated use of the estimate from Lemma 4.17. As we discuss in the preamble to Sect. 4.2, the fact that Lemma 4.17 lies at the core of so much of what we do gives some motivation for the quite tedious geometric constructions from Chap. 3. 3. We shall prove in many cases that certain linear mappings between spaces of sections of real analytic vector bundles are continuous and open onto their image, i.e., homeomorphisms onto their image. One might hope to do this with a general Open Mapping Theorem. Indeed, since the space of real analytic sections of a vector bundle is both webbed and ultrabornological, one is perhaps in a position to use the Open Mapping Theorem of De Wilde [16]. However, since the images of our mappings are not necessarily ultrabornological (even closed subspaces of ultrabornological spaces may not be ultrabornological), we typically prove the openness by a direct argument, by virtue of our having given in Sect. 3.3 relations between iterated covariant derivatives going “both ways.” Moreover, the use of seminorms to prove these results is in keeping with the general tenor of this work.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. D. Lewis, Geometric Analysis on Real Analytic Manifolds, Lecture Notes in Mathematics 2333, https://doi.org/10.1007/978-3-031-37913-0_5

253

254

5 Continuity of Some Standard Geometric Operations

As we have indicated as we have been going along, the results in this chapter are applicable to (and easier in) the smooth case. At various points, we shall indicate the modifications required to adapt our approach to the smooth case.

5.1 Continuity of Algebraic Operations and Constructions In this section we consider topological aspects of two sorts of operations on spaces of sections. First we consider the standard constructions of linear algebra: addition and “multiplication.” Then we consider the relationships between direct sums and tensor products of, on the one hand, vector bundles, and, on the other hand, spaces of sections.

5.1.1 Continuity of Linear Algebraic Operations We begin with a consideration of continuity of standard algebraic operations with vector bundles. Theorem 5.1 (Continuity of Algebraic Operations) Let .πE : E → M and πF : F → M be .Cω -vector bundles. Then the following mappings are continuous:

.

(i) .┌ ω (E) ⊕ ┌ ω (E) ϶ (ξ, η) ⎬→ ξ + η ∈ ┌ ω (E); (ii) .┌ ω (F ⊗ E∗ ) × ┌ ω (E) ϶ (L, ξ ) ⎬→ L ◦ ξ ∈ ┌ ω (F). Also, fixing a vector bundle mapping .L ∈ ┌ ω (F ⊗ E∗ ), the following statements hold: (iii) if L is injective, then the mapping .┌ ω (E) ϶ ξ → ⎬ L ◦ ξ ∈ ┌ ω (F) is a topological monomorphism; (iv) if L is surjective, then the mapping .┌ ω (E) ϶ ξ → ⎬ L ◦ ξ ∈ ┌ ω (F) is a topological epimorphism. Proof We suppose that we have a real analytic affine connection .∇ M on .M, and real analytic vector bundle connections .∇ πE and .∇ πF in .E and .F, respectively. We suppose that we have a real analytic Riemannian metric .GM on .M, and real analytic ω fibre metrics .GπE and .GπF on .E and .F, respectively. This gives the seminorms .pK,a ω ω ω and .qK,a , .K ⊆ M compact, .a ∈ c0 (Z≥0 ; R>0 ), for .┌ (E) and .┌ (F), respectively. ω ⊗p ω , .K ⊆ M compact, We denote the induced seminorms for .┌ ω (F⊗E∗ ) by .qK,a K,a .a ∈ c0 (Z≥0 ; R>0 ). (i) The fibre norms from Sect. 2.3.2 satisfy the triangle inequality, and this readily gives ω ω ω pK,a (ξ + η) ≤ pK,a (ξ ) + pK,a (η),

.

5.1 Continuity of Algebraic Operations and Constructions

255

which immediately gives this part of the result. (ii) Let us make some preliminary computations from which this part of the theorem will follow easily. First, by Lemma 4.11, we have ‖L ◦ ξ(x)‖GM,π ≤ ‖L(x)‖GM,π

.

F

F ⊗πE

‖ξ(x)‖GM,π .

(5.1)

E

Next, by Lemmata 3.15, 4.11, and 4.13, we have ‖D∇k M ,∇ πF (L ◦ ξ(x))‖GM,π

.

F

k ⎛ ⎞ Σ k j k−j ‖D∇ M ,∇ πF ⊗πE L(x)‖GM,π ⊗π ‖D∇ M ,∇ πE ξ(x)‖GM,π ≤ F E E j j =0

for .k ∈ Z>0 . By (1.4) (twice) we have ‖jm (L ◦ ξ )(x)‖GM,π

.

≤

F ,m

m Σ 1 ‖D k π (L ◦ ξ(x))‖GM,π F k! ∇ M ,∇ F k=0

m k ⎛ ⎞ Σ 1 Σ k j k−j ‖D∇ M ,∇ πF ⊗πE L(x)‖GM,π ⊗π ‖D∇ M ,∇ πE ξ(x)‖GM,π ≤ F E E j k! k=0

j =0

j

=

k−j

k ‖D m Σ L(x)‖GM,π ⊗π ‖D∇ M ,∇ πE ξ(x)‖GM,π Σ ∇ M ,∇ πF ⊗πE F E E

j!

(k − j )! ⎫ ⎧ | ⎬ ⎨ ‖D j M π ⊗π L(x)‖GM,π ⊗π || F E ∇ ,∇ F E | j ≤m ≤ (m + 1)2 sup | ⎭ ⎩ j! | ⎧ ⎫ | | ⎨ ‖D k−j ⎬ ξ(x)‖ | G π M,πE ∇ M ,∇ E | j ≤m × sup | ⎩ ⎭ (k − j )! | k=0 j =0

≤ (m + 1)5/2 ‖jm L(x)‖GM,π

F ⊗πE ,m

‖jm ξ(x)‖GM,π

E ,m

.

Noting that .(m + 1)5/2 ≤ 3m+1 , .m ∈ Z>0 , we finally get ‖jm (L ◦ ξ )(x)‖GM,π

.

F ,m

≤ 3m+1 ‖jm L(x)‖GM,π

F ⊗πE ,m

‖jm ξ(x)‖GM,π

E ,m

.

(5.2)

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5 Continuity of Some Standard Geometric Operations

Let .K√⊆ M be compact and let .a ∈ c0 (Z≥0 ; R>0 ). Define .a ' ∈ c0 (Z≥0 ; R>0 ) by ' 3aj , .j ∈ Z≥0 . We then have .a = j ω ω ω ω ω ω ω qK,a (L ◦ ξ ) ≤ qK,a ' ⊗ pK,a ' (L)pK,a ' (ξ ) = (qK,a ' ⊗ pK,a ' ) ⊗ pK,a ' (L ⊗ ξ ).

.

By Jarchow [37, Theorem 15.1.2], this gives continuity of the bilinear map (L, ξ ) ⎬→ L ◦ ξ . To prove the last two parts of the theorem, we first prove a couple of technical lemmata.

.

Lemma 1 Let .U and .V be locally convex topological vector spaces, and let L ∈ L(U; V). If, for every continuous seminorm q for .U, there exists a continuous seminorm p for .V such that

.

q(u) ≤ p ◦ L(u),

.

u ∈ U,

then L is a topological homomorphism. Proof Let us denote by .Q the set of continuous seminorms for .U and by .P the set of continuous seminorms for .V. Our hypothesis is that, for each .q ∈ Q, there exists .pq ∈ P such that .q(u) ≤ pq ◦ L(u) for all .u ∈ U. We fix such a choice of .pq for each .q ∈ Q. We first prove that there are 0-bases .BU for .U and .BV for .V such that, for each .B ∈ BU , there exists .C ∈ BV such that C ∩ image(L) ⊆ L(B).

.

To see this, first let .q ∈ Q and let .pq ∈ P be as hypothesised. Then pq ◦ L(u) < 1 ⇒ q(u) < 1 ⇒ L(u) ∈ L(q −1 ([0, 1))).

.

Thus pq−1 ([0, 1)) ∩ image(L) ⊆ L(q −1 ([0, 1))).

.

Now let .BU be the collection of all 0-neighbourhoods of the form B=

k ∩

.

qj−1 ([0, 1)),

qj ∈ Q, j ∈ {1, . . . , k}, k ∈ Z>0 .

j =1

This is a 0-base for .U. Then, by our above computations, ⎛ .

⎝

k ∩

j =1

⎞

⎛

pq−1 ([0, 1))⎠ ∩ image(L) ⊆ L ⎝ j

k ∩

j =1

⎞ qj−1 ([0, 1))⎠ .

5.1 Continuity of Algebraic Operations and Constructions

257

Thus, the 0-base k ∩ .

pj−1 ([0, 1)),

pj ∈ P, j ∈ {1, . . . , k}, k ∈ Z>0 ,

j =1

for .V has the desired property. Now let .O ⊆ V be open and let .u ∈ O. Let .B ∈ BU be such that .u + B ⊆ O and let .C ∈ BV be such that .C ∩ image(L) ⊆ L(B). Then L(u) + C ∩ image(L) ⊆ L(u) + L(B) = L(u + B) ⊆ L(O).

.

Thus .L(u) + C ∩ image(L) is a neighbourhood of .L(u) in .L(O) which shows that L(O) is open in .image(L). .Δ

.

Lemma 2 If .πE : E → M and .πF : F → M are .Cω -vector bundles, and if .L ∈ ┌ ω (F ⊗ E∗ ), then the following statements hold: (i) if L is injective, then there exists a left-inverse .L' ∈ ┌ ω (E ⊗ F∗ ); (ii) if L is surjective, then the mapping .┌ ω (E) ϶ ξ ⎬→ L ◦ ξ ∈ ┌ ω (F) is surjective. Proof (i) First we note that .image(L) is a .Cω -subbundle of .F and that L is a .Cω -vector bundle isomorphism onto .image(L), cf. [1, Proposition 3.4.18]. Let .G ⊆ F be the .GπF -orthogonal complement to .image(L) which is then itself a .Cω subbundle of .F. Clearly, .F = image(L) ⊕ G. Let L' : image(L) ⊕ G → E .

(L(e), g) ⎬→ e,

and note that .L' is obviously a left-inverse of L. It is also of class .Cω since the projection from .F to the summand .image(L) is of class .Cω . (ii) Note that .ker(L) is a .Cω -subbundle of .E, cf. [1, Proposition 3.4.18]. Using a real analytic fibre metric as in the preceding part of the proof, let .G be a real analytic subbundle of .E for which .E = ker(L) ⊕ G. Then .L|G is a vector bundle isomorphism onto .F. Thus the mapping .┌ ω (G) ϶ ξ ⎬→ L ◦ ξ ∈ ┌ ω (F) is a vector space isomorphism. .Δ (iii) By Lemma 2(i), we suppose that there is a .Cω -vector bundle mapping .L' that is a left-inverse for L. Then, from the first part of the proof, for a compact .K ⊆ M and for .a ∈ c0 (Z≥0 ; R>0 ), let .C ∈ R>0 and .a ' ∈ c0 (Z≥0 ; R>0 ) be such that ω ω pK,a (L' ◦ η) ≤ CqK,a ' (η),

.

η ∈ ┌ ω (F).

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5 Continuity of Some Standard Geometric Operations

We then have, for .ξ ∈ ┌ ω (E), ω ω ω pK,a (ξ ) = pK,a (L' ◦ L ◦ ξ ) ≤ CqK,a ' (L ◦ ξ ).

.

By Lemma 1, this suffices to establish that L is open onto its image. (iv) By Lemma 2(ii) and since the set of real analytic sections of a vector bundle, with the .Cω -topology, is a webbed and ultrabornological locally convex topological vector space (Proposition 2.14), this part of the result follows from the De Wilde Open Mapping Theorem. □ Remarks 5.2 (Adaptation to the Smooth Case) The preceding proof is easily adapted to the smooth case. Indeed, the proof is a little easier since one does not need to carefully keep track of the growth in m of the coefficient of the norm of the m-jet. .◦ The theorem admits the following corollary, where we use the topological notions of monomorphism and epimorphism from Definition 1.7 for exactness of sequences of locally convex topological vector spaces. Corollary 5.3 (Short Exact Sequences of Spaces of Sections Induced by Short Exact Sequences of Vector Bundles) Let .πE : E → M, .πF : F → M, and .πG : G → M be real analytic vector bundles and suppose that we have a short exact sequence .

of real analytic vector bundle mappings. Then the sequence .

is short exact in the category of locally convex topological vector spaces. A consequence of the corollary, and one that one might use without thinking about it, is that isomorphic vector bundles give rise to isomorphic (as locally convex topological vector spaces) spaces of sections.

5.1.2 Direct Sums and Tensor Products of Spaces of Sections Given real analytic vector bundles .πE : E → M and .πF : F → M, we can build two real analytic vector bundles, the direct sum .E ⊕ F and the tensor product .E ⊗ F. We consider in this section topological matters associated with the spaces of sections of these vector bundles.

5.1 Continuity of Algebraic Operations and Constructions

259

First we consider direct sums, where the situation is simple. Theorem 5.4 (Topology for Space of Sections of Direct Sums) If .πE : E → M and .πF : F → M are real analytic vector bundles, then the mapping ┌ ω (E ⊕ F) ϶ (x ⎬→ ξ(x) ⊕ η(x)) ⎬→ ξ ⊕ η ∈ ┌ ω (E) ⊕ ┌ ω (F),

.

ξ ∈ ┌ ω (E), η ∈ ┌ ω (F), is an isomorphism of locally convex topological vector spaces. Moreover, the projections onto the components of the direct sum on the right are topological epimorphisms and the inclusions of the components of the direct sum into the direct sum on the right are topological monomorphisms. ω ω , .K ⊆ M compact, .a ∈ c (Z ; R ), the Proof Denote by .pK,a and .qK,a 0 ≥0 >0 ω ω seminorms for .┌ (E) and .┌ (F), respectively. The mapping given in the statement of the theorem is certainly an algebraic isomorphism. Note that fibre metrics .GE and .GF give rise to the fibre metric

GE⊕F (ex ⊕ fx , ex' ⊕ fx' ) = GE (ex , ex' ) + GF (fx , fx' )

.

ω , for .E ⊕ F. The corresponding set of seminorms for .┌ ω (E ⊕ E) we denote by .rK,a .K ⊆ M compact, .a ∈ c0 (Z≥0 ; R>0 ), for .E ⊕ F. A set of seminorms defining the topology of .┌ ω (E) ⊕ ┌ ω (F) can be taken to be ω ω ξ ⊕ η ⎬→ max{pK,a (ξ ), qL,b (η)},

.

K, L ⊆ M compact, a, b ∈ c0 (Z≥0 ; R>0 ).

Let .K ⊆ M be compact and let .a ∈ c0 (Z≥0 ; R>0 ). The inequality (which makes use of (1.4)) .

ω ω ω max{pK,a (ξ ), qL,b (η)} ≤ rK∪L,c (x ⎬→ ξ(x) ⊕ η(x)),

ξ ∈ ┌ ω (E), η ∈ ┌ ω (F),

with .c ∈ c0 (Z≥0 ; R>0 ) defined by .cm = max{am , bm }, .m ∈ Z≥0 , establishes the continuity of the mapping in the statement of the theorem. For the continuity of the inverse, we let .K ⊆ M be compact and let .a ∈ c0 (Z≥0 , R>0 ), and make use of the inequality ω rK,a (x ⎬→ ξ(x) ⊕ η(x)) ≤

.

√ ω ω 2 max{pK,a (ξ ), qK,a (η)},

ξ ∈ ┌ ω (E), η ∈ ┌ ω (F),

which again uses (1.4). The final assertion of the theorem follows from general facts about finite direct sums [65, §2]. □ For tensor products, the situation is more complicated, and is worked out in the smooth case in [57, Chapter 12]. The first thing one must do is understand the algebraic setting properly, something which is interesting in its own right. In the

260

5 Continuity of Some Standard Geometric Operations

next result we are working with the .Cω (M)-module structure of the set of sections of a vector bundle, and we use .⊗Cω (M) to denote the tensor product of .Cω (M)-modules, as opposed to .⊗ which denotes the tensor product of .R-vector spaces. Proposition 5.5 (Sections of a Tensor Product of Vector Bundles) If .πE : E → M and .πF : F → M are real analytic vector bundles, then the bilinear mapping ┌ ω (E) × ┌ ω (F) ϶ (ξ, η) ⎬→ (x ⎬→ ξ(x) ⊗ η(x)) ∈ ┌ ω (E ⊗ F)

.

defines (by the universal property of tensor products)1 a .Cω (M)-module isomorphism ┌ ω (E) ⊗Cω (M) ┌ ω (M) ≃ ┌ ω (E ⊗ F).

.

Proof We first consider the case of trivial vector bundles, say .E = RkM and .F = RlM . Let .(e1 , . . . , ek ) and .(f 1 , . . . , f l ) be the standard basis for .Rk and .Rl , respectively. We then have the sections .ξj (x) = (x, ej ), .j ∈ {1, . . . , k}, and .ηa (x) = (x, f a ), ω ω k ω l .a ∈ {1, . . . , l}, which are bases for the .C (M)-modules .┌ (R ) and .┌ (R ), M M k l ω respectively. One then readily verifies that .ξj ⊗ ηa ∈ ┌ ((R ⊗ R )M ) defined by ξj ⊗ ηa (x) = (x, ej ⊗ f a ),

.

j ∈ {1, . . . , k}, a ∈ {1, . . . , l},

give a basis of .┌ ω ((Rk ⊗Rl )M ) as a .Cω (M)-module. This establishes the proposition in this case of trivial bundles. If .E and .F are not necessarily trivial, then Corollary 2.5 gives .E as a subbundle of .RkM for a suitable .k ∈ Z>0 and .F as a subbundle of .RlM for a suitable .l ∈ Z>0 . Let .E⊥ and .F⊥ denote the orthogonal complement subbundles using the standard Euclidean fibre metric (for example). We then have the commuting diagram

.

Here the horizontal arrows are those induced by the map from the statement of the proposition; in particular the upper horizontal arrow is an isomorphism of .Cω (M)modules. Note that the vector bundle isomorphism (E ⊕ E⊥ ) ⊗ (F ⊕ F⊥ ) ≃ (E ⊗ F) ⊕ (E ⊗ F⊥ ) ⊕ (E⊥ ⊗ F) ⊕ (E⊥ ⊗ F⊥ )

.

1 By

this we mean that the universal property of tensor products defines a module homomorphism .┌

ω

(E) ⊗Cω (M) ┌ ω (M) → ┌ ω (E ⊗ F).

Thus the assertion of the proposition is that this homomorphism is an isomorphism.

5.1 Continuity of Algebraic Operations and Constructions

261

cf. [35, Theorem IV.5.9] gives rise to the .Cω (M)-module isomorphism ┌ ω ((E⊕E⊥ )⊗(F⊕F⊥ )) ≃ ┌ ω (E⊗F)⊕┌ ω (E⊗F⊥ )⊕┌ ω (E⊥ ⊗F)⊕┌ ω (E⊥ ⊗F⊥ )

.

by Theorem 5.4. By Theorem 5.4 and [35, Theorem IV.5.9] (now for .Cω (M)modules) we have ┌ ω (E ⊕ E⊥ ) ⊗Cω (M) ┌ ω (F ⊕ F⊥ )

.

≃ (┌ ω (E) ⊗Cω (M) ┌ ω (F)) ⊕ (┌ ω (E) ⊗Cω (M) ┌ ω (F⊥ )) ⊕ (┌ ω (E⊥ ) ⊗Cω (M) ┌ ω (F)) ⊕ (┌ ω (E⊥ ) ⊗Cω (M) ┌ ω (F⊥ )). The vertical arrows in the diagram above are then the associated inclusions and projections arising from the preceding two equations. Moreover, the arrows induced by inclusion are injective while those induced by projection are surjective. With all of this in place, we leave to the reader the routine and tedious verification that the diagram makes sense and commutes in the category of .Cω (M)-modules. We also leave to the reader the routine verification that (1) by considering the vertical inclusions, one shows that the bottom horizontal arrow is injective and (2) by consider the vertical projections, the bottom horizontal arrow is surjective. □ To make the preceding algebraic construction have sense topologically, we need to consider the relationship between .┌ ω (E) ⊗Cω (M) ┌ ω (F) and .┌ ω (E) ⊗ ┌ ω (F). For the latter tensor product, we have well-defined locally convex topologies, and it is the projective tensor topology of which we shall make use. To connect the two tensor products, we make a few constructions. It is convenient to make these constructions in a fairly general setting, at least for the algebraic part of the construction. First we have the following definition. Definition 5.6 (Balanced Bilinear Map) Let .R be a commutative unit ring, and let A, .B, and .C be .R-modules. A .Z-bilinear mapping .β : A × B → C is .R-balanced if

.

β(ra, b) = β(a, rb),

.

r ∈ R, a ∈ A, b ∈ B.

◦

The following lemma explains the importance of balanced mappings. Lemma 5.7 (Universal Property of Tensor Product Over Different Rings) Let R be a commutative unit ring, let .S ⊆ R be a subring for which .1 ∈ S, and let .A and .B be .R-modules. Then there exists an .S-module .X and an .R-balanced, .S-bilinear map .γ : A × B → X such that, for every .S-module .C and every .R-balanced, .Sbilinear map .β : A × B → C, there exists a unique .α ∈ HomS (X; C) such that the .

262

5 Continuity of Some Standard Geometric Operations

diagram

.

commutes. Proof The construction of the .S-module .X, and the verification of its uniqueness, is the same as the standard construction of the tensor product .A ⊗R B, e.g., [35, §5], except that we work with .S-modules rather than .Z-modules. Similarly, the universal property as stated for .R-balanced, .S-bilinear maps is verified as in the usual construction. □ As indicated in our outline of the proof, the .S-module .X one gets from the previous lemma is the same as the tensor product .A ⊗R B, except that it has the structure of an .S-module and not simply a .Z-module. In the case .R = S, the previous lemma yields the usual .S-module tensor product .A ⊗S B that is universal for .S-bilinear mappings. We denote the (unique up to .S-module isomorphism) module .X whose existence is asserted in the lemma by .A ⊗R B, with the rôle of .S suppressed in the notation. We also let ⊗R : A × B → A ⊗R B

.

be the associated .R-balanced, .S-bilinear map. The rôle of .S in the constructions is further clarified by the following result. Lemma 5.8 (Tensor Products of Modules Over Different Rings) Let .R be a commutative unit ring, let .S ⊆ R be a subring for which .1 ∈ S, and let .A, .B, and .C be .R-modules. Let .J be the .Z-submodule of .A ⊗S B generated by elements of the form (ra) ⊗S b − a ⊗S (rb),

.

r ∈ R, a ∈ A, b ∈ B.

Then the mapping from .A ⊗R B to .(A ⊗S B)/J defined by a ⊗R b ⎬→ a ⊗S b + J

.

is an isomorphism of .S-modules. Proof First let us check that the statement makes sense in that the domain and codomain of the asserted isomorphism are .S-modules. Since .A⊗R B is an .R-module, it is certainly an .S-module. To show that .(A ⊗S B)/J has an .S-module structure, we

5.1 Continuity of Algebraic Operations and Constructions

263

need only show that .J is an .S-submodule. To see this, we claim that the generators for .J remain in .J upon multiplication with .S. Indeed, s((ra)⊗S b−a⊗S (rb)) = r(sa)⊗S b−(sa)⊗S (rb), r ∈ R, s ∈ S, a ∈ A, b ∈ B.

.

Now let ⊗R : A × B → A ⊗R B,

.

⊗S : A × B → A ⊗S B

be the canonical bilinear maps, the first using the .R-module structure of .A and B, and the second using the .S-module structure. The universal property of the .Smodule tensor product gives a unique .S-module homomorphism .λ as in the diagram

.

.

The diagram also introduces us to the projection .π , which is an homomorphism of S-modules. Note that

.

λ((ra) ⊗S b) = (ra) ⊗R b = a ⊗R (rb) = λ(a ⊗S (rb)),

.

and so .J ⊆ ker(λ). Then the .S-module homomorphism .σ is that induced by the universal property of quotients [35, Theorem IV.1.7]. To define .τ , let .r ∈ R, .a ∈ A, and .b ∈ B, so that .(ra) ⊗S b − a ⊗S (rb) ∈ J. Thus π((ra) ⊗S b) = π(a ⊗S (rb))

.

=⇒

π ◦ ⊗S (ra, b) = π ◦ ⊗S (a, rb);

that is, .π ◦ ⊗S is .R-balanced and .S-bilinear. By Lemma 5.7, there exists an .Smodule homomorphism .τ : A ⊗R S → (A ⊗S B)/J such that .τ ◦ ⊗R = π ◦ ⊗S . We have ⊗R = λ ◦ ⊗S = σ ◦ τ ◦ ⊗S = σ ◦ τ ◦ ⊗R ,

.

and the uniqueness assertion of Lemma 5.7 in the special case .R = S gives .σ ◦ τ as the identity mapping of .S-modules on .(A ⊗S B)/J. Again by the uniqueness assertion of Lemma 5.7 in the case that .R = S, π ◦ ⊗S = τ ◦ ⊗R = τ ◦ λ ◦ ⊗S = τ ◦ σ ◦ π ◦ ⊗S =⇒ π = τ ◦ σ ◦ π.

.

Since .π is surjective, it is right-invertible and so .τ ◦ σ is the identity mapping on the S-module .(A ⊗S B)/J. Therefore, .τ is an .S-module isomorphism. □

.

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5 Continuity of Some Standard Geometric Operations

Now we work our way back from algebra to topology. One can do this in the setting of topological modules over topological rings [57, Chapter 10]. However, we do not undertake this level of generality for the restricted use we shall make of these developments. In any case, a slightly imaginative reader can see how a general theory will work based on our specific application. We let .πE : E → M and .πF : F → M be real analytic vector bundles. The preceding constructions can be applied to the case of .S = R, .R = Cω (M), ω ω .A = ┌ (E), and .B = ┌ (F). Recalling Proposition 5.5, we thus have .R-vector space isomorphisms ┌ ω (E ⊗ F) ≃ ┌ ω (E) ⊗Cω (M) ┌ ω (F) ≃ (┌ ω (E) ⊗ ┌ ω (F))/J,

.

where .J is the subspace of .┌ ω (E) ⊗ ┌ ω (F) generated by elements of the form (f ξ ) ⊗ η − ξ ⊗ (f η).

.

In the topological setting, we do not work with .J, but rather the closure .cl(J) in .┌ ω (E) ⊗ ┌ ω (F). This ensures that the quotient .(┌ ω (E) ⊗ ┌ ω (F))/ cl(J) is Hausdorff [37, Proposition 4.4.2]. We now note that we have the diagram

.

where .⊗π means that we are considering the projective tensor topology on the algebraic tensor product as in Sect. 1.8.5. The unnamed horizontal arrows are the quotient mappings, which are thus continuous. The composition of the three mappings gives a mapping ⊗πCω (M) : ┌ ω (E) × ┌ ω (F) → ┌ ω (E) ⊗πCω (M) ┌ ω (F)  (┌ ω (E) ⊗π ┌ ω (F))/ cl(J).

.

We call .┌ ω (E) ⊗πCω (M) ┌ ω (F) the balanced projective tensor product of .┌ ω (E) and ω .┌ (F). For a locally convex topological vector space .V, let us denote by ω (M)

LC

.

(┌ ω (E), ┌ ω (F); V)

the set of continuous .R-bilinear mappings that are .Cω (M)-balanced. We then have the following result. Lemma 5.9 (Universal Property of Balanced Projective Tensor Product) Let πE : E → M and .πF : F → M be real analytic vector bundles. Then, for any locally ω convex topological vector space .(V, O) and any .β ∈ LC (M) (┌ ω (E), ┌ ω (F); V),

.

5.1 Continuity of Algebraic Operations and Constructions

265

there exists a unique φβ ∈ L(┌ ω (E) ⊗πCω (M) ┌ ω (F); V)

.

for which the diagram

.

commutes. Proof Let .(V, O) be a locally convex topological vector space and let .β ∈ ω LC (M) (┌ ω (E), ┌ ω (F); V). The diagram

.

introduces the players in the proof. The proof, then, consists of understanding why the players are located where they are and what are their properties. By Jarchow [37, Theorem 15.1.2] there exists a unique ψβ' ∈ L(┌ ω (E) ⊗π ┌ ω (F); V)

.

such that the upper triangle in the diagram commutes. Since .β is .Cω (M)-balanced and .R-bilinear, .J ⊆ ker(ψβ' ) (cf. the proof of Lemma 5.8). Thus, by universal properties of quotients [37, Proposition 4.1.2], we have the mapping .ψβ so that the upper two triangles in the diagram commute. As a consequence of the continuity of .β and the fact that (1) the projective tensor topology is the final topology for ω ω .┌ (E) ⊗ ┌ (F) associated with .⊗ (depending on one’s source, this can be the definition of the projective tensor topology) and (2) the topology of .(┌ ω (E) ⊗π ┌ ω (F))/J is the final topology associated with the quotient [37, Proposition 4.1.2], the universal property of final topologies ensures that .ψβ is continuous. Since

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5 Continuity of Some Standard Geometric Operations

J ⊆ ker(ψβ' ) and since .ψβ' is continuous, .cl(J) ⊆ ker(ψβ' ). By universal properties of quotients, this gives rise to the continuous linear mapping .φβ . This mapping is unique by the uniqueness of the continuous linear mappings arising from the various universal properties for tensor products and quotients. □

.

Now let us see how to put all of this together to arrive at a description of the topology for the sections of a tensor product of vector bundles. Theorem 5.10 (Topology for Space of Sections of Tensor Products) If .πE : E → M and .πF : F → M are real analytic vector bundles, then the algebraic isomorphism of Proposition 5.5 induces an isomorphism of locally convex topological vector spaces ┌ ω (E) ⊗πCω (M) ┌ ω (F) ≃ ┌ ω (E ⊗ F).

.

Proof The proof bears some similarities to that for Proposition 5.5 and, as in that proof, we shall leave to the reader the verification of various routine details. First we consider the case of trivial vector bundles, say .E = RkM and .F = RlM . Let .(e1 , . . . , ek ) and .(f 1 , . . . , f l ) be the standard basis for .Rk and .Rl , respectively. We then have the sections .ξj (x) = (x, ej ), .j ∈ {1, . . . , k}, and .ηa (x) = (x, f a ), ω ω k ω l .a ∈ {1, . . . , l}, which are a basis for the .C (M)-modules .┌ (R ) and .┌ (R ), M M respectively. One then readily verifies that .ξj ⊗ ηa ∈ ┌ ω ((Rk ⊗ Rl )M ) defined by ξj ⊗ ηa (x) = (x, ej ⊗ f a ),

.

j ∈ {1, . . . , k}, a ∈ {1, . . . , l},

give a basis of .┌ ω ((Rk ⊗ Rl )M ) as a .Cω (M)-module, exactly as in the proof of Proposition 5.5. The .R-bilinear, .Cω (M)-balanced mapping γ : ┌ ω (RkM ) × ┌ ω (RlM ) → ┌ ω ((Rk ⊗ Rl )M ) .

(ξ, η) ⎬→ (x ⎬→ ξ(x) ⊗ η(x))

is then expanded in bases as ⎛ .

⎝

k Σ

f j ξj ,

j =1

l Σ a=1

⎞ g a ηa ⎠ ⎬→

l k Σ Σ

(x ⎬→ f j (x)g a (x)ξj (x) ⊗ ηa (x)).

j =1 a=1

The continuity of this mapping follows from Theorem 5.1. By Lemma 5.9, there is an induced continuous linear mapping α : ┌ ω (RkM ) ⊗πCω (M) ┌ ω (RlM ) → ┌ ω ((Rk ⊗ Rl )M )

.

5.1 Continuity of Algebraic Operations and Constructions

267

such that the diagram

.

commutes. The mapping β : ┌ ω ((Rk ⊗ Rl )M ) → ┌ ω (RkM ) ⊗πCω (M) ┌ ω (RlM )

.

in the diagram is defined by ⎛ β ⎝x ⎬→

l k Σ Σ

.

⎞ Aj a (x)ξj (x) ⊗ ηa (a)⎠ =

j =1 a=1

l k Σ Σ

Aj a ξj ⊗πCω (M) ηa .

j =1 a=1

Note that .β is the composition of the algebraic isomorphism ⎛ ┌ ω ((Rk ⊗ Rl )M ) ϶ ⎝x ⎬→

l k Σ Σ

.

⎞ Aj a (x)ξj (x) ⊗ ηa (x)⎠

j =1 a=1

⎬→

l k Σ Σ

Aj a ξ ⊗Cω (M) η ∈ ┌ ω (RkM ) ⊗Cω (M) ┌ ω (RlM )

j =1 a=1

with the quotient by .cl(J), and so is continuous by Theorem 5.1 and the definition of the quotient topology. A direct computation using the definitions of the symbols involved shows that .α and .β are inverses of one another. The above establishes the theorem for trivial bundles. If .E and .F are not necessarily trivial, then Corollary 2.5 gives .E as a subbundle of k l ⊥ .R for a suitable .k ∈ Z>0 and .F as a subbundle of .R for a suitable .l ∈ Z>0 . Let .E M M and .F⊥ denote the orthogonal complement subbundles using the standard Euclidean fibre metric (for example). Similarly to what we saw in the proof of Proposition 5.5 (now working with tensor products over .R rather than .Cω (M)), we have the algebraic isomorphism ┌ ω (E ⊕ E⊥ ) ⊗ ┌ ω (F ⊕ F⊥ )

.

≃ (┌ ω (E) ⊗ ┌ ω (F)) ⊕ (┌ ω (E) ⊗ ┌ ω (F⊥ )) ⊕ (┌ ω (E⊥ ) ⊗ ┌ ω (F)) ⊕ (┌ ω (E⊥ ) ⊗ ┌ ω (F⊥ )).

(5.3)

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5 Continuity of Some Standard Geometric Operations

We claim that this algebraic isomorphism is a topological isomorphism with the tensor products over .R equipped with the projective tensor topology. To see this, we note that the algebraic isomorphism (5.3) is the linear mapping associated with the bilinear mapping ┌ ω (E ⊕ E⊥ ) × ┌ ω (F ⊕ F⊥ ) ϶ (ξ ⊕ ξ ⊥ , η ⊕ η⊥ )

.

⎬→ (ξ ⊗ η) ⊕ (ξ ⊗ η⊥ ) ⊕ (ξ ⊥ ⊗ η) ⊕ (ξ ⊥ ⊗ η⊥ ) ∈ (┌ ω (E) ⊗ ┌ ω (F)) ⊕ (┌ ω (E) ⊗ ┌ ω (F⊥ )) ⊕ (┌ ω (E⊥ ) ⊗ ┌ ω (F)) ⊕ (┌ ω (E⊥ ) ⊗ ┌ ω (F⊥ )). This bilinear mapping is continuous by Theorem 5.1. Thus the induced linear mapping is continuous by the universal property of the projective tensor topology [37, Theorem 15.1.2]. Therefore, the induced linear mapping is a topological isomorphism by the De Wilde Open Mapping Theorem, using the fact that the space of sections of a real analytic vector bundle is webbed and ultrabornological by Proposition 2.14. We next claim that the algebraic isomorphism (5.3) induces a topological isomorphism ┌ ω (E ⊕ E⊥ ) ⊗πCω (M) ┌ ω (F ⊕ F⊥ )

.

≃ (┌ ω (E) ⊗πCω (M) ┌ ω (F)) ⊕ (┌ ω (E) ⊗πCω (M) ┌ ω (F⊥ )) ⊕ (┌ ω (E⊥ ) ⊗πCω (M) ┌ ω (F)) ⊕ (┌ ω (E⊥ ) ⊗πCω (M) ┌ ω (F⊥ )).

(5.4)

To see this, we note that each algebraic tensor product .⊗ induces the topological tensor product .⊗πCω (M) by a quotient by an appropriate subspace, denote by .cl(J) in our general development. Here we have a variety of .J’s, so we need to name them:

.

Under the algebraic isomorphism (5.3) we have J0 ≃ J1 ⊕ J2 ⊕ J3 ⊕ J4 ,

.

5.1 Continuity of Algebraic Operations and Constructions

269

as can be directly verified from the definitions. Continuity of the algebraic isomorphism (5.3) and Theorem 5.4 then give .

cl(J0 ) ≃ cl(J1 ) ⊕ cl(J2 ) ⊕ cl(J3 ) ⊕ cl(J4 ).

Now we observe that a direct sum of quotients is isomorphic to the quotient of the direct sums; this is a consequence of the transitivity of final topologies, cf. [34, Proposition 2.12.2]. Thus we have a topological isomorphism

.

┌ ω (E ⊕ E⊥ ) ⊗ ┌ ω (F ⊕ F⊥ ) cl(J0 ) ≃

┌ ω (E) ⊗ ┌ ω (F) ┌ ω (E) ⊗ ┌ ω (F⊥ ) ⊕ cl(J1 ) cl(J2 ) ⊕

┌ ω (E⊥ ) ⊗ ┌ ω (F) ┌ ω (E⊥ ) ⊗ ┌ ω (F⊥ ) ⊕ , cl(J3 ) cl(J4 )

which is exactly the desired conclusion (5.4). Next we note that, just as in the proof of Proposition 5.5, we have the algebraic isomorphism ┌ ω ((E⊕E⊥ )⊗(F⊕F⊥ )) ≃ ┌ ω (E⊗F)⊕┌ ω (E⊗F⊥ )⊕┌ ω (E⊥ ⊗F)⊕┌ ω (E⊥ ⊗F⊥ ),

.

which is a topological isomorphism by Corollary 5.3 since it is the mapping on sections induced by a vector bundle isomorphism. Finally, the first part of the proof gives a topological isomorphism Ф : ┌ ω (E ⊕ E⊥ ) ⊗πCω (M) ┌ ω (F ⊕ F⊥ ) → ┌ ω ((E ⊕ E⊥ ) ⊗ (F ⊕ F⊥ ))

.

Using the definitions of the topological isomorphisms (5.3) and (5.4), one can show that .Ф can be represented as a .4 × 4 “matrix” relative to the two direct sum decompositions, with the matrix having the form ⎡

Ф1 ⎢0 .Ф = ⎢ ⎣0 0

0 Ф2 0 0

0 0 Ф3 0

⎤ 0 0⎥ ⎥. 0⎦ Ф4

The mapping .Ф1 : ┌ ω (E) ⊗πCω (M) ┌ ω (F) → ┌ ω (E ⊗ F) is the asserted mapping in the statement of the theorem, and it is a topological isomorphism since .Ф is, and since the projections and inclusions on the first sums are epimorphisms and monomorphisms, respectively, by Theorem 5.4. □

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5 Continuity of Some Standard Geometric Operations

5.2 Continuity of Operations Involving Differentiation In this section we consider the continuity of operations involving differentiation. First we consider a general version of the assertion that “differentiation is continuous,” and following this we give a collection of consequently elementary results derived from this general fact. Theorem 5.11 (Prolongation of Sections Is Continuous) Let .πE : E → M be a Cω -vector bundle. If .k ∈ Z≥0 , then the map

.

Jkω : ┌ ω (E) → ┌ ω (Jk E) .

ξ ⎬→ jk ξ

is a topological homomorphism. Proof We let .∇ M be a torsion-free (for simplicity) .Cω -affine connection on .M, ω ω π .∇ E be a .C -vector bundle connection in .E, .GM be a .C -Riemannian metric on ω .M, and .GπE be a .C -vector bundle connection in .E. We denote the associated k,ω ω and for .┌ ω (Jk E) by .pK,a , for .K ⊆ M compact seminorms for .┌ ω (E) by .pK,a and .a ∈ c0 (Z≥0 ; R>0 ). We recall from (3.20) that we have the vector bundle mapping .ιE,m,k : Jm+k E → m J Jk E defined by the requirement that .ιE,m,k ◦ jm+k ξ = jm jk ξ . The representation .ιˆE,m,k of this vector bundle mapping with respect to the decompositions of jet bundles as in Sect. 2.3.1 is given by Lemma 3.22. We first perform estimates regarding this vector bundle mapping. Let .j, l ∈ Z>0 . By Lemmata 4.9, 4.13, and 4.14, and by (4.10), ‖θj,l ⊗ idE ‖GM,π ≤

.

E

1 1√ ‖Δj −1,1 ⊗ Δ1,l−1 ‖GM ‖idE ‖GπE ≤ dj l, 2 2

where d is the fibre dimension of .E. By Lemma 2.22, let .C1 , σ1 ∈ R>0 be such that ‖D∇r M ,∇ πE R∇ πE (x)‖GM,π ≤ C1 σ1−r r!,

.

E

r ∈ Z≥0 , x ∈ K.

Without loss of generality, we can assume that .C1 ≥ 1 and .σ1 ≤ 1. For .Aj +l ∈ ┌ ω (Sj +l (T∗x M) ⊗ Ex ) and .Al−1 ∈ ┌ ω (Sl−l (T∗x M) ⊗ Ex ), we now have j −1

‖Aj +l (x) + θj,l ⊗ idE (D∇ M ,∇ πE R∇ πE (Al−1 ))(x)‖GM,π

.

≤ ‖Aj +l (x)‖GM,π

E

E

1√ j −1 dC1 j lσ1 (j − 1)!‖Al−1 (x)‖GM,π . + E 2

5.2 Continuity of Operations Involving Differentiation

271

Then, using (1.4), (4.9), and (4.10),

.

m k Σ 1Σ j −1 ‖Aj +l (x) + θj,l ⊗ idE (D∇ M ,∇ πE R∇ πE (Al−1 ))(x)‖GM,π E l! j =0

l=0

≤

≤ ≤

Σ1Σ 1√ dC1 σ1−m (‖Aj +l (x)‖GM,π + lj !‖Al−1 (x)‖GM,π ) E E 2 l! 1√ dC1 σ1−m 2

m

k

l=0

j =0

k m Σ Σ l=0 j =0

j !(l + j )! (‖Aj +l (x)‖GM,π + ‖Al−1 (x)‖GM,π ) E E (l − 1)!(j + l)!

1√ 2

dC1 σ1−m k!m(m + 1) · · · (m + k) ⎛ ⎞ k m Σ m Σ Σ 1 1 ×⎝ ‖Aj +l (x)‖GM,π + k ‖Al−1 (x)‖GM,π ⎠ E E (j + l)! (l − 1)! l=0 j =0

≤

l=1

1√ dC1 σ1−m k!(m + k)k+1 2 ⎛ ⎞ m+k m+k Σ 1 Σ 1 ‖Aj (x)‖GM,π + k ‖Aj (x)‖GM,π ⎠ × ⎝(m + 1) E E j! j! j =0

j =0

Σ 1 1√ ‖Aj (x)‖GM,π . dC1 σ1−m k!(m + k)k+1 (m + 1 + k) E 2 j! m+k

≤

j =0

(In this computation, we take the convention that .A−1 = 0.) For .σˆ ∈ (0, σ1 ) we have .

1 m√ σˆ dC1 σ1−m k!(m + k)k+1 (m + 1 + k) = 0. m→0 2 lim

Thus let .Nˆ ∈ Z>0 be such that .

1 m√ σˆ dC1 σ1−m k!(m + k)k+1 (m + 1 + k) < 1, 2

ˆ m ≥ N,

and define ˆ = max .C

⎧

| ⎫ | 1√ −m k+1 | dC1 σ1 k!(m + k) (m + 1 + k) | m ∈ {0, 1, . . . , N } . 2

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5 Continuity of Some Standard Geometric Operations

Then, for any .m ∈ Z≥0 , k m Σ 1Σ j −1 ‖Aj +l (x) + θj,l ⊗ idE (D∇ M ,∇ πE R∇ πE (Al−1 ))(x)‖GM,π . E l! j =0

l=0

≤ Cˆ σˆ −m

m+k Σ l=0

1 ‖Aj ‖GM,π . E j!

In particular m k m+k Σ Σ 1 1 Σ l,j j j −m ˆ ‖D ‖D∇ M ,∇ πE (D∇ M ,∇ πE (ξ ))‖GM,π ≤ C σˆ . π (ξ )‖GM,π . E E l! j ! ∇ M ,∇ E l=0

j =0

l=0

By (1.4), we have ‖jm jk ξ(x)‖GM,π

.

E,k ,m

≤

√ k + m + 1Cˆ σˆ −m ‖jm+k ξ(x)‖GM,π

E ,m+k

.

Note that, for .σ < σˆ , we have .

√ lim σ m k + m + 1Cˆ σˆ −m = 0.

m→∞

Now let .N ∈ Z>0 be large enough that √ σ m k + m + 1Cˆ σˆ −m < 1,

.

m ≥ N,

and let C = max

{√

.

| } | k + m + 1Cˆ σˆ −m | m ∈ {0, 1, . . . , N } .

Then, for any .m ∈ Z≥0 , we have ‖jm+k ξ(x)‖GM,π

.

E,k ,m

≤ Cσ −m ‖jm+k ξ(x)‖GM,π

E ,m+k

.

Now let .K ⊆ M be compact and let .a ∈ c0 (Z≥0 ; R>0 ). Define .a ' ∈ c0 (Z≥0 ; R>0 ) by .a0' = a0 , .aj' = C, .j ∈ {1, . . . , k}, and .aj' = σ −1 aj −k , .j ∈ {k + 1, k + 2, . . . }. Then a0 a1 · · · am ‖jm jk ξ(x)‖GM,π

.

E,k ,m

≤ Cσ −m a0 a1 · · · am ‖jk+m ξ(x)‖GM,π

E ,m+k

≤ a0 C k (σ −1 a1 ) · · · (σ −1 am )‖jk+m ξ(x)‖GM,π =

' a0' a1' · · · ak+m ‖jk+m ξ(x)‖GM,π ,m+k , E

E ,m+k

5.2 Continuity of Operations Involving Differentiation

273

since .C ≥ 1. We then immediately have k,ω ω pK,a (jk ξ ) ≤ pK,a ' (ξ ),

.

which gives continuity of .Jkω . Consider the surjective vector bundle mapping .πM,k : Jk E → E and the induced mapping on sections, | |E,k : ┌ ω (Jk E) → ┌ ω (E) .

 ⎬→ πE,k ◦ .

Clearly, .| |E,k ◦ Jkω (ξ ) = ξ . Let .K ⊆ M be compact and let .a ∈ c0 (Z≥0 ; R>0 ). Following the proof of Theorem 5.1, let .C ∈ R>0 and .a ' ∈ c0 (Z≥0 ; R>0 ) be such that k,ω ω pK,a (| |E,k ◦ ) ≤ CpK,a ' (),

.

 ∈ ┌ ω (Jk E).

Then, for .ξ ∈ ┌ ω (E), we have k,ω ω ω pK,a (ξ ) = pK,a (| |E,k ◦ jk ξ ) ≤ CpK,a ' (jk ξ ).

.

Openness of .Jkω onto its image now follows from Lemma 1 from the proof of Theorem 5.1. □ We comment that the proof of the theorem is a little easier in local coordinates since the inclusion .ιE,k,m is simpler in local coordinates than the expression that comes from Lemma 3.22. We use the connection formulation to be consistent with our geometric approach. We can now prove a collection of results regarding standard operations involving differentiation, derived from the preceding result about basic prolongation. Corollary 5.12 (Continuity of Differential) Let .M be a .Cω -manifold. Then the mapping d : Cω (M) → ┌ ω (T∗ M) .

f ⎬→ df

is a topological homomorphism. Proof Note that .J1 (M; R) ≃ RM ⊕ T∗ M and that, under this identification, .j1 f = f ⊕ df . Thus .df = pr2 ◦ j1 f , where .pr2 : J1 (M; R) → T∗ M is the .Cω -vector bundle mapping of projection onto the second factor. The result then immediately follows from Theorems 5.4 and 5.11. □

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5 Continuity of Some Standard Geometric Operations

Corollary 5.13 (Continuity of Lie Derivative) Let .M be a .Cω -manifold. Then the map L : ┌ ω (TM) × Cω (M) → Cω (M) .

(X, f ) ⎬→ L X f

is continuous. Proof We think of X as being a .Cω -vector bundle mapping via X : T∗ M → RM .

αx ⎬→ .

Then the bilinear mapping of the corollary is given by the composition (X, f ) ⎬→ (X, df ) ⎬→ X(df ).

.

The left mapping is continuous since it is the product of the continuous mappings id and .d. The right mapping is continuous by Theorem 5.1(ii), and so the corollary follows. □

.

Corollary 5.14 (Continuity of Covariant Derivative) Let .πE : E → M be a .Cω vector bundle with a .Cω -vector bundle connection .∇ πE . Then the map ∇ πE : ┌ ω (TM) × ┌ ω (E) → ┌ ω (E) .

π

(X, ξ ) ⎬→ ∇XE ξ

is continuous. Proof As in the proof of Lemma 2.15, we have a .Cω -vector bundle mapping 1 π .S∇ πE : E → J E over .idM that determines the connection .∇ E by ∇ πE ξ(x) = j1 ξ(x) − S∇ πE (ξ(x)).

.

The mapping .ξ ⎬→ ∇ πE ξ is continuous by Theorems 5.1 and 5.11. We note that ω π .∇ E ξ is to be thought of as a .C -vector bundle mapping by ∇ πE ξ : TM → E .

π

X ⎬→ ∇XE ξ.

The bilinear mapping of the lemma is then given by the composition (X, ξ ) ⎬→ (X, ∇ πE ξ ) ⎬→ ∇ πE ξ(X).

.

5.3 Continuity of Lifting Operations

275

The left mapping is continuous since it is the product of the continuous mappings id and .ξ ⎬→ ∇ πE ξ . The right mapping is continuous by Theorem 5.1(ii), and so the lemma follows. □

.

Corollary 5.15 (Continuity of Lie Bracket) Let .M be a .Cω -manifold. Then the map [·, ·] : ┌ ω (TM) × ┌ ω (TM) → ┌ ω (TM) .

(X, Y ) ⎬→ [X, Y ]

is continuous. Proof Let .GM be a real analytic Riemannian metric on .M and let .∇ M be the associated Levi-Civita connection. Since M M [X, Y ] = ∇X Y − ∇X Y,

.

the result follows from Corollary 5.14.

□

Corollary 5.16 (Continuity of Linear Partial Differential Operators) Let πE : E → M and .πF : F → M be .Cω -vector bundles and let .Ф ∈ VBω (Jk E; F). Then the kth-order linear partial differential operator .DФ : ┌ ω (E) → ┌ ω (F) defined by .DФ (ξ )(x) = Ф(jk ξ(x)), .x ∈ M, is continuous.

.

Proof The operator .DФ is the composition of the continuous mappings .ξ ⎬→ jk ξ ┌ ω (E) to .┌ ω (Jk E) and . ⎬→ Ф ◦  from .┌ ω (Jk E) to .┌ ω (F). □

.

The reader can no doubt imagine many extensions of results such as the ones we give, and we leave these for the reader to figure out as they need them.

5.3 Continuity of Lifting Operations In Sects. 3.1.1–3.1.4 we introduced a variety of constructions for lifting objects from the base space of a vector bundle to the total space. In Sect. 3.3 we considered how to differentiate these constructions in multiple ways, and how to relate these multiple differentiations. In Sects. 4.1.1–4.1.7 we described fibre norms to give norms for these lifted objects. In this section, we put this all together to prove results that give substantial motivation for all of these quite elaborate constructions. That is, we show that these lift operations are topological homomorphisms. Many of the proofs are similar to one another, so we only give representative proofs. We begin by considering horizontal lifts of functions. We note that continuity of the mapping in the next theorem follows from Theorem 5.26, but openness onto its image does not since the vector bundle projection is not proper. In any case, we give an independent proof of continuity, as it is a model for the proof of subsequent statements for which we will not give detailed proofs.

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5 Continuity of Some Standard Geometric Operations

Theorem 5.17 (Horizontal Lift of Functions Is a Topological Monomorphism) Let .πE : E → M be a .Cω -vector bundle. Then the mapping Cω (M) ϶ f ⎬→ π ∗ f ∈ Cω (E)

.

is a topological monomorphism. Proof It is clear that the asserted map is injective, so we focus on its topological attributes. We let .GM be a .Cω -Riemannian metric on .M, .Gπ be a .Cω -vector bundle connection in .E, .∇ M be the Levi-Civita connection for .GM , and .∇ π be a .Cω -vector bundle connection in .E. Corresponding to this, we have a Riemannian metric .GE on E .E with its Levi-Civita connection .∇ , as in Sect. 3.2.1. We denote the associated ω ω ω ω and .qL,a for .K ⊆ M and .L ⊆ E seminorms for .C (M) and .C (E) by .pK,a compact, and for .a ∈ c0 (Z≥0 ; R>0 ). Let us make some preliminary computations. By Lemma 3.25, we have Symm ◦ ∇ E,m πE∗ f (e) =

.

m Σ

∗ M,s ˆm f (e)). A s (Syms+1 ◦ πE ∇

(5.5)

s=0

By Lemma 4.11, we have m ‖Am s (βs )‖GE ≤ ‖As ‖GE ‖βs ‖GE

.

for .βs ∈ Ts (T∗e E), .m ∈ Z>0 , and .s ∈ {0, 1, . . . , m}. By Lemma 4.13, ‖Syms (A)‖GE ≤ ‖A‖GE

.

for .A ∈ Ts (T∗ N) and .s ∈ Z>0 . Thus, recalling (3.29), m m ˆm ‖A s (Syms (βs ))‖GE = ‖Symm ◦ As (βs )‖GE ≤ ‖As ‖GE ‖βs ‖GE ,

.

for .βs ∈ Ts (πE∗ T∗ M), .m ∈ Z>0 , .s ∈ {1, . . . , m}. Let .L ⊆ E be compact. By Lemmata 4.17 and 4.18 with .r = 0, there exist .C1 , σ1 , ρ1 ∈ R>0 such that −(k−s)

‖Aks (e)‖GE ≤ C1 σ1−k ρ1

.

(k − s)!,

k ∈ Z>0 , s ∈ {0, 1, . . . , k − 1}, e ∈ L.

Without loss of generality, we assume that .C1 ≥ 1 and .σ1 , ρ1 ≤ 1. Thus, using Lemma 4.1 and abbreviating .σ2 = σ1 ρ1 , we have ˆks (π ∗ Syms ◦ ∇ M,s f (e))‖G ≤ C1 σ −k (k − s)!‖Syms ◦ ∇ M,s df (πE (e))‖G ‖A E M E 2

.

5.3 Continuity of Lifting Operations

277

for .k ∈ Z≥0 , .s ∈ {0, 1, . . . , k}, .e ∈ L. Thus, by (1.4), (5.5), and Lemma 4.8, m Σ 1 ‖Symk ◦ ∇ E,k πE∗ f (e)‖GE k! k=0 ‖ k ‖ m ‖ Σ 1 ‖ ‖Σ ˆk ∗ ‖ = As (πE Syms ◦ ∇ M,s f (e))‖ ‖ ‖ ‖ k!

‖jm (πE∗ f )(e)‖GE,m ≤

.

≤

GE

s=0

k=0

k m Σ Σ

C1 σ2−k

k=0 s=0

s!(k − s)! 1 ‖Syms ◦ ∇ M,s f (πE (e))‖GM k! s!

for .e ∈ L and .m ∈ Z≥0 . Now note that .

s!(k − s)! ≤ 1, k!

C1 σ2−k ≤ C1 σ2−m ,

for .s ∈ {0, 1, . . . , m}, .k ∈ {0, 1, . . . , s}, since .σ2 ≤ 1. Then ‖jm (πE∗ f )(e)‖GE,m ≤ C1 σ2−m

.

k m Σ Σ 1 ‖Syms ◦ ∇ M,s f (πE (e))‖GM s! k=0 s=0

≤ C1 σ2−m

m m Σ Σ 1 ‖Syms ◦ ∇ M,s f (πE (e))‖GM s! k=0 s=0

= (m + 1)C1 σ2−m

m Σ 1 ‖Syms ◦ ∇ M,s f (πE (e))‖GM . s! s=0

Now let .σ < σ2 and note that .

lim (m + 1)

m→∞

σ2−m = 0. σ −m

Thus there exists .N ∈ Z>0 such that (m + 1)C1 σ2−m ≤ C1 σ −m ,

.

m ≥ N.

Let ⎧

σ , 2C1 .C = max C1 , C1 σ2

⎛

σ σ2

⎞2

⎛ , . . . , NC1

σ σ2

⎞N ⎫ .

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5 Continuity of Some Standard Geometric Operations

We then immediately have .(m + 1)C1 σ2−m ≤ Cσ −m for all .m ∈ Z≥0 . We then have, by (1.4), ‖jm (πE∗ f )(e)‖GE,m ≤ Cσ −m

.

√

m Σ 1 ‖Syms ◦ ∇ M,s f (πE (e))‖GM s! s=0

≤ C m + 1σ −m ‖jm f (πE (e))‖GM,m . By modifying C and .σ guided by what we did just preceding, we get ‖jm (πE∗ f )(e)‖GE,m ≤ Cσ −m ‖jm f (πE (e))‖GM,m .

.

Now let .a ∈ c0 (Z≥0 ; R>0 ) and define .a ' ∈ c0 (Z≥0 ; R>0 ) be defined by .a0' = Ca0 and .aj' = aj σ −1 , .j ∈ Z>0 . Then we have ω qL,a (πE∗ f ) ≤ pπωE (L),a ' (f ),

.

giving continuity in this case. Now we show that .πE∗ is open onto its image. The idea here is to make some preliminary observations to put ourselves in a position to be able to say, “Now proceed as above.” By Lemma 3.26, we have

.

Symm ◦ πE∗ ∇ M,m f (e) =

m Σ

ˆsm (Syms ◦ ∇ E,s π ∗ f (e)). B E

(5.6)

s=0

For a compact .L ⊆ E we can proceed as above to give a bound ‖jm f (πE (e))‖GM,m ≤ Cσ −m ‖jm (πE∗ f )(e)‖GE,m ,

.

e ∈ L.

We need to choose the compact set .L in a specific way. We let .K ⊆ M be compact and choose a continuous section .ξ ∈ ┌ 0 (E), and then take .L = ξ(K). Then we have the estimate ‖jm f (x)‖GM,m ≤ Cσ −m ‖jm (πE∗ f )(ξ(x))‖GE,m ,

.

x ∈ K.

Now, as advertised, we can proceed as above for continuity to give the bound ω ω ∗ pK,a (f ) ≤ qξ(K),a ' (πE f ),

.

and from this we conclude that .f ⎬→ πE∗ f is indeed open onto its image by Lemma 1 from the proof of Theorem 5.1. □ Now we consider vertical lifts of sections.

5.3 Continuity of Lifting Operations

279

Theorem 5.18 (Vertical Lift of Sections Is a Topological Monomorphism) Let πE : E → M be a .Cω -vector bundle. Then the mapping

.

┌ ω (E) ϶ ξ ⎬→ ξ v ∈ ┌ ω (TE)

.

is a topological monomorphism. Proof This follows in the same manner as Theorem 5.17, using Lemmata 3.31, 3.32, and 4.2. □ One has the similar result for vertical lifts of endomorphisms. Theorem 5.19 (Vertical Lift of Endomorphisms Is a Topological Monomorphism) Let .πE : E → M be a .Cω -vector bundle. Then the mapping ┌ ω (End(E)) ϶ L ⎬→ Lv ∈ ┌ ω (End(TE))

.

is a topological monomorphism. Proof This follows in the same manner as Theorem 5.17, using Lem□ mata 3.46, 3.47, and 4.5. Now we consider horizontal lifts of vector fields. Theorem 5.20 (Horizontal Lift of Vector Fields Is a Topological Monomorphism) Let .πE : E → M be a .Cω -vector bundle. Then the mapping ┌ ω (TM) ϶ X ⎬→ Xh ∈ ┌ ω (TE)

.

is a topological monomorphism. Proof This follows in the same manner as Theorem 5.17, using Lemmata 3.36, 3.37, and 4.3. □ Now we consider vertical lifts of sections of the dual bundle. Theorem 5.21 (Vertical Lift of One-Forms Is a Topological Monomorphism) Let .πE : E → M be a .Cω -vector bundle. Then the mapping ┌ ω (E∗ ) ϶ λ ⎬→ λv ∈ ┌ ω (T∗ E)

.

is a topological monomorphism. Proof This follows in the same manner as Theorem 5.17, using Lemmata 3.41, 3.42, and 4.4. □ Next we consider vertical evaluations of sections of the dual bundle.

280

5 Continuity of Some Standard Geometric Operations

Theorem 5.22 (Vertical Evaluations of One-Forms Is a Topological Monomorphism) Let .πE : E → M be a .Cω -vector bundle. Then the mapping ┌ ω (E∗ ) ϶ λ ⎬→ λe ∈ Cω (E)

.

is a topological monomorphism. Proof Since the given map is clearly injective, we focus on its topological properties. We let .GM be a .Cω -Riemannian metric on .M, .GπE be a .Cω -vector bundle connection in .E, .∇ M be the Levi-Civita connection for .GM , and .∇ πE be a .Cω -vector bundle connection in .E. Corresponding to this, we have a Riemannian metric .GE on .E with its Levi-Civita connection .∇ E , as in Sect. 3.2.1. We denote the associated ω ω seminorms for .┌ ω (E∗ ) and .Cω (E) by .pK,a and .qL,a for .K ⊆ M and .L ⊆ E compact, and for .a ∈ c0 (Z≥0 ; R>0 ). Let us make some preliminary computations. By Lemma 3.51, we have λe (e) = λe (e), ˆ11 ((∇ πE λ)e (e)) + A ˆ10 (λe (e)) + C ˆ10 (λv (e)), ∇ E λe (e) = A .. . .

(Symm ⊗ idT∗ E ) ◦ ∇ E,m λe (e) =

m Σ

M,πE ,s e ˆm λ) (e)) A s ((Syms ⊗ idT∗ E ) ◦ (∇

s=0

+

m−1 Σ

M,πE ,s v ˆm λ) (e)). C s ((Syms ⊗ idT∗ E ) ◦ (∇

s=0

(5.7) Just as in the proof of Theorem 5.17, by Lemmata 4.11 and 4.13, and the appropriate analogue of Eq. (3.33) that would appear in a fully fleshed out proof of Lemma 3.49, we have bounds m m ˆm ‖A s (Syms (βs ))‖GE = ‖Symm ◦ As (βs )‖GE ≤ ‖As ‖GE ‖βs ‖GE

.

and ˆsm (Syms (γs ))‖G = ‖Symm ◦ Csm (γs )‖G ≤ ‖Csm ‖G ‖γs ‖G . ‖C E E E E

.

Let .L ⊆ E be compact. By Lemmata 4.17 and 4.18 with .r = 0, there exist C1 , σ1 , ρ1 ∈ R>0 such that

.

−(k−s)

‖Aks (e)‖GE ≤ C1 σ1−k ρ1

.

(k − s)!,

k ∈ Z≥0 , s ∈ {0, 1, . . . , k}, e ∈ L,

5.3 Continuity of Lifting Operations

281

and −(k−s)

‖Csk (e)‖GE ≤ C1 σ1−k ρ1

.

(k − s)!,

k ∈ Z≥0 , s ∈ {0, 1, . . . , k − 1}, e ∈ L.

Without loss of generality, we assume that .C1 ≥ 1 and .σ1 , ρ1 ≤ 1. Thus, using Lemma 4.6 and abbreviating .σ2 = σ1 ρ1 , we have ˆks ((Syms ⊗ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e))‖G ‖A M,π

.

≤

C1 σ2−k (k

E

− s)!‖(Syms ⊗ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e)‖GM,π

E

for .k ∈ Z≥0 , .s ∈ {0, 1, . . . , k}, .e ∈ L, and ˆks ((Syms ⊗ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e))‖G ‖C M,π

.

≤

C1 σ2−k (k

E

− s)!‖(Syms ⊗ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e)‖GM,π

E

for .k ∈ Z≥0 , .s ∈ {0, 1, . . . , k − 1}, .e ∈ L. Thus, by (1.4) and (5.7), ‖jm λe (e)‖GE,m

.

m Σ 1 ‖ Symk ◦ ∇ E,k λe (e)‖GE k! k=0 ‖ k ‖ m ‖ Σ 1 ‖ ‖Σ ˆk ‖ ≤ As ((Syms ◦ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e))‖ ‖ ‖ k! ‖

≤

k=0

s=0

‖ k m−1 Σ 1 ‖ ‖Σ ˆk + C s ((Syms ‖ k! ‖ k=0

◦

idT∗ E ) ◦ (∇

M,πE ,s

s=0

‖ k m ‖Σ Σ −k s!(k − s)! 1 ‖ ≤ C1 σ2 ‖ (Syms k! s! ‖ k=0

◦

GM,πE

‖ ‖ ‖ λ) (e))‖ ‖ v

idT∗ E ) ◦ (∇

M,πE ,s

s=0

‖ k m−1 ‖Σ Σ −k s!(k − s)! 1 ‖ + C1 σ2 ‖ (Syms k! s! ‖

◦

idT∗ E ) ◦ (∇

s=0

k=0

for .e ∈ L and .m ∈ Z≥0 . Now note that .

s!(k − s)! ≤ 1, k!

GM,πE

C1 σ2−k ≤ C1 σ2−m ,

‖ ‖ ‖ λ) (e)‖ ‖ e

M,πE ,s

GM,πE

‖ ‖ ‖ λ) (e)‖ ‖ v

GM,πE

282

5 Continuity of Some Standard Geometric Operations

for .s ∈ {0, 1, . . . , m − 1}, .k ∈ {0, 1, . . . , s}, since .σ2 ≤ 1. Then ‖jm λe (e)‖GE,m

.

≤ C1 σ2−m

m m Σ Σ 1‖ ‖ ‖(Syms s!

◦

‖ ‖ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e)‖

k=0 s=0

+ C1 σ2−m

1‖ ‖ ‖(Syms s!

◦

‖ ‖ idT∗ E ) ◦ (∇ M,πE ,s λ)v (e)‖

m Σ 1‖ ‖ ‖(Syms s!

◦

‖ ‖ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e)‖

m−1 Σ m−1 Σ k=0 s=0

= (m + 1)C1 σ2−m

GM,πE

s=0

+ (m + 1)C1 σ2−m

m−1 Σ s=0

1‖ ‖ ‖(Syms s!

◦

GM,πE

GM,πE

‖ ‖ idT∗ E ) ◦ (∇ M,πE ,s λ)v (e)‖

GM,πE

.

Now let .σ < σ2 and note that .

lim (m + 1)

m→∞

σ2−m = 0. σ −m

Thus there exists .N ∈ Z>0 such that (m + 1)C1 σ2−m ≤ C1 σ −m ,

.

m ≥ N.

Let ⎧

σ .C = max C1 , 2C1 , 3C1 σ2

⎛

σ σ2

⎞2

⎛ , . . . , (N + 1)C1

σ σ2

⎞N ⎫ .

We then immediately have .(m + 1)C1 σ2−m ≤ Cσ −m for all .m ∈ Z≥0 . We then have, using (1.4), ‖jm λe (e)‖GE,m ⎛ m Σ1‖ ‖ ≤ Cσ −m ‖(Syms s!

.

◦

‖ ‖ idT∗ E ) ◦ (∇ M,πE ,s λ)e (e)‖

s=0

+

m−1 Σ s=0

1‖ ‖ ‖(Syms s!

‖ ‖ M,πE ,s v λ) (e)‖ ◦ idT∗ E ) ◦ (∇

√ ≤ C m + 1σ −m (‖jm λ(πE (e))(e)‖GM,π

E ,m

GM,πE

⎞ GM,πE

+ ‖jm−1 λ(πE (e))‖GM,π

E ,m−1

).

5.3 Continuity of Lifting Operations

283

By modifying C and .σ just as we did in the preceding, we get ‖jm λe (e)‖GE,m ≤ Cσ −m (‖jm λ(πE (e))(e)‖GM,π

.

E ,m

+ ‖jm−1 λ(πE (e))‖GM,π

E ,m−1

).

We take α = max{1, sup{‖e‖GπE | e ∈ L}

.

and then use Lemma 4.11 to arrive at ‖jm λe (e)‖GE,m ≤ 2αCσ −m ‖jm λ(πE (e))(e)‖GM,π

.

E ,m

.

Now, given .a ∈ c0 (Z≥0 ; R>0 ), we define .a ' ∈ c0 (Z≥0 ; R>0 ) by .a0' = 2αCa0 and ' −1 , .j ∈ Z , we then have .a = aj σ >0 j ω qL,a (λe ) ≤ pπωE (L),a ' (λ),

.

and this gives continuity of vertical evaluation. Now we turn to showing that the mapping of the lemma is open onto its image. By Lemma 3.52, we have λe (e) = λe (e), ˆ11 (∇ E λe (e)) + B ˆ10 (λe (e)) + D ˆ10 (λv (e)), (∇ πE λ)e (e) = B ˆ22 (∇ E,2 λe (e)) + B ˆ21 (∇ E λe (e)) + B ˆ20 (λe (e)) (Sym2 ⊗ idTE ) ◦ (∇ M,πE ,2 λ)e (e) = B ˆ21 ((∇ M,πE λ)v (e)) + D ˆ10 (λv (e)), +D .. .

.

(Symm ⊗ idT∗ E ) ◦ (∇ M,πE ,m λ)e (e) =

m Σ

E,s e ˆm λ (e)) B s ((Syms ⊗ idT∗ E ) ◦ ∇

s=0

+

m−1 Σ

E,s v ˆm λ (e)). D s ((Syms ⊗ idT∗ E ) ◦ ∇

s=0

(5.8) Just as in the proof of Theorem 5.17, by Lemmata 4.11 and 4.13, and the appropriate analogue of Eq. (3.33) that would appear in a fully fleshed out proof of Lemma 3.50, we have bounds ˆsm (Syms (βs ))‖G = ‖Symm ◦ Bsm (βs )‖G ≤ ‖Bsm ‖G ‖βs ‖G , ‖B E E E E

.

ˆsm (Syms (γs ))‖G = ‖Symm ◦ Dsm (γs )‖G ≤ ‖Dsm ‖G ‖γs ‖G . ‖D E E E E

284

5 Continuity of Some Standard Geometric Operations

Proceeding analogously to the continuity proof above and using Lemma 4.6, we deduce that there exist .C1 , σ1 ∈ R>0 such that ‖jm λ(πE (e))(e)‖GM,π

.

≤

E ,m

C1 σ1−m (‖jm λe (e)‖GE,m

+ ‖jm−1 λv (e)‖GE,m−1 ),

e ∈ L.

(5.9)

Now let .K ⊆ M be compact and let .a ∈ c0 (Z≥0 , R>0 ). Define L = πE−1 (K) ∩ {e ∈ E | ‖e‖GπE = 1},

.

noting that .L is compact. Let .n = dim(M) and let k be the fibre dimension of .E. By Lemma 4.12, and Eqs. (1.3) and (4.10), we have ‖jm λ(x)‖GM,π

.

≤

E ,m

m Σ j =0

≤

\ ⎛ ⎞ n+j −1 sup{‖jm λ(πE (e))(e)‖GM,π ,m | e ∈ L} k E j

⎞ m ⎛ Σ n+j −1 sup{‖jm λ(πE (e))(e)‖GM,π ,m | e ∈ L} k E j j =0

≤ m2 2n+m sup{‖jm λ(πE (e))(e)‖GM,π

E ,m

| e ∈ L}

for .x ∈ K. For .σ2 < 12 , .

lim m2

m→∞

2m = 0. σ2−m

By by now familiar arguments, one of which the reader can find in the first part of the proof, we can combine this with (5.9) to arrive at .C, σ ∈ R>0 for which ‖jm λ(x)‖GM,π

.

≤ Cσ −m (sup{‖jm λe (e)‖GM,π

E ,m

E ,m

| e ∈ L}

+ sup{‖jm λv (e)‖GE,m | e ∈ L}) for .x ∈ K. Taking .a ' ∈ c0 (Z≥0 ; R>0 ) to be defined by .a0' = Ca0 , .aj' = σ −1 aj , .j ∈ Z>0 , we have ω ω e ω v qK,a (λ) ≤ pL,a ' (λ ) + pL,a ' (λ ).

.

By Lemma 1 from the proof of Theorem 5.1, this shows that the mapping ┌ ω (E∗ ) ϶ λ ⎬→ (λe , λv ) ∈ Cω (E) ⊕ ┌ ω (TE)

.

5.3 Continuity of Lifting Operations

285

is open onto its image. This part of the lemma now follows from the following simple fact. Lemma 1 Let .S, .T1 , and .T2 be topological spaces and let .Ф : S → T1 × T2 be an open mapping onto its image. Then the mappings .pr1 ◦ Ф and .pr2 ◦ Ф are open onto their images. Proof Let .O ⊆ S be open so that .Ф(O) is open in .image(Ф). Then, for each (y1 , y2 ) ∈ O, there exists a neighbourhood .N1 ⊆ image(pr1 ◦ Ф) of .y1 and a neighbourhood .N2 ⊆ image(pr2 ◦ Ф) of .x2 such that .N1 × N2 ⊆ Ф(O). This immediately gives the lemma. .Δ Thus we arrive at the conclusion that the mapping

.

┌ ω (E∗ ) ϶ λ ⎬→ λe ∈ Cω (E)

.

□

is open onto its image, as desired.

Finally, we consider vertical evaluations of sections of the endomorphism bundle. Theorem 5.23 (Vertical Evaluation of Endomorphisms Is a Topological Monomorphism) Let .πE : E → M be a .Cω -vector bundle. Then the mapping ┌ ω (End(E)) ϶ L ⎬→ Le ∈ ┌ ω (TE)

.

is a topological monomorphism. Proof This follows in the same manner as Theorem 5.22, using Lemmata 3.56, 3.57, and 4.7. □ As an illustration of how continuity of these lifts can be helpful, let us consider the continuity of the map that assigns to a vector field on a manifold the tangent lift of that vector field. Precisely, let .M be a real analytic manifold and let .X ∈ ┌ ω (TM) be a real analytic vector field. The tangent lift of X is the vector field T ∈ ┌ ω (TTM) on .TM whose flow is the derivative of the flow for X: .X T

X T ФX t (vx ) = Tx Фt (vx ) ⇒ X =

.

| d || T x ФX t (vx ). dt |t=0

(5.10)

Let us give a formula for the tangent lift that reduces the continuity of the mapping X ⎬→ XT to continuity of familiar operations.

.

Lemma 5.24 (Decomposition of the Tangent Lift via an Affine Connection) Let r ∈ {∞, ω} and let .M be a .Cr -manifold with a .Cr -affine connection .∇ M . Then

.

XT (vx ) = hlft(vx , X(x)) + vlft(vx , ∇vMx X + T (X(x), vx )),

.

where T is the torsion of .∇ M .

286

5 Continuity of Some Standard Geometric Operations

Proof Let .vx ∈ TM and let .Y ∈ ┌ r (TM) be such that .Y (x) = vx . Note that .

| d || ФX ◦ ФYs (x) = Tx ФX t (Y (x)). ds |s=0 t

Also compute .

| | d || d || X Y X Y X Y ◦ Ф Ф = ФY ◦ ФX s t ◦ Ф−t Ф−s ◦ Фt ◦ Фs (x) ds |s=0 t ds |s=0 s ⎛ | ⎞ d || X Y X Y X X X ◦ ◦ ◦ Ф Ф Ф Ф (Ф (x)) . = Y (Фt (x)) + Tx Фt −s t s t ds |s=0 −t

Note that, for .f ∈ Cr (M), Y X Y f ◦ ФX −t ◦ Ф−s ◦ Фt ◦ Фs (x) = f (x) + stL [Y,X] f (x) + o(|st|),

.

by [1, Proposition 4.2.34]. Therefore, .

| d || Y X X ФX ◦ ФY−s ◦ ФX t ◦ Фs (Фt (x)) = t[Y, X](Фt (x)). ds |s=0 −t

Putting the above calculations together gives X X T x ФX t (Y (x)) = Y (Фt (x)) − t[X, Y ](Фt (x)).

.

Thus, making use of (5.10), ФX t

.

h

XT ◦ Фt (Y (x))

X = τγ(t,0) (Y (ФX t (x)) − t[X, Y ](Фt (x))), −

where .γ− is the integral curve of .−X through .ФX t (x) and .τγ− is parallel translation (t,0) (0,t) along .γ− . If .γ is the integral curve of X through x note that .τγ− = τγ . Now we compute .

| | d || d || −Xh XT X Ф Ф (Y (x)) = τ (0,t) (Y (ФX ◦ t (x)) − t[X, Y ](Фt (x))) t dt |t=0 t dt |t=0 γ = ∇X Y (x) − [X, Y ](x) = ∇Y X(x) + T (X(x), Y (x)).

Note that, since .XT and .Xh are both vector fields over X, it follows that t ⎬→ τγ(0,t) (Y (ФX t (x)))

.

5.4 Continuity of Composition Operators

287

is a curve in .Tx M. Thus the derivative of this curve at .t = 0 is in .VY (x) TM. Thus we have shown that | d || h T . Ф−X ◦ ФX t (vx ) = vlft(vx , ∇vx X(x) + T (X(x), vx )). dt |t=0 t Finally, for .f ∈ Cr (M), using the first terms in the Baker–Campbell–Hausdorff formula as in [1, Corollary 4.1.27], we have f ◦ Ф−X t

.

h

XT ◦ Фt (vx )

= f ◦ ФX t

T −X h

+ o(|t|2 ).

Differentiating with respect to t and evaluating at .t = 0 gives the result.

□

Now we can combine Theorems 5.1(i), 5.19, 5.20, and 5.23, and Corollary 5.14 to give the following result. Corollary 5.25 (Continuity of Tangent Lift) If .M is a .Cω -manifold, then the mapping ┌ ω (TM) ϶ X ⎬→ XT ∈ ┌ ω (TTM)

.

is continuous.

5.4 Continuity of Composition Operators In this section we consider continuity of the various sorts of compositions. For real analytic manifolds .M and .N, there are three sorts of composition operators we can consider: CФ : Cω (N) → Cω (M) .

Sf : Cω (M; N) → Cω (M)

f ⎬→ Ф∗ f,

Ф ⎬→ f ◦ Ф,

CM,N : Cω (M; N) × Cω (N) → Cω (M) (Ф, f ) ⎬→ f ◦ Ф, the first being defined for fixed .Ф ∈ Cω (M; N) and the second for fixed .f ∈ Cω (N). We call these the composition operator associated with .Ф, the superposition operator associated with f , and the joint composition operator, respectively. The superposition operator is also known as the “nonlinear composition operator” or the “Nemytskii operator.” In general, one studies these mappings for classes of function spaces, e.g., Lebesgue spaces or Hardy spaces. The questions one can ask for such operators include the following.

288

5 Continuity of Some Standard Geometric Operations

1. Well-definedness: Here, for instance, one wishes to know for which f ’s or .Ф’s do the operators .Sf or .CФ maps one function space into another. 2. Continuity: The continuity of the linear composition operator .CФ is often fairly easily established, and also often coincides with the well-definedness of the operator. The continuity of the nonlinear superposition operator .Sf , however, is often quite difficult to establish. Moreover, there are important cases where continuity of this operator does not coincide with its well-definedness, a wellknown example of this being in the Lipschitz class [21]. 3. Boundedness: Of course, in the linear case, continuity of .CФ implies boundedness, although not necessarily the converse if the function spaces in question are not metrisable or, better, not bornological [37, Theorem 13.1.1]. For example, in the real analytic case in which we are interested here, these spaces are not metrisable, but are bornological. The boundedness and continuity of .Sf are not generally logically comparable, e.g., this mapping is nonlinear. 4. Real analyticity: It is sometimes the case that the superposition operator, though nonlinear, admits a convergent power series expansion, in which case it is said to be “real analytic.” We mention this here mostly because this is not what we are considering here; here we are considering operators on spaces of real analytic mappings, not operators which are themselves real analytic. A reader interested in a detailed discussion of superposition operators for various classes of function spaces is referred to [3].

5.4.1 The Real Analytic Composition Operator We first consider the continuity of the linear composition operator. This continuity is established is a rather ad hoc way during the course of the proof of their Lemma 2.5 by Jafarpour and Lewis [36] using a local description of the real analytic topology. Here we give an intrinsic proof using our more systematic analysis. Theorem 5.26 (Composition Induces a Continuous Map Between Function Spaces) Let .M and .N be .Cω -manifolds. If .Ф ∈ Cω (M; N), then the mapping Ф∗ : Cω (N) → Cω (M) .

f ⎬→ f ◦ Ф

is continuous. Moreover, if .Ф is a proper surjective submersion or a proper embedding, then .Ф∗ is open onto its image. In case .Ф is a proper embedding, for any compact .K ⊆ M and any .a ∈ c0 (Z≥0 ; R>0 ), there exists .a ' ∈ c0 (Z≥0 ; R>0 ) such that ω ω ∗ qФ(K),a (f ) ≤ pK,a ' (Ф f ),

.

f ∈ Cω (N),

5.4 Continuity of Composition Operators

289

ω ω where .pK,a and .qL,a are the seminorms for .Cω (M) and .Cω (N), respectively, associated to .K ⊆ M and .L ⊆ N compact, and .a ∈ c0 (Z≥0 ; R>0 ).

Proof We let .∇ M and .∇ N be .Cω -affine connections on .M and .N, respectively, and let .GM and .GN be .Cω -Riemannian metrics on .M and .N, respectively. From Lemma 3.63 we have the formula .

Symm ◦ ∇ M,m Ф∗ f =

m Σ

∗ N,s ˆm f ). A s (Syms ◦ Ф ∇

(5.11)

s=0

By Lemma 4.11, we have m ‖Am s (βs )‖GM ≤ ‖As ‖GM ,GN ‖βs ‖GN

.

for .βs ∈ Ts (T∗x M), .m ∈ Z>0 , and .s ∈ {0, 1, . . . , m}. By Lemma 4.13, ‖Syms (A)‖GM ,GN ≤ ‖A‖GM ,GN

.

for .A ∈ Ts (T∗ N) and .s ∈ Z>0 . Thus, recalling (3.29) (and its analogue that would arise in a spelled out proof of Lemma 3.63), m m ˆm ‖A s (Syms (βs ))‖GM = ‖Symm ◦ As (βs )‖GM ≤ ‖As ‖GM ,GN ‖βs ‖GM ,

.

for .βs ∈ Ts (Ф∗ T∗ N), .m ∈ Z≥0 , .s ∈ {1, . . . , m}. Let .K ⊆ M be compact. By Lemmata 4.17 and 4.18 with .r = 0, there exist .C1 , σ1 , ρ1 ∈ R>0 such that ‖Aks (x)‖GM ,GN ≤ C1 σ1−k ρ1−(k−s) (k − s)!,

.

k ∈ Z≥0 , s ∈ {0, 1, . . . , k}, x ∈ K.

By Lemma 4.8, let .C2 ∈ R>0 be such that ‖Ф∗ ∇ N,m f (x)‖GM ≤ C2m ‖∇ N,m f (Ф(x))‖GN ,

.

x ∈ K, m ∈ Z≥0 .

Without loss of generality, we assume that .C1 , C2 ≥ 1 and .σ1 , ρ1 ≤ 1. Thus, abbreviating .σ2 = σ1 ρ1 , we have ˆks (Ф∗ Syms ◦ ∇ N,s f (x))‖G ≤ C1 C s σ −k (k − s)!‖Syms ◦ ∇ N,s f (Ф(x))‖G ‖A 2 2 M N

.

290

5 Continuity of Some Standard Geometric Operations

for .k ∈ Z≥0 , .s ∈ {0, 1, . . . , k}, .x ∈ K. Thus, by (1.4) and (5.11), m Σ 1 ‖Symk ◦ ∇ M,k Ф∗ f (x)‖GM k! k=0 ‖ k ‖ m ‖ Σ 1 ‖ ‖Σ ˆk ∗ ‖ = As (Ф Syms ◦ ∇ N,s f (Ф(x)))‖ ‖ ‖ ‖ k!

‖jm (Ф∗ f )(x)‖GM,m ≤

.

≤

GM

s=0

k=0

k m Σ Σ

C1 σ2−k

k=0 s=0

s!(k − s)! C2s ‖Syms ◦ ∇ N,s f (Ф(x))‖GN k! s!

for .x ∈ K and .m ∈ Z≥0 . Now note that .

s!(k − s)! ≤ 1, k!

C1 σ2−k C2s ≤ C1 C2m σ2−m ,

for .s ∈ {0, 1, . . . , m}, .k ∈ {0, 1, . . . , s}, since .σ2 ≤ 1. Then ‖jm (Ф∗ f )(x)‖GM,m ≤ C1 C2m σ2−m

.

k m Σ Σ 1 ‖Syms ◦ ∇ N,s f (Ф(x))‖GN s! k=0 s=0

≤ C1 C2m σ2−m

m m Σ Σ 1 ‖Syms ◦ ∇ N,s f (Ф(x))‖GN s! k=0 s=0

= (m + 1)C1 C2m σ2−m

m Σ 1 ‖Syms ◦ ∇ N,s f (Ф(x))‖GN . s! s=0

Now let .σ < C2−1 σ2 and note that .

lim (m + 1)

m→∞

C2m σ2−m = 0. σ −m

Thus there exists .N ∈ Z>0 such that (m + 1)C1 C2m σ2−m ≤ C1 σ −m ,

.

m ≥ N.

Let ⎧

σ , 3C1 C22 .C = max C1 , 2C1 C2 σ2

⎛

σ σ2

⎞2

⎛ , . . . , (N

+ 1)C1 C2N

σ σ2

⎞N ⎫ .

5.4 Continuity of Composition Operators

291

We then immediately have .(m + 1)C1 C2m σ2−m ≤ Cσ −m for all .m ∈ Z≥0 . We then have, using (1.4), ∗

‖jm (Ф f )(x)‖GM,m ≤ Cσ

.

−m

√

⎞ ⎛ m Σ1 N,s ‖Syms ◦ ∇ f (Ф(x))‖GN s! s=0

= C m + 1σ −m ‖jm f (Ф(x))‖GN,m . By modifying C and .σ guided by what we did just preceding, we get ‖jm (Ф∗ f )(x)‖GM,m ≤ Cσ −m ‖jm f (Ф(x))‖GN,m .

.

Now, for .a ∈ c0 (Z≥0 ; R>0 ), let .a ' ∈ c0 (Z≥0 ; R>0 ) be defined by .a0' = Ca0 and ' −1 , .j ∈ Z . Then we have .a = aj σ >0 j ω ω pK,a (Ф∗ f ) ≤ qФ(K),a ' (f ),

.

and this gives continuity of .Ф∗ . Now we turn to the final assertion concerning the openness of .Ф∗ in particular cases. First we note that, by Lemma 3.64, we have

.

Symm ◦ Ф∗ ∇ N,m f (x) =

m Σ

ˆsm (Syms ◦ ∇ M,s Ф∗ f (x)). B

s=0

First consider the case where .Ф is a proper surjective submersion. For .L ⊆ N compact and for .y ∈ L, since .Ф is surjective, there exists .x ∈ M such that .Ф(x) = y. Also, since .Ф is proper, .Ф−1 (L) is compact. We can now reproduce the steps from the proof above, now making use of the second part of Lemma 4.8, to prove that ω ω ∗ qL,a (f ) ≤ pФ −1 (L),a ' (Ф f ),

.

which suffices to prove the openness of .Ф∗ by Lemma 1 from the proof of Theorem 5.1. Finally consider the case where .Ф is a proper embedding. Here, Lemma 1 from the proof of Theorem 2.25 immediately gives openness of .Ф∗ in the case that .Ф is a proper embedding. For the final assertion, we can follow the same argument as was sketched for the openness of .Ф∗ when .Ф is a proper surjective submersion to give ω ω ∗ qФ(K),a (f ) ≤ pK,a ' (Ф f ),

.

as desired.

□

292

5 Continuity of Some Standard Geometric Operations

The matter of determining general conditions under which .Ф∗ is an homeomorphism onto its image or has closed image are taken up in [18, 19]. Remarks 5.27 (Adaptation to the Smooth Case) The preceding proof can be adapted to the smooth case. Indeed, much of the elaborate work of the proof can be simplified by not having to pay attention to the exponential growth of m-jet norms as .m → ∞. In the smooth case, one works with fixed orders of derivatives. .◦

5.4.2 The Real Analytic Superposition Operator Note that our definition of the weak-PB topology ensures continuity of the superposition operator .Sf : Cω (M; N) → Cω (M) for .f ∈ Cω (N). Indeed, the weak-PB topology is defined expressly so that these mappings are continuous. Thus the most meaningful assertions regarding the superposition operator result from the other descriptions of the topology for the space of real analytic mappings as given in Theorems 2.25 and 2.32, and where continuity of superposition is not a tautology; thus these assertions are really about the various equivalent characterisations of the weak-PB topology, rather than about the continuity of the superposition operator. What we shall do in this section is prove continuity of the joint composition operator. In our development of the continuity results in this chapter up to this point, we have made dedicated use of the seminorms for the real analytic topology developed in Sect. 2.4. Our results in this section have to do with continuity involving spaces of mappings, and so one may like to use the semimetrics (2.20) that define the uniformity for this topology. We do not, however, take this approach, instead proving the result in the holomorphic case, and then using the descriptions from Sect. 2.5.4 for the real analytic topology for spaces of mappings derived from germs of holomorphic extensions. The reader may wish to explore using the semimetrics (2.20) to prove Theorem 5.29 below. The first step, then, is to prove continuity for the joint composition operator in the holomorphic case. Theorem 5.28 (Continuity of the Holomorphic Joint Composition Operator) If M and .N are holomorphic manifolds, then the mapping

.

CM,N : Chol (M; N) × Chol (N) → Chol (M) .

(Ф, f ) ⎬→ f ◦ Ф

is continuous, using the compact-open topology for .Chol (M; N). Proof We use the semimetrics (2.21) for the topology of .Chol (M; N) and the seminorms (2.2) for the topology of .Chol (M) and .Chol (N). We use metrics .dM and .dN whose metric topologies agree with the topologies for .M and .N, respectively. We hol for .K ⊆ M compact, and we denote the denote the seminorms for .Chol (M) by .pK hol for .L ⊆ N compact. Let .K ⊆ M be compact and let seminorms for .Chol (N) by .qL

5.4 Continuity of Composition Operators

293

∈ ∈ R>0 . Let .f0 ∈ Chol (N) and .Ф0 ∈ Chol (M; N). Let .y ∈ Ф0 (K) and let .ry ∈ R>0 be such that

.

1. .BdN (ry /2, y) and .BdN (ry , y) are precompact neighbourhoods of y, 2. .cl(BdN (ry /2, y)) ⊆ BdN (ry , y), and 3. .|f0 (y ' ) − f0 (y)| < 6∈ for .y ' ∈ BdN (ry , y). Let .y1 , . . . , yk ∈ Ф0 (K) be such that .Ф0 (K) ⊆ V  ∪kj =1 BdN (ryj /2, yj ). Denote k k .L = ∪ j =1 cl(BdN (ryj /2, yj )), which is compact. Note that .L ⊆ ∪j =1 BdN (ryj , yj ). By the Lebesgue Number Lemma [11, Theorem 1.6.11], let .δ ∈ R>0 be such that, if .z1 , z2 ∈ L satisfy .dN (z1 , z2 ) < δ, then .z1 , z2 ∈ BdN (ryj , yj ) for some .j ∈ {1, . . . , k}. Let O = B(K, L) ∩ {Ф ∈ Chol (M; N) | dN (Ф(x), Ф0 (x)) < δ, x ∈ K}.

.

Let .P ⊆ Chol (L) be a neighbourhood of .f0 such that hol qL (f − f0 )
0 ). Let .K ⊆ M and .L ⊆ N be compact and such that .C ⊆ K × L. Then the inequality ω ω ω pC,a (Ф ⊕ f ) ≤ pK,a (Ф) + pL,a (f )

.

gives continuity of (5.12). Now let .K ⊆ M and .L ⊆ N be compact, and let .a, b ∈ c0 (Z≥0 ; R>0 ). Define .c ∈ c0 (Z≥0 ; R>0 ) by .cm = max{am , bm }, .m ∈ Z≥0 . Then the inequality ω ω ω pK,a (Ф) + pL,b (f ) ≤ pK×L,c (Ф ⊕ f )

.

establishes the openness of (5.12) by Lemma 1 from the proof of Theorem 5.1. The continuity and openness of (5.13) is proved similarly. Since our preceding constructions have led to mappings with domain .M × N, let us understand an aspect of the direct limit topology for .Cω (M × N; P) for a real analytic manifold .P. In particular, let us understand the directed set .N M×N of neighbourhoods of .M × N in the product .M × N of the complexifications. We claim that the subset of neighbourhoods of the form .U × V, .U ∈ N M , .V ∈ N N , is cofinal in .N M×N . To see this, note that the projections .

pr1 : M × N → M,

pr2 : M × N → N

are open. Therefore, if .W ∈ N M×N , then .U  pr1 (W) ∈ N M and .V  pr2 (W) ∈ N N . One moreover easily verifies that .U×V ⊆ W, giving our claim. It then follows that the direct limit topology for .Cω (M × N; P) is given by the direct limit .

lim − →

(U,V)∈N

M ×N N

Chol,R (U × V; P).

5.4 Continuity of Composition Operators

297

Now the preceding constructions give rise to the diagram

.

The upper left vertical arrow is the bijection whose continuity we wish to establish. The upper right vertical is a topological isomorphism, as we have seen. The lower two vertical mappings are those giving rise to the final topologies, which is the direct limit topology on the right. The upper and lower horizontal arrows are the homeomorphisms onto their image as described above. The middle horizontal arrow is defined so that the upper square commutes, and one readily verifies that the lower square commutes, essentially because the large outer square commutes. For this diagram, we shall first show that the middle horizontal arrow, .ιM,N , is an homeomorphism onto its image. First of all, given the continuity of the bottom diagonal arrow and the universal property of the final topology, we conclude that .ιM,N is continuous. For openness, we first note that, for [(Ф, f )]M×N ∈

lim − →

.

(U' ,V' )∈N

Chol,R (U' ; N) × Chol,R (V' ), M ×N N

we have ιU×V (r −1

.

(U,V),M×N

([(Ф, f )]M×N )) = r −1

(ι ([(Ф, f )]M×N )); U×V,M×N M,N

this is easily directly verified from the definitions. Now let O⊆

.

lim − →

(U' ,V' )∈N

Chol,R (U' ; N) × Chol,R (V' ) M ×N N

be open. Then r −1

.

(ι (O)) U×V,M×N M,N

= ιU×V (r −1

(U,V),M×N

(O)),

and this latter set is open since .ιU×V is an homeomorphism onto its image and since r(U,V),M×N is continuous. Now we note that openness of

.

r −1

.

(ι (O)) U×V,M×N M,N

298

5 Continuity of Some Standard Geometric Operations

for every .U ∈ N M and .V ∈ N N is precisely the openness of .ιM,N (O). Now, given that .ιM,N is an homeomorphism onto its image, this is precisely the assertion that the final topology for .

lim − →

(U' ,V' )∈N

Chol,R (U' ; N) × Chol,R (V' ) M ×N N

is the initial topology associated with .ιM,N . Therefore, the continuity of the upper dashed diagonal arrow gives the desired continuity of the upper vertical arrow by the universal property of the initial topology. This completes the proof of the assertion that the product topology of ω ω .C (M; N) × C (N) is the same as the final topology for .

lim − →

(U' ,V' )∈N

Chol,R (U' ; N) × Chol,R (V' ). M ×N N

With this at hand, it is relatively straightforward to prove the continuity of the joint composition map. For .U ∈ N M and .V ∈ N N , we have the diagram

.

where the vertical arrows are the joint composition operators. The left vertical arrow is continuous by Theorem 5.28. Since the final topology for .Cω (M; N) × Cω (N) is equal to the product topology, the universal property of the final topology, along with the continuity of the dashed diagonal arrow, shows that the right vertical arrow is continuous if its domain has the product topology. □

List of Symbols

(x |→ f (x)) ⊗∗ πE : ⊗∗ E → N A∗ E∗ ⊗∗ η ⊗∗ ∇ πF V∗ V' [ξ ]A [X, Y ] AOB U ⊗π V L ω (E) ⊗πCω (M) L ω (F) q ⊗π p

The mapping sending x to f (x), 1 Pull-back bundle, 6 Dual of linear map A, 2, 103 Dual bundle of E, 6 Pull-back of section, 116 Pull-back connection, 117 The algebraic dual of V, 2 Topological dual of V, 17 Germ of ξ , 41 Lie bracket of X and Y , 7 Symmetric tensor product of A and B, 3 Projective tensor product, 25 Balanced projective tensor product, 264 Seminorm for projective tensor product, 25

∪ Xi

Disjoint union, 1

Sk Sk,l Sk|l Ak,m L r (E) L hol,R (E) ιE,k,m ιˆE,k,m ⊗X t πE,k k πE,l

Symmetric group on {1, . . . , k}, 2 Ordered permutations, 2 Product of Sk and Sl , 2 Inclusion for symmetric tensors, 131 Cr -sections of E, 6 Real sections of an holomorphic vector bundle, 41 Jet bundle inclusion, 129 Jet bundle inclusion, 135 Flow of vector field X, 7 Projection from k-jets to 0-jets, 11 Projection from k-jets to l-jets, 11, 63

nS π T∗ M πTM w ⊗

Second fundamental form of S, 109 Cotangent bundle projection, 6 Tangent bundle projection, 6 Vector bundle mapping induced by ⊗, 182

◦

i∈I

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. D. Lewis, Geometric Analysis on Real Analytic Manifolds, Lecture Notes in Mathematics 2333, https://doi.org/10.1007/978-3-031-37913-0

299

300

List of Symbols

∇A ∇E ∇M ∇ M,πE ∇ πE ∇ πE,m ∇ πE ⊗πF ∇Uver V Ak (V) E|A f |A ||·|| ||·||1 ||·||GM,π ||·||∞ ann(S) Am ∇E

E ,m

Am s wm A s Aπ B∇mE Bsm wm B s BπE B(r, x) B(r, x) card(X) C cl(A) CM,N C⊗ Csm wm C s

Covariant differential of a tensor A, 10 Induced affine connection on E, 105 Affine connection on M, 9, 39 Linear connection in tensor products of TM and E, 53 Linear connection in E, 8, 39 Connection induced in the bundle of m-jets, 127 Linear connection in E ⊗ F, 10 Vertical covariant derivative, 106 k-fold alternating tensors of V, 3 Restriction of vector bundle E to A, 6 Restriction of f to A, 1 Generic norm on a vector space, especially the Euclidean norm on Rn , 4 The 1-norm, 4 Norm associated with GM,πE ,m , 62 The ∞-norm, 4 The annihilator of S, 2 Vector bundle mappings induced by decomposition of jets, 144, 151, 155, 159, 163, 169, 180, 191, 198 Tensors relating covariant differentials, 138, 146, 152, 156, 160, 165, 170, 187, 193 Tensors relating symmetrised covariant differentials, 141, 149, 154, 158, 162, 167, 176, 190, 196 Fundamental tensor for π : F → M, 106 Vector bundle mappings induced by decomposition of jets, 144, 151, 155, 159, 163, 169, 180, 191, 198 Tensors relating covariant differentials, 139, 147, 153, 157, 161, 166, 173, 188, 195 Tensors relating symmetrised covariant differentials, 143, 150, 154, 158, 162, 168, 179, 197 Tensor for derivation, 117

C hol

Closed ball, 4, 13 Open ball, 4, 13 Cardinality of X, 1 Complex numbers, 2 Closure of A, 12 Joint composition operator, 287 Composition operator, 287 Tensors relating covariant differentials, 165, 170 Tensors relating symmetrised covariant differentials, 167, 176 Germs of holomorphic functions about A, 41

CAhol (M; N)

Germs of real holomorphic mappings about A, 82

hol,R C A,M hol,R CA (M; N) r CM

Germs of real holomorphic functions about A, 41

A,M

Germs of real holomorphic mappings about A, 82 Sheaf of Cr -functions on M, 8

List of Symbols

301

C∗m E c0 (Z≥0 ; R>0 ) c0 (Z≥0 ; R>0 ; ρ)

Jets of vertical evaluations of vector bundle mappings, 170 Positive sequences converging to 0, 67 Positive sequences converging to 0 and bounded above by ρ, 69

c0 (Z≥0 ; R>0 )

Positive sequences converging monotonically to 0, 69

C0 (X; Y) Chol Chol,R (M; N) Cr Cr (M) Cr (M; N) df DI O DO Da O Dsm wm D s

Positive sequences converging monotonically to 0 and bounded above by ρ, 69 Continuous mappings from X to Y, 12 Holomorphic regularity, 32 Real mappings of holomorphic manifolds M and N, 82 Class C∞ or class Cω , 6 Cr -functions on M, 6 Cr -mappings from M to N, 6 Differential of f , 6 Derivative of O for a multi-index I , 5 Derivative of O, 5 ath partial derivative of O, 5 Tensors relating covariant differentials, 166, 173 Tensors relating symmetrised covariant differentials, 168, 179

↓

↓ c0 (Z≥0 ; R>0 ; ρ)

D∇k,m M ,∇ πE

Derivative induced by connections, 127

D∇mM ,∇ πE

Derivative induced by connections, 53

Dn (r, x) Dn (r, x) D∗m E dG dhol K d∞ K,m,f

Closed polydisk, 4 Open polydisk, 4 Jets of vertical evaluations of dual sections, 164 Distance function associated to Riemannian metric, 10 Semimetric for Chol (M; N), 81 Semimetric for C∞ (M; N), 74

dm K,f

Semimetric for Cm (M; N), 74

dωK,a,f

Semimetric for Cω (M; N), 79

EndR (V) Ej (K) E (K) πE : E → M Ex Ae Le λe F∗m E GE Gb G# GM GM,πE GM,πE ,m GπE

Linear mappings from V to V, 2 Component of direct limit defining E (K), 63 Direct limit of infinite jets, 63 Vector bundle, 6 Fibre of E at x, 6 Vertical evaluation of tensor A, 100 Vertical evaluation of vector bundle mapping L, 98 Vertical evaluation of dual section λ, 95 Jets of vertical lifts of dual sections, 156 Induced Riemannian metric on E, 105 The isomorphism induced by the inner product G, 3 The inverse of Gb , 3 Riemannian metric on M, 8, 39 Induced fibre metric on tensor products of TM and E, 61 Induced fibre metric on Jm E, 62 Fibre metric on E, 8, 39

302

List of Symbols

GEr

Sheaf of Cr -sections of E, 8

G hol A,E hol,R G A,E

Germs of holomorphic sections about A, 41

hlft(e, α) hlft(e, v) HomR (U, V; W) HomR (U; V) hor HE H∗m E Ah fh Xh idX Insj int(A) I! aI

Horizontal lift of α to e, 97 Horizontal lift of v to e, 95 Bilinear mappings from U × V to W, 2 Linear mappings from U to V, 2 Projection onto horizontal bundle, 10 Horizontal bundle of E, 10 Jets of horizontal lifts of vector fields, 152 Horizontal lift of tensor A, 100 Horizontal lift of function f , 95 Horizontal lift of vector field X, 96 Identity map for a set X, 1 Insertion operator, 101 Interior of A, 12 i1 ! · · · in !, 5 a i1 · · · anin , 5 √1 −1, 2 Jet of function, section, or mapping, 11, 63 m-jets of sections, 11, 62 m-jets of functions, 11 Connector for a connection, 8 Direct limit, 17

i jm Jm E Jm (M; R) K ∇ πE lim (Xi , Oi ) − →i∈I lim (Vi , Oi ) − →i∈I lim (Xi , Oi ) ← −i∈I lim (Vi , Oi ) ← −i∈I lG LX

L∗m E Linr (E) L(U; V) L(U, V; W) NA NA hol pstuff ∞ pK,m pK,j m pK ω pK,a

'ω pK,a pra P∗m E

Germs of real holomorphic sections about A, 42

Direct limit, 20 Inverse limit, 16 Inverse limit, 19 Length function for curves, 9 Lie derivative with respect to X, 7 Jets of vertical lifts of vector bundle mappings, 160 Fibre-linear functions, 94 Continuous linear maps from U to V, 17 Continuous bilinear mappings from U × V to W, 25 Neighbourhoods of A, 43 Neighbourhoods of A in a complexification, 41 Seminorm for holomorphic topology, 41, 45 Seminorm for smooth topology, 70 Norm for Ej (K), 63 Seminorm for continuous (m = 0) or finitely differentiable (m ∈ Z>0 ) topology, 70 Seminorm for Cω topology, 67 Local seminorm for Cω -topology, 246 Projection onto the ath factor, 1 Jets of pull-backs of functions, 137

List of Symbols

303

pushj,k R RkM R>0 rA,B R ∇ πE Sf SM SπE

Argument swapping operator, 101 Real numbers, 1 Trivial bundle M × Rk , 6 Positive real numbers, 2 Restriction mapping, 42, 46 Curvature tensor, 10 Superposition operator, 287 Tensor relating different affine connections, 192 Tensor relating different linear connections, 192

Sk (V) S≤m (V) Symk LT T⊗ Tx ⊗ Tπ XT T∗ M T∗m M TM T∗m ⊗ M

k-fold symmetric tensor product of V, 3 Symmetric tensors of degree at most m, 3 Symmetrising operator, 3 Transpose of L, 3 Derivative of ⊗, 6 Derivative of ⊗ at x, 6 Fundamental tensor for π : F → M, 106 Tangent lift of X, 285 Cotangent bundle of M, 6 m-jets of functions with value 0, 11 Tangent bundle of M, 6 Jets of pull-backs of functions, 182

Tk (V) T≤m (V) Trs (V) T∇ E VBr (E; F) vlft(e, e' ) vlft(e, λ) ver VE V∗m E Av Lv λv ξv Z Z≥0 Z>0

k-fold tensor product of V, 2 Tensors of degree at most m, 3 r-contravariant, s-covariant tensors on V, 2 Torsion tensor, 10 Cr -vector bundle mappings from E to F, 7 Vertical lift of e' to e, 95 Vertical lift of λ to e, 97 Projection onto vertical bundle, 10 Vertical bundle of E, 10 Jets of vertical lifts of sections, 145 Vertical lift of tensor A, 100, 101 Vertical lift of vector bundle mapping L, 98 Vertical lift of dual section λ, 98 Vertical lift of section ξ , 95 Integers, 1 Nonnegative integers, 1 Positive integers, 1

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Index

affine connection, see Connection algebraic dual, see Dual alternating tensor, see Tensor analytic continuation, see Identity Theorem annihilator, 2

balanced bilinear map, 261 balanced projective tensor topology, see Projective tensor topology balanced subset, 23 base for a topology, 12 bounded linear mapping, see Mapping boundedly retractive direct limit, see Direct limit boundedness of composition operators, 288 of linear map, 18 of subset of locally convex space, 18 bundle dual, 6, 94 horizontal, 10, 105, 106 jet, 11, 53, 60, 62, 126, 127, 129, 135, 144, 151, 155, 159, 163, 168, 180, 191, 198 pull-back, 6, 116 vector, 6, 53, 60, 224 holomorphic, 40, 45 real analytic, 32, 37, 39, 40 vertical, 10, 105, 106

Cartan’s Theorem B, see Sheaf theory Cauchy estimate, 29, 71 cofinal subset, 43, 45, 46, 83

compact direct limit, see Direct limit compact exhaustion, 12, 49, 81 compact mapping, see Mapping compact-open topology, see Topology complexification, 39, 41, 42, 65, 83 connection affine, 9, 37, 39, 53, 60, 105, 116, 137, 192, 238 induced in tensor product, 10, 53, 116 Levi-Civita, 9, 105, 107, 109, 137, 238 linear, 8, 32, 37, 39, 53, 60, 96–98, 100, 105, 116, 130, 137, 192, 193, 195–198, 224, 238 pull-back, 117, 182 connector, 8, 37, 130, 182 continuity of addition, 254 of composition, 288, 292, 294 of covariant derivative, 274 of differential, 273 of horizontal lift, 276, 279 of Lie bracket, 275 of Lie derivative, 274 of partial differential operators, 275 of prolongation, 270 of tangent lift, 287 of tensor evaluation, 254 of vertical evaluation, 280, 285 of vertical lift, 279 continuous linear mapping, see Mapping cotangent bundle, 6 covariant differential, 10 curvature tensor, 10, 129, 130

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 A. D. Lewis, Geometric Analysis on Real Analytic Manifolds, Lecture Notes in Mathematics 2333, https://doi.org/10.1007/978-3-031-37913-0

309

310 De Wilde Open Mapping Theorem, see Open Mapping Theorem differentiable mapping, see Mapping differentiation of lifts, 96, 105, 106, 116, 117, 137–139, 141, 143, 144, 146, 147, 149–163, 165–168, 170, 173, 176, 179, 180, 238 of pull-back, 185, 187, 188, 190, 191 direct limit, 16, 19, 42, 46, 62, 63, 68, 83, 87 boundedly retractive, 20, 46, 47 compact, 20, 47 regular, 20, 47, 63 direct sum of vector bundles, 259 disjoint union, 1 dual algebraic, 2 topological, 17 dual bundle, see Bundle, 97

Index horizontal lift of function, 95, 96, 138, 139, 141, 143, 144, 202, 276 mapping, 95, 97 of one-form, 97 of tensor, 100, 117 of vector field, 96, 96, 106, 152–155, 204, 279

Identity Theorem, 32, 66 immersion, 6, 211 induced topology, see Topology initial topology, see Topology insertion operator, 101, 102, 103, 112,113, 221, 223 integral curve, 7 inverse limit, 15, 18, 49, 62, 69, 87

jet embedding, 6, 288 embedding theorem Grauert’s Embedding Theorem, 36, 37, 75 Remmert’s Embedding Theorem, 37 Whitney’s Embedding Theorem, 36 Euclidean norm, see Norm exterior covariant derivative, 130

fibre metric, see Metric final topology, see Topology flow, 7 Fréchet space, see Locally convex space function fibre linear, 94 holomorphic, 32 real analytic, 32

geodesic, 9 germ, 41, 42, 62, 65, 82 Grauert’s Embedding Theorem, see Embedding theorem

Hadamard’s Lemma, 33 holomorphic extension, 27, 39, 40, 42 holomorphic function, see Function holomorphic manifold, see Manifold holomorphic mapping, see Mapping holomorphic section, see Section holomorphic vector bundle, see Bundle horizontal bundle, see Bundle

of a function, 11 of a section, 11, 33, 63 jet bundle, see Bundle

Leibniz’ Rule, 114 length of a curve, 9 Lie derivative, 7, 274 linear connection, see Connection locally compact topological space, 12 locally convex space, 17 Fréchet, 21, 47 metrisable, 21 nuclear, 24, 46, 52 ultrabornological, 23, 46, 51, 52, 253 webbed, 23, 46, 51, 52, 253 Lusin space, 13

manifold holomorphic, 32, 39 real analytic, 32 Stein, 37, 80, 81 mapping bounded linear, 18 compact, 18, 45 continuous linear, 17 differentiable, 5 holomorphic, 32, 33 nuclear, 24 proper, 12 real analytic, 6, 27, 28, 32, 33 smooth, 6

Index topological epimorphism, 23 topological homomorphism, 23 topological monomorphism, 23 vector bundle, 7, 97, 254, 257 metric, 13 distance associated to a Riemannian metric, 9 fibre, 8, 37, 39, 60, 61, 105, 137, 224, 238, 242 Hermitian, 40 for jet bundles, 62 Hermitian, 40 Riemannian, 8, 37, 39, 60, 61, 105, 109, 137, 224, 238 on tensor product, 61 metric topology, see Topology metrisable locally convex space, see Locally convex space

neighbourhood base, 12 norm, 4, 215 1-, 4 .∞-, 4, 45 Euclidean, 4 of evaluation, 216 fibre, 60, 62, 201–204, 206–210, 238 for jet bundles, 62, 202, 203, 205–210, 242, 248 of identity map, 215 of insertion operator, 221, 223 of linear mapping, 217 of symmetrisation, 217 on tensor product, 215, 238 nuclear locally convex space, see Locally convex space nuclear mapping, see Mapping

Open Mapping Theorem, 22, 51, 66, 256 De Wilde, 23, 51, 66, 76, 253, 258, 268

parallel transport, 9, 111 Polish space, 13 polydisk, 5 power series, 26 convergent, 26 precompact set, 12 projective tensor topology, 25, 264 balanced, 264, 264, 266 prolongation, 126, 270 proper mapping, see Mapping pull-back bundle, see Bundle

311 pull-back connection, see Connection pull-back section, see Section

real analytic diffeomorphism, 29 real analytic manifold, see Manifold real analytic mapping, see Mapping real analytic section, see Section real analytic topology, see Topology real analytic vector bundle, see Bundle regular direct limit, see Direct limit relatively compact set, see Precompact set Remmert’s Embedding Theorem, see Embedding theorem Riemannian metric, see Metric Riemannian submersion, 106, 106

second fundamental form, 109 section, 6 finitely differentiable, 70 holomorphic, 32, 36 pull-back, 116 real analytic, 32, 36, 42 smooth, 70 semimetric, 13, 74, 79, 81 seminorm, 17 for finitely differentiable topology, 70 for holomorphic topology, 41, 45, 71 for real analytic topology, 67, 72, 214, 238, 246, 248 for smooth topology, 70, 71, 238 sequential space, 14, 46 sheaf theory, 7, 33, 76, 118 Cartan’s Theorem B, 33, 36 smooth mapping, see Mapping Stein manifold, see Manifold subbase for a topology, 12 submersion, 6, 211, 288 Suslin space, 13, 46 direct limits of, 21 inverse limits of, 21 symmetric group, 2 symmetric tensor, see Tensor

tangent bundle, 6 tangent lift, 285, 287 Taylor series, 27, 43 tensor, 2 alternating, 3 symmetric, 3 tensor product, 261, 262 of vector bundles, 260, 264, 266

312 topological dual, see Dual topological epimorphism, see Mapping topological homomorphism, see Mapping topological monomorphism, see Mapping topology compact-open, 12, 41, 47, 70, 79–81 final, 14, 17, 20 finitely differentiable, 70 of germs about a compact set, 46, 50 of germs about a subset, 42 holomorphic, 41, 79–81 induced, 15, 75, 77, 80, 81 initial, 14, 16, 19, 74, 75 metric, 13 real analytic, 51, 72, 75, 77, 88, 246, 248 direct limit, 43, 50–52, 87, 88 inverse limit, 50–52, 87, 88 smooth, 70, 74, 246 weak-PB, 75, 75, 77, 79–81, 88 torsion tensor, 10, 129 totally geodesic submanifold, 106, 109 totally real subspace, 39, 41 transpose, 3 ultrabornological locally convex space, see Locally convex space

Index uniform space, 13, 17, 31

vector bundle, see Bundle vector bundle mapping, see Mapping vertical bundle, see Bundle vertical evaluation of dual section, 95, 96, 165–168, 208, 280 of tensor, 100, 117 of vector bundle mapping, 98, 170, 173, 176, 179, 180, 209, 285 vertical lift of dual section, 98, 156–159, 206, 279 mapping, 95, 97 of section, 95, 96, 146, 147, 149–151, 203, 279 of tensor, 100,101, 117 of vector bundle mapping, 98, 160–163, 207, 279

web, 23 webbed locally convex space, see Locally convex space Whitney’s Embedding Theorem, see Embedding theorem

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