Differential Geometry and its Applications: Proceedings of the 10th International Conference Dga 2007 Olomouc, Czech Republic 27-31 August 2007 9789812790606, 9812790608

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DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS Proceedings of the 10th International Conference on DGA2007 Copyright © 2008 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-279-060-6 ISBN-10 981-279-060-8

Printed in Singapore.

v

PREFACE

The International Conference on Differential Geometry and its Applications is a series of scientific meetings which have been held in the Czech Republic every three years since 1980. The most recent of these, DGA 2007, took place in Olomouc on August 27-31, and was dedicated to the 300th anniversary of the birth of Leonhard Euler. This book begins with an address by Prof. R.J. Wilson, Leonhard Euler – 300 Years On, surveying the life, labours and legacy of the great mathematician. The remainder of the book contains selected conference contributions covering contemporary differential geometry, global analysis, and geometric methods in physics, presented in three sessions: A Riemannian geometry and submanifolds (chair O. Kowalski), B Geometric structures (chair J. Slov´ak), C Global analysis and Geometric methods in physics (chair D. Krupka) or in the poster session. We have also included a list of participants and a list of previous conference proceedings. The main organizer of the Conference was Palack´ y University in Olomouc. The Scientific Committee comprised Demeter Krupka (chairman); Olga Krupkov´ a and Josef Mikeˇs (Palack´ y University in Olomouc); Josef Janyˇska, Ivan Kol´ aˇr, Jana Musilov´a and Jan Slov´ak (Masaryk University in Brno); Oldˇrich Kowalski (Charles University in Prague); and Jiˇr´ıVanˇzura (Mathematical Institute of the Czech Academy of Sciences, branch Brno). The editors would like to thank all the authors for their contributions, and also the referees for their hard work under strict time limits. They also appreciate the indispensable help of Petr Voln´ y in organizing the refereeing process and preparing the electronic version of the manuscript.

Olomouc April 10, 2008

The Editors

vi

vii

ACKNOWLEDGMENTS

A number of people and institutions participated in the organization of the conference, and I would like to thank all of them on behalf of the Scientific Committee of the DGA 2007. The conference took place under the auspices of Prof. RNDr. Lubom´ır Dvoˇr´ ak, CSc., Rector of the Palack´ y University in Olomouc, RNDr. Ivan Kosat´ık, President of the Olomouc Region, and Martin Novotn´ y, Mayor of the City of Olomouc. Our thanks should be extended to all of them for their support and interest in this international scientific event. We especially acknowledge the awards presented by the Rector to five conference participants, S. Gindikin, I. Kol´aˇr, O. Kowalski, D. Krupka, A.M. Vinogradov, appreciating their research in differential geometry, initiatives connected with the establishing of the journal Differential Geometry and its Applications, as well as long lasting activities related to previous conference organization. Special thanks should go to Prof. R.J. Wilson for his address Leonhard Euler – 300 Years On. In particular, I appreciate very much the work of the session chairmen Oldˇrich Kowalski and Jan Slov´ak. I thank all the members of the Local Organizing Committee, but especially young researchers from the Research Group of Global Analysis and its Applications, Jan Brajerˇc´ık, Marie Chodorov´a, Zdenˇek Duˇsek, Dana Smetanov´ a, Zbynˇek Urban and Petr Voln´ y for their effort and work and many useful ideas making the conference successful. The help of the Economical Department of the Faculty of Science, namely by Ing. Dagmar Kopeck´a, is also highly appreciated. The organizers gratefully acknowledge a financial support of the conference activities from the following institutions: • Olomouck´ y kraj (the Olomouc district) • Statut´ arn´ı mˇesto Olomouc (the City of Olomouc) • Laboratory Imaging s.r.o., Prague

viii

• • • •

Inˇzen´ yring dopravn´ıch staveb a.s., Prague Stavoprojekt Olomouc a.s. Alfaprojekt a.s., Olomouc Ecological consulting a.s.

Last but not least, we thank to the Czech Science Foundation and the Czech Ministry of Education, Youth and Sports for their grant support of many research articles published in this book. We also thank the Publisher, World Scientific Publishing Company, for an outstanding collaboration during the preparation of this volume to print.

Olomouc April 10, 2008

Demeter Krupka Chairman of the Scientific Committee and the Organizing Committee

ix

CONTENTS

Preface

v

Acknowledgments Leonhard Euler – 300 Years On R. Wilson

Part 1

Riemannian geometry, submanifolds

vii 1

11

Curvature logarithmic derivatives of curves and isometric immersions T. Adachi

13

Global behaviour of maximal surfaces in Lorentzian product spaces A.L. Albujer

23

Invariant Einstein metrics on certain Stiefel manifolds A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov

35

Constant mean curvature submanifolds in (α, β)−Finsler spaces V. Balan

45

Generalized Einstein manifolds C.-L. Bejan and T.Q. Binh

57

Canonical almost geodesic mappings of type π ˜1 onto pseudoRiemannian manifolds V.E. Berezovski, J. Mikeˇs and A. Vanˇzurov´ a

65

x

On minimal hypersurfaces of hyperbolic space H4 with zero Gauss-Kronecker curvature U. Dursun

77

Structure of geodesics in the flag manifold SO(7)/U (3) Z. Duˇsek

89

Global properties of the Ricci flow on open manifolds J. Eichhorn

99

Gauss maps and symmetric spaces J.-H. Eschenburg

119

Ricci flow on almost flat manifolds G. Guzhvina

133

Compact minimal CR submanifolds of a complex projective space with positive Ricci curvature M. Kon

147

Unique determination of domains A.P. Kopylov

157

Invariance of g-natural metrics on tangent bundles O. Kowalski and M. Sekizawa

171

Some variational problems in geometry of submanifolds H. Li

183

Generalized Veronese manifolds and isotropic immersions S. Maeda and S. Udagawa

197

Integral formulae for foliations on Riemannian manifolds V. Rovenski and P. Walczak

203

Part 2

Geometric structures

The prequantization of Tk1 Rn A.M. Blaga

215 217

xi

Extension of connections M. Doupovec and W.M. Mikulski Conformally-projectively flat statistical structures on tangent bundles over statistical manifolds I. Hasegawa and K. Yamauchi

223

239

Morphisms of almost product projective geometries J. Hrdina and J. Slov´ ak

253

Flows of Spin(7)-structures S. Karigiannis

263

Connections on principal prolongations of principal bundles I. Kol´ aˇr

279

The (κ, µ, ν)-contact metric manifolds and their classification in the 3-dimensional case Th. Koufogiorgos, M. Markellos and V.J. Papantoniou

293

Invariants and submanifolds in almost complex geometry B. Kruglikov

305

Hirzebruch signature operator for transitive Lie algebroids J. Kubarski

317

Quadratic Nambu-Poisson structures N. Nakanishi

329

Bochner-Kaehler metrics and connections of Ricci type M. Pan´ ak and L.J. Schwachh¨ ofer

339

Special n-forms on a 2n-dimensional vector space J. Vanˇzura

353

Symmetries of almost Grassmannian geometries L. Zalabov´ a

371

xii

Part 3

Global analysis

Zeta regularized integral and Fourier expansion of functions on an infinite dimensional torus A. Asada On the direction independence of two remarkable Finsler tensors S. B´ acs´ o and Z. Szilasi Lie systems and integrability conditions of differential equations and some of its applications J.F. Cari˜ nena and J. de Lucas Transverse Poisson structures: The subregular and minimal orbits P.A. Damianou, H. Sabourin and P. Vanhaecke On first-order differential invariants of the non-conjugate subgroups of the Poincar´e group P (1, 4) V.M. Fedorchuk and V.I. Fedorchuk On the flag curvature of the dual of Finsler metric D. Hrimiuc

383 385

397

407

419

431

445

A note on the height of the canonical Stiefel–Whitney classes of the oriented Grassmann manifolds J. Korbaˇs

455

Differential invariants of velocities and higher order Grassmann bundles D. Krupka and Z. Urban

463

Like jet prolongation functors of affine bundles J. Kurek and W.M. Mikulski

475

Two analytical Goodwillie’s theorems R. L´eandre

489

De Rham like cohomology of geometric structures M.A. Malakhaltsev

501

xiii

Submanifolds in pseudo-Euclidean spaces and Dubrovin– Frobenius structures O.I. Mokhov The Cartan form, 20 years on D.J. Saunders Calculus along the tangent bundle projection and projective metrizability J. Szilasi Distance functions of Finsler spaces and distance spaces L. Tam´ assy

Part 4

Geometric methods in physics

515

527

539

559

571

Differential forms relating twistors to Dirac fields I.M. Benn and J.M. Kress

573

The gravitational field of the Robertson-Walker spacetime W. Borgiel

581

Lie algebra pairing and the Lagrangian and Hamiltonian equations in gauge-invariant problems M. Castrill´ on L´ opez and J. Mu˜ noz Masqu´e Non-holonomic constraint forces in field theory in physics L. Czudkov´ a and J. Musilov´ a On a generalization of the Dirac bracket in the De DonderWeyl Hamiltonian formalism I. Kanatchikov

595

603

615

Constrained Lepage forms O. Krupkov´ a, J. Voln´ a and P. Voln´y

627

Variationality of geodesic circles in two dimensions R.Ya. Matsyuk

635

xiv

Noether identities in Einstein–Dirac theory and the Lie derivative of spinor fields M. Palese and E. Winterroth

643

Hilbert–Yang–Mills functional: Examples A. Pat´ ak

655

Hamiltonian formalism on Lie algebroids and its applications L. Popescu

665

Variational theory of balance systems S. Preston

675

Constraint Lepage one forms in higher order mechanics with nonholonomic constraints on the dynamical space M. Swaczyna

689

A connection-theoretic approach to the analysis of symmetries of nonholonomic field theories J. Vankerschaver

703

List of Participants

713

Previous DGA Conference Proceedings

717

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 1–9

1

Leonhard Euler – 300 Years On Robin Wilson The Open University, United Kingdom E-mail: [email protected] 2007 marked the 300th anniversary of the birth of Leonhard Euler. This article presents a selection of his achievements.

Euler was the most prolific mathematician of all time. He wrote more than 500 books and papers during his lifetime, with 400 further publications appearing posthumously. His collected works and correspondence, still not completely published, fill over seventy large volumes comprising tens of thousands of pages. He worked in an astonishing variety of areas, ranging from pure mathematics (the theory of numbers, the geometry of a circle and musical harmony), via infinite series, logarithms, the calculus and mechanics, to practical topics (optics, astronomy, the motion of the Moon and the sailing of ships). He originated so many ideas that his successors have been kept busy trying to follow them up ever since. Indeed, his influence was such that several concepts were later named after him: Euler’s constant, Euler’s polyhedron formula, the Euler line of a triangle, Euler’s equations of motion, Eulerian graphs, Euler’s pentagonal formula for partitions, to name but a few. His life can be divided into four periods. He was born in Basel, Switzerland, on 15 April 1707, where he grew up and went to university. At the

2

R. Wilson

age of 20 he went to Russia, to the St Petersburg Academy, where he became head of the mathematics division. In 1741 he went to Berlin, where he stayed for twenty-five years. In 1766 he returned to St Petersburg where he spent the rest of his life, dying on 7 September 1783. Basel, Switzerland Leonhard Euler’s father was a Calvinist pastor who wished his son to follow him into the ministry. On entering the University of Basel at the age of 14, the young Euler duly studied theology and Hebrew, law and philosophy. While there, he encountered Johann Bernoulli, possibly the finest mathematician of his day, who was impressed with his mathematical abilities and gave him private teaching every Saturday, quickly realising that his pupil was highly talented. Euler also became close friends with Johann’s sons, Daniel and Nicholas. Euler took his Master’s degree in 1724, at the age of 17, and entered divinity school to train for the ministry, but made little progress since mathematics was proving to be such a distraction. Eventually, Bernoulli persuaded Euler’s reluctant father that his son was destined to become a great mathematician, and Euler abandoned his theological training. Euler’s first significant mathematical achievement occurred when he was just 20. The Paris Academy had proposed a prize problem involving the placing of masts on a sailing ship in order to combine speed with stability. Euler’s memoir, while not gaining the prize, received an honourable mention; later, he won the Paris prize on twelve occasions. Euler next applied for the Chair of Mathematics at the University of Basel. Because of his young age he was unsuccessful, but meanwhile Daniel Bernoulli had taken up a position at the St Petersburg Academy in Russia, and invited Euler to join him there. The only available position was in medicine and physiology, but jobs were scarce so Euler learned these subjects, and his study of the ear led him to investigate the mathematics of sound and the propagation of waves. St Petersburg, Russia Unfortunately, on the very day that Euler arrived in Russia, Empress Catherine I, who had set up the Academy, died. Her heir was still a boy, and the faction that ruled on his behalf regarded the Academy as a luxury. Euler quietly got on with his work, while working closely with Daniel Bernoulli. In 1733, Daniel Bernoulli had had enough of the problems of the

Leonhard Euler – 300 Years On

3

Academy and returned to an academic position in Switzerland. Euler, still aged only 26, replaced him in the Chair of Mathematics, and determined to make the best of a difficult situation. The 1730s were indeed very productive years for him, with substantial advances in number theory, the summation of series, and mechanics. At the same time he was acting as a scientific consultant to the government – preparing maps, advising the Russian navy, testing designs for fire engines, and writing textbooks for the Russian schools. An area to which Euler contributed throughout his life was the theory of numbers. In December 1729, he received a letter from his St Petersburg colleague Christian Goldbach, who is best remembered for the still-unproved Goldbach conjecture that every even number can be written as the sum of two prime numbers. Goldbach’s letter was concerned with the Fermat numbers: 21 + 1 = 3, 22 + 1 = 5, 24 + 1 = 17, 28 + 1 = 257, 216 + 1 = 65, 537, . . . . Are all such numbers prime? Fermat had conjectured that they are, but Euler found an ingenious argument to prove that the next one (232 + 1), a ten-digit number, is divisible by 641. Since then, no other ‘Fermat number’ has been shown to be prime, so Fermat’s conjecture was unfortunate. Euler’s prodigious calculating abilities were legendary. One day, two students were trying to sum a complicated progression and disagreed over the 50th decimal place: Euler simply calculated the correct value in his head to settle the argument. Another challenge given to Euler was to find four different numbers, the sum of any two of which is a perfect square: he produced the quartet: 18530, 38114, 45986 and 65570. A different preoccupation in the 1730s was his work on infinite series. He first became interested in the ‘harmonic series’ 1+

1 2

+

1 3

+

1 4

+

1 5

+ ...,

which does not converge, and noticed that the sum of the first n terms is very close to log n. In particular, he proved that as n becomes large, the difference between them tends to a fixed number 0.5772 . . . , now known as Euler’s constant. Another problem on infinite series, known as the Basel problem, exercised many minds at the time. It was to find the sum of the reciprocals of the perfect squares: 1+

1 4

+

1 9

+

1 16

+

1 25

+ ... .

4

R. Wilson

One of Euler’s earliest achievements was to show that this sum is π 2 /6, and this brought him international fame. He also extended his calculations to find the sum of the reciprocals of the 4th powers, the 6th powers, and so on, up to the 26th powers. This led him to investigate what is now known as the Riemann zeta function. We next turn to a recreational puzzle that Euler solved in 1735: the problem of the bridges of K¨ onigsberg. The medieval city of K¨onigsberg consisted of four areas of land linked by seven bridges, and the problem was to find a route crossing each bridge just once and returning to the starting point. Euler discovered a counting argument involving the number of bridges emerging from each land area, and proved that no such route exists. Furthermore, he obtained for any arrangement of land areas and bridges a corresponding rule for deciding when such a route is possible – namely, • if there are no areas with an odd number of bridges, then a route exists starting anywhere and ending in the same place; • if there are two areas with an odd number of bridges, then a route exists, starting in one area and ending in the other; • if there are more than two such areas (as in K¨onigsberg), then there is no such route. Euler’s solution of the K¨ onigsberg bridges problem is considered as the earliest contribution to graph theory, and is now solved by looking at a network with four points representing the land areas and seven lines representing the bridges. But Euler never did this – the network that represents this puzzle was not drawn for 150 years. Around the same time, Euler published Mechanica, his first treatise on the dynamics of a particle. However, his most important work in this area came in 1750 with his work on the motion of rigid bodies – free, or rotating about a point. By choosing the point as the origin of coordinates, and with axes aligned along the principal axes of inertia of the body, he obtained what are now called Euler’s equations of motion; the concept of moment of inertia was also due to him. Even later, in 1776, he proved that any rotation of a rigid body about a point is equivalent to a rotation about a line through that point. Much of his work in this area used differential equations, an area to which he had himself contributed a great deal. In the late 1730s, Euler went blind in his right eye. Although he attributed this to overwork, it was probably due to an eye infection. However, this did not diminish his productivity: he continued to write on acoustics, musical harmony, ship-building, prime numbers, and much more besides.

Leonhard Euler – 300 Years On

5

Berlin, Germany In 1741, with his fame preceding him, Euler received an invitation from Prussia’s Frederick the Great to join the newly vitalised Berlin Academy. With the political situation in Russia still uncertain, he accepted it and remained in Berlin for 25 years. At first, he got on well with Frederick, but later, especially after the seven years war between Germany and Russia, things began to cool as Frederick started to take more and more interest in the workings of the Academy. Frederick considered himself cultured and witty, and found Euler unsophisticated; in return, Euler found Frederick pretentious, petty and rude. Even so, Euler managed to work on a dazzling range of topics, writing works in the 1740s and 1750s on the theory of tides, the calculus of variations, the motion of the moon, hydrodynamics, and the wave motion of vibrating strings. His most important work from this period was the Introductio in Analysin Infinitorum (‘Introduction to the analysis of the infinite’), published in 1748. It was here that he presented some of his earlier work on the number e = 2.718281 . . . , defined as
n 1 1 1 + 2! + 3! + . . . or as lim 1 + n1 , 1 + 1! n→∞

x

and the related exponential function e . In the Introductio, Euler expressed certain well-known functions as infinite series, and then introduced his great masterstroke. He knew how the functions ex , sin x and cos x could be expanded in powers of x, but at first they seem to have nothing in common. However, on introducing the complex number i, and manipulating the power series, he deduced the fundamental formula linking them, eix = cos x + i sin x.

Although he did not explicitly write down the simple consequence eiπ = −1, often described as his most famous result, he would surely have known it. There were many other interesting things in the Introductio. Over the 100 years since Descartes there had been a gradual movement from geometry towards algebra, and this reached its climax when Euler actually defined the conic sections (ellipse, parabola and hyperbola) by their algebraic equations, rather than geometrically as sections of a cone. In particular, starting with the equation y 2 = α + βx + γx2 , he showed that we get an ellipse if γ < 0, a parabola if γ = 0, and a hyperbola if γ > 0. He then extended

6

R. Wilson

his algebraic arguments to three dimensions, to the seven types of quadrics, and discovered the hyperbolic paraboloid in the process. Yet another interesting topic in the Introductio is partitions, or ‘divulsions of integers’, as Leibniz had called them in a letter to Bernoulli. In how many ways can we split up a positive integer n into smaller ones? Let p(n) be this number – for example, p(4) = 5, corresponding to the five partitions 4, 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1 (the order doesn’t matter). So we can draw up a table of values – but how can we show that p(200) = 3, 972, 999, 029, 388? To investigate partitions, Euler introduced the generating function p(x) = 1 + p(1)x + p(2)x2 + p(3)x3 + . . . , and used it to derive what is now called Euler’s pentagonal number formula involving the ‘generalised pentagonal numbers’ 12 k(3k ± 1): p(n) = p(n − 1) + p(n − 2) − p(n − 5) − p(n − 7) + p(n − 12) + . . . . This yields p(n) by iteration, and is still the most efficient way of finding p(n). A particularly nice result on partitions, which appears in the Introductio, concerns odd and distinct partitions. In an odd partition all the parts are odd, and in a distinct partition all of them are different: for example, 9 has eight odd partitions (9, 7+1+1, 5+3+1, etc.) and eight distinct partitions (9, 8 + 1, 7 + 2, etc.). Using generating functions, Euler proved that for any number, the number of odd partitions is the same as the number of distinct partitions. Another preoccupation was mentioned in a letter to Goldbach, in 1750. Euler had been looking at polyhedra, and observed that the numbers of vertices, edges and faces are always related by the formula: (no. of faces) + (no. of vertices) = (no. of edges) + 2. This formula, now known as Euler’s polyhedron formula, has been incorrectly credited to Descartes, who did not have the terminology or motivation to derive it: indeed, it was Euler who introduced the concept of an edge. However, Euler’s proof was deficient – a complete proof was not given until 40 years later, by the algebraist and number-theorist Legendre. Euler’s most popular and best-selling book was his Letters to a German Princess. Euler was always an extremely clear writer, and this was a multi-volume masterpiece of exposition that he produced when he was asked to give elementary science lessons to the Princess of Anhalt-Dessau. The resulting collection had over 200 ‘letters’ that Euler wrote on a range

Leonhard Euler – 300 Years On

7

of scientific topics, including gravity, astronomy, light, sound, magnetism, logic, and much else besides. He wrote about why the sky is blue, why the moon looks larger when it rises, and why the tops of mountains are cold (even in the tropics). It was one of the best books ever written on popular science. The last of Euler’s Berlin books was his 1755 massive tome on the differential calculus. This contained all the latest results, many due to him, and presented the calculus in terms of the basic idea of a function – indeed, it was Euler who introduced the notation f for a function. Other notations P he introduced at various times were (for summation), i (the square root of −1) and e (the exponential number). He also popularised the notation for π, although that had actually been introduced by William Jones in 1706. He followed his book on differentiation in 1768–1770 with a three-volume treatise on the integral calculus. St Petersburg, revisited After his difficulties with Frederick the Great, Euler must have felt very relieved when in 1766, at the age of 59, he received an invitation from Catherine the Great of Russia to return to St Petersburg. Things there had improved greatly, thanks to the enlightened Empress, and he was received royally. He continued to work enthusiastically, soon producing a delightful result in pure geometry. In any triangle three particular points of interest are the orthocentre (the meeting point of the perpendiculars from the vertices to the opposite sides), the centroid (the meeting point of the three lines joining a vertex to the midpoint of the opposite side), and the circumcentre (the centre of the circle passing through the vertices of the triangle). By calculating their coordinates, Euler proved the attractive result that these three points always lie in a straight line – now called the Euler line of the triangle – and that the centroid always lies exactly one-third of the distance between the other two. Euler’s life-long interest in number theory continued into his later years, when he extended some results associated with Fermat – in particular, Fermat’s last theorem that, for any n > 2, there are no positive numbers a, b, c satisfying an + bn = cn . In his number theory book of 1770, Euler proved for the first time that the sum of two cubes cannot equal another cube (n = 3). Another connection with Fermat was Fermat’s little theorem, which states that, if a is any number that is not divisible by a given prime number

8

R. Wilson

p, then p divides ap−1 − 1; for example, on taking p = 29 and a = 48, we deduce that 4828 − 1 is divisible by 29. In 1760 Euler extended this result to numbers other than primes, introducing the Euler ϕ-function and proving that, for any numbers a and n, aϕ(n) − 1 is always divisible by n. Yet another result in number theory concerned perfect numbers – numbers whose proper divisors add up to the number itself; for example, 28 is a perfect number because its proper divisors are 1, 2, 4, 7 and 14, which sum to 28. In his Elements, Euclid had proved that every number of the form 2n−1 × (2n − 1) is perfect, when 2n − 1 is prime. Euler proved that every even perfect number has this form – but it is still not known whether any odd perfect numbers exist. The last few years of Euler’s life, though more peaceful than his earlier ones, saw many personal tragedies. In 1771 his house burned down, with the loss of his library and almost his life, but fortunately his manuscripts were saved. Shortly after this, his beloved wife died and he remarried. And finally he lost most of the sight in his other eye – but again, his productivity remained undiminished as he wrote on slates with his sons and friends as amenuenses; indeed, shortly after he went blind, he produced a 700-page volume on the motion of the moon. In a book on recreational mathematics in 1725, Jacques Ozanam had shown how to lay out the sixteen court cards so that each row and column contains each suit and each value (J, Q, K, A). It is also possible to make a similar arrangement with 25 cards (five suits and five values), but what about 36 cards? In the year before he died, in a paper mainly on magic squares (another interest of his), Euler posed this as the 36 officers problem: Arrange 36 officers, one each of 6 ranks from 6 regiments, in a square array so that each row or column has one officer of each rank and one officer of each regiment. Euler believed that this cannot be done, and this was eventually confirmed around 1900 by Gaston Tarry, essentially by enumerating all the possibilities. Euler also claimed that the corresponding problem has a solution for any number of ranks and regiments, except when this number is of the form 4k + 2: that is, 6, 10, 14, 18, . . . . He was right about 6, but wrong about all the others, although his error was not demonstrated until almost 300 years later.

Leonhard Euler – 300 Years On

9

Conclusion Euler worked until the very end. In a Eulogy by the Marquis de Condorcet, we read about his final afternoon: On the 7th of September 1783, after amusing himself with calculating on a slate the laws of the ascending motion of air balloons, the recent discovery of which was then making a noise all over Europe, he dined with Mr Lexell and his family, talked of Herschel’s planet (Uranus), and of the calculations which determine its orbit. A little after, he called his grandchild, and fell a playing with him as he drank tea, when suddenly the pipe, which he held in his hand, dropped from it, and he ceased to calculate and to breathe. The great Euler was no more. Acknowledgments This article is adapted from the author’s article ‘Read Euler, read Euler, he is the master of us all’, Plus (Online mathematics magazine), Mathematics Millennium Project (University of Cambridge, England), 42, March 2007. Further reading A good introductory book on Euler’s life and works is: William Dunham, Euler: the master of us all, Mathematical Association of America, 1999. To celebrate the 300th anniversary of his death, the Mathematical Association of America has also published a series of five books in 2007 on the life and works of Leonhard Euler. Further information can also be found in standard reference works, such as: The Dictionary of Scientific Biography, Scribner, New York, 1970–1990.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 13–21

13

Curvature logarithmic derivatives of curves and isometric immersions Toshiaki Adachi Department of Mathematics, Nagoya Institute of Technology, Nagoya 466-8555, Japan E-mail: [email protected] In this note we study isometric immersions by extrinsic shapes of some curves and extend Nomizu-Yano’s characterization of extrinsic spheres. We give a characterization of non-totally geodesic isotropic immersions in the sense of O’Neill by the amount of curvature logarithmic derivatives of curves preserved by these isometries. Also we give a characterization of totally umbilic submanifolds with parallel normalized mean curvature vector in terms of curvature logarithmic derivatives of curves of order 2 preserved by immersions. Keywords: Isotropic immersion, curvature logarithmic derivative, totally umbilic, parallel normalized mean curvature vector, extrinsic shapes of curves, derived maps. MS classification: 53C40.

1. Introduction f be an isometric immersion. For a smooth curve γ : I → Let f : M → M M parameterized by its arc-length, we call the curve f ◦γ the extrinsic shape of γ. It is an interesting problem to characterize isometric immersions by some properties of extrinsic shapes of smooth curves. In this area NomizuYano’s characterization on extrinsic spheres by extrinsic shapes of circles is one of the most fundamental results. A Riemannian submanifold M is said f if it is totally umbilic and has parallel mean to be an extrinsic sphere in M curvature vector. We call a smooth curve γ parameterized by its arc-length circle if it satisfies ∇γ˙ γ˙ = κYγ , ∇γ˙ Yγ = −κγ˙ with some positive constant κ and a unit vector field Yγ along γ. In Ref. 5, Nomizu and Yano showed f if and only if the extrinsic shape of that M is an extrinsic sphere in M each circle is also a circle. In this note we give characterizations of isotropic immersions and totally umbilic immersions by extending their idea from a bit different point of view. This also refines Sugiyama and the author’s

14

T. Adachi

results (Refs. 1,7,8). 2. Derived maps of curves

 For a Riemannian manifold M we consider a family C(M ) = γ : Iγ → M of smooth curves on M which are parameterized by their arc-length f causes and whose domains contain the origin. An isometry f : M → M f a correspondence F : C(M ) ∈ γ 7→ f ◦ γ ∈ C(M ) between the families of curves. In previous papers which characterized some isometric immersions (Refs. 3,5,6, for example), properties of F are taken into account: An isometry is characterized by the extrinsic shape F (S) of a “nice” family S. In this note by changing our viewpoint we shall consider maps derived from the extrinsic shape correspondence F . For a smooth curve γ ∈ C(M ), we define a function κγ along γ by κγ = k∇γ˙ γk ˙ and call it the first geodesic curvature of γ. When it does not have inflection points, that is κγ > 0, we set ℓγ = κ′γ /κγ and call it the curvature logarithmic derivative of γ. For each circle, this function is identically zero. As a first level we consider the product U M × [0, ∞) of the unit tangent bundle of M and a half line.
We then have a canonical projection ̟1 given as C(M ) ∋ γ 7→ γ(0), ˙ κγ (0) ∈ U M × [0, ∞) which shows initial vector and initial geodesic curvature. For f we see by use of Gauss formula that an isometric immersion f : M → M κ2F (γ) = κ2γ + σf (γ, ˙ γ) ˙ 2

(1)

with the second fundamental form σf of f . We hence find that f induces a f × [0, ∞) of the first level. derived map f1 : U M × [0, ∞) → U M As for a second level we consider the product O2 M × [0, ∞) × R of the set O2 M of orthonormal pairs of tangent vectors and a half plane. If we restrict ourselves on the subfamily C2 (M ) of smooth curves whose initials are not inflection points, we have a canonical projection ̟2 which is given as  
C2 (M ) ∋ γ 7→ γ(0), ˙ Yγ (0) , κγ (0), ℓγ (0) ∈ O2 M × [0, ∞) × R,

where Yγ is a unit vector field along γ defined by Yγ = (1/κγ )∇γ˙ γ. ˙ By differentiating both sides of the equality (1) we have the following: Lemma 2.1 (Ref. 1). Curvature logarithmic derivatives of a curve γ and its extrinsic shape F (γ) are related as follows:

˙ γ)k ˙ 2 = κ2F (γ) ℓF (γ) − ℓγ + ℓγ kσ(γ, ˙ γ)k ˙ 2 κ2γ ℓF (γ) − ℓγ + ℓF (γ)kσ(γ, (2)





˙ γ), ˙ σ(γ, ˙ γ) ˙ + 2 σ ∇γ˙ γ, ˙ γ˙ , σ(γ, ˙ γ) ˙ . = ∇γ˙ σ (γ,

Curvature logarithmic derivatives of curves and isometric immersions

15

Here the covariant differentiation ∇ of σf is defined by
(∇X σf )(Y, Z) = ∇⊥ X σf (Y, Z) − σf (∇X Y, Z) − σf (Y, ∇X Z) for vector fields X, Y, Z on M .

This lemma guarantees that an isometry f induces a derived map f × [0, ∞) × R of the second level. Instead f2 : O2 M × [0, ∞) × R → O2 M of properties of extrinsic shape correspondence, we make use of properties of these derived maps. Since the derived map of the first level is too simple to take its properties, we shall study the derived map of the second level. We pay attention to the last component of the second level, which corresponds to curvature logarithmic derivatives of curves, and characterize some isometries by properties of the derived map of the second level. 3. Isotropic immersions f is said to be isotropic at x ∈ M An isometric immersion f : M → M in the sense of O’Neill if the norm kσf (u, u)k of the second fundamental form does not depend on the choice of a unit tangent vector u ∈ Ux M . It is clear that if f is umbilic at x then it is isotropic at this point. We say an isometric immersion f is geodesic at x if σf (u, u) = 0 for every u ∈ Ux M . Trivially, if f is geodesic at x then it is isotropic at this point. In this section we characterize isotropic immersions. For an orthonormal pair
(u, v) ∈ O2 Mx of tangent vectors at x ∈ M , we denote by C(u, v) ⊂ C(M ) the subfamily of curves whose initial velocity vector is u and initial principal normal is v. We define the sets of curvature logarithmic derivatives preserved by f as  Af (u, v) = ℓγ (0) γ ∈ C(u, v), ℓF (γ)(0) = ℓγ (0) , \ [ Af (u, v). Af (u, v), A− A+ f (u) = f (u) = v (⊥u)

v (⊥u)

If we need tospecify the geodesic curvature at initial, we consider the family C(u, v; κ) = γ ∈ C(u, v) κγ (0) = κ for a positive constant κ, and define − the sets Af (u, v; κ), A+ f (u; κ), Af (u; κ) in the same manner for this family. If we interpret the set Af (u, v; κ) in terms of the derived map of the second level, it is the set of fixed values of the function f2 ((u, v), κ, ∗) : R → R. It should be noted that Af (u, v; κ) 6= ∅ for each (u, v) ∈ O2 M and each positive κ for an arbitrary isometric immersion f . Hence the condition of existence of curvature logarithmic derivatives preserved by an isometry is

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too weak. We hence consider the quantity of such curvature logarithmic derivatives and characterize isotropic immersions in the following manner: f the following Theorem 3.1. For an isometric immersion f : M → M conditions at a point x ∈ M are mutually equivalent to each other: 1) 2) 3) 4)

At this point f is isotropic in the sense of O’Neill and is not geodesic; The set Af (u, v) consists of a single value for every (u, v) ∈ O2 Mx ; the set A+ f (u) consists of a single value for every u ∈ Ux M ; For every u ∈ Ux M there is a positive κu satisfying that the set A+ f (u; κu ) consists of a single value; 5) The set A− f (u) consists of a single value for every u ∈ Ux M ; 6) For every u ∈ Ux M there is a positive κu satisfying that the set A− f (u; κu ) consists of a single value.

f the following Theorem 3.2. For an isometric immersion f : M → M conditions at a point x ∈ M are mutually equivalent to each other:

1) f is geodesic at this point; 2) There exist an orthonormal basis {u1 , . . . , un } of the tangent space Tx M and vj ∈ Ux M (j = 1, . . . , n) satisfying vj ⊥ uj and Af (uj , vj ) = R for all j; 3) There exist a basis {u1 , . . . , un } of the tangent space Tx M by unit vectors and pairs (vj , κj ) (j = 1, . . . , n) of a unit tangent vector and a positive constant satisfying vj ⊥ uj and Af (uj , vj ; κj ) = R for all j.

Proof of Theorems 3.1 and 3.2. By Lemma 2.1, we see γ ∈ C(u, v) satisfies ℓF (γ)(0) = ℓγ (0) if and only if 




 ℓγ (0) = kσf (u, u)k−2 ∇u σf (u, u), σf (u, u) +2κγ (0) σf (u, u), σf (u, v)

when σf (u, u) 6= 0. Hence the feature of the set Af (u, v) is classified into the following 3 cases: 1) Af (u, v) = R if and only if σf (u, u) = 0; 2) Af (u, v) consists of a single value if and only if σf (u, u) 6= 0 and hσf (u, u), σf (u, v)i = 0; 3) Af (u, v) is an open half line if and only if hσf (u, u), σf (u, v)i = 6 0.

It is clear that in the second case the value Af (u, v) does not depend on the choice of v. Since f is isotropic at x if and only if hσf (u, u), σf (u, v)i = 0 for every (u, v) ∈ O2 Mx , we get our conclusions.

Curvature logarithmic derivatives of curves and isometric immersions

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Remark 3.1. Our proof of Theorem 3.1 also shows that f is isotropic at x if and only if for every (u, v) ∈ O2 Mx there is a positive κu,v with Af (u, v : κu,v ) ∩ Af (u, −v; κu,v ) 6= ∅ (see Ref. 1). We are now interested in the single value of preserved curvature logarithmic derivatives. When f is isotropic everywhere, we just call it isotropic and define a function of isotropy λf : M → R as the norm of the second fundamental form at each point. f is isotropic, then we have A+ (u) = A− (u) = Remark 3.2. If f : M → M f f  u(λf )/λf (x) for every u ∈ Ux M at an arbitrary point x ∈ M .

As a consequence of Theorem 3.1 we can characterize Veroneze embeddings. For a positive integer k, we denote by fk : CP n (c/k) → CP N (k) (c) the k-th Veronese embedding which is given by i hq    k! (k0 ! · · · kn !) z0k0 · · · znkn zi 0≤i≤n 7→ k0 +···+kn =k

with homogeneous coordinate [∗], where N (k) = (n + k)!/(n!k!) − 1. This embedding is isotropic whose function of isotropy is a constant function; λfk ≡ c(k − 1)/(2k).

Theorem 3.3 (c.f. Ref. 1). Let f : M → CM N(c) be a non-totally geodesic K¨ ahler isometric full immersion of a K¨ ahler manifold M of complex dimension n ≥ 2 into a complex space form. Then the following conditions are mutually equivalent: 1) There is a positive integer k satisfying that N = N (k), the ambient space is CP N(c), the manifold M is locally congruent to CP n (c/k) and f is locally equivalent to fk ; − 2) The set A+ f (u) (resp. Af (u)) consists of a single value for every unit tangent vector u ∈ T M . 3) The set Af (u, v) consists of a single value for every (u, v) ∈ O2 M ; 4) For every u ∈ Ux M there is a positive κu satisfying that the set − A+ f (u; κu ) (resp. Af (u; κu )) consists of a single value. We should note that such characterization does not hold for the case dimC M = 1 because all holomorphic curves are isotropic. 4. Totally umbilic immersions In this section we study isometric immersions with umbilic points. When γ ∈ C(M ) does not have inflection points, the differential of Yγ is of the form

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T. Adachi

∇γ˙ Yγ = −κγ γ˙ + Zγ with some vector field Zγ along γ which is orthogonal to both γ˙ and Yγ . We say γ is of (proper) order 2 at initial if κγ 6= 0 and Zγ (0) = 0. Clearly every circle is of order 2 at each point. For an orthonormal pair (u, v) ∈ O2 Mx we define a subset Bf (u, v) of Af (u, v) as   γ ∈ C(u, v), ℓγ˜ (0) = ℓγ (0), . Bf (u, v) = ℓγ (0) F (γ) is of order 2 at f (x) S We put Bf (u) = v (⊥u) Bf (u, v). The set Bf (u, v) can be interpreted in the following manner. We consider a subfamily  S = γ ∈ C2 (M ) F (γ) is of order 2 at F (γ)(0)

and study the restriction of the derived map f2 of the second level onto the set ̟2 (S). The set Bf (u, v) consists of fixed values of this map with respect to the last component. We can also explain this set in another way. We denote by O3 M the set of triplets of orthonormal tangent vectors and set
] O M = O M ⊕ O M×{0} . As for a third level we consider the product 3 3 2 ] O 3 M × [0, ∞) × R × [0, ∞) and define a projection ̟3 of C2 (M ) onto this level by  
˙ Yγ (0), Zγ (0)/kZγ (0)k , κγ (0), ℓγ (0), kZγ (0)k , γ 7→ γ(0),

where in case kZγ (0)k = 0 we regard Zγ (0)/kZγ (0)k as a null tangent vector. By the following Lemma 4.1 we get a derived map ] f ] f3 : O 3 M ×[0, ∞)×R×[0, ∞) → O3 M ×[0, ∞)×R×[0, ∞)

f. Since γ ∈ C2 (M ) belongs of the third level from an isometry f : M → M
to S if and only if the last component of f3 ̟3 (γ) is 0, we can interpret Bf (u, v) in terms of this derived map f3 . Lemma 4.1. For γ ∈ C2 (M ) the vector field ZF (γ) along F (γ) is given as
κF (γ) ZF (γ) = kσf (γ, ˙ γ)k ˙ γ˙ − Aσf (γ, ˙ + κγ ℓγ − ℓF (γ) Yγ + κγ Zγ ˙ γ) ˙ γ
(3) − ℓF (γ) σf (γ, ˙ γ) ˙ + 3κγ σf (γ, ˙ Yγ ) + ∇γ˙ σf (γ, ˙ γ). ˙

We now study isometric immersions with umbilic points. The quantity of Bf (u, v) shows the umbilic property of isometries. f be an isometric immersion. Theorem 4.1. Let f : M → M

(1) If Bf (u, v) 6= ∅ for every (u, v) ∈ O2 Mx , then f is umbilic at x ∈ M . (2) If we suppose moreover there is a basis {u1 , . . . , un } of Tx M by unit vectors which satisfies the property that Bf (uj ) contains at least two values for all j, then f is geodesic at this point x.

Curvature logarithmic derivatives of curves and isometric immersions

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Proof. If F (γ) for γ ∈ C(u, v) is of order 2 at the origin, by paying attention on the tangential and normal components of the equality (3), we find the following hold: (
κγ (0) ℓγ (0) − ℓF (γ)(0) = hσf (u, u), σf (u, v)i,
(4) ℓF (γ)(0)σf (u, u) = 3κγ (0)σf (u, v) + ∇u σf (u, u). Thus if Bf (u, v) 6= ∅ then hσf (u, u), σf (u, v)i = 0. Since we also have Bf (u, −v) 6= ∅ there is a smooth curve ρ ∈ C(u, −v) whose extrinsic shape F (ρ) is of order 2 at f (x). We therefore have
ℓF (ρ) (0)σf (u, u) = −3κρ (0)σf (u, v) + ∇u σf (u, u). Thus we obtain



ℓF (γ)(0) − ℓF (ρ) (0) σf (u, u) = 3 κγ (0) + κρ (0) σf (u, v),

which shows σf (u, v) = 0 holds. As σf (u, v) = 0 holds for an arbitrary (u, v) ∈ O2 Mx , we can conclude x is an umbilic point. Since the condition in Theorem 4.1 is not a necessary and sufficient condition, we here study more about it. We shall say a submanifold has parallel normalized mean curvature vector if its mean curvature vector h is of the form h = Hξ with its mean curvature H = khk and some unit vector field ξ which is parallel with respect to ∇⊥ . f is totally umbilic Theorem 4.2. An isometric immersion f : M → M and has parallel normalized mean curvature vector on the outside of the set of geodesic points if and only if Bf (u, v) 6= ∅ for every orthonormal pair (u, v) ∈ O2 M . Proof. By Equality (3) we can conclude the following: When f is totally umbilic, for a smooth curve γ we have
(5) κF (γ) ZF (γ) = κγ ℓF (γ) − ℓγ Yγ + κγ Zγ + ℓF (γ) hγ − ∇⊥ γ˙ h.

If Bf (u, v) 6= ∅ for every (u, v) ∈ O2 M , we see by Theorem 4.1 that f is totally umbilic. On the outside of the set of geodesic points, we consider a unit vector field ξ = (1/H)h. If F (γ) of γ ∈ C(u, v), (u, v) ∈ O2 Mx is of order 2 at f (x), we find by Equation (5) that ℓF (γ)(0)hx = ∇⊥ u h, which ⊥ is equivalent to ℓF (γ)(0)Hx ξx = u(H)ξx + Hx ∇⊥ u ξ. As h∇u ξ, ξx i = 0, we see ∇⊥ u ξ = 0. Thus we obtain M has parallel normalized mean curvature vector on the outside of the set of geodesic points.

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On the other hand, we suppose f is totally umbilic and has parallel normalized mean curvature vector on the outside of the set of geodesic points. For (u, v) ∈ O2 Mx at non-geodesic point x ∈ M we take a smooth curve γ ∈ C(u, v) which is of order 2 at x and satisfies ℓγ (0) = u(H)/Hx . We then see by Lemma 2.1 that
κγ (0) ℓF (γ) (0) − ℓγ (0) + ℓγ (0)Hx2 = h∇u h, hi = u(H)Hx ,

hence ℓF (γ) (0) = ℓγ (0). Therefore we find by Equality (5) that F (γ) is of order 2 at f (x). Thus we see u(H)/Hx ∈ Bf (u, v).

We here study isometric immersions of K¨ahler manifolds into real space forms. We denote by RM n (c) a real space form of constant sectional curvature c, which is either a standard sphere S n (c), a Euclidean space Rn , or a real hyperbolic space RH n (c). Theorem 4.3. Let f : (M, J) → RM N(˜ c) be an isometric immersion of a connected K¨ ahler manifold of complex dimension n with complex structure J into a real space form. (1) If Bf (u, Ju) 6= ∅ and Bf (u, −Ju) 6= ∅ hold for every u ∈ U M , then f has parallel second fundamental form. (2) If we suppose moreover that Bf (u, Ju) ∪ Bf (u, −Ju) contains a nonzero value for every u ∈ U M , then f is totally geodesic. That is, it is locally equivalent to an embedding Cn → RN . Proof. By the proof of Theorem 4.1 we see σf (u, Ju) = 0 for every u ∈ U M . Thus for arbitrary u, w ∈ T M , considering (u + w)/ku + wk we get σf (u, Jw) = −σf (Ju, w). Since M is K¨ahler, this leads us to (∇X σf )(Y, JZ) = −(∇X σf )(JY, Z) for arbitrary vector fields X, Y, Z on M . By use of Codazzi equation we can conclude (∇X σf )(Y, JZ) = 0 (see Ref. 7). When Bf (u, Ju)∪Bf (u, −Ju) contains a non-zero value, the second equality in (4) shows that σf (u, u) = 0. Thus we get the conclusion. At the last stage, we shall mention that there are some studies on the geometry of frame bundles (Refs. 2,4). The author suspects that there might be some relationship between the geometry of O2 M and that of a base manifold M through the geometry of curves of order 2. Acknowledgments The author is partially supported by Grant-in-Aid for Scientific Research (C) (No. 17540072) JSPS.

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References 1. T. Adachi and T. Sugiyama, A characterization of isotropic immersions by extrinsic shapes of smooth curves, to appear in Diff. Geom. its Appl. 2. O. Kowalski and M. Sekizawa, On the geometry of orthonormal frame bundles, to appear in Math. Nachr. 3. S. Maeda, A characterization of constant isotropic immersions by circles, Arch. Math. (Basel) 81 (2003) 90–95. 4. K.P. Mok, On the differential geometry of frame bundles of Riemannian manifolds, J. reine angew Math. 302 (1978) 16–31. 5. K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann. 210 (1974) 163–170. 6. K. Sakamoto, Planer geodesic immersions, Tˆ ohoku Math. J. 29 (1977) 25–56. 7. T. Sugiyama, Totally geodesic K¨ ahler immersions in veiw of curves of order two, to appear in Geom. Dedicata. 8. T. Sugiyama and T. Adachi, Totally umbilic isometric immersions and curves of order 2, Monatsh. Math. 150 (2007) 73–81.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 23–33

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Global behaviour of maximal surfaces in Lorentzian product spaces Alma L. Albujer Departamento de Matem´ aticas, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain E-mail: [email protected] In this paper we report on some recent results about maximal surfaces in a Lorentzian product space of the form M 2 × R1 , where M 2 is a connected Riemannian surface and M 2 × R1 is endowed with the Lorentzian metric h, i = h, iM − dt2 . In particular, if the Gaussian curvature of M is non-negative, we establish new Calabi-Bernstein results for complete maximal surfaces immersed into M 2 × R1 and for entire maximal graphs over a complete surface M . We also construct counterexamples which show that our Calabi-Bernstein results are no longer true without the hypothesis KM ≥ 0. Finally, we introduce two local approaches to our global results. We do not provide here with detailed proofs of our results. For further details, we refer the reader to the original papers Refs. 1–3. Keywords: Maximal surfaces, Lorentzian product spaces, Parabolicity. MS classification: 53C42, 53C50.

1. Introduction The study of maximal surfaces, that is, spacelike surfaces with zero mean curvature, has been of increasing interest in recent years from both physical and mathematical points of view. Here by spacelike we mean that the induced metric from the ambient Lorentzian metric is a Riemannian metric on the surface and the term maximal comes from the fact that these surfaces locally maximize area among all nearby surfaces having the same boundary. One of the most important global results in Lorentzian geometry is the Calabi-Bernstein theorem for maximal surfaces in the Lorentz-Minkowski space R31 . The Calabi-Bernstein theorem states, in a parametric version, that the only complete maximal surfaces in R31 are the spacelike planes. This result can also be seen in a non-parametric form, and it establishes

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A.L. Albujer

that the only entire maximal graphs in R31 are the spacelike planes; that is, the only entire solutions to the maximal surface equation ! Du = 0, |Du|2 < 1 Div p 1 − |Du|2

on the Euclidean plane R2 are affine functions. The Calabi-Bernstein theorem was first proved by Calabi,7 and extended later to the general ndimensional Lorentz-Minkowski space by Cheng and Yau.8 The aim of this paper is to report on some of our recent results, obtained jointly with Al´ıas, about maximal surfaces immersed into a Lorentzian product space of the form M 2 × R, where M 2 is a connected Riemannian surface and M 2 × R is endowed with the Lorentzian metric ∗ h, i = πM (h, iM ) − πR∗ (dt2 ).

Here πM and πR denote the projections from M × R onto each factor, and h, iM is the Riemannian metric on M . For simplicity, we will write h, i = h, iM − dt2 , and we will denote by M 2 × R1 the 3-dimensional product manifold M 2 ×R endowed with that Lorentzian metric. In the first section, we introduce some Calabi-Bernstein type results for maximal surfaces in M 2 × R1 . In that sense, our first main result, Theorem 2.1, establishes that any complete maximal surface Σ immersed into a Lorentzian product space M 2 × R1 , where M is a (necessarily complete) Riemannian surface with non-negative Gaussian curvature, is totally geodesic. Moreover, if M is non-flat then Σ must be a slice M ×{t0}, t0 ∈ R. Here by complete it is meant, as usual, that the induced Riemannian metric on Σ from the ambient Lorentzian metric is complete. We also present a non-parametric version of this result. In concrete, Theorem 2.2 states that any entire maximal graph in M 2 × R1 must be totally geodesic and, as a consequence, the only entire solutions to the maximal surface equation on any complete non-flat Riemannian surface M with non-negative Gaussian curvature are the constant functions. Observe that in these results the assumption on the Gaussian curvature is necessary as shown by the fact that when M = H2 is the hyperbolic plane there exist examples of complete maximal surfaces in H2 × R1 which are non-totally geodesic, as well as examples of non-trivial entire maximal graphs over H2 (see examples in Section 3). These examples are obtained in two different ways. Firstly, we can obtain some examples in an implicit way via a simple but nice

Global behaviour of maximal surfaces in Lorentzian product spaces

25

duality result between solutions to the minimal surface equation in a Riemannian product M 2 × R and solutions to the maximal surface equation in a Lorentzian product M 2 × R1 , Theorem 3.1. On the other hand, we can also obtain new examples of non-trivial complete and non-complete entire maximal graphs over H2 by solving directly the maximal surface equation for particular type of functions. In the last two sections, we introduce local approaches to these CalabiBernstein type results. In concrete in Section 4 we make an incursion on the study of parabolicity properties of Riemannian surfaces with non-empty boundary, giving in Theorem 4.1 an interesting parabolicity criterium for these surfaces in M 2 × R1 , being M 2 a complete Riemannian surface with non-negative Gaussian curvature. Finally, in Section 5 we establish a local integral inequality for the squared norm of the second fundamental form of a maximal surface in M 2 × R1 , Theorem 5.1. As a consequence of these local results, when we consider complete entire maximal graphs we provide alternative proofs of Theorem 2.2. We do not provide here with detailed proofs of our results. For further details, we refer the reader to the original papers.1–3 The author would like to heartily thank Luis J. Al´ıas for some useful comments during the preparation of this work. 2. A Calabi-Bernstein result for maximal surfaces Firstly, it is worth pointing out the following remarkable topological property of M 2 × R1 . If M 2 × R1 admits a complete spacelike surface f : Σ2 →M 2 × R1 , then M has to be necessarily complete and the projection Π = πM ◦ f : Σ→M is a covering map. This follows from the fact that Π is a local diffeomorphism which increases the distance between the Riemannian surfaces Σ and M , that is Π∗ (h, iM ) ≥ h, i. Then the assertion follows recalling that if a map, from a connected complete Riemannian manifold M1 into another connected Riemannian manifold M2 of the same dimension, increases the distance, then it is a covering map and M2 is complete. In particular, if M 2 × R1 admits a compact spacelike surface, then M is necessarily compact (see Ref. 6, Proposition 3.2 (i)). Under completeness assumption, we establish a parametric version of a Calabi-Bernstein result for maximal surfaces in M 2 × R1 . Theorem 2.1 (Ref. 2, Theorem 3.3). Let M 2 be a (necessarily complete) Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Then any complete maximal surface Σ2 in M 2 × R1 is totally geodesic.

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In addition, if KM > 0 at some point on M , then Σ is a slice M × {t0 }, t0 ∈ R. As a direct consequence of Theorem 2.1 we have the following. Corollary 2.1 (Ref. 2, Corollary 3.4). Let M 2 be a complete non-flat Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Then the only complete maximal surfaces in M 2 × R1 are the slices M × {t0 }, t0 ∈ R. Take into account that if M 2 = R2 is the flat Euclidean plane, then M × R1 = R31 is nothing but the 3-dimensional Lorentz-Minkowski space, and any spacelike affine plane in R31 which is not horizontal determines a complete totally geodesic surface which is not a slice. On the other hand, the assumption KM ≥ 0 is necessary as shown by the fact that there exist examples of non-totally geodesic complete maximal surfaces in H2 × R1 , where H2 is the hyperbolic plane (see examples in Section 3). Although we are not going to give here a proof of Theorem 2.1, we want to remark that the main idea of the proof is to show that Σ is a nonhyperbolic surface in the sense that any non-positive subharmonic function on the surface must be constant. Then, considering a suitable function we derive the conclusions of the theorem, (for the details, see the proof of Theorem 3.3 in Ref. 2). Let Ω ⊆ M 2 be a connected domain. Every smooth function u ∈ C ∞ (Ω) determines a graph over Ω given by Σ(u) = {(x, u(x)) : x ∈ Ω} ⊂ M 2 × R1 . The metric induced on Ω from the Lorentzian metric on the ambient space via Σ(u) is given by 2

h, i = h, iM − du2 .

(1)

Therefore, Σ(u) is a spacelike surface in M 2 × R1 if and only if |Du|2 < 1 everywhere on Ω, where Du stands for the gradient of u in Ω and |Du| denotes its norm, both with respect to the original metric h, iM on Ω. If Σ(u) is a spacelike graph over a domain Ω, then it can be seen that the mean curvature H(u) of Σ(u) is given by ! Du 2H(u) = Div p , 1 − |Du|2

where Div stands for the divergence operator on Ω with respect to the metric h, iM . In particular, Σ(u) is a maximal graph if and only if the function u satisfies the following partial differential equation on the domain

Global behaviour of maximal surfaces in Lorentzian product spaces

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Ω, Maximal[u] = Div

Du p 1 − |Du|2

!

= 0,

|Du|2 < 1.

(2)

A graph is said to be entire if Ω = M . It is known that when M is a complete Riemannian surface which is simply connected, then every complete spacelike surface in M 2 × R1 is an entire graph. In fact, since M is simply connected then the projection Π is a diffeomorphism between Σ and M , and hence Σ can be written as the graph over M of the function u = h ◦ Π−1 ∈ C ∞ (M ). Here h ∈ C ∞ (Σ) is defined as the height function of the surface Σ, that is, h = πM ◦ f : Σ2 →M 2 × R1 . However, in contrast to the case of graphs into a Riemannian product space, an entire spacelike graph in M 2 × R1 is not necessarily complete. To see it, in Section 3 we construct examples of non-complete entire maximal graphs in H2 × R1 . For that reason, the Calabi-Bernstein result given at Theorem 2.1 does not imply in principle that, under the same hypothesis on M , any entire maximal graph in M 2 × R1 must be totally geodesic. However, we can prove the following non-parametric version of our Calabi-Bernstein theorem. Theorem 2.2 (Ref. 2, Theorem 4.3). Let M 2 be a complete Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Then any entire maximal graph Σ(u) in M 2 × R1 is totally geodesic. In addition, if KM > 0 at some point on M , then u is constant. And as a direct consequence we get, Corollary 2.2 (Ref. 2, Corollary 4.4). Let M 2 be a complete non-flat Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Then the only entire solutions to the maximal surface equation (2) are the constant functions. The proof of Theorem 2.2 is obtained as a consequence of the following result, (we refer the reader to Ref. 2, Section 4 for a detailed proof). Theorem 2.3 (Ref. 2, Theorem 4.1). Let M 2 be a (non necessarily complete) Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Then any maximal surface Σ2 in M 2 × R1 which is complete with respect to the metric induced from the Riemannian product M 2 × R is totally geodesic. In addition, if KM > 0 at some point on Σ, then M is necessarily complete and Σ is a slice M × {t0 }.

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3. Non-trivial examples of entire maximal graphs in H2 × R1 In this section we give examples of complete and non-complete entire maximal graphs, different to slices, in H2 × R1 . These graphs provide counterexamples which show that our Calabi-Bernstein results are no longer true without the assumption KM ≥ 0. On the other hand, is well known that every complete spacelike surface with constant mean curvature which is a closed subset in the Lorentz-Minkowski space R31 is necessarily complete. Therefore, the existence of non-complete entire maximal graphs in H2 × R1 points out a curious difference between the topology of entire maximal graphs in R31 and in H2 × R1 . Calabi7 first gave a simple but nice duality result between solutions to the minimal surface equation in the Euclidean space R3 and solutions to the maximal surface equation in the Lorentz-Minkowski space R31 . By regarding R3 as the Riemannian product space R2 × R and R31 as the Lorentzian product space R2 × R1 , we observe that the same duality holds in general between solutions to the minimal surface equation in a Riemannian product space M × R and solutions to the maximal surface equation in a Lorentzian product space M × R1 . First of all, recall that a smooth function u on a connected domain Ω ⊆ M 2 defines a minimal graph Σ(u) in M × R if and only if u satisfies the following partial differential equation on Ω, ! Du Minimal[u] = Div p = 0, (3) 1 + |Du|2 where, as in (2), Div and Du stand for the divergence operator and the gradient of u in Ω with respect to the metric h, iM , respectively. Following the ideas in Ref. 4 we can prove the following general result.

Theorem 3.1 (Ref. 2, Theorem 5.1). Let Ω ⊆ M 2 be a simply connected domain of an oriented Riemannian surface M 2 . There exists a nontrivial solution u to the minimal surface equation on Ω, Minimal[u] = 0, if and only if there exists a non-trivial solution w to the maximal surface equation on Ω, Maximal[w] = 0,

|Dw|2 < 1.

Here non-trivial solutions correspond to non-totally geodesic graphs. The interest of Theorem 3.1 relays on the fact that it allows us to construct new solutions to the maximal surface equation from known solutions

Global behaviour of maximal surfaces in Lorentzian product spaces

29

to the minimal surface equation, and vice versa. Let us consider the halfplane model of the hyperbolic plane H2 ; that is, H2 = {x = (x1 , x2 ) ∈ R2 : x2 > 0} endowed with the complete metric h, iH2 =

1 (dx21 + dx22 ), x22

conformal to the flat Euclidean metric. This allows us to express the hyperbolic differential operators in terms of the Euclidean ones, so the minimal surface equation of a graph over the hyperbolic plane yields Minimal[u]= p

x22 ∆o u 1+x22 |Do u|2o

−

x22 3 (1+x22 |Do u|2o ) 2


x2 ux2 |Do u|2o+x22 Q(u) , (4)

where Do and ∆o stand for the Euclidean gradient and Laplacian, respectively, | · |o denotes the Euclidean norm and Q(u) = u2x1 ux1 x1 + 2ux1 ux2 ux1 x2 + u2x2 ux2 x2 .

Example 3.1 (Ref. 2, Example 5.2). From (4) is a straightforward computation to show that the function u(x1 , x2 ) = log(x21 + x22 )

(5)

defines a non-trivial entire minimal graph in H × R. Therefore, from our Theorem 3.1 and the entire minimal graph defined by the function (5) we know that there exists a smooth function w ∈ C ∞ (H2 ) which determines a non-trivial entire maximal graph in H2 ×R1 . This shows that the assumption KM ≥ 0 in Theorem 2.2 and Corollary 2.2 is necessary. Moreover, the entire maximal graph determined by w is also complete. Actually, if we denote by h, i the induced metric on H2 via the graph, which is given by 2

h, i = h, iH2 − dw2 ,

then it holds that 1 h, i 2 . 5 H As a consequence, the metric h, i is complete on H2 . This shows that the assumption KM ≥ 0 in Theorem 2.1 and Corollary 2.1 is also necessary. h, i ≥

Example 3.2 (Ref. 2, Example 5.3). It is also an immediate fact that the function x1 u(x1 , x2 ) = 2 (6) x1 + x22

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satisfies (4). Therefore, it defines another non-trivial entire minimal graph in H2 × R which gives rise via Theorem 3.1 to another non-trivial entire maximal graph in the Lorentzian product H2 × R1 . In contrast to Example 3.1, this example is non-complete. To see it, let w ∈ C ∞ (H2 ) stands for the smooth function defining this entire maximal graph, which we denote by Σ(w). Then, the curve α : (0, 1) → Σ(w) given by α(s) = (0, s, w(0, s)) defines a divergent curve in Σ(w) with finite length. As a consequence, Σ(w) is non-complete. The above examples are given in an implicit way, since we do not obtain any explicit expression for the functions which determine both graphs. However, in Ref. 1 we provide new examples of complete and non-complete entire maximal graphs in H2 × R1 by solving directly the maximal surface equation in H2 for particular types of functions, 4. A parabolicity criterium for maximal surfaces. A Riemannian surface (Σ, g) with non-empty boundary, ∂Σ 6= ∅, is said to be parabolic if every bounded harmonic function on Σ is determined by its boundary values, which is equivalent to the existence of a proper nonnegative superharmonic function on Σ. Fern´andez and L´opez9 have recently proved that properly immersed maximal surfaces with non-empty boundary in the Lorentz-Minkowski spacetime R31 are parabolic if the Lorentzian norm on the maximal surface in R31 is eventually positive and proper. Motivated by their work, we study similar parabolicity criteria for maximal surfaces immersed into M 2 × R1 . Given any Riemannian surface M 2 , consider the function rˆ : M →R defined by rˆ(x) = distM (x, x0 ) where x0 ∈ M is a fixed point. Then, a natural generalization of the Lorentzian norm on a surface in R31 to immersed surfaces into M 2 × R1 is the function φ = r2 − h2 ∈ C ∞ (Σ) where the function r : Σ→R is given by r = rˆ ◦ πM ◦ f . In that context, we prove the following result (see Ref. 3 for details of the proofs). Theorem 4.1 (Ref. 3, Theorem 2). Let M 2 be a complete Riemannian surface with non-negative Gaussian curvature, KM ≥ 0. Consider Σ a maximal surface in M 2 × R1 with non-empty boundary, ∂Σ 6= ∅, and assume that the function φ : Σ→R defined by φ(p) = r2 (p) − h2 (p)

Global behaviour of maximal surfaces in Lorentzian product spaces

31

is eventually positive and proper. Then Σ is parabolic. As is usual, by eventually we mean here a property that is satisfied outside a compact set. It is interesting to look for some natural conditions under which the assumptions of Theorem 4.1 are satisfied. In that sense we can state the following proposition. Proposition 4.1 (Ref. 3, Corollary 4). Let M 2 be a complete Riemannian surface with non-negative Gaussian curvature. Then every proper maximal immersion f : Σ2 →M 2 × R1 with non-empty boundary which eventually lies in Wa = {(x, t) ∈ M 2 × R1 : |t| ≤ aˆ r(x)}, for some 0 < a < 1, is parabolic. It is worth pointing out that a Riemannian surface (Σ, g) without boundary, ∂Σ = ∅, is non-hyperbolic if and only if for every non-empty open set O ⊂ Σ with smooth boundary Σ \ O is parabolic (as a surface with boundary). This follows from the observation that a Riemannian manifold without boundary is non-hyperbolic precisely when almost all Brownian paths are dense in the manifold. Therefore, as a direct consequence of Theorem 4.1 we derive Corollary 4.1 (Ref. 3, Corollary 5). Let M 2 be a complete Riemannian surface with non-negative Gaussian curvature, KM ≥ 0, and let Σ be a maximal surface in M 2 × R1 without boundary, ∂Σ = ∅. If the function φ(p) = r2 (p) − h2 (p) is eventually positive and proper on Σ, then Σ is non-hyperbolic. Another interesting situation is when we consider entire maximal graphs Σ(u). Observe that we can always assume, up to a vertical translation which is an isometry in M 2 × R1 , that for a fixed point x0 ∈ M , u(x0 ) = 0. In that case, we can also conclude that φ : M →R is a positive function for every x 6= x0 and proper. Therefore, Corollary 4.2 (Ref. 3). Every entire maximal graph defined over a complete Riemannian surface M 2 with non-negative Gaussian curvature is nonhyperbolic. As a consequence of Corollary 4.2 we can derive an alternative proof of Theorem 2.2, (see Ref. 3 for the details).

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5. A local estimate for maximal surfaces Al´ıas and Palmer5 introduced a new approach to the Calabi-Bernstein theorem in the Lorentz-Minkowski space R31 , based on a local integral inequality for the Gaussian curvature of a maximal surface in R31 which involved the local geometry of the surface and the image of its Gauss map. In this section, we generalize that local approach to the case of maximal surfaces in a Lorentzian product space M 2 × R1 . Theorem 5.1 (Ref. 2, Theorem 6.1). Let M 2 be an analytic Riemannian surface with non-negative Gaussian curvature, KM ≥ 0, and let f : Σ2 →M 2 × R1 be a maximal surface in M 2 × R1 . Let p be a point of Σ and R > 0 be a positive real number such that the geodesic disc of radius R about p satisfies D(p, R) ⊂⊂ Σ. Then for all 0 < r < R it holds that Z L(r) kAk2 dΣ ≤ cr 0≤ , (7) r log (R/r) D(p,r) where L(r) denotes the length of the geodesic circle of radius r about p, and cr =

π 2 (1 + α2r )2 > 0. 4αr arctan αr

Here αr = sup cosh θ ≥ 1, D(p,r)

where θ denotes the hyperbolic angle between N and ∂t along Σ. In particular, when Σ is complete then it is not difficult to prove that the local integral inequality (7) implies Theorem 2.1, (see Ref. 2, Section 6). Acknowledgments A.L. Albujer was partially supported by FPU Grant AP2004-4087 from Secretar´ıa de Estado de Universidades e Investigaci´on, MEC Spain and MEC/FEDER Grant MTM2004-04934-C04-02. References 1. A.L. Albujer, New examples of entire maximal graphs in H2 × R1 , to appear in Differential Geom. Appl. 2. A.L. Albujer and L.J. Al´ıas, Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces, preprint 2006; available at http://arxiv.org/abs/0709.4363.

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3. A.L. Albujer and L.J. Al´ıas, Parabolicity of maximal surfaces in Lorentzian product spaces, preprint 2007. 4. L.J. Al´ıas and B. Palmer, A duality result between the minimal surface equation and the maximal surface equation, An. Acad. Bras. Ciˆenc. 73 (2001) 161–164. 5. L.J. Al´ıas and B. Palmer, On the Gaussian curvature of maximal surfaces and the Calabi-Bernstein theorem, Bull. London Math. Soc. 33 (2001) 454–458. 6. L.J. Al´ıas, A. Romero and M. S´ anchez, Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 27 (1995) 71–84. 7. E. Calabi, Examples of Bernstein problems for some nonlinear equations. 1970 In: Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968, Amer. Math. Soc., Providence, R.I.) 223–230. 8. S.Y. Cheng and S.T. Yau, Maximal space-like hypersurfaces in the LorentzMinkowski spaces, Ann. of Math. (2) 104 (1976) 407–419. 9. I. Fern´ andez and F. L´ opez, Relative parabolicity of zero mean curvature in R3 and R31 , preprint. 10. A. Grigor’yan, Analytic and geometric background of recurrence and nonexplosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999) 135–249. 11. J.E. Marsden and F.J. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Phys. Rep. 66 (1980) 109–139. 12. W. Meeks and J.P´erez, Conformal properties on classical minimal surface theory, In: Surveys in Differential Geometry (Volume IX: Eigenvalues of Laplacians and other geometric operators, (A. Grigor’yan and S.T. Yau, Eds.) International Press). 13. J. P´erez, Parabolicity and minimal surfaces, Proceedings of the 2002 Summer School“The global theory of minimal surfaces” (Clay Mathematical Institute & University of California at Berkeley, MSRI), to appear.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 35–44

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Invariant Einstein metrics on certain Stiefel manifolds A. Arvanitoyeorgos Department of Mathematics, University of Patras, Rion, GR-26500, Greece E-mail: [email protected] www.math.upatras.gr/e arvanito V.V. Dzhepko and Yu.G. Nikonorov Rubtsovsk Industrial Institute, ul. Traktornaya, 2/6, Rubtsovsk, 658207, Russia E-mail: J Valera [email protected], [email protected] A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on the Stiefel manifolds SO(2k + l)/SO(l). Keywords: Riemannian manifolds, Homogeneous spaces, Einstein metrics, Stiefel manifolds. MS classification: 53C25, 53C30.

1. Introduction A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c·ρ for some real constant c. A detailed exposition on Einstein manifolds can be found in the book of A. Besse,1 and more recent results on homogeneous Einstein manifolds can be found in the survey of M. Wang.2 General existence results are hard to obtain. Among the first important attempts are the works of G. Jensen3 and M. Wang, W. Ziller.4 Recently, a new existence approach was introduced by C. B¨ohm, M. Wang, and W. Ziller.5,6 In the present work we prove existence of invariant Einstein metrics on certain Stiefel manifolds SO(n)/SO(n − k). The simplest case S n−1 = SO(n)/SO(n − 1) is an irreducible symmetric space, therefore it admits up to scale a unique invariant Einstein metric. It was S. Kobayashi7

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A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov

who proved first the existence of an invariant Einstein metric on the unit sphere bundle T1 S n = SO(n)/SO(n − 2). In Ref. 8 A. Sagle proved that the Stiefel manifolds SO(n)/SO(n − k) admit at least one homogeneous invariant Einstein metric. For k ≥ 3 G. Jensen9 found a second metric. Einstein metrics on SO(n)/SO(n − 2) are completely classified. If n = 3 the group SO(3) has a unique Einstein metric. If n ≥ 5 it was shown by A. Back and W.Y. Hsiang10 that SO(n)/SO(n − 2) admits exactly one homogeneous invariant Einstein metric. The same result was obtained by M. Kerr.11 The Stiefel manifold SO(4)/SO(2) admits exactly two invariant Einstein metrics which follows from the classification of 5-dimensional homogeneous Einstein manifolds due to D.V. Alekseevsky, I. Dotti, and C. Ferraris.12 We also refer to Ref. 5, p.727-728 for further discussion. For k ≥ 3 there is no obstruction for existence of more than two homogeneous invariant Einstein metrics on Stiefel manifolds SO(n)/SO(n − k). In this paper we investigate G-invariant metrics on G/H with additional symmetries. Let G be a compact Lie group and H a closed subgroup so that G acts almost effectively on G/H. Let K be a closed subgroup of G with H ⊂ K ⊂ G, and suppose that K = L′ × H ′ , where {eL′ } × H ′ = H. It is clear that K ⊂ NG (H), the normalizer of H in G. If we denote L = e = G×L acts on G/H by (a, b)·gH = agb−1 H, L′ ×{eH ′ }, then the group G e = {(a, b) : ab−1 ∈ H}. and the isotropy subgroup at eH is H e e e H e is a It will be shown that the set MG of G-invariant metrics on G/ G subset of M , the set of G-invariant metrics on G/H. Therefore, it would e be simpler to search for invariant Einstein metrics on MG . In this way we obtain existence results for Einstein metrics for certain quotients. We apply the above method for the homogeneous space SO(k1 + k2 + k3 )/SO(k3 ), and we show the following: Theorem 1.1. If l > k ≥ 3 then the Stiefel manifold SO(2k + l)/SO(l) admits at least four SO(2k+l)×SO(k)×SO(k)-invariant Einstein metrics, two of which are Jensen’s metrics. Our method can be extended to other homogeneous spaces of classical Lie groups (cf. Ref. 13). 2. The main construction Let G be a compact Lie group and H a closed subgroup so that G acts almost effectively on G/H. Let g, h be the Lie algebras of G and H, and let g = h ⊕ p be a reductive decomposition of g with respect to some Ad(G)-invariant inner product of g. The orthogonal complement p

Invariant Einstein metrics on certain Stiefel manifolds

37

can be identified with the tangent space TeH G/H. Any G-invariant metric ρ of G/H corresponds to an Ad(H)-invariant inner product (·, ·) on p and vice-versa. For G semisimple, the negative of the Killing form B of g is an Ad(G)-invariant inner product on g, therefore we can choose the above decomposition with respect to this form. We will use such a decomposition later on. Moreover, the restriction h·, ·i = −B|p is an Ad(G)-invariant inner product on p, which generates a G-invariant metric on G/H called standard. The normalizer NG (H) of H in G acts on G/H by (a, gH) 7→ ga−1 H. For a fixed a this action induces a G-equivariant diffeomorphism ϕa : G/H → G/H. Note that if a ∈ H this diffeomorphism is trivial, so the action of the gauge group NG (H)/H is well defined. However, it is simpler from technical point of view to use the action of NG (H). Let ρ be a Ginvariant metric of G/H with corresponding inner product (·, ·). Then the diffeomorphism ϕa is an isometry of (G/H, ρ) if and only if the operator Ad(a)|p is orthogonal with respect to (·, ·). Let K be a closed subgroup of G with H ⊂ K ⊂ G such that K = L′ ×H ′ , where {eL′ }×H ′ = H, and consider L = L′ ×{eH ′ }. It is clear that e = G × L acts on G/H by (a, b) · gH = agb−1 H, K ⊂ NG (H). The group G and the isotropy at eH is given as follows: e is isomorphic to K. Lemma 2.1. The isotropy subgroup H

e = {(a, b) ∈ G × L : ab−1 ∈ H}. Let i : K ֒→ G be Proof. It is clear that H the inclusion of K in G. Then i({eL′ } × H ′ ) = H and i(L′ × {eH ′ }) = L. Let (a, b) ∈ G × L be such that ab−1 = h ∈ H. Then a = hb, so (a, b)=(hb, b)=(i(b′, eH ′ )i(eL′ , h′ ), i(b′, eH ′ ))=((b′, h′ ), b′ )∈K×L′=L′ ×H ′ ×L′. e is identified with a subgroup of L′ × H ′ × L′ , and it is then obvious Thus H e is isomorphic to L′ × H ′ = K. that H The set MG of G-invariant metrics on G/H is finite dimensional. We consider the subset MG,K of MG corresponding to Ad(K)-invariant inner products on p (and not only Ad(H)-invariant). Let ρ ∈ MG,K and a ∈ K. The above diffeomorphism ϕa is an isometry e on (G/H, ρ) is isometric, so any metric form of (G/H, ρ). The action G e G,K M can be identified a metric in MG and vice-versa. Therefore, we may e think of MG as MG,K , which is a subset of MG . Since metrics in MG,K correspond to Ad(K)-invariant inner products on p, we call these metrics Ad(K)-invariant metrics on G/H.

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A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov

We will apply the above construction for G/H = SO(k1 + k2 + k3 )/SO(k3 ), and prove existence of Einstein metrics in the set MG,K , where K = L′ × H ′ = (SO(k1 ) × SO(k2 )) × SO(k3 ). 3. Ad(K)-invariant metrics on SO(k1 + k2 + k3 )/SO(k3 ) Let n = k1 + k2 + k3 , and pi be the subalgebra so(ki ) in g = so(n), i = 1, 2. Note that the submodules pi of p are Ad(K)-invariant and Ad(K)irreducible submodules. For 1 ≤ i < j ≤ 3 we denote by p(i,j) the Ad(K)invariant and Ad(K)-irreducible submodule of p which is determined by the equality so(ki + kj ) = so(ki ) ⊕ so(kj ) ⊕ p(i,j) , where p(i,j) is orthogonal to so(ki ) ⊕ so(kj ) with respect to the Killing form B. Denote by di and d(i,j) the dimensions of the modules pi and p(i,j) respectively. It is easy to obtain that di = ki (k2i −1) , d(i,j) = ki kj . We have a decomposition of p into a sum of Ad(K)-invariant and Ad(K)irreducible submodules: p = p1 ⊕ p2 ⊕ p(1,2) ⊕ p(1,3) ⊕ p(2,3) .

(1)

Lemma 3.1 (Ref. 13). Assume that k1 , k2 , k3 ≥ 2, and at most one of k1 , k2 is equal to 2. Then there are no pairwise Ad(K)-isomorphic submodules among p1 , p2 and p(i,j) (1 ≤ i < j ≤ 3). If the assumptions of Lemma 3.1 are satisfied, then we have a complete description of all Ad(K)-invariant metrics on G/H. Let ρ be any Ad(K)-invariant metric on G/H with corresponding Ad(K)-invariant inner product (·, ·) on p. Lemma 3.2. If there are no pairwise Ad(K)-isomorphic submodules among pi and p(i,j) , then (·, ·) = x1 · h·, ·i|p1 + x2 · h·, ·i|p2 + x(1,2) · h·, ·i|p(1,2) + x(1,3) · h·, ·i|p(1,3) + x(2,3) · h·, ·i|p(2,3)

(2)

for positive constants x1 , x2 > 0 and x(i,j) > 0, where h·, ·i = −B|p . Therefore, the set of Ad(K)-invariant metrics on SO(k1 + k2 + k3 )/SO(k3 ) depends on 5 parameters. 4. The scalar curvature and the Einstein condition Let {ejα } be an orthonormal basis of pα with respect to h·, ·i, where 1 ≤ j ≤ dα (here α means any of the symbols of type i or (k, i)). We define

Invariant Einstein metrics on certain Stiefel manifolds

39

the numbers (cf. Ref. 4) [αβγ] by the equation i X h 2 [αβγ] = h eiα , ejβ , ekγ i , i,j,k

where i, j, k vary from 1 to dα , dβ , dγ respectively. The symbols [αβγ] are symmetric with respect to all three indices, as follows from the Ad(G)invariance of h·, ·i. According to Ref. 4, the scalar curvature S of (·, ·) is given by S((·, ·)) =

1 X xγ 1 X dα − , [αβγ] 2 α xα 4 xα xβ α,β,γ

where α, β and γ are arbitrary symbols of the type i (i = 1, 2) or of the type (i, j) (1 ≤ i < j ≤ 3). By performing explicit computations of [αβγ] (cf. Ref. 13), the scalar curvature for the metric (2) takes the following form. Proposition 4.1. The scalar curvature S of an Ad(K)-invariant metric (2) is given by k1 (k1 − 1)(k1 − 2) 1 k2 (k2 − 1)(k2 − 2) 1 + · · 8(n − 2) x1 8(n − 2) x2   1 x1 k1 k3 k2 k3 1 k1 k2 − k1 k2 (k1 − 1) 2 + + + 2 x(1,2) x(1,3) x(2,3) 8(n − 2) x(1,2) ! x1 x2 x2 +k1 k3 (k1 − 1) 2 + k2 k3 (k2 − 1) 2 + k1 k2 (k2 − 1) 2 x(1,3) x(2,3) x(1,2)   x(1,2) x(1,3) x(2,3) 1 . k1 k2 k3 + + − 4(n − 2) x(1,3) x(2,3) x(1,2) x(2,3) x(1,2) x(1,3)

S=

(3)

Denote by MG 1 the set of all G-invariant metrics with a fixed volume element on the space G/H. The following variational principle for invariant Einstein metrics is well known. Proposition 4.2 (Ref. 1). Let G/H be a homogeneous space, where G and H are compact. Then the G-invariant Einstein metrics on the homogeneous space G/H are precisely the critical points of the scalar curvature functional S restricted to MG 1. For the general construction as described in Section 2, the above variational principle implies the following:

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A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov

Proposition 4.3. Let MG,K be the subset of MG,K with fixed volume 1 G,K is Einstein if and only if it is a critical element. Then a metric in M1 point of the scalar curvature functional S restricted to MG,K . 1 e Proof. The set MG,K is precisely the set of G-invariant metrics with fixed 1 e e volume element on G/H. The volume condition for the metric (2) takes the form d

d

d

(1,2) (1,3) (2,3) xd11 xd22 x(1,2) x(1,3) x(2,3) = constant.

(4)

By using Proposition 4.3 the problem of searching for Ad(K)-invariant Einstein metrics on G/H reduces to a Lagrange-type problem for the scalar curvature functional S under the restriction (4). 5. Invariant Einstein metrics on Stiefel manifolds In this section we will investigate SO(k1 + k2 + k3 ) × SO(k1 ) × SO(k2 )invariant Einstein metrics on the space SO(k1 +k2 +k3 )/SO(k3 ). Here L′ = SO(k1 ) × SO(k2 ). By Lemma 3.2 these metrics depend on 5 parameters. We apply Proposition 4.1, and by the Lagrange method we obtain the following system of equations    x2 x223 (k1 − 2)x212 x213 + k2 x21 x213 + k3 x21 x212 = x1 x213 (k2 − 2)x212 x223  + k3 x22 x212 + k1 x22 x223 ,    x13 (k2 − 2)x212 x223 + k3 x22 x212 + k1 x22 x223 = x2 x23 2(k1 + k2 + k3

− 2)x12 x13 x23 − (k1 − 1)x1 x13 x23 − (k2 − 1)x2 x13 x23 + k3 x312  − k3 x12 x213 − k3 x12 x223 ,  x13 2(k1 + k2 + k3 − 2)x12 x13 x23 − (k1 − 1)x1 x13 x23 − (k2 − 1) (5)   · x2 x13 x23 + k3 x312 − k3 x12 x213 − k3 x12 x223 = x12 2(k1 + k2 + k3  − 2)x12 x13 x23 − (k1 − 1)x1 x12 x23 + k2 x313 − k2 x212 x13 − k2 x13 x223 ,  x23 2(k1 + k2 + k3 − 2)x12 x13 x23 − (k1 − 1)x1 x12 x23 + k2 x313   − k2 x212 x13 − k2 x13 x223 = x13 2(k1 + k2 + k3 − 2)x12 x13 x23  − (k2 − 1)x2 x12 x13 + k1 x323 − k1 x212 x23 − k1 x213 x23 .

Invariant Einstein metrics on certain Stiefel manifolds

41

If x13 = x23 = z then system (5) reduces to the following:   x12 z 2 (k1 − k2 )x12 + (k2 − 1)x2 − (k1 − 1)x1 = 0,   z 2 ((k1 − 2)x2 − (k2 − 2)x1 )x212 + (k2 x1 − k1 x2 )x1 x2 z 2  + k3 (x1 − x2 )x1 x2 x212 = 0,  z (k2 − 2)x212 + (k1 + k2 − 1)x22 − 2(k1 + k2 − 2)x2 x12   + (k1 − 1)x1 x2 z 2 + k3 (x2 − x12 )x2 x212 = 0,   z 2(k1 + k2 − 2)x12 − (k2 − 1)x2 − (k1 − 1)x1 z 2    + (k2 + k3 )x12 − 2(k3 + k2 + k1 − 2)z + (k1 − 1)x1 x212 = 0.

(6)

From the first equation of system (6) we obtain that x1 = and substituting to the two other equations we obtain

(k1 −k2 )x12 +(k2 −1)x2 , k1 −1

  z(x2 − x12 ) ((2k2 + k1 − 2)x2 − (k2 − 2)x12 )z 2 + k3 x2 x212 = 0,    z (k1 + 3k2 − 4)x12 − 2(k2 − 1)x2 z 2 + (k3 + k1 )x12   − 2(k1 + k2 + k3 − 2)z + (k2 − 1)x2 x212 = 0,  (k1 − k2 )z 2 (x12 − x2 ) (1 − k1 )(k2 − 2)x212 + (k1 (k1 − 2)  + k2 (k1 − k2 ) + 1)x12 x2 + (k1 (k2 − 1) + k2 (k2 − 2) + 1)x22 z 2    + k3 (k1 − k2 )x12 + (k2 − 1)x2 x2 x212 = 0.

(7)

If x2 = x12 then system (7) reduces to the equation 

 (k1 + 3k2 − 4)x2 − 2(k2 − 1)x2 z 2 − 2(n − 2)x22 z + (n − 1)x32 = 0,

which has two solutions, that were found by G. Jensen.9 So assume that x2 6= x12 . Then from the first equation of system (7) we get x2 =

42

A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov

(k2 −2)x12 z 2 . k3 x212 +(2k2 +k1 −2)z 2

Substituting to (7) we obtain the following system:

  z 4 x312 (k2 − 1)(k2 − 2)(k1 − k2 ) (k1 + k2 − 1)z 2 + k3 x212 2  · (k1 + k2 )z 2 + k3 x212 = 0,

(k32 + k1 k3 )x412 + (4k3 − 2k2 k3 − 2k1 k3 − 2k32 )zx312 + (k12 − 6k3 + 5k2 k3 + 2k1 k3 − 2k1 +

k22

− 6k1 k2 − 4k2 k3 + 12k2 −

+ 2k1 k2 − 3k2 +

2k12

+

4k3 − 4k22 4

2)z 2 x212

(8)

+ (8k1

− 2k1 k3 − 8)z 3 x12

+ (5k1 k2 − 8k2 + k12 + 4k22 − 6k1 + 4)z = 0.

From the first equation of (8) we see that a solution exists only when k1 = k2 . Let k1 = k2 = k (k ≥ 3), k3 = l, x1 = x2 = x, x12 = 1, x13 = x23 = z. Then the original system (5) reduces to the following:   z(x − 1) ((3k − 2)x − k + 2)z 2 + lx = 0,   (9) z (2kx − 2x − 4k + 4)z 2 + (4k + 2l − 4)z − (k − 1)x − k − l = 0.

If x = 1 we obtain Jensen’s solutions, so assume that x 6= 1. Then from the first equation of (9) we find that x=

(k − 2)z 2 , (3k − 2)z 2 + l

which implies in particular, that 0 < x < 1 for any positive z. Substituting this expression to the second equation of (9), the Einstein equation reduces to F (z) = 0, where F (z) = 2(5k 2 − 7k + 2)z 4 − 2(6k 2 + 3kl − 10k − 2l + 4)z 3

+ (4k 2 + 7kl − 5k − 6l + 2)z 2 − 22k + l − 2)z + l(k + l).

(10)

If the equation F (z) = 0 has a positive solution, then we obtain a new SO(2k +l)×SO(k)×SO(k)-invariant Einstein metric on SO(2k +l)/SO(l). The numbers of new Einstein metrics for some values of k, l are shown in Table 1. However, we can show that there exists an infinite series of new homogeneous Einstein manifolds, as the next proposition shows. Proposition 5.1. If l > k ≥ 3 then the Stiefel manifold SO(2k + l)/SO(l) admits at least four SO(2k+l)×SO(k)×SO(k)-invariant Einstein metrics.

Invariant Einstein metrics on certain Stiefel manifolds

43

Table 1. The number of positive solutions of the equation FSO (z) = 0 for various (k, l). (New SO(2k + l) × SO(k) × SO(k)-invariant Einstein metrics on SO(2k + l)/SO(l)). k l

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 2

0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2

Proof. We consider the polynomial (10). Then F (0) = l(k+l) > 0, F (z) → ∞ as z → ∞, and F (1) = 2k 2 −2kl+k−2−l2 +2l < 0 for l > k, so F (z) = 0 has two positive solutions. From the above discussion these solutions are Einstein metrics, which are different from Jensen’s Einstein metrics. Thus the result follows.

Acknowledgments The first author was partially supported by the C. Carath´eodory grant # C.161 (2007-10), University of Patras. The second and the third author are supported in part by RFBR (grant N 05-01-00611-a) and by the Council on grants of the President of Russian Federation for supporting of leading scientific schools of Russian Federation (grants NSH-8526.2006.1) The third author is supported by the Council on grants of the President of Russian Federation for supporting of young Russian scientists (grant MD-5179.2006.1)

44

A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov

References 1. A. Besse, Einstein Manifolds (Springer, Berlin Heidelberg New York, 1987). 2. M. Wang, Einstein metrics from symmetry and bundle constructions, In: Surveys in differential geometry: essays on Einstein manifolds (Surv. Differ. Geom., VI, Int. Press, Boston, MA 1999) 287–325. 3. G. Jensen, The scalar curvature of left invariant Riemannian metrics, Indiana J. Math. 20 (1971) 1125–1144. 4. M. Wang and W. Ziller, Existence and Non-existence of Homogeneous Einstein Metrics, Invent. Math. 84 (1986) 177–194. 5. C. B¨ ohm, M. Wang and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Func. Anal. 14 (2004) 681–733. 6. C. B¨ ohm, Homogeneous Einstein metrics and simplicial complexes, J. Diff. Geom. 67 (2004) 79–165. 7. S. Kobayashi, Topology of positive pinched K¨ ahler manifolds, Tˆ ohoku Math. J. 15 (1963) 121–139. 8. A. Sagle, Some homogeneous Einstein manifolds, Nagoya Math. J. 39 (1970) 81–106. 9. G. Jensen, Einstein metrics on principal fiber bundles, J. Diff. Geom. 8 (1973) 599–614. 10. A. Back and W.Y. Hsiang, Equivariant geometry and Kervaire spheres, Trans. Amer. Math. Soc. 304 (1987) 207–227. 11. M. Kerr, New examples of homogeneous Einstein metrics, Michigan J. Math. 45 (1998) 115–134. 12. D.V. Alekseevsky, I. Dotti and C. Ferraris, Homogeneous Ricci positive 5manifolds, Pacific J. Math. 175 (1996) 1–12. 13. A. Arvanitoyeorgos, V.V. Dzhepko and Yu.G. Nikonorov, Invariant Einstein metrics on some homogeneous spaces of classical Lie groups, to appear in Canadian J. Math.; arXiv:math.DG/0612504v2.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 45–55

45

Constant mean curvature submanifolds in (α, β)−Finsler spaces V. Balan Department of Mathematics I, Faculty of Applied Sciences, University Politehnica of Bucharest, Bucharest, RO-060042, Romania E-mail: [email protected] The explicit equations of CMC submanifolds immersed in certain (α, β) Finsler spaces - which extend the known results from3 and include the Randers, Kropina and Euclidean cases, are determined. In particular, the minimal surfaces of revolution and the graphs immersed in Kropina Finsler spaces, are characterized. Keywords: CMC hypersurface; Minimal surface; Mean curvature, Finsler structure, Randers metric, Kropina metric. MS classification: 53C60, 53B40, 53C42.

1. Introduction Recent advances in the theory of Finsler surfaces have been provided after 2002 by Z.Shen - who has intensively studied 2-dimensional Finsler metrics and their flag curvature. Recently Z. Shen,1 and further M. Souza and K. Tenenblat2,3 have investigated minimal surfaces immersed in Finsler spaces from differential geometric point of view. Still, earlier rigorous attempts using functional analysis exist in the works of G. Bellettini and M. Paolini after 1995. In 1998, based on the notion of Hausdorff measure, Z. Shen1 has introduced the notion of mean curvature on submanifolds of Finsler spaces as follows. ˜ , F˜ ) be a Finsler structure, and let Theorem 1.1 (Shen, 1998). Let (M ˜ ˜ ϕ : (M, F ) → (M , F ) be an isometric immersion, with F induced by F˜ . Then the mean curvature of M is given by  1  Hϕ (X) = G;xi − G;zi zj ϕj;ua ub − G;xj zai ϕj;ua X i , a b G where lower indices stand for corresponding partial derivatives and: • (ua , v b )a,b∈1,n are local coordinates in T M (dim M = n);

46

V. Balan

˜ (dim M ˜ = m); • (xi , y j )i,j∈1,m are local coordinates in T M i • za are the entries of the Jacobian matrix [J(ϕ)] = (∂ϕi /∂ua )a=1,n,i=1,m ; ˜ , t ∈ (−ε, ε), ϕ0 = ϕ, is the variation of the surface; • ϕt : M → M ∂ϕt • X is the vector field Xx = | (x) induced along ϕ attached to the ∂t t=0 variation; • G is the Finsler induced volume form Ge˜(z) =

vol[B n ] , vol{(v a ) ∈ Rn | F˜ (v a zai e˜i ) ≤ 1}

(1)

where z = (zai )a=1,n,i=1,m ∈ GLm×n (R), e˜ = {˜ ei }i=1,m is an arbitrary basis in Rm and B n ⊂ Rn is the standard Euclidean ball. It was proved that the variation of the volume of M reaches a minimum for Hϕ = 0.1 Recent advances in constructing minimal surfaces (n = 2) based on (1) were provided in Ref. 2,3, by characterizing the minimal sur˜ = R3 , F˜ ) with the Randers faces of revolution in Finsler (α, β)−spaces (M fundamental function q F˜ (x, y) = α(x, y) + β(x, y), α(x, y) = aij (x)i y j , β(x, y) = bi (x)y i , for the particular case when aij = δij (the Euclidean metric) and β = b·dx3 , with b ∈ [0, 1).

In the following, we determine the Busemann-Hausdorff mean curvature for hypersurfaces of (α, β) Finsler spaces which allow algebraic indicatrix closure of second order. In particular, we shall explicitly derive the equations of CMC and minimal graphs and surfaces of revolution for Finsler Kropina spaces. ˜ = m = n + 1. Let F˜ : We further consider the case when dim M ˜ T M → R be a positive 1-homogeneous locally Minkowski Finsler fundamental function.4,5 Continuing the research from Refs. 6,7 we shall study the Busemann-Hausdorff mean curvature for certain (α, β)-metrics, which include the subcases: ˜ , F˜ ) with the fundamental function a) the Randers space (M q F˜ (x, y) = δij y i y j + bs y s , bs ∈ R, s = 1, n, which is reducible to the case studied by M. Souza and K. Tenenblat,2,3 v un+1 uX ˜ (2) F (x, y) = t (y i )2 + by n+1 , b ∈ [0, 1). i=1

Constant mean curvature submanifolds in (α, β)−Finsler spaces

47

˜ , F ) with the fundamental function b) the Kropina space (M δij y i y j , F˜ (x, y) = bs y s

bs ∈ R, s = 1, n,

n+1 X

b2i > 0,

i=1

which is reducible to F˜ (x, y) =

n+1 X

(y i )2 /(by n+1 ),

i=1

b ∈ (0, 1).

(3)

2. The volume form of a Finsler hypersurface ˜ = Rn+1 be a simple hypersurface. Let H = Im ϕ, ϕ : D ⊂ Rn → M i ∂ϕ , u = (u1 , . . . , un ) ∈ D. We shall further determine We denote zαi = ∂uα the volume of the body Q ⊂ Tϕ(u) H bounded by the induced on Tϕ(u) H ˜ indicatrix from M ∂Q = Tϕ(u) H ∩ {y ∈ Tϕ(u) Rn+1 |F˜ (y) = 1}. α ∂ i α ∂ If v = v ∈ Tϕ(u) H and hence ∈ Tu D, then ϕ∗,u (v) = zα v α i ∂u ∂y ϕ(u) at some fixed point u ∈ D, Q is given by Q = {v ∈ Tu D | F˜ (ϕ(u), ϕ∗,u (v)) ≤ 1}.

In particular, Q is explicitly described as follows: a) In the Randers case (2) we have ( 1 2 (y ) + · · · + (y n+1 )2 ≤ (1 − by n+1 )2 ⇔ QR : y i = zαi v α , i = 1, n + 1 ⇔

n X (zαi v α )2 + (1 − b2 )(zαn+1 v α )2 + 2bzαn+1 v α − 1 ≤ 0. i=1

b) In the Kropina case (3) we obtain ( 1 2 n+1 X (y ) + . . . (y n+1 )2 − by n+1 ≤ 0 (zαi v α )2 − bzαn+1 v α ≤ 0. QK : ⇔ i i α y = zα v , i = 1, n + 1 i=1 We note that in both cases body Q has the generic form Q:

n X i=1

(zαi v α )2 + µ(zαn+1 v α )2 + 2νzαn+1 v α + ρ ≤ 0,

(4)

48

V. Balan

where µ, ν, ρ ∈ R and that one may specify Q to the Randers body QR (for µ = 1 − b2 , ν = b, ρ = −1) and to the Kropina body QK (for µ = 1, ν = −b/2, ρ = 0). ˜ , we need to determine first In order to find the volume form of H ⊂ M vol(B n ) . To this aim, we use Ge˜(z) = vol[Q] ˜ = (h ˜ ij ) ˜ Lemma 2.1. Consider the matrix H i,j∈1,n , hij = hij + li lj , with H = (hij )i,j∈1,n non-degenerate. Then: ˜ ij = hij − (1 + ls ls )−1 li lj , ˜ has the coefficients h a) The inverse of H where li = his ls . ˜ = det(H) · (1 + ls ls ). b) We have det(H) We note that (4) briefly rewrites ˜ + 2νbv + ρ ≤ 0 Q : v t Hv ˜ where v = (v 1 , . . . , vn )t , H

⇔

= C t C, C

(z1n+1 , . . . , znn+1 ) and ˆ = H



˜ νbt H νb ρ



=

  v ≤0 (5) 1  i  (zα )i,α=1,n = , b = √ µb

ˆ (v t , 1)H

n+1 ˜ ab ) (h )α a,b=1,n ν(zα

ν(zαn+1 )αt

ρ

!

.

(6)

After performing an orthogonal transformation, the bounding quadric ∂Q has a canonic analytic equation; hence Q:

n X

a=1

λa (v a )2 +

n X (v a )2 ∆ p ≤0⇔ ≤ 1, (v 1 , . . . , vn ) ∈ Rn , 2 δ∗ ( −∆/(δ λ )) ∗ a a=1

˜ δ∗ = λ1 · · · · · λn = det H ˜ and where λ1 , . . . , λn are the eigenvalues of H, ˆ ∆ = det H. Hence s V ol(Bn ) · (−∆)n/2 (−∆)n = (7) V ol(Q) = V ol(Bn ) · (n+1)/2 δ∗n · λ1 · · · · · λn δ∗ ˜ αβ , or, denoting η∗ = ρ − ν 2 zαn+1 zβn+1 h V ol(Q) =

V ol(Bn ) · (−η∗ )n/2 √ . δ∗

(8)

Constant mean curvature submanifolds in (α, β)−Finsler spaces

49

˜ which provide δ∗ and conWe note that the coefficients of the matrix H tribute to η∗ are of the form ˜ hab =

n X

zai zbi + µzan+1 zbn+1 = hab + la lb ,

i=1

where hab =

n+1 X

zai zbi and la =

i=1

ing

√ µ − 1zan+1 . Using the Lemma, and denot-

τ = la la , δ = det(hab )a,b=1,n ,

(9)

we get  τ = (µ − 1)zan+1 zbn+1 hab    ˜ ab = hab − (1 + τ )−1 la lb h    ˜ = δ(1 + τ ), δ∗ = det H

and hence a more refined form of the volume of the body bounded by Q given by (4) is ˜ αβ − ρ)n/2 V ol(Bn ) · (ν 2 zαn+1 zβn+1 h p . V ol(Q) = δ(1 + τ )

(10)

Two distinct cases occur:

a) For µ 6= 1 (and hence for τ 6= 0) we infer


η∗ = ρ − ν 2 zan+1 zbn+1 hab − (1 + τ )−1 la lb =   ν2 ν2τ τ2 =ρ− =ρ− · la lb hab − . µ−1 1+τ (µ − 1)(1 + τ )

(11)

b) For µ = 1 we have τ = 0 and

η∗ = ρ − ν 2 zan+1 zbn+1 hab .

(12)

Hence, making use of (11) and (12) in (8), we finally obtain: Theorem 2.1. The volume of the body Q in (4) is given by   2 n/2  V ol(Bn ) ν τ   · − ρ(1 + τ ) ,for µ 6= 1 √  δ · (1 + τ )(n+1)/2 µ − 1 V ol(Q) = (13)   V ol(Bn ) · (−ρ + ν 2 zan+1 zbn+1 hab )n/2   √ , for µ = 1,  δ

where τ and δ are given by (9).

50

V. Balan

In particular, we obtain the following result: Corollary 2.1. a) In the Randers case (2) we obtain the known result (Ref. 2, p.627; Ref. 3), V ol(B n ) V ol(QR ) = √ . δ(1 − b2 zαn+1 zβn+1 hαβ )(n+1)/2 b) In the Kropina case (3) we have V ol(QK ) =

V ol(B n ) √ δ



b2 n+1 n+1 αβ z zβ h 4 α

n/2

.

Proof. We use (13) with µ = 1−b2 , ν = b, ρ = −1, τ = −b2 zαn+1 zβn+1 hαβ in the Randers case, and with µ = 1, ν = −b/2, ρ = τ = 0, δ ∗ = δ, ˜hαβ = hαβ in the Kropina case. Remark 2.1. In the Kropina case, the function G from (1) has the expression √ n  δ 2 V ol(Bn ) √ = n+1 n+1 ab 2 n/2 = C · , GK = V ol(QK ) (za zb h · b /4) B where we have used the notations B = b2 zan+1 zbn+1 hab ,

C=

√ δ,

δ = det(hαβ ),

hαβ =

n+1 X

zαi zβj . (14)

i=1

Then the mean curvature field has the components ! ∂2G 1 ∂ 2 ϕj ¯ , i = 1, n + 1, Hi = · G ∂zεi zηj ∂uε ∂uη

(15)

and the volume form of a hypersurface H is dVF = GK du1 ∧ · · · ∧ dun . 3. CMC and minimal surfaces in Kropina spaces Having in view the previous remark, we are able to state the following main result Theorem 3.1. The mean curvature field of a hypersurface H in the ˜ = Rn+1 with the fundamental function (3) has the folKropina space M lowing expression in terms of B and C given by (14)

Constant mean curvature submanifolds in (α, β)−Finsler spaces

nB 2

∂2C

n(n + 2) ∂B ∂B C i j− 4 ∂zε ∂zη (16) !# ∂C ∂B ∂ 2 ϕj ∂2B ∂C ∂B , i = 1, n + 1. + j i +C ∂zεi ∂zηj ∂uε ∂uη ∂zη ∂zε ∂zεi ∂zηj

1 Hi = − 2 B C −

"

51

∂zεi ∂zηj

B2 +

Proof. We subsequently obtain   ∂ ∂C −n/2 n ∂B −(n+2)/2 ∂2G εη n =2 · j Kij ≡ = B − C iB 2 ∂zε ∂zεi ∂zηj ∂zη ∂zεi " n(n + 2) ∂B ∂B ∂2C B2 + = 2n B −(n+4)/2 C i j− j i 4 ∂zε ∂zη ∂zε ∂zη !# nB ∂C ∂B ∂2B ∂C ∂B − , + j i +C j i 2 ∂zε ∂zη ∂zη ∂zε ∂zεi ∂zηj whence, using G = 2n CB −n/2 , the claim follows. Corollary 3.1. The mean curvature field of a surface H in the Kropina ˜ = R3 with the fundamental function (3) has the following expresspace M sion " ∂E ∂E ∂2E −1 ∂C ∂C − Hi = 2 2 6E 2 i j + 2C 2 i j − C 2 E C ∂zε ∂zη ∂zε ∂zη ∂zεi ∂zηj ! # 2 ∂E ∂C ∂E ∂C ∂ C ∂ 2 ϕj 2 −3CE +3E C , i = 1, 3, + ∂zεi ∂zηj ∂zηj ∂zεi ∂zεi ∂zηj ∂uε ∂uη

(17)

where E = b2

3 2 X X

(−1)α+β zαk˜ zβk˜ zα3 zβ3 ,

k=1 α,β=1

α ˜ = 3 − α.

(18)

Proof. For n = 2, we infer B = EC −2 . Then tedious computation leads to the stated result. ! 2 2 ∂ C ∂C ∂2C ∂C = (2C)−2 − 2 i j , we infer Remark 3.1. Using ∂zε ∂zη ∂zεi ∂zηj ∂zεi ∂zηj ˜ = R3 that the mean curvature field of a surface H in the Kropina space M

52

V. Balan

with the fundamental function (3) is H∗ = Hi X i , with " ∂C ∂C −1 ∂E ∂E ∂2E Hi = 2 2 3E 2 i j + 2C 2 i j − C 2 − E C ∂zε ∂zη ∂zε ∂zη ∂zεi ∂zηj (19) # ! ∂E ∂C ∂ 2 ϕj ∂E ∂C 3 2 ∂2C 2 + j i + E −3CE , i = 1, 3, ∂zεi ∂zηj 2 ∂zεi ∂zηj ∂uε ∂uη ∂zη ∂zε where C and E are provided respectively by (14) and (18), −1 i ijk X = ||N ||F (G∗ Z 1 )j (G∗ Z 2 )k , ˜ ,X · N, N = ε

where Z 1 = (z11 , z12 , z13 ), Z 2 = (z21 , z22 , z23 ), and G∗ v is defined by the equality (G∗,X v)(v ′ ) = hv, v ′ iF,X =

1 ∂F 2 i ′j vv , 2 ∂y i ∂y j

“*” is the Euclidean Hodge operator and εijk is the skew-symmetrization symbol. We note that N ≡ ∗((G∗ Z 1 ) ∧ (G∗ Z 2 )) and ˜ | F˜ (y) = 1}. X ∈ Ker(G∗ Z 1 ) ∩ Ker(G∗ Z 2 ) ∩ {y ∈ Tϕ(u) M

(20)

Remark 3.2. The relation (19), leads to the general form of the mean curvature: ∂C i ∂C ∂ 2 ϕj 3 2 ∂ 2 C 2 ∂ 2 ϕj + Xi X · E ∂zεi 2 ∂zεi ∂zηj ∂uε ∂uη ∂zηj ∂uε ∂uη ! ∂E i ∂C ∂ 2 ϕj ∂C i ∂E ∂ 2 ϕj −3CE X · j + iX · j (21) ∂zεi ∂zε ∂zη ∂uε ∂uη ∂zη ∂uε ∂uη ! 2 j 2 ∂ 2 ϕj 2 ∂E 2 ∂ E i i ∂E ∂ ϕ +2C −C X . X ∂zεi ∂zηj ∂uε ∂uη ∂zεi ∂zηj ∂uε ∂uη

Hi X i =

−1 E2C 2

3E 2

3.1. CMC surfaces of revolution Corollary 3.2. Let M = Σ = Im ϕ be a surface of revolution described by ϕ(t, θ) = (f (t) cos θ, f (t) sin θ, t), (t, θ) ∈ D = R × [0, 2π). Then M is of constant mean curvature iff the function f satisfies the ODE   −1 3f 3 b2 (1 + 2g 2 ) 2 2 2 2 2 h + 2g (b f − 2) + 3b f − 2 = k, k ∈ R. (22) f 3 b2 1 + g2

Constant mean curvature submanifolds in (α, β)−Finsler spaces

53

Proof. Let v be the transversal vector to Σ = Im r, X = (X 1 , X 2 , X 3 ) = (− sin θ, − cos θ, f ′ (t)), such that Z1 , Z2 and X are linear independent. Then ¯ = the minimality of M is equivalent with the vanishing of the value of H Hi dy i |ϕ(u) on X. Assuming f (t) > 0, ∀t ∈ R, for the surface of revolution Σ we have   p 1 + g2 0 A = (hαβ )α,β=1,2 = , C = det A = f 1 + g 2 , E = b2 f 2 , 2 0 f where we have denoted for brevity f = f (t), g = f ′ (t), h = f ′′ (t). Then, according to Ref. 2, Lemma 5, p.632, we have:  2 j   ∂E X i = 2b2 f 2 gδε1 , ∂E ∂ ϕ = 2b2 f gδε1 ,  j  i ∂zε  ∂zη ∂uε ∂uη       ∂C i ∂ 2 E ∂ 2 ϕj   X i = 2b2 f (1 + 2g 2 ), X =0   ∂z i ∂zηj ∂uε ∂uη ∂zεi ε  ∂ 2 C 2 ∂ 2 ϕj   X i = 2f (−f h + 1 + g 2 ),   i ∂z j ∂uε ∂uη  ∂z  η ε      ∂ 2 ϕj g ∂C   p (f h + 1 + g 2 )δε1  j ∂uε ∂uη = ∂zη 1 + g2

and hence from (21) we infer   −1 3f 3 b2 (1 + 2g 2 ) i 2 2 2 2 2 Hi X = 3 2 h + 2g (b f − 2) + 3b f − 2 , f b 1 + g2 whence the claim follows. Remark 3.3. As consequence of (22), the ODE which characterizes the generator curve of minimal surfaces of revolution in the 3-dimensional Kropina space with metric (3), has the explicit form: 3f 3 b2 (1 + 2g 2 ) h + 2g 2 (b2 f 2 − 2) + 3b2 f 2 − 2 = 0. 1 + g2

(23)

3.2. CMC graphs In the case of a graph f : D ⊂ R2 → R immersed in the Kropina ˜ = R3 with metric (3), the relation which defines the mean curspace M vature considerably simplifies. Considering the associated parametrization ϕ : D → R3 , ϕ(u1 , u2 ) = (u1 , u2 , f (u1 , u2 )) = (u1 δi1 + u2 δi2 + f δi3 )i=1,3 ,

(24)

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we yield ϕiε = zεi = δ1ε δi1 + δi3 fε and ϕjεη = δj3 fεη , where the lower indices denote partial derivatives w.r.t. the corresponding domain-variables. As well, we shall choose X = ϕ1 × ϕ2 = (−δi1 f1 − δi2 f2 + δi3 )i=1,3 .

(25)

we obtain the following computational result: Corollary 3.3. Let M = Σ = Im ϕ be a graph described by (24) immersed in the 3-dimensional Kropina space with metric (3). Then M is of constant mean curvature iff the function f satisfies the ODE −1 {[3b4 (C 2 −1)2 −2b2 C 2 ][(1+f12 )f22 −2f1 f2 f12 +(1+f2)2 f11 ] E2C 2 (26) + [8b2 C 2 − 6b4 (C 2 − 1)](f12 f11 + 2f1 f2 f12 + f22 f22 )} = k, k ∈ R, p √ where C = det A = 1 + f12 + f22 , E = zα3 zβ3 hαβ = BC 2 and where 2 2 2 −2 B = b (f1 + f2 )C . Proof. Computational, using the shape of the terms within the CMC equation infered by (21), for the case of graphs,8 namely: ∂E i X = 2b2 (δε1 f1 + δε2 f2 ), ∂zεi

∂E ∂ 2 ϕj = 2b2 (f1 f1ε + f2 f2ε ), ∂zηj ∂uε ∂uη

∂C ∂ 2 ϕj = (f1 f1ε + f2 f2ε )C −1 , ∂zηj ∂uε ∂uη ∂2C 2 ∂zεi ∂zηj 2

∂C i X = 0, ∂zεi

∂ 2 ϕj X i = 2[(1 + f12 )f22 − 2f1 f2 f12 + (1 + f22 )f11 ] ∂uε ∂uη

∂ 2 ϕj X i = 2b2 [(1 + f12 )f22 − 2f1 f2 f12 + (1 + f22 )f11 ]. ∂zεi ∂zηj ∂uε ∂uη ∂ E

Remark 3.4. Equation (26) yields as consequence of (22), the PDE which characterizes the function of a minimal graph in the 3-dimensional Kropina space with metric (3): {[3b4 (C 2 − 1)2 − 2b2 C 2 ][(1 + f12 )f22 − 2f1 f2 f12 + (1 + f2 )2 f11 ] +[8b2 C 2 − 6b4 (C 2 − 1)](f12 f11 + 2f1 f2 f12 + f22 f22 )} = 0.

(27)

Acknowledgments The author is grateful to Yi-Bing Shen and to V.S. Sab˘au for useful discussions and support in developing the present topic. Warm thanks are addressed to the organizers of the Conference DGA-2007 for their concern and hospitality.

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References 1. Z. Shen, On Finsler geometry of submanifolds, Mathematische Annalen 311 (1998) 549–576. 2. M. Souza, Superficies Minimas en Espacos de Finsler com Uma Metrica de Randers, Ph.D. Thesis, University of Brasil, 2001. 3. M. Souza and K. Tenenblat, Minimal surface of rotation in Finsler space with a Randers metric, Math. Ann. 325 (2003) 625. 4. A. Bejancu, Special immersions of Finsler spaces, Stud.Cerc.Mat. 39 (1987) 463. 5. R. Miron and M. Anastasiei, The Geometry of Vector Bundles. Theory and Applications (Kluwer Acad. Publishers, FTPH, no.59, 1994). 6. V. Balan, DPW method for CMC surfaces in Randers spaces, In: Modern Trends in Geometry and Topology (Cluj Univ. Press, Romania, 2006). 7. V. Balan, Moving frames of Riemannian surfaces in GL3 spaces, Algebras, Groups and Geometries, Hadronic Press 17 (2000) 273. 8. M. Souza, J. Spruck and K. Tenenblat, A Bernstein type theorem on a Randers space, Math. Annalen 329 (2004) 291–305.

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Generalized Einstein manifolds Cornelia-Livia Bejan Seminar matematic, Universitatea “Al.I. Cuza”, Iasi, 700506, Romˆ ania E-mail: [email protected] Tran Quoc Binh Department of Mathematics, University of Debrecen, H-4010, Debrecen, P.O. Box 12, Hungary E-mail: [email protected] We give here necessary and sufficient conditions for a quasi–Einstein manifold to be η–Einstein. Then we deal with the locally decomposable Riemannian manifolds and we generalize a result obtained by K. Yano and M. Kon in Ref. 15. The special case of quasi–constant sectional curvature is considered. Keywords: Quasi–Einstein manifolds, quasi–constant curvature, locally decomposable Riemannian manifolds. MS classification: 53C25, 53C15.

1. Some classes of generalized Einstein manifolds The field equations for the interaction of gravitation with other fields, proposed by A. Einstein in Lorentzian framework, take on a Riemannian manifold (M, g) the form: Q − αI = T,

(1)

where Q, I and T are respectively the Ricci operator, the identity and the energy–momentum tensor. The self–adjoint operator T is particularized on certain manifolds as follows: I). The “Ricci constant condition” (Q = αI) yields T = 0, which defines Einstein metrics. When dimM > 2, the function α is necessarily constant. II). Einstein manifolds can be generalized by Dedicated to Prof. Lajos Tamassy for the 85th anniversary of his birthday.

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Definition 1.1. A Riemannian manifold (M, g) is called quasi–Einstein, provided it carries a globally defined unit vector field ξ (which we refer to as the generator) and its dual form η w.r.t. g η = g(ξ, ·)

(2)

T = βη ⊗ ξ,

(3)

such that

6

where α, β are functions on M .

In particular, if α, β are constant, it is natural to say that M is η–Einstein, by extending this notion from the context of contact geometry.13 From (3), or equivalently: g(T X, Y ) = βη(X)η(Y ), ∀X, Y ∈ Γ(T M ), one can see that T is self-adjoint and the Ricci (0,2)–tensor field Ric satisfies Ric(X, Y ) = g(QX, Y ) = αg(X, Y ) + βη(X)η(Y ), ∀X, Y ∈ Γ(T M ), which gives the scalar curvature r: r = nα + β.

(4)

As any η–Einstein manifold is quasi–Einstein, we study now when the converse is true. We recall that on a Riemannian manifold (M, g), a 1–form η is called harmonic if η is both closed (dη = 0) and coclosed (δη = trace(∇· η)· = 0). Theorem 1.1. Let (M n , g, ξ), n ≥ 2, be a quasi–Einstein manifold and assume that the dual form η of ξ is harmonic. Then M is η–Einstein if and only if its scalar curvature is constant. Proof. As M is quasi–Einstein, from (1) and (3) we have Q = αI + βη ⊗ ξ.

(5)

We apply the Levi–Civita connection ∇ to (5) and we obtain: (∇Z Q)(W ) = (∇Z α)(W ) + (∇Z β)η(W )ξ + β[(∇Z η)(W )ξ + η(W )∇Z ξ], ∀Z, W ∈ Γ(T M ). We may chose an orthonormal frame {ei , i = 1, n} such that e1 = ξ. Then we sum over repeated indices, as follows: g((∇ei Q)(W ), ei ) = g((∇ei α)(W ), ei ) + (∇ei β)η(W )η(ei ) + β[(∇ei η)(W )η(ei ) + η(W )g(∇ei ξ, ei )], ∀W ∈ Γ(T M ).

Generalized Einstein manifolds

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Equivalently, we have: g((∇ei Q)(W ), ei ) = ∇W α + (∇ξ β)η(W )

+ β[(∇ξ η)(W ) − η(W )η(∇ei ei )], ∀W ∈ Γ(T M ).

(6)

The second Bianchi identity gives:

W r = 2g((∇ei Q)(W ), ei ).

(7)

Now we use that η is coclosed: 0 = δη = trace(∇· η)· = (∇ei η)ei and that η is closed:

= ei (η(ei )) − η(∇ei ei ) = −η(∇ei ei )

(∇ξ η)(W ) = ξη(W ) − η(∇ξ W )

= ξη(W ) − η(∇W ξ + [ξ, W ]) = ξη(W ) − η([ξ, W ])

(8)

(9)

= dη(ξ, W ), ∀W ∈ Γ(T M ),

since from (2) we have on one side η(ξ) = 1 and therefore W (η(ξ)) = 0 and on the other side η(∇W ξ) = g(∇W ξ, ξ) = 0. By replacing (7), (8) and (9) in (6) we obtain: W r = 2W α + 2(ξβ)η(W ), ∀W ∈ Γ(T M ).

(10)

If we suppose that M is η–Einstein, it follows from (10) that r is constant. Conversely, if we suppose that r is constant, then from (4) and (10) we obtain the system:  nW α + W β = 0 and (11) W α + (ξβ)η(W ) = 0, ∀W ∈ Γ(T M ) If we take W = ξ, then 

nξα + ξβ = 0 and ξα + (ξβ) = 0,

which imply ξα = ξβ = 0. If in (11) we take W orthogonal to ξ, then we get W α = W β = 0, ∀W ∈ ξ ⊥ . From the last two relations, we obtain W α = W β = 0, ∀W ∈ Γ(T M ), that is α and β are constant and therefore M is η–Einstein. Remark 1.1. If (M, g, ξ) is an η–Einstein manifold, then from (5) it follows that any two of the following conditions imply the third: (i) η is coclosed; (ii) Q is divergence–free (i.e. δQ = trace(∇· Q)· = 0) and (iii) ξ is autoparallel (i.e. ∇ξ ξ = 0).

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Example 1.1. In our terminology, the main theorem in Ref. 5 states that the unit tangent bundle of a 4–dimensional Einstein manifold, equipped with the canonical contact metric structure, is η–Einstein if and only if the base manifold is a space of constant sectional curvature 1 or 2. Therefore, the unit tangent bundle T1 (S 4 ) of the unit sphere S 4 , endowed with the canonical contact metric structure stands for an example of Theorem 1.1. III). A K¨ ahler manifold (M, J, g) is called K¨ahler–Einstein if its metric g is Einstein. This notion has a natural generalization, as follows: IV). A K¨ ahler manifold (M, J, g) endowed with a generator ξ and its dual form η is called K¨ ahler η–Einstein, provided T = β[η ⊗ ξ − (η ◦ J) ⊗ Jξ], where α, β are functions on M .1 The first author and V. Oproiu gave in Ref. 3 the conditions when some natural metrics obtained on T M by lifting procedure, provide a particular class of K¨ ahler η–Einstein metrics, where η is the dual form of the Liouville vector field. For the study of T M and classification theorems, see Ref. 10. V). From now on, let M = M1 × M2 be a locally decomposable Riemannian manifold, that is a class of Riemannian almost product manifolds (M, P, g).12 Precisely, P 2 = I, P 6= ±I, P is self adjoint w.r.t. g and parallel w.r.t. the Levi–Civita connection of g. For this type of manifolds, we introduce now a notion which corresponds to the above cases I, II, III, IV, i.e. we consider another particular case for the energy–momentum tensor T . 2. Product Einstein manifolds Definition 2.1. Let (M, P, g) be a locally decomposable Riemannian manifold. We say that M is: (a) product–Einstein, if in (1.1) we have T = βP,

(12)

with α, β constant; (b) product quasi–Einstein, provided M carries a generator ξ and its dual form η such that T = βP + γ[η ⊗ ξ + (η ◦ P ) ⊗ P ξ] + ν[η ⊗ P ξ + (η ◦ P ) ⊗ ξ], where α, β, γ, ν are functions on M .

(13)

Generalized Einstein manifolds

61

If moreover these functions are constant, then we call M a product η–Einstein manifold. Examples (1) Any Riemannian product M = M (c1 ) × M (c2 ) of two manifolds of constant sectional curvature is a product–Einstein manifold. (2) If M1 is an Einstein manifold and M2 is a canal hypersurface in Rn , i.e. the envelope of a 1–parameter family of hyperspheres,8 then the Riemannian product M = M1 × M2 is a product quasi–Einstein manifold which is not product–Einstein. Theorem 2.1. Let M = M1 × M2 be a locally decomposable Riemannian manifold with a generator ξ. Then M is product quasi–Einstein (resp. product η–Einstein) if and only if: (i) both components are quasi–Einstein (resp. η–Einstein) provided ξ is nowhere tangent to only one of them; (ii) M1 is quasi–Einstein (resp. η–Einstein) and M2 is Einstein, provided ξ is tangent to M1 only, while dimM2 > 2. Proof. Some geometric objects on M are decomposed on M1 , M2 as follows: Q = Q1 × Q2 , ξ = ξ ′ + ξ ′′ , η = η ′ + η ′′ , ′

′′

′

(14)

′′

where ξ = (ξ+P ξ)/2, ξ = (ξ−P ξ)/2 and η , η are their dual, respectively. (i) Since in the first case ξ ′ and ξ ′′ are nowhere vanishing, we may take ξ1 = ξ ′ /kξ ′ k, ξ2 = ξ ′′ /kξ ′′ k and let ηi be the dual of ξi , i = 1, 2. If we suppose that M is a product quasi–Einstein (resp. product η– Einstein) manifold, then Q = αI + βP + γ[η ⊗ ξ + (η ◦ P ) ⊗ P ξ] + ν[η ⊗ P ξ + (η ◦ P ) ⊗ ξ], (15) where α, β, γ, ν are functions (resp. constant). From (14) and (15) one gets: Qi = αi I + βi ηi ⊗ ξ, i = 1, 2,

(16)

where α1 = α + β, α2 = α − β, β1 = 2(γ + ν), β2 = 2(γ − ν) are functions (resp. constant). Conversely, let suppose M1 , M2 are quasi–Einstein (resp. η–Einstein). From (16) one gets (15), where α = (α1 + α2 )/2, β = (α1 − α2 )/2, γ = (β1 + β2 )/4, ν = (β1 − β2 )/4 are functions (resp. constant) and

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therefore M is a product quasi–Einstein (resp. product η–Einstein) manifold. (ii) In this case ξ1 = ξ ′ = ξ, η1 = η, ξ ′′ = 0, η ′′ = 0. We obtain the equivalence between (15) and (16), where the coefficients are related as in the case (i), but here α2 is constant (since dimM2 > 2), γ = ν, hence β2 = 0, which complete the proof. Theorem 2.1 generalizes Theorem 2.4 obtained by K. Yano, M. Kon in Ref. 15 (pp. 421), which in view if Definition 2.1 can be restated as follows: Corollary 2.1. A locally decomposable Riemannian manifold M = M1p × M2q (p, q > 2) is a product–Einstein manifold if and only if M1 and M2 are both Einstein. 3. Quasi–constant sectional curvature The following notion was defined independently by different authors: Definition 3.1. A Riemannian manifold (M, g) endowed with a generator ξ and its dual form η is called of quasi–constant sectional curvature if it satisfies one of the following conditions: (a) The curvature of any plane σ depends only on the point and on the angle between σ and ξ; (b) The curvature (0,4)–tensor field R takes the form R = αU + βS,

(17)

where α, β are functions on M and U , S are (0,4)–tensor fields defined by U (X, Y, Z, V ) = g(X, V )g(Y, Z) − g(Y, V )g(X, Z),

(18)

S(X, Y, Z, V ) = g(X, V )η(Y )η(Z) − g(X, Z)η(Y )η(V )

+ g(Y, Z)η(X)η(V ) − g(Y, V )η(X)η(Z),

∀X, Y, Z, V ∈ Γ(T M ).

(c) M is of almost constant curvature. Comment. The notion of quasi–constant sectional curvature was introduced geometrically in Boju and Popescu’s paper4 by the statement (a). Independently, Chen and Yano defined the same notion in Ref. 7, by the algebraic condition (b). On the other hand, Vranceanu defined in Ref. 14 the almost constant curvature. It turns out that (a), (b) and (c) are equivalent.

Generalized Einstein manifolds

63

The equivalence (a) ⇔ (b) (resp. (b) ⇔ (c)) was proved in Ref. 9 (resp. in Ref. 11). Some types of manifolds of quasi–constant sectional curvature are discussed in Ref. 2. Theorem 3.1. Let M = M1 × M2 be a locally decomposable Riemannian manifold with Mi endowed by the unit form ηi for i = 1, 2. Then M has both components of quasi–constant sectional curvature if and only if its (0,4)– tensor field R is of the form: R(X, Y, Z, V ) = α[U (X, Y, Z, V ) + U (P X, P Y, Z, V )] + β[U (P X, P Y, P Z, V ) + U (P X, P Y, Z, P V )] + γ[S(X, Y, Z, V ) + S(P X, P Y, P Z, P V ) + S(P X, P Y, Z, V ) + S(X, Y, P Z, P V )]

(19)

+ ν[S(X, Y, Z, P V ) + S(X, Y, P Z, V ) + S(P X, P Y, Z, P V ) + S(P X, P Y, P Z, V )], ∀X, Y, Z, V ∈ Γ(T M ), where α, β, γ, ν are functions on M and η = (η1 + η2 )/2. Proof. If we suppose that Mi is of quasi–constant sectional curvature, then from (17), Ri takes the form Ri = αi Ui + βi Si , i = 1, 2,

(20)

where Ui and Si satisfy (18) with gi and ηi , i = 1, 2. Then R = R1 + R2 will satisfy (19) with α = (α1 + α2 )/4, β = (α1 − α2 )/4, γ = (β1 + β2 )/8 and ν = (β1 − β2 )/8. Conversely, if we suppose that R is given by (19), then R = R1 + R2 , where Ri satisfies (17) on Mi , i = 1, 2, with α1 = 2(α + β), α2 = 2(α − β), β1 = 4(γ + ν) and β2 = 4(γ − ν), which complete the proof. Acknowledgments The authors are indebted to professor O. Kowalski for valuable suggestions. References 1. C.L. Bejan, K¨ ahler manifolds of quasi–constant holomorphic sectional curvature, to appear in J. Geom.

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2. C.L. Bejan, Types of manifolds of quasi–constant curvature, Sci. Annals UASVM Iasi, Tom XLIX, V. 2 (2006) 155–158. 3. C.L. Bejan and V. Oproiu, Tangent bundles of quasi–constant holomorphic sectional curvatures, J. Geom. Appl. 11 (1) (2006) 11–22. 4. V. Boju and M. Popescu, Espaces ` a courbure quasi–constante, J. Diff. Geom. 13 (1978) 373–383. 5. Y.D. Chai, S.H. Chun, J.H Park and K. Sekigawa, Remarks on η–Einstein unit tangent bundles, 10 Aug 2007; arXiv:0708.1400v1[math.DG]. 6. M. Chaki and R. Maity, On quasi Einstein manifolds, Publ. Math. Debrecen 57 (2000) 297–306. 7. B.Y. Chen and K. Yano, Hypersurfaces of a conformally flat space, Tensor N.S. 26 (1972) 318–322. 8. B.Y. Chen and K. Yano, Special conformally flat spaces and canal hypersurfaces, Tohoku Math. J. 25 (1973) 177–184. 9. G. Ganchev and V. Mihova, Riemannian manifolds of quasi–constant sectional curvatures, J. Reine Angew. Math. 522 (2000) 119–141. 10. O. Kowalski and M. Sekizawa, Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles – a classification –, Bull. Tokyo Gakugei Univ. 40 (4) (1988) 1–29. 11. A.L. Mocanu, Les vari`et`es ` a courbure quasi–constant de type Vranceanu (Lucr. Conf. Nat. Geom., Targoviste, 1987). 12. A.M. Naveira, A classification of Riemannian almost product manifolds, Rend. Mat. Appl. Math., VII, ser. 3 (1983) 577–592. 13. M. Okumura, On infinitesimal conformal and projective transformations of normal contact spaces, Tohoku Math. J. II, ser. 14 (1962) 394–412. 14. Gh. Vranceanu, Le¸cons des Geometrie Differential (vol. 4, Ed. Acad., Bucharest, 1968). 15. K. Yano and M. Kon, Structures on manifolds (World Scientific, Singapore, 1984).

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 65–75

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Canonical almost geodesic mappings of type π ˜ 1 onto pseudo-Riemannian manifolds V.E. Berezovski Department of Mathematics, University of Uman, Institutskaya 1, Uman, Ukraine E-mail: [email protected] J. Mikeˇs and A. Vanˇ zurov´ a Department of Algebra and Geometry, Faculty of Science, Palack´ y University, Tomkova 40, 779 00 Olomouc, Czech Republic E-mail: [email protected], [email protected] Our aim is to examine almost geodesic mappings of affine manifolds and give necessary and sufficient conditions for existence of the so-called π ˜ 1 mappings (canonical almost geodesic mappings of type π according to Sinyukov) of a manifold endowed with a linear connection onto pseudo-Riemannian manifolds. The conditions take the form of a closed system of PDE’s of first order of Cauchy type. Our result is a generalization of some previous theorems of N.S. Sinyukov. Keywords: Linear connection, affine manifold, pseudo-Riemannian space, geodesic curve, almost geodesic curve, geodesic mapping, almost geodesic mapping, deformation tensor. MS classification: 53B20, 53B30, 53B35.

1. Introduction Unless otherwise specified, all objects under consideration are supposed to be differentiable of a sufficiently high class (mostly, differentiability of the class C 3 is sufficient). Let An = (M, ∇) be an n-dimensional (C k , C ∞ or C ω ) manifold endowed with a linear connection ∇ (an “affine manifold”). Let c : I → M , t 7→ c(t) defined on an open interval I ⊂ R be a (C k , or smooth) curve on M satisfying the regularity condition c′ (t) = dc(t)/dt 6= 0 for all t ∈ I.

Denote by ξ the corresponding (C k−1 , or smooth) tangent vector field

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V.E. Berezovski, J. Mikeˇs and A. Vanˇzurova´

along c (“velocity field”), ξ(t) = (c(t), c′ (t)) , t ∈ I, and let ξ1 = ∇(ξ; ξ) = ∇ξ ξ,

ξ2 = ∇2 (ξ; ξ, ξ) = ∇ξ ξ1 .

(1)

Geodesics c(s), parametrized by canonical affine parameter (given up to affine transformations s 7→ as + b), are characterized by ∇ξ ξ = 0 while unparametrized geodesic curves (i.e. arbitrarily parametrized, called also pregeodesics in the literature) can be characterized by the formula ∇ξ ξ = λξ where λ(t) : I → R is a real function. Let D = span (X1 , X2 ) (i.e. vector fields X1 , X2 along c form a basis of D). Recall that D is parallel (along c) if and only if covariant derivatives along c of basis vector fields belong to the distribution (the property is independent on reparametrization of the curve).1–3 As a generalization of (an unparametrized) geodesic, let us introduce an almost geodesic curve as a curve c satisfying: there exists a two-dimensional (differentiable) distribution D parallel along c (relative to ∇) such that for any tangent vector of c, its parallel translation along c (to any other point) belongs to the distribution D. Equivalently, c is almost geodesic if and only if there exist vector fields X1 , X2 parallel along c (i.e. satisfying ∇ξ Xi = aj Xj for some differentiable functions aji (t) : I → R) and differentiable real functions bi (t), t ∈ I along c, such that ξ = b1 X1 + b2 X2 holds. For almost geodesic curves, the vector fields ξ1 and ξ2 belong to the corresponding distribution D. If the vector fields ξ and ξ1 are independent at any point (and hence the (local) curve c is not a geodesic one), we can write D = span (ξ, ξ1 ). So we get another equivalent characterization: a curve is almost geodesic if and only if ξ2 ∈ span (ξ, ξ1 ). 2. Almost geodesic mappings Geodesic mappings of manifolds with linear connection (in short, affine manifolds) are (C k )-diffeomorphisms characterized by the property that all geodesics are send onto (unparametrized in general) geodesic curves. The classification of geodesic mappings is more or less known. Recall that even for Riemannian spaces, there is a lack of a nice simple criterion for decision when a given Riemannian space admits non-trivial geodesic mappings. ¯ , ∇) ¯ be n-dimensional affine manifolds, Let An = (M, ∇), A¯n = (M n > 2, endowed with torsion-free linear connections. We may ask which (C k -)diffeomorphisms of affine manifolds send almost geodesic curves onto almost geodesic ones again. The answer is: such map-

Canonical almost geodesic mappings of type π ˜1 onto pseudo-Riemannian manifolds

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pings reduce to geodesic ones, since there are “too many” almost geodesic curves. It appears that the following definition is more acceptable. ¯ is almost geodesic if any We say that a (C k -)diffeomorphism f : M → M geodesic curve of (M, ∇) is mapped under f onto an almost geodesic curve ¯ , ∇). ¯ in (M

This concept of an almost geodesic mapping was introduced by N.S. Sinyukov,1 and before by V.M. Chernyshenko,4 from a rather different point of view. The theory of almost geodesic mappings was treated in Ref. 1–3. Due to the fact that f is a diffeomorphism we can accept the useful ¯ are in fact defined on the convention that both linear connections ∇ and ∇ same underlying manifold M , so that we can consider their difference P = ¯ ∇−∇. That is, P is a (1, 2)-tensor, called sometimes a deformation tensor of ¯ the given connections under f ,2 given by ∇(X, Y ) = ∇(X, Y )+P (X, Y ) for X, Y ∈ X (M ). Since the connections are symmetric, P is also symmetric in X, Y . Of course, we identify objects on M with their corresponding objects ¯ : a curve c on M identifies with its image c¯ = f ◦ c, its tangent vector on M ¯ = T f (ξ(t)) etc. field ξ(t) with the corresponding vector field ξ(t) Besides the deformation tensor, we will use type (1, 3) tensor field, denoted by the same symbol P , introduced by X ∇Z P (X, Y ) + P (P (X, Y ), Z), X, Y, Z ∈ X (M ), P (X, Y, Z) = CS(X,Y,Z)

P

where CS( , , ) means the cyclic sum on arguments in brackets (i.e. symmetrization without coefficients). ¯ are characterized Almost geodesic diffeomorphisms f : (M, ∇) → (M, ∇) by the following condition on the type (1, 3) tensor P : P (X1 , X2 , X3 ) ∧ P (X4 , X5 ) ∧ X6 = 0,

Xi ∈ X (M ), i = 1, . . . , 6;

X ∧ Y means the exterior product of X and Y , the decomposable bivector. N.S. Sinyukov1–3 distinguished three kinds of almost geodesic mappings, namely π1 , π2 , and π3 , characterized, respectively, by the conditions for the deformation tensor: π1 : ∇X P (X, X)+P (P (X, X), X) = a(X, X)·X+b(X)·P (X, X), X ∈ X (M ), where a ∈ S 2 (M ) is a symmetric type (0, 2) tensor field and b is a 1-form; π2 : P (X, X) = ψ(X) · X + ϕ(X) · F (X),

X ∈ X (M ),

where ψ and ϕ are 1-forms, and F is a type (1, 1) tensor field satisfying ∇X F (X) + ϕ(X) · F (F (X)) = µ(X) · X + ̺(X) · F (X),

X ∈ X (M )

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for some 1-forms µ, ̺; π3 : P (X, X) = ψ(X) · X + a(X, X) · Z,

X ∈ X (M )

where ψ is a 1-form, a ∈ S 2 (M ) is a symmetric bilinear form and Z ∈ X (M ) is a vector field satisfying ∇X Z = h · X + θ(X) · Z for some scalar function h: M → R and some 1-form θ. Remark that the above classes are not disjoint. 3. Canonical almost geodesic mappings π ˜1 We are interested here in a particular subclass of π1 -mappings, the socalled π ˜1 -mappings, or canonical almost geodesic mappings, distinguished by the condition b = 0. That is, π ˜1 -mappings are just morphisms satisfying ∇X P (X, X) + P (P (X, X), X) = a(X, X) · X,

a ∈ S 2 (M ), X ∈ X (M ).

In local coordinates, the condition reads α h h h . Pjk) − Pα(i = a(ij δk) P(ij,k)

(2)

Here and after the comma “,” denotes covariant derivative with respect to ∇, δih is the Kronecker delta, the round bracket denote the cyclic sum on indices. Any geodesic mapping is a π1 -mapping (the characterizing condition can be checked), and any π1 -mapping can be written as a composition of a geodesic mapping followed by a π ˜1 -mapping. So we can consider geodesic mappings as trivial almost geodesic mappings, and we will omit them in further considerations; they have been analysed in Ref. 5. It was proven by Sinyukov2 that the basic partial differential equations (PDE’s) of π ˜1 ¯ Ric ¯ = 0, mappings of an affine manifold (M, ∇) onto Ricci-symmetric (∇ ¯ , g¯) (of arbitrary the Ricci tensor is parallel) pseudo-Riemannian spaces (M signature) can be transformed into (an equivalent) closed system of PDE’s of first order of Cauchy type. Hence the solution (if it exists) depends on a finite set of parameters. Consequently, for an affine manifold with a symmetric connection admitting π ˜1 -mappings onto Ricci-symmetric spaces, ¯ , g¯) which can serve as images of the the set of all Ricci-symmetric spaces (M given affine manifold (M, ∇) under π ˜1 -mappings is finite. The cardinality r of such a set is bounded by the number of free parameters. On the other hand, geodesic mappings form a subclass among π ˜1 mappings (they obey the definition). Basic equations describing geodesic

Canonical almost geodesic mappings of type π ˜1 onto pseudo-Riemannian manifolds

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mappings of affine manifolds do not form a closed system of Cauchy type (the general solution depends on n arbitrary functions; if the given manifold admits geodesic mappings, the cardinality of the set of possible images is big). It follows that the conditions (2) describing π˜1 -mappings of affine manifolds, in general, cannot be transformed into a closed system of Cauchy type. But if we choose a suitable subclass of images and restrict ourselves (for the given manifold) only onto mappings with co-domain in the appropriate subclass we might succeed to get an equivalent closed system of Cauchy type. If this is the case then the given manifold admits either non (if the system is non-integrable) or a finite number of π ˜1 -images in the given class. Our aim is to analyse π ˜1 -mappings of affine manifolds onto affine manifolds in general, and to use the reached results for examining π ˜1 -mappings of affine manifolds onto (pseudo-)Riemannian spaces (in general, without any restrictive conditions onto Ricci tensor), which will generalize the above result by Sinyukov. In the rest, we will omit “pseudo”. All π ˜1 -mappings f : M → M can be described by the following system of (differential) equations:2,3 3(∇Z P (X, Y ) + P (Z, P (X, Y ))) = X

CS(X,Y )

¯ (R(Y, Z)X − R(Y, Z)X) +

X

a(X, Y )Z.

CS(X,Y,Z)

In the rest, we prefer to express our equalities in local coordinates (with respect to a map (U, ϕ) on M ) since the invariant formulas are rather complicated. The above formula has the local expression h h h h ¯h 3(Pij,k + Pkα Pijα ) = R(ij)k −R (ij)k + a(ij δk) ,

(3)

h ¯ ¯ h are local components of tensors P , a, R, and R. ,R where Pijh , aij , Rijk ijk h Assuming (3) as a system of PDE’s for functions Pij on M , the corresponding integrability conditions read α α ¯h h h h h ¯h R (ij)[k,ℓ] = R(ij)[k,ℓ] + δ(i ajk),ℓ − δ(i ajℓ),k + 3(Pij Rαkℓ − Pα(j Ri)kℓ )

h α ¯ α δ α ajk) ) . ¯ α δ α ajℓ) ) + P h (Rα − R −R − Pαk (R(ij)ℓ αℓ (ij)k (i (ij)k (ij)ℓ (i

¯ to ∇ ¯R ¯ on the left hand side we get integrability conditions Passing from ∇R of the system (3) in the form h h h ¯h R (ij)[k;ℓ] = δ(i ajk),ℓ − δ(i ajℓ),k + Θijkℓ ,

(4)

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where we denoted h α α h h h α ¯ αkℓ ) − Pαk (R(ij)ℓ + δ(i ajℓ) ) Θhijkℓ = R(ij)[k,ℓ] + 3(Pijα R − Pα(j Ri)kℓ

α ¯h α ¯h α ¯h h α α α ¯h R|α|j)ℓ +Pk(i Rj)αk +Pk(i R|α|j)k −Pℓ(i +Pαℓ (R(ij)k +δ(i ajk) )−Pℓ(i Rj)αℓ ,

¯ where “;” denotes covariant derivative with respect to ∇. ¯ to the integrabilIf we apply covariant differentiation with respect to ∇ ity conditions (4) of the system (3), and then pass from covariant derivation ¯ to ∇, we get ∇ h h h ¯h ¯h R (ij)k;ℓm − R(ij)ℓ;mk = δ(i ajk),ℓm − δ(i ajℓ),km + Tijkℓm ,

(5)

where we denoted h ¯α ¯h ¯α ¯h ¯α ¯h ¯h R ¯α Tijkℓm =R αmk (ij)ℓ − Rℓmk R(ij)α − Rjmk R(iα)ℓ − Rimk R(jα)ℓ α α h α h h α h − Pmα δ(i aαk),ℓ − Pmi δ(i ajk),ℓ − Pmj δ(α ajk),ℓ − Pmk δ(α aij),ℓ

α h h α α h α h δ(i ajℓ),k + Pmi δ(α ajℓ),k + Pmj δ(i aαℓ),k δ(i ajk),α − Pmα − Pml

α h h α h α h δ(i ajα),k − Θhijkℓ,m + Pαm Θα + Pmk δ(i ajℓ),α − Pml ijkℓ − Pmi Θαjkℓ

α α α − Pmj Θhiαkℓ − Pmk Θhijαℓ − Pmℓ Θhijkα .

Alternating (5) in ℓ, m yields h h h ¯h ¯h R (ij)m;ℓk − R(ij)ℓ;mk = δ(i ajm),kℓ − δ(i ajℓ),km + Tijk[lm]

¯α ¯h ¯α ¯α ¯h ¯α ¯h ¯h +R (i|αk| Rj)mℓ + R(ij)α Rkmℓ − R(ij)k Rαmℓ + Rα(i|k| Rj)mℓ

(6)

α α h α h α h h ajk) Riℓm + δ(α ajk) Rαℓm . ajα) Rkℓm − δ(i aik) Rjℓm + δ(i + δ(α

Using properties of the Riemannian tensor, we rewrite (6) as h ¯ himℓ;jk + R ¯ hjmℓ;ik = δ h ajℓ),km − δ h ajm),kℓ − Nijkℓm R , (i (i

(7)

where the last term is h h ¯α ¯h ¯α ¯h ¯α R ¯h +R Nijkℓm = Tijk[ℓm] imℓ (αj)k + Rjmℓ R(αi)k + Rkmℓ R(ij)α α h α h α h h ¯h R ¯α −R αmℓ (ij)k + δ(α ajk) Riℓm + δ(α aik) Rjℓm + δ(α aij) Rkℓm − a(ij Rk)ℓm .

Alternating (7) in j, k we get ¯ hjmℓ;ik − R ¯ hkmℓ;ij = δ h ajℓ),km − δ h ajm),kℓ − δ h akℓ),jm + δ h akm),jℓ R (i (i (i (i h ¯ hαmℓ R ¯α ¯h ¯α ¯h ¯α ¯α ¯h − Ni[jk]ℓm +R ikj + Riαℓ Rmkj + Rimα Rℓkj − Rimℓ Rαkj .

(8)

Let us change mutually i and k in (7), and then use (8). We evaluate h h h h ¯ jmℓ;ik 2R = δ(i ajℓ),km − δ(i ajm),kℓ − δ(k ajm),iℓ

h h h + δ(i akm),jℓ − δ(i akℓ),jm + δ(jℓ ak),im + Ωhijkℓm ,

(9)

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where we used the notation h h ¯h ¯α ¯h ¯α ¯h R ¯α −R Ωhijkℓm = −Nijkℓm + Nk[ij]kℓm αmℓ (kj)i + Rjαℓ Rmik + Rjmα Rℓik

¯h ¯α ¯h ¯α ¯α ¯h ¯h ¯α ¯h ¯α ¯α ¯h R −R αi(j k)mℓ + Rjαℓ Rmik + Rjmα Rℓik − Rαmℓ Rikj − Riαℓ Rmkj + Rim[ℓ Rα]kj .

¯ to ∇: On the left side of (9), let us pass from covariant derivation ∇ h h h h ¯ jmℓ,ik 2R = δ(i ajℓ),km − δ(i ajm),kℓ − δ(k ajm),iℓ h h h h ajℓ),im + Sijkℓm , akℓ),jm − δ(k akm),jℓ − δ(i + δ(i

(10)

where h h α h α h α ¯ αmℓ,i ¯ jαℓ,i ¯ jmℓ,i Pℓk −R Pjk −R Pmk Sijkℓm = Ωhijkℓm − 2 [R α h α h ¯ jmα,i ¯ jmℓ,α Pik −R Pℓk −R

¯ h P α )P h ¯α P β − R ¯h P α − R ¯h P α − R + (R βk jαℓ im jmα iℓ αmℓ ij jmℓ αi

¯α P h − R ¯h P α − R ¯ h P α )P β ¯h P α − R − (R αmℓ βj jmℓ αβ jmα βℓ jαℓ βm ik −

¯α P h (R βmℓ αi

−

¯h P α R αmℓ βi

−

¯h P α R βαℓ im

−

(11)

¯ h P α )P β R βmα iℓ jk

β h α h h α h α ¯ αβℓ ¯ jβℓ ¯ jαℓ ¯ jβα Pαi −R − (R Pji −R Pβi −R Piℓα )Pkm

β α h h α h α h α ¯ jmβ ¯ αmβ ¯ jαβ ¯ jmα − (R Pαi −R Pji −R Pmi −R Pβi )Pkℓ ].

Let there exist a π ˜1 -mapping of an affine manifold An = (M, ∇) onto a Riemannian manifold V¯n = (M, g¯) where g¯ ∈ T20 M is a metric tensor ¯ hijk = R ¯ α g¯αh with components g¯ij . Recall that the Riemannian tensor R ijk of type (0, 4) satisfies ¯ hijk + R ¯ ihjk = 0. R (12) In (9), let us apply the metric tensor g¯hβ and then use symmetrization with respect to h and j. According to (12) we get g¯ih (am[k,j]l + al[j,k]m ) + g¯ij (am[k,h]l + al[h,k]m ) + g¯kh (am[i,j]l + al[j,i] ) + g¯kj (am[i,h]l + al[h,i]m ) + g¯mh (ak[i,j]l − aij,kl )

+ g¯mj (ak[i,h]l − aih,kl ) + g¯lj (akh,il − ai(h,k)m )

(13)

¯α|h) . + 2¯ gjh (ak(l,i)m − am(i,k)l ) + g¯lh (ak[j,i]m − aij,km ) = −Ωα i(j|klm g

Contraction of the last formula with the dual tensor g¯jh (k¯ g ij k = k¯ gij k−1 ) gives 2 akl,im − aim,kl − akm,il + ail,km = − Ωα . (14) n + 1 iαklm Let us symmetrize the above formula in k and l. From (14) we get α α α α 2akl,im − 2aim,kl = 2aαm Rlik + aαi Rmlk + aαk Rmil + aαl Rmik 2 (Ωα − Ωα + iα(kl)m ). n + 1 lαkim

(15)

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Using (14) and (15) the equation (13) reads 2¯ gih (akm,jl − ajm,kl ) + 2¯ gij (akm,hl − ahm,kl ) + 2¯ gkh (aim,jl − ajm,il ) + 2¯ gkj (aim,hl − ahm,il ) + g¯mk (aki,jl − akj,il − aij,kl )

+ g¯mj (aki,hl − akh,il − aih,kl ) + g¯lj (akh,im − ai(h,k)m )

(16)

+ g¯lh (akj,im − ai(k,j)m ) = Cijkhl ,

where

¯α|h) + Cijkhl = −Ωα i(j|klm g

2 α Ωα g¯jh − g¯kh aαl Rmij n + 1 iαklm

2 α α α α − aαm Rlkj − aαl Rmkj ) − aαj R(l|k|m) Ωα − aαk R(ml)j n + 1 mαljk 2 α α α α − aαm Rlkh − aαl Rmkh ) + g¯ij ( Ωα − aαk R(ml)h − aαh R(l|k|m) n + 1 mαlhk 2 α α α α + g¯kh ( Ωα − aαi R(ml)j − aαj R(l|i|m) − aαm Rlij + aαl Rmij ) n + 1 mαlji 2 α α α α − aαm Rlih + aαl Rmih ). − aαh R(l|i|m) Ωα − aαi R(ml)h + g¯kj ( n + 1 mαlhi + g¯ih (

If we contract (16) with the dual g¯ij of the metric tensor, use (15) and the Ricci identity we get akm,hl − akl,hm =

1 ghm µkl 2(n+3) (¯

− g¯hl µkm ) + Bkmhl ,

(17)

where µkm = aαβ,km g¯αβ , and 3 α α α α + akα Rmhl + alα Rmhk ) Bkmhl = Cαβkmhl g¯αβ + 3amα Rlhk + (ahα Rmkl 2 1 3 α α α α α + (Ωα lαkhm −Ωhα(kl)m )− (amα Rlkm +akα Rmhl +ahα Rmkl +alα Rmkh ) n+1 2 1 1 α α α α − (Ωα −Ωα kα(hl)m )−aα(h Rk)lm + (akα Rlmh +ahα Rlkm +amα Rlkh ). n+1 lαhkm 2 Now contract (16) with g¯ih . According to (17) we get n+3 g¯kl µjm − g¯jl µkm + g¯km µjl − g¯jkm µkl = Ckljm , n+1 where

(18)

α Ckljm = Cαjkl(m|β|l) g¯αβ − 2(n + 1)(Bk(ml)j − aα(l Rm)jk α α ). + akα R(lm)j + ajα R(m|k|l)

Contracting (18) with g¯kℓ and using the notation K = µαβ g¯αβ we obtain components of the tensor µ: n+3 1 Cαβjm g¯αβ . (19) µjm = K g¯jm + n n(n + 1)

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Using (19) we can rewrite (17) in the form akm,hl − ahm,kl =

K (¯ gmh g¯kl − g¯lh g¯km ) + Akmhl , 2n(n + 3)

(20)

where Akmhl = Bkmhl +

1 (¯ ghm Cαβkl g¯αβ − g¯hl Cαβkm g¯αβ ). 2n(n + 1)

Combining (16) and (20) we get g¯jl aih,km + g¯hl aij,km − g¯jm aih,kl − g¯hm aij,kl = K (¯ gih g¯kl g¯jm − g¯ih g¯km g¯jl + g¯ij g¯kl g¯hm − g¯ij g¯km g¯hl − n(n + 3)

(21)

+ 3¯ gkh g¯il g¯jm − 3¯ gkh g¯jl g¯im + 3¯ gkj g¯il g¯hm − 3¯ glh g¯jk g¯im ) + Aijkmhl , where we have denoted Aijkmhl =Cijkmhl −2(¯ gi(h A|km|j)l +¯ gk(h A|im|jl) −¯ gm(h A|ki|j)l −¯ gl(h A|k|j)im) ). Finally, symmetrization of (21) in indices i, j, followed by contraction with g¯ℓh , enables us to express second covariant derivatives of the tensor a, aij,km =

K (¯ gij g¯km + 3¯ gk(j g¯i)m ) + A(ij)kmαβ g¯αβ . n(n + 3)

(22)

Now we can consider (22) as the system of PDE’s (of first order) of Cauchy type relative to the tensor ∇a (i.e. in aij,k ), find the integrability conditions and contract them with g¯ij and g¯km , respectively. We calculate ∇K, K,β =

n(n + 3) Aβ , n2 + 5n − 6

where we denoted  α α A̺ = aα(j,|k Ri)m̺ + aij,α Rkm̺ −

(23)

K (¯ gij,[̺ g¯m]k + g¯ij g¯k[m,̺ n(n + 3)

+ 3¯ gkj,[̺ g¯m]i + 3¯ gkj g¯i[m,̺] + 3¯ gki,[̺ g¯m]j + 3¯ gki g¯j[m,̺] ) i +A(ij)k[m|αβ|,̺] g¯αβ + A(ij)k[m|αβ| g¯αβ,̺] g¯ij g¯km .

¯ h = Γh + P h and get We use Γ ij ij ij

α α g¯ij,k = Pik g¯αj + Pjk g¯αi .

(24)

¯ and denote their components by aijk := Assume the tensors ∇a and ∇R, h h ¯ ¯ aij,k and Rijkℓ := Rijk,ℓ , respectively. Then (10) and (22) take the form h h h h h 2Rjmli,k = δ(i ajl)k,m − δ(i ajm)k,l + δ(k ajl)i,m − δ(k ajm)i,l h h h + δ(i akm)j,l − δ(i akl)j,m + Sijklm ,

(25)

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aijk,m =

K (¯ gij g¯km + 3¯ gk(j g¯i)m ) + A(ij)kmαβ g¯αβ , n(n + 3)

(26)

where covariant derivatives of the tensor aijk in (25) are supposed to be expressed according to (26), the tensor S was introduced componentwise in (11). The formulas (3), (23)–(26) represent a closed system of Cauchy type for unknown functions ¯ h (x), Rh (x), g¯ij (x), Pijh (x), aij (x), aijk (x), K(x), R ijk ijkl

(27)

which, moreover, must satisfy a finite set of algebraic conditions h h ¯h gij (x)k 6= 0. (28) = a[ij] = a[ij]k = R g¯[ij] = P[ij] i(jk) = Ri(jk)l = 0, detk¯

So we have proven: Theorem 3.1. The given affine manifold An = (M, ∇) admits π ˜1 mappings (i.e. canonical almost geodesic mappings of type π1 ) onto Riemannian spaces V¯n = (M, g¯) if and only if there exists solution of the mixed system of Cauchy type (3), (23)-(26), (28) for functions (27). As a consequence of the additional algebraic conditions, we get an upper boundary for the number r of possible solutions: Corollary 3.1. The family of all Riemannian manifolds V¯n which can serve as images of the given affine manifold An = (M, ∇), depends on at most 1 2 2 n (n − 1) + n(n + 1)2 + 1 2 parameters. The above Theorem generalizes the result of Sinyukov3 already mentioned as well as his results on geodesic mappings of Riemannian spaces. Acknowledgments Supported by grant No. 201/05/2707 of The Grant Agency of Czech Republic and by the Council of Czech Government MSM 6198959214. References 1. N.S. Sinyukov, Almost geodesic mappings of affine-connected and Riemannian spaces. DAN SSSR 151 (4) (1963) 781–782 (in Russian).

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2. N.S. Sinyukov, Geodesic mappings of Riemannian spaces (Nauka, Moscow, 1979). 3. N.S. Sinyukov, Almost geodesic mappings of affine-connected and Riemannian spaces. (Russian) Itogi Nauki Tekh., Ser. Probl. Geom. 13 (1982) 3–26; Transl. in J. Sov. Math. 25 (1984) 1235–1249. 4. V.M. Chernyshenko, Affine-connected spaces with a correspondent complex of geodesics. Collection of Works of Mech.-Math. Chair of Dnepropetrovsk Univ. 6 (1961) 105–118. 5. V. Berezovsky and J. Mikeˇs, On a classification of almost geodesic mappings of affine connection spaces. Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 35 (1996) 21–24.

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On minimal hypersurfaces of hyperbolic space H4 with zero Gauss-Kronecker curvature U. Dursun Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics, 34469 Maslak, Istanbul, Turkey E-mail: [email protected] We determine minimal hypersurfaces of the hyperbolic space H4 (−1) with identically zero Gauss-Kronecker curvature. Such a hypersurface is the image of a subbundle spanned by a timelike vector field of the normal bundle of a totally geodesic surface of the de Sitter space S41 (1) under the normal exponential map. We also give some examples. Keywords: Hyperbolic space, Minimal hypersurfaces, Gauss-Kronecker Curvature. MS classification: 53C42.

1. Introduction Minimal hypersurfaces in 4-dimensional space forms with vanishing Gauss-Kronecker curvature curvature has been studied in Refs. 1–5. Hasanis, Savas-Halilaj and Vlachos5 studied the classification of complete minimal hypersurfaces in the 4-dimensional hyperbolic space H4 (−1) with identically zero Gauss-Kronecker curvature. The Gauss-Kronecker curvature of a hypersurface is the product of the principal curvatures. Such hypersurfaces are closely related to stationary (maximal) spacelike surfaces in the de Sitter space S41 , which is the Lorentzian unit sphere in the flat Lorentzian space R51 . More precisely, if f : M → S41 is spacelike stationary immersion, where M is a 2-dimensional manifold, and N 1 is the timelike unit normal bundle of f , then the “polar map” ψf (p, w) = w, (p, w) ∈ N 1 , defines a minimal hypersurface in H4 (−1) with identically zero GaussKronecker curvature, see Ref. 5. Also they provided a way to produce all spacelike stationary surfaces in S41 with normal curvature identically zero and without totally geodesic points.

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In Ref. 6, Kimura determined minimal hypersurfaces M foliated f4 that given by M = by geodesics of a 4-dimensional space forms M {expp (tξ)|p ∈ Σ, t ∈ R}, where Σ is a minimal surface of a 4-dimensional f4 and ξ is a local unit normal vector field on Σ. In this work, space form M motivated by Kimura’s work we use totally geodesic spacelike surfaces of S41 to determine minimal hypersurfaces of the hyperbolic space H4 (−1) with identically zero Gauss-Kronecker curvature. We firstly determine minimal hypersurfaces of Hm+2 (−1), that is, if f : M → Sm+2 is a totally geodesic 1 isometric immersion from a connected m-dimensional Riemannian M into Sm+2 , and ξ is a non-parallel, timelike unit local normal vector field on 1 M in Sm+2 , then by using normal exponential mapping of M in Sm+2 1 1 in the direction ξ we define a map F : M × (R − {0}) → Hm+2 (−1) by F (x, t) = exp(x, tξ), (x, t) ∈ M × (R − {0}), which is also called the suspension of f in Hm+2 (−1),5 The image F (M × (R − {0})) is a spacelike hypersurface of Hm+2 (−1) foliated by the geodesics of Hm+2 (−1). We prove that the immersion F is minimal under some conditions on the normal connection form of f . Then we show that if m = 2, then the minimal immersion F : M 2 × (R − {0}) → H4 (−1) has vanishing Gauss-Kronecker curvature. We also give some examples. 2. Preliminaries fqm be an m-dimensional pseudo-Riemannian manifold with Let M pseudo-Riemannian metric tensor g˜ of index q. Denoting by h , i the asfm fm , a tangent vector X to M sociated nondegenerate inner product on M q q is said to be spacelike if hX, Xi > 0 ( or X = 0), timelike if hX, Xi < 0 or lightlike (null) if hX, Xi = 0 and X 6= 0. fm+n . Let M m be a submanifold of a pseudo-Riemannian manifold M q fqm+n induces a pseudoIf the pseudo-Riemannian metric tensor g˜ of M Riemannian metric g on M m , then M m is called a pseudo-Riemannian fm . If the index of g is zero, then M is called a spacelike submanifold of M q submanifold. Let X and Y be tangent vector fields on M m and let ξ be a normal fm+n . Then the Gauss formula and the Weingarten vector field on M m in M q formula are, respectively, given as e X Y = ∇X Y + h(X, Y ) ∇

and

e X ξ = −Aξ (X) + ∇⊥ ∇ X ξ,

e is the Riemannian connection of M fm+n , ∇ and ∇⊥ are the induced where ∇ q fqm+n , Riemannian connection of M and the normal connection of M m in M

On minimal hypersurfaces of hyperbolic space H4

79

fm+n and Aξ is the shape h is the second fundamental form of M in M q operator of M with respect to the normal vector ξ. Also the Gauss and Weingarten formulas yield hAξ (X), Y i = hh(X, Y ), ξi.

m

(1) fqm+n . M

Let M be a submanifold of a pseudo-Riemannian manifold Let ξ1 , . . . , ξn be an orthonormal local basis for T ⊥ M . Then the mean curvature vector is given by n 1 X εi (traceAξi )ξi , H= m i=1 fm+n is called stationary if where εi = hξi , ξi i = ±1. A submanifold M of M q H = 0 on M . A stationary spacelike submanifold is called maximal. Let Rm 1 be an m-dimensional Lorentz-Minkowski space with metric tensor given by m X (dxi )2 , g˜ = −(dx1 )2 + i=2

where (x1 , . . . , xm ) is a rectangular coordinate system of Rm 1 . The de Sitter m-space Sm (1) is a pseudo-Riemannian m-manifold of constant sectional 1 m+1 curvature 1 that can be realized as the hyperquadratic in R1 : m+1 Sm | hx, xi = 1}. 1 (1) = {x ∈ R1

The hyperquadratic Hm (−1) = {x ∈ Rm+1 | hx, xi = −1 and x1 > 0}, 1 is the simply connected hyperbolic m-space of constant sectional curvature −1, where x1 is the first component in Rm+1 . 1 Let f : M m → Sm+2 (1) be a smooth isometric immersion from an 1 m m-dimensional connected Riemannian manifold M into an (m + 2)dimensional de Sitter space Sm+2 (1). Let ξ, η be a local orthonormal normal 1 basis of M m in Sm+2 (1) with signatures ε1 = hξ, ξi and ε2 = hη, ηi. Let 1 X1 , . . . , Xm be a local orthonormal tangent basis on M and s be the normal connection form for ∇⊥ defined by s(Xi ) = h∇⊥ Xi ξ, ηi. Since hξ, ηi = 0, then ⊥ η = −ε ξ = ε s(X )η and ∇ we see that ∇⊥ 1 s(Xi )ξ. Here it is seen that 2 i Xi Xi if either ξ or η is parallel in the normal space, then the normal connection form for ∇⊥ is zero. We therefore suppose that ξ and η are nonparallel. Denoting by si the components of the connection form s, the covariant derivative of the 1-form s is defined by sij = (∇Xj s)(Xi ) = Xj (si ) − s(∇Xj Xi ).

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Then it is easily seen that ⊥ ⊥ sij = h∇⊥ Xj ∇Xi ξ − ∇∇X

j

Xi ξ, ηi.

As the ambient space is a space form, then the Ricci equation can be written as hR⊥ (X, Y )ξ, ηi = h[Aξ , Aη ]X, Y i, (Ref. 7 [p. 125]), where R⊥ denotes the normal curvature tensor of the normal connection ∇⊥ and [Aξ , Aη ] = Aξ Aη − Aη Aξ . So we can express the Ricci equation as sji − sij = hR⊥ (Xi , Xj )ξ, ηi = h[Aξ , Aη ]Xi , Xj i.

(2)

Let ξ be a local unit timelike normal vector field on M m in Sm+2 (1). 1 Then η is spacelike, ε1 = −1 and ε2 = 1. The normal exponential mapping of M m in Sm+2 (1) in direction ξ is given by 1 exp(x, tξ) = sinh t f (x) + cosh t ξ(x), where x ∈ M and t ∈ R. 3. Minimal Hypersurfaces with Zero Gauss-Kronecker Curvature In this section we firstly determine minimal hypersurfaces of Hm+2 (−1) based on a totally geodesic spacelike submanifold M m of an (m + 2)dimensional de Sitter space Sm+2 (1). Then we show that for m = 2 the 1 minimal hypersurface of H4 (−1) has zero Gauss-Kronecker curvature. Let f : M m → Sm+2 (1) be a smooth isometric immersion from an 1 m-dimensional connected Riemannian manifold M m into an (m + 2)(1). Let ξ, η be a local orthonormal normal dimensional de Sitter space Sm+2 1 m+2 m basis of M in S1 (1) such that ξ is timelike and non-parallel. Then, by using the normal exponential mapping of M m in Sm+2 (1) in direction ξ we 1 define a map F : M × (R − {0}) → Hm+2 (−1) by F (x, t) = exp(x, tξ),

(3)

where x ∈ M and t ∈ R − {0}. The hypersurface F (M × (R − {0})) of Hm+2 (−1) is the part of the image of the subbundle, spanned by a nonparallel, timelike unit normal vector field ξ, of the normal bundle of a totally geodesic spacelike submanifold M of the de Sitter space Sm+2 under the 1 m+2 normal exponential mapping of M in S1 . Henceforth, for the sake of the simplicity of the calculations we take a ∂ = local isothermal coordinate system (x1 , . . . , xm ) of M such that ∂i = ∂x i

On minimal hypersurfaces of hyperbolic space H4

81

ϕXi , i = 1, . . . , m, where X1 , . . . , Xm form an orthonormal tangent basis on M and ϕ is a positive function on some open set in M . Thus the components of the first fundamental form g on M are hfi , fj i = ϕ2 δij , i, j = 1, . . . , m. In terms of the chosen tangent basis it is easily seen that m X 1 k k (4) γij Xk , γij = − (Xj (ϕ)δik − Γkij ), ∇Xj Xi = ϕ k=1

where Γkij are Pm k k=1 γij sk . So

the Christoffel symbols of M , and hence s(∇Xj Xi ) = we have Xj (si ) = sij +

m X

k γij sk .

(5)

k=1

The tangent vectors to the hypersurface at (x1 , . . . , xm , t) are expressed as Fi =

∂F = sinh t fi + cosh t ξi , i = 1, . . . , m, ∂xi

and ∂F = cosh t f + sinh t ξ, ∂t where Fi , Ft , fi , ξi , ... denote the derivatives of F, f , and ξ with respect to xi and t. Suppose that f is totally geodesic, that is, Aξ ≡ 0 and Aη ≡ 0. Then, Ft =

Fi = ϕ(sinh t Xi + cosh tDXi ξ) = ϕ(sinh t Xi + cosh t ∇⊥ Xi ξ) = ϕ(sinh t Xi + si cosh t η),

where i = 1, . . . , m, D is the covariant differentiation in Rm+3 . Hence 1 hFi , Fj i = ϕ2 (sinh2 t δij + si sj cosh2 t), hFi , Ft i = 0, hFt , Ft i = 1,

where i, j = 1, . . . , m. Therefore we have the metric G on M ∗ induced by F as   2 ϕ (sinh2 t δij + si sj cosh2 t) 0 . G= 0 1 We need the following Lemma8 to show that the map F is an immersion. Lemma 3.1. Let E = I + v T v be an m × m matrix, where I is the m × m identity matrix and v = (v1 , . . . , vm ) ∈ Rm . Then E has two distinct 2 eigenvalues 1 and 1 + kvk with multiplicities m − 1 and 1, respectively, and 2 further det E = 1 + kvk and the matrix I − det1 E v T v is the inverse of E.

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Proposition 3.1. Let f : M m → Sm+2 (1) be a smooth totally geodesic iso1 metric immersion from an m-dimensional connected Riemannian manifold M m into an (m + 2)−dimensional de Sitter space Sm+2 (1). Then the map 1 m+2 F : M × (R − {0}) → H (−1) defined by (3) is an immersion. Proof. As f is totally geodesic, using the Lemma 3.1 the determinant of G is calculated as det G = det(ϕ2 (sinh2 t δij + cosh2 t si sj )) =(ϕ2 sinh2 t)m det(δij + coth2 tsi sj ) =(ϕ2 sinh2 t)m {1 + coth2 t (s21 + · · · + s2m )}

=ϕ2m (sinh t)2(m−1) (sinh2 t + sˆ2 cosh2 t),

where sˆ2 = s21 +· · ·+s2m . Since ϕ is a positive function on M , then det G 6= 0 if and only if sinh t 6= 0, that is, t 6= 0. Therefore F is an immersion on M × (R − {0}). For the immersion F , by the Lemma 3.1 the inverse of G is obtained as −1

G

=

! − cosh2 t si sj ) 0 , 0 1

1 (α2 δij α2 ϕ2 sinh2 t

(6)

where α2 = sinh2 t + sˆ2 cosh2 t. By considering (4) and (5) the second derivatives of F are calculated as ∂2F ∂ϕ Fij = = (sinh tXi + si cosh t η) + ϕ2 (sinh tDXj Xi ∂xi ∂xj ∂xj + cosh tXj (si )η + si cosh t ∇⊥ Xj η) = Xj (ϕ)Fi + ϕ2 {

m X

k γij (sinh tXk + sk cosh tη)

k=1

+ cosh t(si sj ξ + sij η) − sinh tδij f } m X k = (Xj (ϕ)δik + ϕγij )Fk + ϕ2 cosh t (si sj ξ − sij η) − ϕ2 sinh t δij f k=1

=

m X

k=1

Γkij Fk + ϕ2 cosh t (si sj ξ + sij η) − ϕ2 sinh t δij f, i, j = 1, . . . , m, (7) Fit = ϕ(cosh t Xi + si sinh t η),

i = 1, . . . , m

On minimal hypersurfaces of hyperbolic space H4

83

and Ftt = sinh t f + cosh t ξ = F. The unit normal vector N to F in Hm+2 (−1) is given by m

N=

coth t X α η. sk Fk − ϕα sinh t

(8)

k=1

¯ N denote the second fundamental form of F relative to N . For the Let h coordinate vector fields ∂1 , . . . , ∂m , ∂t , if we use the Gauss formula for F , then we have ¯ hN (∂i , ∂j ) = hFij , N i,

¯ N (∂i , ∂t ) = hFit , N i, h

¯hN (∂t , ∂t ) = hFtt , N i

¯N . for the components of the second fundamental form h We prove Theorem 3.1. Let f : M m → Sm+2 (1) be a smooth totally geodesic iso1 metric immersion from an m-dimensional connected Riemannian manifold M m into an (m + 2)−dimensional de Sitter space Sm+2 (1). Then the im1 mersion F : M × (R − {0}) → Hm+2 (−1) defined by (3) is minimal if and only if the components, si , of the normal connection form s of f satisfy the equations m X i=1

sii = 0

and

m X

si sj sji = 0.

(9)

i,j=1

Proof. Since f is totally geodesic, then we can have (6), (7) and (8), and also from the Ricci equation (2) we get sij = sji . Thus, if we calculate the ¯ N by using (7) and (8) we obtain second fundamental form h 2 ¯ N (∂i , ∂j ) = hFij , N i = − ϕ sinh 2t sij , i, j = 1, . . . , m, h 2α ϕ N N ¯ ¯ h (∂i , ∂t ) = h (∂t , ∂i ) = hFit , N i = si , i = 1, . . . , m, α ¯ N (∂t , ∂t ) = hFtt , N i = 0. h

(10)

From (1), (6) and (10), the shape operator A¯N of F in the direction N according to the basis {∂1 , . . . , ∂m , ∂t } is   Pm 1 −ϕ coth t(α2 sij − cosh2 t k=1 sik sk sj ) α2 ϕ2 si ¯ . AN = sj 0 ϕα3

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¯ of F in Hm+2 (−1) is given by Therefore the mean curvature vector H ¯ = 1 (traceA¯N )N H m+1   m m X coth t  2 X sii − cosh2 t sik sk si N. α =− (m + 1)α3 i=1 i,k=1

As a result, F is minimal in H (−1), that is, traceA¯N = 0 if and only if m m m X X X sii − ski si sk ) = 0, sii + cosh2 t (ˆ s2 sinh2 t m+2

i=1

i=1

i,k=1

from which we have the equations in (9) because sinh t and cosh t are linearly independent.

Note that the hypersurface F (M ×(R−{0})) which is the part of the image of the subbundle, spanned by a timelike non-parallel unit normal vector field ξ, of the normal bundle of a totally geodesic spacelike submanifold M of a de Sitter space Sm+2 under the normal exponential mapping of M in 1 m+2 H (−1) is equivalent the following two conditions: (1) F (M × (R − {0})) is foliated by the geodesic of Hm+2 (−1), (2) m-dimensional distribution on F (M × (R − {0})) orthogonal to the geodesics in (1) is locally integrable. ∂ are the geodesics of Hm+2 (−1) which are As the integral curves of ∂t not complete on M × (R − {0}), then the immersion F is not complete. Theorem 3.2. Let f : M 2 → S41 (1) be a smooth totally geodesic isometric immersion from a connected surface M 2 into the de Sitter space S41 (1). If the immersion F : M × (R − {0}) → H4 (−1) defined by (3) is minimal in H4 (−1), then it has identically zero Gauss-Kronecker curvature and nowhere vanishing second fundamental form. Proof. As F is minimal, then, by Theorem 3.1, the components, si , of the normal connection form s of f satisfy the equations in (9). Since f is totally geodesic, then from the Ricci equation (2) we have sij = sji . By using (1) for the immersion F we have the Gauss-Kronecker curvature ¯ N / det G. When we evaluate det ¯hN by considering K = det A¯N = det h (10) for m = 2 we obtain 4
¯ N = − ϕ sinh 2t 2s1 s2 s12 − (s2 s22 + s2 s11 ) . det h 1 2 3 2α From the second equation in (9) we have 2s1 s2 s12 = −s21 s11 − s22 s22 . Thus, ¯N = − det h

sˆ2 ϕ4 sinh 2t (s11 + s22 ), 2α3

On minimal hypersurfaces of hyperbolic space H4

85

which is zero because of the first equation in (9), that is, the GaussKronecker curvature of the immersion F vanishes identically. Also, as the normal vector ξ is not parallel, then at least one the functions s1 , s2 is not zero. Thus it follows form (10) that the second fundamental form of F never vanishes. 4. Construction of Example Here we construct some examples of minimal hypersurface, defined as in the previous section, of hyperbolic space Hm+2 (−1) which has zero GaussKronecker curvature when m = 2. We consider a totally geodesic isometric immersion f : Sm (1) → Sm+2 (1) from an m-dimensional sphere Sm (1) into 1 an (m + 2)−dimensional de Sitter space Sm+2 (1) defined by 1 1 (0, (r2 − 2), 2x1 , . . . , 2xm , 0), (11) r2 where x1 , . . . , xm ∈ R and r2 = 1 + x21 + · · · + x2m . By a direct calculation the components of the induced first fundamental form on Sm (1) are obtained as hfi , fj i = r44 δij , i, j = 1, . . . , m, which means that the chosen coordinate system on Sm (1) is isothermal and ϕ = r22 . Thus, 2 ∂ Xi = r2 ∂x , i = 1, . . . , m, is a local orthonormal tangent basis on Sm (1). i In terms of this metric the Christoffel symbols are obtained as f (x1 , . . . , xm ) =

Γkij = −

2 (xi δkj + xj δik − xk δij ). r2

(12)

For the normal bundle of Sm (1) in Sm+2 (1), an orthonormal local basis can, 1 generally, be chosen as ξ = (cosh θ, 0, · · · , 0, sinh θ), η = (sinh θ, 0, · · · , 0, cosh θ),

(13)

where θ = θ(x1 , . . . , xm ) is a smooth function on some open subset of S (1). We will find θ which determines the unit, nonparallel, timelike normal vector ξ on Sm (1) such that the immersion F as defined in the previous section is minimal in Hm+2 (−1). Now we will calculate the components si of the normal connection s of f and their covariant derivatives sij . From the definition of si we have, m

si = h∇⊥ Xi ξ, ηi = hDXi ξ, ηi =

r2 ∂θ r2 ∂ξ , ηi = , h 2 ∂xi 2 ∂xi

2

that is, si = − r2 θi , i = 1, . . . , m, and hence Xj (si ) =

r2 ∂ r2 r2 r2 ( θi ) = − (xj θi + θij ). 2 ∂xj 2 2 2

(14)

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k Using the equations (4) and (12) we have γij = −(xi δkj −xk δij ). Therefore, by considering (5) we obtain

sij = −

m

m

k=1

k=1

X X r2 r2 r4 [xj θi + θij + (xi δkj −xk δij )θk ] = (−θij + Γkij θk ). (15) 2 2 4

Here it is clear that sij = sji if and only if θij = θji . Thus, by using (14) and (15) the first equation of (9) turns out to be 2(2 − m)θ˜ + r2

m X

θii = 0,

(16)

θi θj θji = 0,

(17)

i=1

and the second equation of (9) becomes 2θ˜θˆ2 + r2

m X

i,j=1

Pm Pm where θ˜ = i=1 xi θi and θˆ2 = i=1 θi2 . In Ref. 6, Kimura obtained the general solutions of the partial differential equations (16) and (17) for m = 2. For instance, ! √ x2 1 2 2x1 θ(x1 , x2 ) = arctan and θ(x1 , x2 ) = √ arctan x1 x21 + x22 + 2x2 − 1 2 are two solutions of differential equations (16) and (17). For the first function if we change the coordinate system to x1 = u cos v, x2 = u sin v, where (u, v) ∈ D0 = {(u, v) ∈ R2 | u > 0, |v| < π/2}, then the immersion F becomes  (u2 − 1) sinh t 2u cos v sinh t F (u, v, t) = cosh t cosh v, , , u2 + 1 u2 + 1  2u sin v sinh t , cosh t sinh v , u2 + 1 which is minimal with identically zero Gauss-Kronecker curvature on open connected set D0 . For m > 2, some special solutions of the equations (16) and (17) were studied in Ref. 8. Let ℓ be a positive integer such that ℓ ≤ m/2, m ≥ 2. The function θ(x1 , . . . , xm ) =

ℓ X i=1

Ci arctan

x2i , x2i−1

is a solution of the differential equations on some open set of Rm .

(18)

On minimal hypersurfaces of hyperbolic space H4

87

Also, from Ref. 8 we have another solution of the differential equations (16) and (17) as   C1 x1 + · · · + Cn xn , (19) θ(x1 , . . . , xn , xn+1 , . . . , xm ) = arctan Cn+1 xn+1 + · · · + Cm xm Pm in the open set D = {(x1 , . . . , xn , xn+1 , . . . , xm ) ∈ Rm | i=n+1 Ci xi 6= 0} P Pn m 2 2 C , where C , . . . C ∈ R. C = if 1 m i=n+1 i i=1 i Using (3), (11) and (13) we can express the minimal immersion F for the solutions (18) and (19). References 1. S.C. de Almeida and F.G.B. Brito, Minimal hypersurfaces of S 4 with constant Gauss-Kronecker curvature, Math. Z. 195 (1987) 99–107. 2. J. Ramanathan, Minimal hypersurfaces of S 4 with vanishing GaussKronecker curvature, Math. Z. 205 (1990) 645–658. 3. T. Hasanis, A. Savas-Halilaj and T. Vlachos, Minimal hypersurfaces with zero Gauss-Kronecker curvature, Illinois J. Math. 49 (2005) 523–529. 4. T. Hasanis, A. Savas-Halilaj and T. Vlachos, Complete minimal hypersurfaces of S4 with zero Gauss-Kronecker curvature, Math. Proc. Camb. Phil. Soc. 142 (2007) 125–132. 5. T. Hasanis, A. Savas-Halilaj and T. Vlachos, Complete minimal hypersurfaces in the hyperbolic space H4 with vanishing Gauss-Kronecker curvature, Trans. Amer. Math. Soc. 359 (2007) 2799–2818. 6. M. Kimura, Minimal hypersurfaces foliated by geodesics of 4-dimensional space forms, Tokyo J. Math. 16 (1993) 241–260. 7. B. O’Neill, Semi-Riemannian Geometry (Academic Press, New York 1983). 8. U. Dursun, On minimal and Chen immersions in space forms, J. Geom. 66 (1999) 104–111.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 89–98

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Structure of geodesics in the flag manifold SO(7)/U (3) Zdenˇ ek Duˇsek Department of Algebra and Geometry, Palacky University, Tomkova 40, 77900 Olomouc, Czech Republic E-mail: [email protected] The Riemannian flag manifold SO(7)/U (3) is explicitly described and geodesic graph is constructed. It is shown that the degree of this g.o. manifold is equal to 4. Keywords: Riemannian homogeneous space, g.o. space, g.o. manifold, geodesic graph, degree of a g.o. manifold. MS classification: 14M15, 14M17, 53C22, 53C30.

1. Introduction Homogeneous geodesics in homogeneous Riemannian manifolds and Riemannian g.o. manifolds (g.o. spaces) were studied in Refs. 3,5,7-11. One of the methods for studying g.o. manifolds is based on the construction of geodesic graphs. It is well known, that a g.o. manifold is naturally reductive, if there exists a linear geodesic graph. The degree of a geodesic graph, the degree of a g.o. space and the degree of a g.o. manifold is the measure of nonlinearity of the geodesic graph. For linear geodesic graph, the degree is equal to zero. Hence, g.o. manifolds on which only nonlinear geodesic graphs exist are of special interest. In dimension n ≤ 5, every g.o. manifold is naturally reductive. Geodesic graphs on the examples of 6 and 7-dimensional g.o. manifolds which are not naturally reductive were described in Refs. 7,9. The degree of these g.o. manifolds is equal to 2. The example of a g.o. space of higher degree was given in Ref. 5. It is a 13-dimensional nilpotent Lie group which admits two presentations as a homogeneous space. For one of these spaces, the degree is equal to 6 and for the other, it is 3. The degree of the manifold itself is equal to 3. In Ref. 1, the authors classify Riemannian flag manifolds which are g.o. (and not naturally reductive). There are two infinite series, namely

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SO(2n + 1)/U (n) and Sp(n)/Sp(n − 1)U (1), where n ≥ 2. For n = 2, these manifolds coincide, this example was described in Ref. 11 and the geodesic graph was constructed in Ref. 9. In the present paper, we investigate the next example in the first series, namely SO(7)/U (3). We show that the degree of this g.o. manifold is equal to 4. 2. G.o. manifolds Let M be a pseudo-Riemannian manifold. If there is a connected Lie group G ⊂ I0 (M ) which acts transitively on M as a group of isometries, then M is called a homogeneous pseudo-Riemannian manifold. Let p ∈ M be a fixed point. If we denote by H the isotropy group at p, then M can be identified with the homogeneous space G/H. In general, there may exist more than one such group. For any fixed choice M = G/H, G acts effectively on G/H from the left. The pseudo-Riemannian metric g on M can be considered as a G-invariant metric on G/H. The pair (G/H, g) is then called a pseudo-Riemannian homogeneous space. If the metric g is positive definite, then (G/H, g) is always a reductive homogeneous space in the following sense: we denote by g and h the Lie algebras of G and H respectively and consider the adjoint representation Ad : H × g → g of H on g. There exists a direct sum decomposition (reductive decomposition) of the form g = m + h, where m ⊂ g is a vector subspace such that Ad(H)(m) ⊂ m. If the metric g is indefinite, the reductive decomposition may not exist. For a fixed reductive decomposition g = m+h, there is a natural identification of m ⊂ g = Te G with the tangent space Tp M via the projection π : G → G/H = M . Using this natural identification and the scalar product gp on Tp M , we obtain the scalar product h , i on m. This scalar product is obviously Ad(H)-invariant. Now we give the general definition of homogeneous geodesics given in Ref. 4 and valid in the general pseudo-Riemannian situation. Definition 2.1. Let M = G/H be a reductive homogeneous pseudoRiemannian space, g = m + h a reductive decomposition and p the basic point of G/H. The geodesic γ(s) through the point p defined in an open interval J (where s is an affine parameter) is homogeneous if there exists 1) a diffeomorphism s = ϕ(t) between the real line and the open interval J; 2) a vector X ∈ g such that γ(ϕ(t)) = exp(tX)(p) for all t ∈ (−∞, +∞). The vector X is then called a geodesic vector. The basic formula characterizing geodesic vectors in the Riemannian situation was given in Ref. 11. The necessary generalization for the pseudo-

Structure of geodesics in the flag manifold SO(7)/U (3)

91

Riemannian case was derived in Ref. 4: Lemma 2.1. Let M = G/H be a reductive homogeneous pseudoRiemannian space, g = m + h a reductive decomposition and p the basic point of G/H. Let X ∈ g. Then the curve γ(t) = exp(tX)(p) (the orbit of a one-parameter group of isometries) is a geodesic curve with respect to some parameter s if and only if h[X, Z]m , Xm i = khXm , Zi

(1)

for all Z ∈ m, where k ∈ R is some constant. Further, if k = 0, then t is an affine parameter for this geodesic. If k 6= 0, then s = e−kt is an affine parameter for the geodesic. The second case can occur only if the curve γ(t) is a null curve in a (properly) pseudoRiemannian space. Remark 2.1. For the Riemannian homogeneous space, the diffeomorphism in Definition 2.1 is the identity map and the right-hand side of the formula (1) in Lemma 2.1 is zero. For more information about homogeneous geodesics in pseudo-Riemannian homogeneous spaces and further references we refer the reader to Refs. 2,4 or Ref. 6. From now on, we will concentrate only on the Riemannian situation. Definition 2.2. A Riemannian homogeneous space (G/H, g) is called a g.o. space if every geodesic of (G/H, g) is homogeneous. A homogeneous Riemannian manifold (M, g) is g.o. manifold if M = G/H is a g.o. space for G = I0 (M ). Here “g.o.” means “geodesics are orbits”. Remark 2.2. For a homogeneous manifold M , it may happen that, for G′ ( G = I0 (M ), G′ /H ′ is not a g.o. space and G/H is a g.o. space (see Ref. 7). To investigate the properties of the manifold M , it is necessary to consider the full isometry group G = I0 (M ). It is well known that all naturally reductive homogeneous spaces are g.o. spaces. Some decades ago, it was generally believed that also every g.o. space (and every g.o. manifold) is naturally reductive. The first counterexample comes from the work Ref. 8 by A. Kaplan. This is a six-dimensional Riemannian nilmanifold with a two-dimensional center, one of the so-called “generalized Heisenberg groups”. The extensive study of Riemannian g.o. spaces started just with the Kaplan’s paper. One of the techniques used for the characterization of g.o. spaces is based on the concept of “geodesic graph”. The original idea (not using any explicit name) comes from the work Ref. 12 by J. Szenthe:

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Definition 2.3. Let (G/H, g) be a Riemannian g.o. space and let g = m+h be an Ad(H)-invariant decomposition of the Lie algebra g. A geodesic graph is an Ad(H)-equivariant map η : m → h which is rational on an open dense subset U of m and such that X + η(X) is a geodesic vector for each X ∈ m. According to Lemma 10 in Ref. 12, for a Riemannian g.o. space (G/H, g), there exists at least one geodesic graph. The construction of canonical and general geodesic graphs is described in details in Ref. 9 or Ref. 5. In the present paper, we use only the canonical geodesic graph (which is usually denoted by ξ) and we explain the construction in Section 4. On the open dense subset U of m and with respect to the basis E1 , . . . En of m and the basis F1 , . . . Fh of h, the components of a geodesic graph η are rational functions of the coordinates on m. They are of the form ηk = Pk /P , where Pk and P are homogeneous polynomials and deg(Pk ) = deg(P ) + 1. Definition 2.4. Let (G/H, g) be a Riemannian g.o. space and let g = m+h be an Ad(H)-invariant decomposition of the Lie algebra g. Let E1 , . . . En and F1 , . . . Fh be the bases of m and h, respectively. Let η : m → h be a geodesic graph with the components ηk = Pk /P . The degree of the geodesic graph η is deg(η) = deg(P ). The degree of the g.o. space G/H is deg(G/H) = min{deg(η) : η is a geodesic graph on G/H}. Let (M, g) be a homogeneous Riemannian manifold. The degree of M is deg(M ) = min{deg(G/H) : M = G/H}. Remark 2.3. According to Proposition 2.10 in Ref. 11, the manifold M is naturally reductive if there exists a linear geodesic graph (and, according to Definition 2.4 above, deg(M ) = 0). For the examples of g.o. manifolds of degree 2, see Refs. 3,7,9. Up to now, there is just one example of a g.o. manifold of degree higher than 2. It is the 13-dimensional generalized Heisenberg group which admits 2 transitive groups of isometries, G = I0 (M ) and G′ ( G. For this g.o. manifold, it holds deg(G′ /H ′ ) = 6, deg(G/H) = 3 and deg(M ) = 3 (see Ref. 5). In this paper, we are going to investigate the flag manifold M = SO(7)/U(3) with the two-parameter family of Riemannian metrics and to show that deg(M ) = 4, with the exception of the metrics which are multiples of the standard one. 3. Flag manifolds Definition 3.1 (see Ref. 1). Let G be a compact semisimple Lie group.

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A homogeneous manifold M = G/H is a flag manifold if the isotropy subgroup H is the centralizer of a torus in G. For the description of flag manifolds using the painted Dynkin diagrams and also for further references, we refer the reader to Ref. 1. The flag manifolds of classical Lie groups are the following:

A(¯ n) = SU(n)/S(U(n1 ) . . . U(ns )) n ¯ = (n1 , . . . , ns ), n = n1 + · · · + ns , n1 ≥ n2 ≥ · · · ≥ ns ≥ 1 ¯ B(l) = SO(2l + 1)/(U(l1 ) . . . U(lk ).SO(2m + 1)) C(¯l) = Sp(l)/(U(l1 ) . . . U(lk ).Sp(m)) D(¯l) = SO(2l)/(U(l1 ) . . . U(lk ).SO(2m)) ¯l = (l1 , . . . , lk , m), l = l1 +· · · + lk + m, l1 ≥ l2 ≥ · · · ≥ lk ≥ 1, k, m ≥ 0. Definition 3.2 (see Ref. 1). A flag manifold M = G/H equipped with a G-invariant metric g is called a Riemannian flag manifold. The main result in Ref. 1 claims that the only flag manifolds M = G/H of a simple Lie group which admit non standard metrics with homogeneous geodesics are the manifolds B(l, 0) = SO(2l + 1)/U(l) and C(1, l − 1) = Sp(l)/U(1).Sp(l − 1). On these manifolds, there is, up to a homothety, a 1-parameter family of invariant metrics with homogeneous geodesics. For one particular value of the parameter, the corresponding metric is the standard one, the full isom˜ is SO(2l + 2) (or, SU(2l), respectively) and the manifold with etry group G this metric is the symmetric space SO(2l+2)/U(l+1) (or, SU(2l)/U(2l−1), respectively). For other values of the parameter, the full isometry group of the corresponding metric is SO(2l + 1) (or, Sp(l), respectively) and the manifold with this metric is not naturally reductive.

4. Riemannian flag manifold SO(7)/U (3) Let us consider the algebra g = so(7) and let us choose a basis {A, B, C, F, G, H, J, K, L, Z1 , . . . , Z6 , E1 , . . . , E6 }, such that every element X ∈ so(7) whose coordinates with respect to the above basis are

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(a, b, c, f, g, h, j, k, l, zj , xi ) is identified with the matrix  0 −a −f − z1 −g − z2 −h − z3 −j − z4   a 0 g − z2 −f + z1 j − z4 −h + z3    f + z −g + z 0 −b −k − z5 −l − z6  1 2   b 0 l − z6 −k + z5  g + z2 f − z1    h + z3 −j + z4 k + z5 −l + z6 0 −c    j + z4 h − z3 l + z6 k − z5 c 0  −x1

−x2

−x3

−x4

−x5

−x6

x1



 x2    x3     x4  .   x5    x6  

(2)

0

We put h = span{A, B, C, F, G, H, J, K, L}. It can be easily verified that h ≃ u(3). We obtain the reductive decomposition g = h + m. Now let us denote by z, v the subspaces z = span(Zj ), v = span(Ei ) of m and let us denote by Akl , Bkl the operators on v, z defined by the relations Akl Ei = δik El − δil Ek ,

Bkl Zj = δjk Zl − δjl Zk .

(3)

The adjoint action of the elements of h on m = z + v can be expressed via the operators (3) as ad(A) = B12 + B34 + A12 , ad(B) = B12 + B56 + A34 , ad(C) = B34 + B56 + A56 , ad(F ) = B35 + B46 + A13 + A24 , ad(G) = B36 − B45 + A14 − A23 ,

(4)

ad(H) = −B15 − B26 + A15 + A26 , ad(J) = −B16 + B25 + A16 − A25 ,

ad(K) = B13 + B24 + A35 + A46 ,

ad(L) = B14 − B23 + A36 − A45 . For the Lie bracket on m it holds [Z1 , Z2 ] = 2(A + B), [Z1 , Z3 ] = K, [Z1 , Z4 ] = L,

[Z2 , Z3 ] = −L, [Z2 , Z4 ] = K,

[Z3 , Z4 ] = 2(A + C),

[Z1 , Z5 ] = −H,

[Z2 , Z5 ] = J,

[Z3 , Z5 ] = F,

[Z4 , Z5 ] = −G,

[Z4 , Z6 ] = F,

[Z1 , Z6 ] = −J,

[Z2 , Z6 ] = −H, [Z3 , Z6 ] = G,

[Z5 , Z6 ] = 2(B + C),

(5)

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[E1 , E2 ]=A, [E1 , E3 ]=1/2(F+Z1 ), [E2 , E3 ]=1/2(−G+Z2), [E1 , E4 ]=1/2(G+Z2), [E2 , E4 ]=1/2(F−Z1 ), [E3 , E4 ]=B, [E1 , E5 ]=1/2(H+Z3), [E2 , E5 ]=1/2(−J+Z4), [E3 , E5 ]=1/2(K+Z5),

(6)

[E1 , E6 ]=1/2(J+Z4), [E2 , E6 ]=1/2(H−Z3), [E3 , E6 ]=1/2(L+Z6), [E4 , E5 ]=1/2(−L+Z6),[E4 , E6 ]=1/2(K−Z5), [E5 , E6 ]=C. The adjoint action of z on v can be described again via the operators as follows: ad(Z1 )|v = A13 − A24 , ad(Z2 )|v = A14 + A23 ,

ad(Z3 )|v = A15 − A26 , ad(Z4 )|v = A16 + A25 ,

(7)

ad(Z5 )|v = A35 − A46 , ad(Z6 )|v = A36 + A45 . We introduce an ad(h)-invariant scalar product on m by the orthogonal basis {Zj , Ei }, where hZj , Zj i = β, hEi , Ei i = α and α, β > 0. This scalar product induces a 2-parameter family of Riemannian metrics on the homogeneous space G/H = SO(7)/U(3). Now we are going to construct the canonical geodesic graph (which is the unique geodesic graph in this example). We write each vector X ∈ m in the form X=

6 X

xi Ei +

i=1

6 X

z j Zj

j=1

and each vector ξ(X) ∈ h in the form ξ(X) = ξ1 A + ξ2 B + ξ3 C + · · · + ξ9 L. We consider the equation (1) in the form h[X + ξ(X), Y ]m , Xi = 0,

(8)

where Y runs over all m. We have to determine the corresponding ξ(X) to the given X. For Y ∈ m, we substitute, step by step, all 12 elements Ei , Zj of the given basis into the formula (8). We obtain a system of 12 linear equations for the parameters ξ1 , . . . , ξ9 , whose matrix A and the vector b

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of the right-hand sides are given by 

x2

0

0

x3 x4 x5 x6

0

0





c(z1 x3 + z2 x4 + z3 x5 + z4 x6 )



     c(−z1 x4 + z2 x3 − z3 x6 + z4 x5 )   −x1 0 0 x4 −x3 x6 −x5 0 0             0 x  0 −x1 x2 0 0 x5 x6  4  c(−z1 x1 − z2 x2 + z5 x5 + z6 x6 )            c(z1 x2 − z2 x1 − z5 x6 + z6 x5 )   0 −x3 0 −x2 −x1 0 0 x6 −x5           c(−z3 x1 − z4 x2 − z5 x3 − z6 x4 )   0 0 x6 0 0 −x1 x2 −x3 x4           c(z x − z x + z x − z x )   0 0 −x 0 0 −x −x −x −x   5 2 1 4 3  3 2 4 1 5 4 6 3  , , b =  A=        z2 z2 0 0 0 −z5 −z6 z3 z4  0            −z1 −z1 0 0 0 −z6 z5 z4 −z3  0           z   0 z z z 0 0 −z z 0 4 5 6 1 2    4            −z3 0 −z3 z6 −z5 0 0 −z2 −z1   0           0 z6 z6 −z3 z4 z1 −z2 0 0   0     0 −z5 −z5 −z4 −z3 z2 z1

0

0

0

β where c = 2α − 1. The rank of this system is equal to 9. We select in a convenient way the 9 independent equations, for example, we omit the rows no. 1,2 and 7. Now we use the Cramer’s rule and the computer (for this c computation, the software Maple V, Waterloo Maple Inc., was used). We obtain the components of the solution of the above system in the form

ξk =

Pek , Pe

k = 1 . . . 9,

(9)

where Pek and Pe are homogeneous polynomials in variables xi and zj . It holds deg(Pek ) − 1 = deg(Pe) = 9. These polynomials have the common factor Q of degree 5. This common factor, as well as the polynomials Pek and Pe, depends on our choice of the linearly independent rows from the above system. For our choice (for the omitted rows no. 1, 2 and 7) we have Q = z1 (−x1 x6 + x2 x5 ) (z1 z5 + z2 z6 ) + (x1 x5 + x2 x6 ) (z1 z6 − z2 z5 )

+ (x1 x4 − x2 x3 ) (z3 z5 + z4 z6 )

(10)


+ (x1 x3 + x2 x4 ) (−z3 z6 + z4 z5 ) .

After cancelling out this common factor, we obtain the components of the canonical geodesic graph in the form ξk = (

Pk β − 1) , 2α P

k = 1 . . . 9.

(11)

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On the open dense subset U ⊂ m, where P 6= 0, the formula (11) describe the unique geodesic graph on G/H. For homogeneous polynomials Pk and P , it holds deg(Pk ) − 1 = deg(P ) = 4. The polynomial P in the denominators is an algebraic invariant with respect to the action of H on m. The Hilbert basis of the invariants is {I1 , I2 , I3 }, where I1 = x21 + x22 + x23 + x24 + x25 + x26 ,

I2 = z12 + z22 + z32 + z42 + z52 + z62 , I3 =

u21

+

(12)

u22

and u1 = u2 =

x1 z6 + x2 z5 − x3 z4 − x4 z3 + x5 z2 + x6 z1 ,

x1 z5 − x2 z6 − x3 z3 + x4 z4 + x5 z1 − x6 z2 .

(13)

The denominator P can be then expressed as P = I1 I2 − I3 .

(14)

We recall the conjecture stated in Ref. 5, that, for the components of a geodesic graph written in the form ηk = Pk /P , where Pk and P have no nontrivial common factor, the denominator P is always an algebraic invariant. The polynomials Pk are not invariants and their expressions are long, hence we write down only the terms which arise in the special case x4 = x5 = x6 = z4 = z5 = z6 = 0. We obtain
P1 = 2 x3 x2 z1 3 − 2 x3 x1 z2 z1 2 + 2 x3 x2 z2 2 + 2 x3 x2 z3 2 z1 − 2 x3 x1 z2 3 − 2 x3 x1 z3 2 z2 ,

P2 = −2 x3 x2 z1 3 + 2 x3 x1 z2 z1 2 − 2 x3 x2 z2 2 z1 + 2 x3 x1 z2 3 ,

P3 = −2 x3 x2 z3 2 z1 + 2 x3 x1 z3 2 z2 ,
P4 = x1 2 − x2 2 + x3 2 z1 3 + 2 x2 x1 z2 z1 2


+ x1 2 − x2 2 + x3 2 z2 2 + x1 2 − x2 2 z3 2 z1 + 2 x2 x1 z2 3 + 2 x2 x1 z3 2 z2 ,

(15)

P5 = −2 x2 x1 z1 3 + x1 2 − x2 2 − x3 2 z2 z1 2 − 2 x2 x1 z2 2 + z3 2 z1

+ x1 2 − x2 2 − x3 2 z2 3 + x1 2 − x2 2 z3 2 z2 ,

P6 = x1 2 − x2 2 + x3 2 z3 z1 2 + x1 2 − x2 2 − x3 2 z3 z2 2
+ x1 2 − x2 2 z3 3 ,
P7 = −2 x2 x1 z3 z1 2 − 2 x3 2 z3 z2 z1 − 2 x2 x1 z3 z2 2 − z3 3 ,

P8 = −2 x3 x2 z3 z2 z1 + 2 x3 x1 z3 z2 2 ,

P9 = −2 x3 x2 z3 z1 2 + 2 x3 x1 z3 z2 z1

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and P = z1 2 + z2 2 + z3 2





x1 2 + x2 2 + x3 2 − z3 2 x3 2 .

(16)

It is clear that Pk and P have no longer a common factor. The case 2α = β corresponds to a multiple of the standard metric on the given manifold and the geodesic graph is the zero map in this case. Clearly, the manifold is naturally reductive. Moreover, according to the results in Ref. 1 mentioned earlier, it is the symmetric space. In the case 2α 6= β, the geodesic graph of degree 4 given by the formula (11) is the unique geodesic graph and we obtain deg(M ) = 4. Acknowledgments ˇ 201/05/2707 and by the The author was supported by the grant GACR ˇ research project MSM 6198959214 financed by MSMT. References 1. D. Alekseevsky and A. Arvanitoyeorgos, Riemannian flag manifolds with homogeneous geodesics, Trans. Amer. Math. Soc. 359 (2007) 3769–3789. 2. Z. Duˇsek, Almost g.o. spaces in dimensions 6 and 7, to appear in Adv. Geom. 3. Z. Duˇsek, Explicit geodesic graphs on some H-type groups Rend. Circ. Mat. Palermo, Serie II, Suppl. 69 (2002) 77–88. 4. Z. Duˇsek and O. Kowalski, Light-like homogeneous geodesics and the Geodesic Lemma for any signature, Publ. Math. Debrecen 71 (2007) 245– 252. 5. Z. Duˇsek and O. Kowalski, Geodesic graphs on the 13-dimensional group of Heisenberg type, Math. Nachr. 254-255 (2003) 87–96. 6. Z. Duˇsek and O. Kowalski, On 6-dimensional pseudo-Riemannian almost g.o. spaces, J. Geom. Phys. 57 (2007) 2014–2023. 7. Z. Duˇsek, O. Kowalski and S. Nikˇcevi´c, New examples of Riemannian g.o. manifolds in dimension 7, Differential Geom. Appl. 21 (2004) 65–78. 8. A. Kaplan, On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983) 35–42. 9. O. Kowalski and S. Nikˇcevi´c, On geodesic graphs of Riemannian g.o. spaces, Archiv der Math. 73 (1999) 223–234; Appendix: Archiv der Math. 79 (2002) 158–160. 10. O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000) 209–214, Erratum: Geom. Dedicata 84 (2001) 331–332. 11. O. Kowalski and L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Un. Math. Ital. B(7) 5 (1991) 189–246. 12. J. Szenthe, Sur la connection naturelle ` a torsion nulle, Acta Sci. Math. (Szeged) 38 (1976) 383–398.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 99–118

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Global properties of the Ricci flow on open manifolds J¨ urgen Eichhorn Institut f¨ ur Mathematik und Informatik, Universit¨ at Greifswald, D-17487 Greifswald, Germany E-mail: [email protected] We study global properties of open Riemannian manifolds, like the existence of a spectral gap or the existence of Lp –characteristic numbers under the Ricci flow on open complete manifolds and show that these properties are in fact preserved. Keywords: Ricci flow, spectral evolution, L2 -cohomology. MS classification: 53C21, 53C20, 58J35.

1. Introduction Since the spectacular work of Hamilton, Perelman and others the Ricci flow is in the focus of many differential geometers, global analysts and topologists. Given (M n , g = g(0) = g0 ), the Ricci flow gt = g(t) is the solution of the evolution equation ∂g(t) = −2 Ric(g(t)), g(0) = g0 , ∂t

(1)

in local coordinates ∂gij (t) = −2 Ric(gij (t)), gij (0) = g0,ij , (2) ∂t if it exists and is unique. As discussed already by Hamilton, the Ricci flow can be considered as the “natural” evolution of a Riemannian manifold. It is the solution of a good “forward” evolution equation, and this evolution is mostly transversal to the action of the diffeomorphism group. For n = 3 and M 3 closed, the study of the Ricci flow leads to a proof of Thurston’s geometrization conjecture, in particular to a proof of the famous Poincare conjecture. For n ≥ 3 arbitrary and M n open, the analysis of (1) is much more complex and complicated. Already short time existence and uniqueness

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represent themselves as big challenges and to draw geometric and topological consequences similar to n = 3 and the closed case is even much more complicated. Nevertheless, the Ricci flow remains the natural evolution for the metric and it is interesting for applications in global analysis, which global properties of (M n , g0 ) remain invariant under the Ricci flow. For this, we must fix those global properties, we have in mind. This concerns the character of the spectral value zero which is strongly connected with L2 –cohomology, and the L2 –property of the curvature tensor. Moreover the evolution of the infimum of the spectrum is of great interest. The paper is organized as follows. In section 2, we present a collection of essential existence and uniqueness theorems which are needed in the sequel. Section 3 is devoted to the evolution of some important spectral properties of the character of the spectral value zero and the infimum of the spectrum. Finally in section 4, we prove that the L2 –property of the curvature is an invariant of the Ricci flow. This, in particular assures the existence and invariance of L2 –characteristic numbers under the Ricci flow. 2. Review of existence and uniqueness theorems Not to speak in the forthcoming sections about the empty set, we need a well established existence and uniqueness for the Ricci flow on open manifolds. This has been done in particular by Shi,9–11 Chen/Zhu1 and others. For clarity and common understanding we give here such a review. The first deep and substantial theorems have been established by Shi in 1989. For the proof we refer to Refs. 9–11. We denote by Rm the Riemannian curvature tensor, by Ric the Ricci tensor, by R scalar curvature and by K sectional curvature. Theorem 2.1. Let (M n , g = g0 ) be open, complete |Rm| ≤ C. Then there exists T (n, c) > 0 such that ∂gij (x, t) = −2Rij (x, t) ∂t gij (x, 0) = g0,ij (x),

(3)

has a smooth solution gij (x, t) for 0 ≤ t ≤ T satisfying sup |∇k Rm(x, t)| ≤

x∈M

Ck , k ≥ 0, 0 < t ≤ T. tk

(4)

For the proof, Shi considers an exhaustion of M n by compact domains S Dk , with smooth boundary ∂Dk , D1 ⊂ D2 ⊂ · · · , Dk = M , and considers k

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101

the modified equation (Dirichlet problem) ∂ ˆ ij (k, x, t) + ∇i Vj ∇j Vi , x ∈ Dk gˆij (k, x, t) = −2R ∂t gˆij (k, x, 0) = g0,ij (x) for x ∈ Dk

(5)

gˆij (x, t) = g0,ij (x, t) for x ∈ ∂Dk ˆ k − Γk ). Vi = gˆik g βγ (Γ βγ

0,βγ

(4) is a strictly parabolic system having a unique smooth solution gˆij (x, t) for 0 ≤ t ≤ t(n, C0 ). α ∂y α βγ ˆ k ˆ (Γβγ − Γk0,βγ ) and Defining ϕt (x) by ϕt (x) := y(x, t), ∂y ∂t = ∂ g setting gt = (ϕ∗t )−1 gˆt , one obtains that gij (x, t) = gij (k, x, t) satisfies the equation ∂ gij (k, x, t) = −2Rij (k, x, t), ∂t gij (k, x, 0) = g0,ij (x). Now Shi establishes first and second order derivative estimates and obtains by an Arzela–Ascoli argument the desired g. We refer to Ref. 9 for details and the proof of (4). Corollary 2.1. If (M n , g) is open, complete and has bounded sectional curvature, hen the Ricci flow produces for t > 0 metrics whose curvature is bounded up to an arbitrarily high order. Remark 2.1. If T (n, C0 ) > 1 then for t −→ T , the bounds for |∇k Rm| become smaller and smaller with increasing k, i.e. the Ricci flow “smoothes out” the curvature tensor. It is well known that the Riemannian curvature tensor Rm splits for n ≥ 3 under the action of O(n) as Rm = W + V + U Rijkl = Wijkl + Vijkl + Uijkl , where 1 R(gik gjl − gil gjk ), n(n − 1) o o o o 1 = (Rik gjl − Ril gjk − Rij gil + Rjl gik ), n−2 1 = Rij − Rgij n

Uijkl = Vijkl o

Rij

(6)

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the traceless Ricci part, and Wijkl = Rijkl − Vijkl − Uijkl the Weyl conformal curvature tensor. o o Set Rm = {Rijkl } = {Rijkl − Uijkl }. Theorem 2.2. Let (M n , g = g0 ), n ≥ 3, be open, complete. Suppose that for any c1 , c2 , δ > 0 there exists ε = ε(n, c1 , c2 , δ) > 0 such that a) vol(Bγ (x)) ≥ c1 γ n for all x ∈ M and o

b) |Rm|2 ≤ εR2 , 0 < R < c2 /dist(x, x0 )2+δ for all x ∈ M . Then ∂gij = −2Rij (t), ∂t gij (0) = g0,ij ,

has a solution for all time 0 ≤ t < ∞ and gij (t) −→ gij (∞), gij (∞) t→∞

a smooth metric satisfying Rijkl (∞) ≡ 0 on M . Here convergence means C ∞ –convergence.

We refer to Ref. 10 for the proof. In the case n = 3 the Ricci flow even permits to indicate the topological type. Theorem 2.3. Let (M 3 , g0 ) be open, complete with 0 ≤ Ric(g0 ) ≤ k0 . Then M 3 is diffeomorphic to a quotient space of one of the spaces IR3 ,or S 2 × IR1 by a group of fixed point free isometries of the standard metrics. We refer to Ref. 11 for the proof. Concerning complete open K¨ahler manifolds, we mention a short time and a long time existence result, established by Shi in Ref. 12. Theorem 2.4. Let (M n , g0 ) be open, complete with |Rm(g0,αβ )| ≤ k0 . Then the equation ∂gαβ

(x, t) = −Rαβ (x, t), x ∈ M, t > 0 ∂t gαβ (x, 0) = g0,αβ x), x ∈ M,

has a maximal solution gαβ (·, t) on [0, tmax [ with tmax > 0 which remains K¨ ahler. If additionally the initial metric has positive holomorphic bisectional curvature, the evolving metric has still positive holomorphic bisectional curvature.

Global properties of the Ricci flow on open manifolds

103

Theorem 2.5. Suppose (M, gα,β ) is a complete noncompact K¨ ahler manifold with bounded and positive bisectional curvature. And suppose there exist positive constants C2 and 0 < θ < 2 such that Z C2 1 R(x, 0)dx ≤ volBr (x0 , g(0)) (1 + r)θ Br (x0 ,g(0))

for all x0 ∈ M , 0 ≤ r < ∞. Then the Ricci flow has a solution for all t ∈ [0, ∞[. Concerning the uniqueness, Chen and Zhu proved in Ref. 1 the following theorem Theorem 2.6. Let (M n , g0 ) be open, complete with bounded sectional curvature. If g(t) and g(t) are two solutions of the Ricci flow on M n ×[0, T ] with g(0) = g0 = g(0) and with bounded sectional curvature, then g(t) = g(t) on M × [0, T ] 3. Evolution of some spectral properties Before studying evolution of spectral properties, we need some facts concerning the evolution of curvature sign. Hamilton proved already in Ref. 3, that on closed 3–manifolds the Ricci flow preserves the nonnegativity of the Ricci curvature and the sectional curvature. But in Ref. 8 Ni gives examples of complete manifolds (M n , g0 ), n ≥ 4, such that the Ricci flow does not preserve the nonnegativity of the sectional curvature. In the very general case (admitting arbitrary solutions g(t)), a certain control of sectional curvature Kg(t) is given by the following theorem of Kapovitch.5 Theorem 3.1. Let (M n , g0 ) be open, complete with |sectional curvature Kg0 | ≤ 1. Then for the Ricci flow g(t) on [0, T ] there holds inf Kg0 − C(n, T )t ≤ Kgt ≤ sup Kg0 + C(n, T )t. It is absolutely clear and can be supported by simple examples that the spectrum σ(∆q ) of the Laplace operators changes under the Ricci flow. On open manifolds, it is quite another question, whether the appearance of certain components of the spectrum like point spectrum σp , its purely discrete part σpd , continuous spectrum σc , absolutely continuous spectrum σac , essential spectrum σe , is an invariant property under the Ricci flow. For application in PDE theory, the property of the spectral value 0 is of

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particular meaning.2 For the sake of clarity, we recall a definition. Denote by σc,R (∆q ) := σ(∆q ) \ σp (∆q ) the resolvent continuous spectrum. It can be different from σc (∆q ) = σ(∆q |(L2 )c ), the spectrum of the Laplace operator restricted to the continuous subspace of L2 which is defined as those ω ∈ L2 for which dhEλ ω, ωi is a purely continuous measure. There holds σc \ (σc ∩ σ p ) ⊂ σc,R . Theorem 3.2. Let (M n , g0 = g(0)) be open, complete, |Rm(g0 )| ≤ k0 , and let g(t) be the Ricci flow according to 2.1 and 2.5, 0 ≤ t ≤ T (n, k0 ). Then there holds a) 0 ∈ σp (∆q (g0 )) if and only if 0 ∈ σp (∆q (g(t)), b) ∆q (g(o)) has a spectral gap above zero if and only if this holds for ∆q (g(t)), c) 0 ∈ σc,R ∆q (g(0)) if and only if this holds for ∆q (g(t)). Proof. According to theorem 2.1, sup |∇m Rm(g(t))|x ≤ Cm /tm , 0 < t ≤ T (n, k0 ).

x∈M

In particular, |Rm(g(t))|2 ≤ C0 , 0 ≤ t ≤ T (n, k0 ). Moreover 2 ∂ gij = 4|Rij |2 ≤ 4n2 |Rm(g(t))|2 ≤ 4n2 C0 , ∂t p p p ∂ gij ≤ 2n C0 , −2n C0 gij ≤ ∂ gij ≤ 2n C0 gij , 0 ≤ t ≤ T, (7) ∂t ∂t √ C0 t

e−2n

√

gij (x, 0) ≤ gij (x, t) ≤ e2n

C0 t

gij (x, 0), 0 ≤ t ≤ T,

in the sense of quadratic forms, i.e. g(0) and g(t) are quasi isometric. This implies for non–reduced and reduced L2 –cohomology H ∗,2 (M, g(0)) = H ∗,2 (M, g(t)), 0 ≤ t ≤ T, H

∗,2

(M, g(0)) = H

∗,2

(M, g(t)), 0 ≤ t ≤ T,

Global properties of the Ricci flow on open manifolds

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and (im (dq : Ddq −→ L2 (Ωq+1 ))/im dq )(g(0))

= (im (dq : Ddq −→ L2 (Ωq+1 ))/im dq )(g(t)), 0 ≤ t ≤ T. q,2

But 0 ∈ σp (∆q ) if and only if H 6= 0. This proves a). ∆q has a spectral gap above zero if and only if im ∆q = im ∆q |(ker ∆q )⊥ is closed. But this is the case if and only if im dq−1 , im dq are closed, hich is equivalent to q,2 q+1,2 H q,2 = H , H q+1,2 = H . This proves b). λ ∈ σc,R (∆q ) if and only if im ∆q − λ = L2 and im (∆q − λ) is properly contained in L2 . But for λ = 0 this is equivalent to ker ∆q = 0, im ∆q not closed and im ∆q = L2 . q,2 The latter is equivalent to H = 0 and at least one of im dq−1 /im dq−1 , im dq /im dq is 6= {0}. These 3 conditions are independent of t, according to (7), c) is done. As we already mentioned, for n ≥ 4 the Ricci flow does not preserve positive sectional curvature (for open M n ). For applications in this section, we must consider this question more carefully and add some assumptions. Following Ref. 10, we formulate Assumption A. Suppose ∂gij (x, t) = −2Rij (x, t) on M × [0, T ], ∂t 0 < Rijij (x, 0) ≤ k0 , x ∈ M, 0 ≤ t ≤ T,

|Rijkl (x, t)|2 ≤ c1 , x ∈ M, 0 ≤ t ≤ T. ZT |∇p Rijkl (x, t)|dt ≤ c2 , x ∈ M 0

Proposition 3.1. Under the assumption A. √ 0 < R(x, t) ≤ n2 c1 on M × [0, T ], i.e. the scalar curvature remains positive and bounded. Main idea of proof. Apply the maximum principle 4.6 of Ref. 10, p. 318 for open manifolds to ∂R = ∆R + 2| Ric |2 . ∂r We refer to Ref. 10, p. 327 for details. Remark 3.1. Unfortunately this does not hold for the sectional curvature.

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We recall the definition of the curvature operator Rop m, X (Rm(el , ek )ej , ei )g e∗i ∧ e∗j , Rop m(e∗l ∧ e∗k ) = 1≤i,j≤n

or, what is equivalent, (Rop mϕ, ψ)g = Rijkl ϕij ψkl . op denote Rop m ≥ 0 by Rijkl ≥ 0 or simply Rijkl ≥ 0.

Remark 3.2. Rop m ≥ c implies K ≥ 0. The converse is wrong: P n (| C), P n (IHI), P 2 (Ca) have 1 ≥ K ≥ 14 but the smallest eigenvalue of Rop m is 0. For later use, we recall some inequalities from Ref. 4 and Ref. 10. op Proposition 3.2. Assume assumption A and Rijkl (g(0)) ≥ 0 on op op n (M , g(0)). Then Rijkl (g(x, t)) ≥ 0 on M × [0, T ]. If even Rijkl (0) ≥ 0 op then Rijkl (t) ≥ 0 on M × [0, T ].

We refer to Ref. 10 for the proof. o

Remind Rm = Rm − U . o

2 R2 , where ε > 0, δ3 > 0, δ4 = 15 , Lemma 3.1. If |Rm|2 ≤ δn (1 − ε)2 n(n−1) 1 2 δ5 = 10 and δn = (n−2)(n+1) for n ≥ 6, then

Rop m ≥ 2ε

R . n(n − 1)

We refer to Ref. 4 for the proof. Corollary 3.1. Let βn ≤

δn 2n(n−1)

o

and suppose |Rm|2 ≤ βn R2 . Then

Rop m ≥

R . n(n − 1)

o

Lemma 3.2. |Rm|2 ≤ βn R2 implies |Rm|2 ≤ [βn + on M .

2 ]R2 n(n − 1)

Proof. o

|Rm|2 = |Rm|2 +

2 R2 . n(n − 1)

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Theorem 3.3. Let (M n , g0 ) be open, complete, o

|Rm|2 ≤ βn R2 , 0 < R ≤ c0 . Then the Ricci flow exists for 0 < t < ∞. Outline of proof. First one has |Rm(x, 0)|2 ≤ [βn +

2 ]c2 n(n − 1) 0

on M . Corollary 3.1 implies Rop m(x, 0) > 0, altogether 0 < Rijij (x, 0) ≤ [βn +

1 2 ] 2 n 2 c0 . n(n − 1)

From theorem 2.1 we infer the existence of a short time solution on 0 ≤ t ≤ T0 , sup |∇m Rijkl (x, t)|2 ≤ cm+1 (n, c0 )/tm . x∈M

Lemma 3.3. This solution satisfies assumption A. Thereafter one performs iteration, the procedure does not stop. Corollary 3.2. The solution according to lemma 3.3 still satisfies op Rijkl (x, t) > 0,

0 < R(x, t) ≤ C, o

|Rm|2 ≤ βn R2 ,

sup |∇m Rijkl (x, t)|2 ≤ Cm .

x∈M

Outline of the proof of theorem 2.2. One shows: There exists c3 (n, ε, c1 , c2 ) such that  1+ε1 1 R(x, t) ≤ c3 , 0 ≤ t < ∞, 1+t where ε1 =

nδ 8(2+δ)

o

> 0 and |Rm| ≤ εR2 by assumption. Hence R(x, t) 0 , r→∞

Rm(x, t) 0 . r→∞ We come now to some questions concerning the spectral evolution. Denote λ0 ≡ λ0 (∆(g(t))) := inf σ(∆(t))) = smallest spectral value of the Laplace operator acting on functions. We ask: How evolves λ0 (t)? This is an extraordinary complicated question and we present here only some results for small n. We proved in theorem 3.1. Proposition 3.3. λ0 (∆q (0)) = 0 if and only if λ0 (∆q (t))0.

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Corollary 3.3. If λ0 (t∗ ) > 0 or some t∗ then λ0 (t) > 0 for all t, 0 ≤ t ≤ T . By the Raleigh–Ritz variational principle, λ0 =

inf ∞

ϕ∈Cc (M ) |ϕ|L =1 2

h∆ϕ, ϕiL2 = |ϕ|2L2

inf ∞

ϕ∈Cc (M ) |ϕ|L =1 2

|∇ϕ|2L2 . |ϕ|2L2

Theorem 3.4. Suppose (M 2 , g0 ) open, complete, K(M 2 , g0 ) ≤ −c, c > 0, {g(t)}t≥0 the bounded Ricci flow and suppose λ0 (0) > 0. Then there exist ε > 0 such that λ0 (t) is decreasing for t ∈ [0, ε]. Proof. Let Ω ⊂ M 2 be a bounded domain with smooth boundary, λ0 (0) = λ0 (Ω, g0 ) the first non–zero eigenvalue with Dirichlet boundary conditions, ∆ϕ0 = λ0 ϕ0 , |ϕ0 |L2 = 1. Consider the equation ∆(t)ϕ0 (t) = λ0 (t)ϕ0 (t). An easy calculation (cf. Ref. 7) yields for Z λ′0 = −2 Rij ϕ0ij ϕ0 dvol(g(t)). Ω

In the case n = 2, Rij = 12 Rgij , R = 2K, hence Z Z Z λ′0 = − Rgij ϕ0ij ϕ0 dvol = 2K∆ϕ0 · ϕ0 dvol = 2λ0 K|ϕ0 |2 dvol, Ω

Ω

Ω

which is ≥ 0 if K ≥ 0, ≤ 0 if K ≤ 0. But, according to proposition 3.1, inf K(g0 ) − C(n, T )t ≤ K(g(t)) ≤ sup K(g0 ) + C(n, T ). In our case, we have K(g(t)) ≤ 0 if −c + C(n, T )t ≤ 0, i.e. t ≤ Under this condition, λ0 (Ω, g(t)) is decreasing for all Ω. But

c C(n,T )

= ε.

λ0 (M 2 , g(t)) = inf λ0 (Ω, (t)). Ω

Next we consider the case n = 3. o

1 Theorem 3.5. Let (M 3 , g0 ) be open, complete, |Rm(g0 )| ≤ 12 R2 (g0 ), 0 < R ≤ c0 , and let {g(t)}t be the bounded Ricci flow. Then Rij (t) − R3 gij ≥ 0. If in some time intervall [t1 , t2 ]

Rij (g(t)) −

R 1 gij (t) ≥ Rgij (t) 3 6

then λ0 (t) is increasing, t1 ≤ t ≤ t2 .

Global properties of the Ricci flow on open manifolds

109

Proof. We obtain from the preceding statements that Rop m ≥ 16 R, 0 ≤ t < ∞, 0 < R < C. Let again Ω ⊂ M 3 be a compact domain with smooth boundary, λ0 (t) = λ0 (Ω, g(t)). According to Ref. 6, Z Z λ0 R λ′0 = Rϕ20 dvol − (Rij − gij )ϕ0i ϕ0j dvol, 2 2 2 Ω

Ω

R 1 R Rij − gij = (Rij − gij ) − Rgij . 2 3 6 The assumption yields λ0 (Ω, g(t)) is increasing, the some holds for λ0 (M 3 , g(t)). Remark 3.3. For n, q arbitrary, Ω ⊂ M n compact with smooth boundary and standard boundary conditions, the following is clear. ∆q ω0 = ∆ω0 = λ0 ω0 immediately implies λ′0

=

Z

(∆′ ω0 , ω0 )g(t) dvol.

Ω

The point is to find appropriate estimates for (∆′ ω, ω)g(t) . In the case q = 1,we have in local coordinates ∆′ = (−g kl ∇k ∇l + Rii )′ = −g ′kl ∇k ∇l − g kl ∇′k ∇l − g kl ∇k ∇′l + (Rji )′ ′ , = −2Rkl ∇k ∇l − g kl ∇′k ∇′l − g kl ∇k ∇′l + 2Rik Rjk − 2g ik Rjk

where ∇′k = (Γjik )′ = g jγ (∇γ Rik − ∇i Rjγ − ∇k Rjγ . These expressions already indicate, how difficult it would be to establish meaningful results concerning monotonicity of λ0 (∆q ), q ≥ 1. We conclude this section with some considerations concerning the evolution of L2 –cohomology classes and their norm. Consider (M n , g) open, complete, and the reduced and unreduced Lp – cohomology H q,p (g) = Z q,p (g)/B q,p (g), H q,p (g) = Z q,p (g)/B q,p (g). Suppose that (M n , g0 ) has bounded curvature. Then there exists the bounded Ricci flow {g(t)}0≤t≤T , and there holds H q,p (M n , g(0)) = H q,p (M n , g(t)).

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For [ϕ] ∈ H q,p (g(t)), |[ϕ]|p,g(t) = inf |ψ|p,g(t) ψ∈[ϕ]

is a norm in the Banach space H q,p (g(t)). In the case p = 2 |[ϕ]|2,g(t) = |ϕh |2,g(t) , where ϕh ∈ [ϕ] is the unique L2 –harmonic representative of [ϕ], H q,p (g(t)) ∼ = Hq,2 (g(t)) canonically. We remark that the single L2 –cohomology classes of H q,p remain invariant under the bounded Ricci flow since B q,2 L2 (g0 ) = B q,2 L2 (g(t)) , i. e. the classes do not evolve. But the minimizer ϕh ∈ [ϕ] ∈ H q,p can change. If ϕh = ϕh (g(t1 )) is harmonic, i.e. dϕh (g(t1 )) = 0 = ∗g(t1 ) d ∗g(t1 ) ϕh (g(t1 )) = 0, then there is no argument visible that ∗g(t2 ) d ∗g(t2 ) ϕh (g(t2 )) = 0. Moreover the norm |[ϕ]|2,g(t) can evolve. We construct an evolution for each single ϕ ∈ [ϕ] and control its L2 – norm. This yields a control of the L2 –norm of [ϕ]. Lemma 3.4. Let ϕ0 = ϕ(0), [ϕ0 ] ∈ H q,2 (g(0)), (M n , g(0)) with bounded curvature and {g(t)}0≤t≤T the bounded Ricci flow. Then the equation ∂ϕ = −∆(g(t))ϕ, ϕ(0) = ϕ0 ∂t

(8)

has a unique solution in L2 and [ϕ(t)] = [ϕ(0)]. Proof. Consider the equation for the (q − 1)–form f , ∂f = −∆(g(t))f − δ(g(t))ϕ0 , f (0) = 0. ∂t According to proof of theorem 4.1, this equation has a unique solution f which is ∈ L2 . Set ϕ = df + ϕ0 . Then   ∂ ∂ ∂f ∂f = d((−∆)f − δϕ0 ) = (ϕ0 − df ) = df = d ∂t ∂t ∂t ∂t = (−∆)(df + ϕ0 ) = (−∆)ϕ

and ϕ(0) = 0, [ϕ] = [ϕ0 + df ] = [ϕ0 ]. Corollary 3.4. The map [ϕ0 ] ∋ ψ −→ ψ(t) ∈ [ϕ0 ] is an affine isomorphism.

Global properties of the Ricci flow on open manifolds

111

Lemma 3.5. Let {g(t)}0≤t≤T be the bounded Ricci flow as above, ϕ0 ∈ L2 (Λq , g(0)) and ϕ = ϕ(t) satisfying the differential equation ∂ ϕ = (−∆)ϕ, ϕ(0) = ϕ0 . ∂t If u(t) = |ϕ(t)|2g(t) and q = 1, then u(t) satisfies the differential inequality ∂u ≤ ∂t

  1 − ∆ u. 2

If q > 1 then u(t) = |ϕ(t)|2g(t) satisfies this differential inequality if Rop m(g(t)) > 0. Proof. We recall the Weitzenboeck formulas, q = 1, (∆ϕ)i = −∇r ∇r ϕi + Rϕ.

X X q > 1, (∆ϕ)i1 ...iq = −∇r ∇r ϕi1 ...iq + Rirh ϕi1 ...r...iq + Rirhsil ϕi1 ...r...s...iq . h

h 1, (∆ϕ, ϕ)g(t) = (∇∗ ∇ϕ, ϕ)g(t) + Fq,1 (ϕ) + Fq,2 (ϕ), where F1 (ϕ) = Rij ϕi ϕj = Rij ϕi ϕj , 1 Rrs ϕri2 ...iq ϕsi2 ...iq Fq,1 (ϕ) = (q − 1)! q − 1 rsjk rsi3 ...iq jh and Fq,2 (ϕ) = ϕi3 ...iq . R ϕr 2 We set for q > 1 Fq = Fq,1 + Fq,2 . For any q–form ϕ there holds 1 ∆(|ϕ|2 ) = (∆ϕ, ϕ) − |∇ϕ|2 − Fq (ϕ). 2

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Now we obtain for q > 1 1 ∂ 1 ∂ i1 j1 1 ∂ · · · g iq jq ϕi1 ...iq ϕj1 ...jq ) = (ϕ, ϕ)g(t) = (g 2 ∂t 2 ∂t 2 ∂t =Ri1 j1 g i2 j2 · · · g iq jq ϕi1 ...iq ϕj1 ...jq + · · · +Riq jq g i1 j1 · g iq−1 ,jq−1 ϕi1 ...iq ϕj1 ...jq   ∂ + ϕ, ϕ = Fq,1 (ϕ) + Fq,2 (ϕ) − Fq,2 (ϕ) − (∆ϕ, ϕ)g(t) ∂t g(t)

1 = −[(∆ϕ, ϕ) − Fq (ϕ)] − Fq,2 (ϕ) = − ∆|ϕ|2 − |∇ϕ|2 − Fq,2 (ϕ) 2 1 1 ≤ − ∆|ϕ|2 = − ∆u, 2 2 if Rop m(g(t)) ≥ 0 (since then Fq,2 (ϕ) ≥ 0). In the case q = 1, we obtain the same inequality by setting Fq,2 = 0. Proposition 3.4. Assume |Rm(g(0))| bounded, {g(t)}0≤t≤T the bounded Ricci flow, R(g(t)) ≥ 0. If u(·, t) is a non–negative solution of the scalar differential inequality ∂u ≤ −∆u ∂t and u(·, 0) is ∈ L1 , then u(·, t) remains ∈ L1 and u(t) ≤ u(s) for t > s. We refer to Ref. 6 for the proof. Remark 3.4. R(g(t)) ≥ 0 is satisfied if we assume R(g(0)) ≥ 0 and assumption A. Theorem 3.6. Suppose (M n , g(0)) open, complete, |Rm(g(0))| bounded, R(g(0)) ≥ 0. Let {g(t)}0≤t≤T be the bounded Ricci flow and assume addi¯ q,2 (M n (g(0))) be an L2 –cohomology tionally R(g(t)) ≥ 0. Let 0 6= [ϕ] ∈ H class. [ϕ] remains invariant under the Ricci flow. a) If q = 1, then |[ϕ]|L2 (g(t)) ≤ |[ϕ]|L2 (g(s)) for t > s. b) If q > 1 and Rop m(g(t)) ≥ 0, then |[ϕ]|L2 (g(t)) ≤ |[ϕ]|L2 (g(s)) for t > s.

Global properties of the Ricci flow on open manifolds

113

c) If 0 6= [ϕ] ∈ H q,2 (g(0)), ϕh ∈ [ϕ] is the unique harmonic representative and ϕh evolves according to ∂ ϕh (t) = −∆(g(t))ϕh (t), ϕh (0) = ϕh , ∂t then |ϕh (t)|L2 (g(t)) ≤ |ϕh (s)]|L2 (g(s)) ≤ |ϕh |L2 (g(0)) for t > s. Proof. According to corollary 3.4, the map [ϕ] ∋ ψ(0) −→ ψ(t) ∈ [ϕ] is an affine isomorphism, which according to lemma 3.5, implies the differential 1 2 inequality ∂u ∂t ≤ − 2 ∆u for u(t) = |ψ(t)|g(t) . Proposition 3.4 then yields the norm inequality |ψ(t)|L2 (g(t)) ≤ |ψ(s)|L2 (g(s)) for t > s. Taking the infimum over all elements of [ϕ] finally yields the assertions. Corollary 3.5. The assertions of theorem 3.6 are true in the following cases a) |Rm(g(0))|2 ≤ βn R2 (g(0)), 0 < R2 (g(0)) ≤ c. Then theorem 3.6 holds for 0 ≤ t < ∞. b) n = 2, 0 < K ≤ c. 4. Evolution of the L2 –property of the curvature If M n is closed, oriented then we have well–defined characteristic numbers. By the Chern–Weil construction, these can be expressed as an integral over an n–form with curvature coefficients. In the case n = 2m even, the Euler number χ(M n ) can be expressed e.g. as Z 2 e(g), χ(M n ) = χ(M n , g) = vol(S n ) where e(g) =

1 X i1 ...in i1 i Ωi2 ∧ Ωii34 ∧ · · · ∧ Ωin−1 ε , n n!

εi1 ...in = sign of the permutation, Ωij the curvature form, R(·, ·)Ei = Ωij Ej . We use that f ∈ Lp and bounded implies f ∈ Lq for q ≥ p. It is well known

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that in the closed case characteristic number (M n , g) = characteristic number (M n , g) . In the open case, this is wrong. At first the integral version does not exist in general and moreover it can depend on g. Cheeger/Gromov gave an example of metrics g0 , g1 on IR2m , m ≥ 2 such that χ(IR2m , g0 ) = 1 6= 0 = χ(IR2m , g1 ).

There arise the following natural questions. 1) If for (M n , g0 ) characteristic numbers are defined and {g(t)}0≤t≤T is the bounded Ricci flow, are they defined also for (M n , g(t)) and 2) do they coincide? For n ≥ 4 these questions are essentially equivalent to the following question. ∈ L2 implyR Rm(g(t)) ∈ L2 , with other words, does R Does Rm(g(0)) |Rm(g(0))|2 dvol(g(0)) imply |Rm(g(t))|2 dvol(g(t)) < ∞? M

M

n

Theorem 4.1. Let (M , g(0)) be open, complete, Rm(g(0)) bounded and ∈ L2 , {g(t)}0≤t≤T the bounded Ricci flow. Then Rm(g(t)) ∈ L2 . Proof. The evolution equations for |Rm| or |Rm|2 , resp., are

∂ mp mp |Rm|2 = [−∆|Rm|2 −2|∇Rm|2+8Rempq Rejqh Rjh +2Reqmp Reqih Rih ] (9) ∂t and 1 1 ∂ |Rm| = [ ]. (10) ∂t 2 |Rm|

We introduce as usual the following abbreviation. If A and B are tensors, then we denote by A∗B a linear combination of terms formed by contraction on Ai...j Bk...l using the g ik . Then one can write more general X ∂ k ∇ Rm = −∆(∇k Rm) + ∇i Rm ∗ ∇j Rm, (11) ∂t i+j=k

X ∂ k |∇ Rm|2 =−∆|∇k Rm|2−2|∇k+1 Rm|2+ ∇i Rm∗∇j Rm∗∇k Rm.(12) ∂t i+j=k

All these equations (9) – (12) are non–linear evolution equations of the type ∂ u = −∆(t)u + non–linear terms. ∂t If Rm(g(0)) is bounded, then there exists a solution of (9) – (12), according to theorem 2.1. Hence the point for us is not the existence of a solution but

Global properties of the Ricci flow on open manifolds

115

its properties. The whole problem consists in he fact that ∆ = ∆(g(t)), i.e. in local coordinates the coefficients of the elliptic operator on the right hand side still (strongly) depend on t, i.e. the standard non–linear parabolic theory is not applicable. To prove theorem 4.1, we must very briefly establish an appropriate framework. Let H be a Hilbert space and V ֒→ H ֒→ V ′ be a Gelfand triple. Set W21 (0, T ) = {f ∈ L2 (]0, T [, V )|

df ∈ L2 (]0, T [, V ′ )} dt

and hf, giW :=

ZT 0

hf (t), g(t)iV dt +

ZT 0

dg h df dt , dt iV ′ dt.

Suppose for t ∈]0, T [ to be given a sesquilinear form a(t, ϕ, ψ) with a) a(t, ϕ, ψ) is for fixed ϕ, ψ ∈ V measurable on [0, T ], b) there exists c > 0 (independent of t) such that |a(t, ϕ, ψ)| ≤ c · |ϕ|V · |ϕ|V for all t, ϕ, ψ, c) there exist α > 0, k0 > 0 (independent of t) such that Rea(t, ϕ, ψ) + k0 |ϕ|2H ≥ a|ϕ|2V . We infer from b) that there exist a representation L(t) such that a(t, ϕ, ψ) = hL(t)ϕ, ψiH . Lemma 4.1. The assumptions a), b) imply that L(t) : L2 (]0, T [, V ) −→ L2 (]0, T [, V ′ ) is linear and continuous. Let be f ∈ L2 (]0, T [, V ′ ), y0 ∈ H. Proposition 4.1. The problem L(t)y +

dy = f, y(0) = y0 , dt

(13)

has for T < ∞ a unique solution y ∈ W21 (0, T ) which continuously depends on f and y0 .

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We refer to Ref. 13 for the proof which relies on a Picard–Lindel¨of technique or the Banach fixed point theorem. Now we set y(t) = |Rm(g(x, t))|, f (t) = 0, L(t)y(t) = − r.h.s. of (10), H = L2 (M, g(0)), H(t) = L2 (M, g(t)), 0 ≤ t ≤ T . The quasiisometry of g(0) and g(t) imples that there are constants c1 , c2 independent of t, such that (14) c1 | |H ≤ | |H(t) ≤ c2 | |H , 0 ≤ t ≤ T, Set o

V (t) = W 12 (M, ∆(t)) = W21 (M, ∆(t)) = W21 (M, ∇g(t) ) ≡ W21 (t),

V ′ (t) = W2−1 (M, ∆(t)) = W2−1 (M, ∇g(t) ).

We need that W21 (M, ∇g(t) ) is independent of t, i.e. there exist d1 , d2 independent of t such that d1 | |W21 (t1 ) ≤ | |W21 (t2 ) ≤ d2 | |W21 (t1 , 0 ≤ t1 , t2 ≤ T.

(15)

Having in mind (14) and the quasiisometry of the g(t)s, (15) would follow if we could prove |∇g(t) − ∇g(0) | ≡ |∇(t) − ∇(0)| bounded, 0 ≤ t ≤ T.

(16)

It remains to prove (16). ∇(t) − ∇(0) = Γkij (t) − Γkij (c). The Γkij (t) satisfy the system of ordinary differential equations ∂ ∂ k (Γ (t) − Γkij (0)) = Γkij = −g kl (∇i Rjl + ∇j Ril − ∇l Rij ). (17) ∂t ij ∂t The right hand side of (17) is bounded on 0 ≤ t ≤ T and satisfies a Lipschitz condition w.r.t. t. The latter follows from ∂ Rij = −∆Rij + 2Rpiqj Rpq − 2g pq Rpi Rqj . ∂t Hence the system (4.9) for Γkij (t) − Γkij (0) has a unique bounded solution 0 ≤ t ≤ T . Set Z Z a(t, ϕ, ψ) = ϕψdµ(t) + (∇ϕ, ∇ψ)g(t) dµ(t). M

M

There exist constants c3 , c4 such that c3 dµ(0) ≤ dµ(t) ≤ c4 dµ(0).

(18)

Then we obtain from (14), (15), (18) with H = L2 (M, g(0)), V = W21 (M, g(0)) |a(t, ϕ, ψ)| ≤ |ϕ|V (t) · |ψ|V (t) ≤ c5 |ϕ|V (0) · |ψ|V (0) = c5 |ϕ|V · |ψ|V

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and for any k0 > 0 a(t, ϕ, ψ) + k0 hϕ, ϕiL2 (g(t)) ≥ |ϕ|2V (t) ≥ c6 |ϕ|2V (0) = c6 |ϕ|2V .

Hence the problem (13) has a unique solution ∈ W21 (0, T ) and the solution of (10) is ∈ L2 . Corollary 4.1. Suppose (M n , g(0)) open, complete, Rm(g(0)) bounded and ∈ L2 and let {g(t)}0≤t≤T be the bounded Ricci flow. Then L2 –characteristic numbers are defined and independent of t, 0 ≤ t ≤ T . Proof. The existence immediately follows from theorem 4.1. The invariance follows from several other papers of the author. Applying exactly the same considerations to equation (12), we obtain Theorem 4.2. Suppose the hypotheses of theorem 4.1 and additionally ∇i (0)Rm(g(0)) ∈ L2 , 1 ≤ i ≤ k. Then ∇i (g(t))Rm(g(t)) ∈ L2 , 1 ≤ i ≤ k, 0 ≤ t ≤ T. Remark 4.1. In the same manner we get the unique solution of equation (2 in section 3) and ϕ ∈ L2 . References 1. B.-L. Chen, X.-P. Zhu, Uniqueness of the Ricci flow on complete noncompact anifolds; arXiv:math.DG/0505447v3. 2. J. Eichhorn, The zero–in–the–spectrum conjecture and its modifications, preprint Greifswald University, to appear. 3. R.S. Hamilton, Three–manifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982) 255–306. 4. G. Huisken, Ricci deformation of the metric on an Riemannian manifold, J. Diff. Geom. 21 (1985) 47–62. 5. V. Kapovitch, Curvature bounds via Ricci smoothing; arXiv:math.DG/ 0405569v1. 6. L. Ma, Y. Yang, L2 –forms and Ricci flow with bounded curvature on complete non–compact manifolds; arXiv:math.DG/0509237v1. 7. L. Ma, Y. Yang, Eigenvalue monotonicity for the Ricci Hamilton flow, Ann. Glob. Anal. Geometr. 29 (2006) 287–297. 8. L. Ni, Ricci flow and nonnegativity of curvature; arXiv:math.DG/0305246v1. 9. W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Diff. Geom. 30 (1989) 223–301. 10. W.-X. Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Diff. Geom. 30 (1989) 303–394. 11. W.-X. Shi, Complete noncompact three–manifolds with nonnegative curvature, J. Diff. Geom. 29 (1989) 353–360.

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12. W.-X. Shi, Ricci flow and the uniformization on complete noncompact K¨ ahler manifolds, J. Diff. Geom. 45 (1997) 94–220. 13. Wloka, Partielle Differentialgleichungen (Teubner Publishers, Stuttgart 1982).

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 119–132

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Gauss maps and symmetric spaces J.-H. Eschenburg Institut f¨ ur Mathematik, Universit¨ at Augsburg, D-86135 Augsburg, Germany E-mail: [email protected] www.math.uni-augsburg.de How much of the geometry of a full submanifold M m ⊂ Rn is encoded by the induced metric and the Gauss map? We discuss this question in a number of cases. In particular we give a new proof for a theorem of Nikolaevskii claiming that M m ⊂ Rn is rigid if Rn contains also a full, irreducible extrinsic symmetric submanifold Mo of the same dimension m < n − 1 and the Gauss image of M is a subset of the Gauss image of Mo : then M (up to translation and scaling) is an open subset of Mo . Keywords: Submanifolds, Fundamental Forms, Symmetric R-Spaces, Group Representations. MS classification: 53A07, 53A10, 53C35, 53C24.

1. Introduction In the last three decades, the theory of submanifolds in euclidean nspace gained momentum by two sources: the relationship between holonomy and symmetry (cf. Ref. 1) and the loop group methods in connection with the theory of integrable systems (cf. Ref. 16). In the present article we deal mainly with the first one which is connected to a classical problem: Give geometric characterizations of distinguished objects. One of the classical tools for the theory of m-dimensional submanifolds M in euclidean n-space E = Rn is the Gauss map N . It assigns to each point x ∈ M ⊂ E its normal space Nx = (Tx )⊥ and takes values in the Grassmannian Gr = Grk (Rn ) of all (possibly oriented) k-dimensional linear subspaces in Rn where k = n − m is the codimension of M . We briefly discuss the question if the submanifold can be recovered from its Gauss map, and we ask for classes of submanifolds where this happens. Then we restrict attention to a class of submanifolds where much more is valid: these submanifolds are recovered just from the image of their Gauss maps.

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In the hypersurface case k = 1, the range of the Gauss map is the m-sphere which has the same dimension as M . In higher codimension we always have dim M < dim Gr, but sometimes the values of N actually lie in an m-dimensional totally geodesic subspace of Gr; e.g. this happens for complex hypersurfaces of Cp where N takes values in CPp−1 ⊂ Gr2 (R2p ). Further, if M ⊂ E is an extrinsic symmetric space, i.e. it is invariant under the reflection at each of its normal spaces, then the Gauss image of M is also totally geodesic. According to Nikolaevskii,15 there are no other totally geodesic Gauss images in Gr, and the last mentioned case is completely rigid: Theorem 1.1. Let Mo ⊂ E be an m-dimensional (full) irreducible extrinsic symmetric submanifold other than the sphere Sm , and let No : Mo → Gr be its Gauss map. Let M ⊂ E be any submanifold of the same dimension m with non-constant Gauss map N : M → Gr. Assume N (M ) ⊂ No (Mo ). Then up to translations and scaling, M is an open subset of Mo . In the present paper we give a conceptual proof of this part of Nikolaevskii’s theorem (the original proof involves extended matrix computations with many different cases). This purely local theorem is very remarkable: It determines a (small piece of a) submanifold only from its Gauss image. A similar statement for real or complex hypersurfaces would be obviously false; the Gauss image is always contained in the sphere or complex projective space which does not restrict the shape of the submanifold. The paper is organized as follows. In sections 2–5 we introduce the “main players”: fundamental forms, Gauss maps and extrinsic symmetric spaces. The theorem is proved in Sections 6 and 7. One of the main tools for the proof is Lemma 2 which was also used in Ref. 2 and goes back to Naitoh (cf. Ref. 14). Note that all our considerations are local, therefore we do not have to distinguish between embeddings and immersions. It is a pleasure to thank V. Matveev for hints and discussion. 2. Fundamental forms and Gauss map Differential geometry began with the study of submanifolds, curves and surfaces in euclidean 3-space, and even today these are the main objects used to explain general geometric ideas. The theory of submanifolds M in euclidean n-space E = En (cf. Ref. 1 for a recent approach) has two different aspects: inner and outer geometry. Inner geometry is based on the interior distance between points in M where the distance is measured by arc length of curves within M while outer geometry describes how tangent or normal spaces move from point to point inside the ambient space. They correspond

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to the two main invariants, the first and second fundamental forms g and α: g(v, w) = hv, wi, αξ (v, w) = h∂v w, ξi = −hw, ∂v ξi = hAξ v, wi for any normal field ξ and any two tangent vector fields v, w on M . For the simplest objects of differential geometry, the planar curves, the two invariants are just numbers, arc length and curvature. C.F. Gauss derived a relation between the two fundamental forms which is now called Gauss equation: R=α∧α (1) where R is the Riemannian curvature tensor (a second order expression in g). It was created by Gauss’ student B. Riemann who isolated the first aspect and created an inner geometry on manifolds which no longer need to be embedded. The existence and uniqueness theorem for submanifolds says that one only needs to prescribe the two fundamental forms on M in order to encode the full submanifold geometry; in fact, the relations between the fundamental forms discovered by Gauss, Codazzi and Ricci are necessary and sufficient to warrant an embedding which is unique up to rigid motions. Yet in some sense there is a more fundamental invariant for submanifolds in euclidean space, M ⊂ E, expressing directly the point-dependence of the normal spaces. This is the Gauss map which has been studied before by Euler and others. It assigns to each point x ∈ M its normal space Nx := Nx M = (Tx M )⊥ ⊂ E. For surfaces in E3 and more generally for mdimensional submanifolds of Em+1 (hypersurfaces) the normal space is just a line, i.e. an element of real projective m-space RPm , and if the normal lines are orientated, the Gauss map takes values in the m-sphere Sm , the 2-fold cover of RPm . Gauss had a practical reason to study this map: Being the astronomer Royal of the kingdom of Hannover (now the German state Niedersachsen), he was responsible for the surveying of this country 1821– 1825, and the Gauss map of the earth surface (the zenith direction) is important for surveying since it determines the geographical coordinates. The triangulation used by Gauss was shown on the last German 10 Mark bill.

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In general, the Gauss map of a submanifold M m ⊂ En is a map into the Grassmannian Gr = Grk (En ) of k-dimensional linear subspaces of En where k = n − m is the codimension; if M is oriented, one may put Gr the oriented Grassmannian.a Since the derivative of this map N : M → Gr measures how the normal space is moving inside the ambient space, it should be essentially the second fundamental form. In fact, the tangent vectors of Gr at Nx ∈ Gr are the linear maps f : Nx → Nx⊥ = Tx ,b and in particular, the tangent vector ∂v N ∈ TNx Gr for any v ∈ Tx is the linear map ∂v N : Nx ∋ ξ 7→ (∂v ξ)T = −Aξ v ∈ Tx .

(2)

3. “Inverting” the Gauss map? We want to consider the following problem: To which extend does the Gauss map of a submanifold M ⊂ E determine its shape? More precisely, given the Gauss map N : M → Gr together with the induced metric on M (first fundamental form), can we recover the embedding M ֒→ E? On the first sight, the answer seems easy: Since the second fundamental form is the derivative of N , both fundamental forms are given and the submanifold is determined by the existence and uniqueness theorem. But this is false! Counterexamples are obtained from minimal surfaces, i.e. surfaces in E3 with H := 12 trace α = 0. They allow an isometric deformation preserving the Gauss map; the best known example is the deformation of the catenoid into the helicoid.c Hence the same metric and Gauss map on the parameter manifold may allow several non-congruent isometric embeddings.d What was wrong with the argument? The given data consist of an abstract Riemannian manifold M and a smooth map N : M → Gr. In order a The oriented Grassmannian consists of the k-dimensional oriented subspaces of E (each subspace appears doubly, with the two possible orientations). It is a 2-fold cover of the ordinary Grassmannian; for k = 1 this is the covering Sm → RP m . b Subspaces near N are graphs of linear maps f : N → N ⊥ = T . x x x x

Tx

f(ξ ) Nx’ ξ

Nx

c http://en.wikipedia.org/wiki/Catenoid d However,

this cannot happen for non-minimal surfaces in E3 : The Weingarten map A (being a 2 × 2-matrix) satisfies 0 = A2 − (trace A)A + (det A)I = A2 − 2HA + KI. From g and N we obtain K and A2 = dN t dN and H 2 = 41 trace (A2 ) + 21 K, and if H 6= 0, we 1 (A2 + KI). Thus the embedding is uniquely determined by g and N . recover A = 2H

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to obtain the second fundamental form from (2), we need to identify the abstract tangent bundle T M with the subbundle N ⊥ ⊂ M × E. The corresponding orthogonal isomorphism F : T M → N ⊥ needs to be parallel with respect to the Levi-Civita connection on T M and the projection connection on N ⊥ , and moreover, the expression (∂v (F w))⊥ = α(v, w) must be symmetric for any two tangent vector fields v, w on M .e For generic data it is impossible to construct such a map. The precise obstructions for a map N : M → Gr to be the Gauss map of an embedding of M seem to be unknown. We can find positive answers for certain classes of submanifolds. One such class is formed by the surfaces in E3 with fixed nonzero constant mean curvature H, say H = 12 . Their Gauss map is harmonic,18 and for each S2 valued harmonic map N on a simply connected two-dimensional Riemann surface M there is (up to translations) exactly one immersion M → E3 with H = 12 and Gauss map N . Ironically it is the same isometric deformation which provides the counterexamples in the minimal surface case (H = 0) and which in the case H = 12 is used to reconstruct the surface from its Gauss map.f In higher dimensions and codimensions we only know one class of examples which we are going to describe next. If M ⊂ E is a hypersurface (k = 1), then M and Gr = Sm have the same dimension m, a very important geometric property: recall that the Gauss-Kronnecker curvature of M is the determinant of dN which has many implications. This is no longer true for codimension k ≥ 2 since dim Gr = dim Hom(N, T ) = km > m = dim M . But in some cases we are able to find a totally geodesic submanifold Q ⊂ Gr with dim Q = m and N (M ) ⊂ Q. E.g. if E = R2p = Cp and M ⊂ Cp is a complex submanifold with dimension m = 2p − 2 (complex hypersurface), then Nx is always a complex line and thus N takes values in complex projective space CPp−1 which is a totally geodesic m-dimensional submanifold of Gr2 (R2p ). In fact, Nikolaevskii15 has classified the situation where such Q exists. Besides real and complex hypersurfaces there is a third class which is related e It is easy to see that these two requirements are also sufficient since the Rn -valued 1-form F is closed and can be integrated. If (M, g) allows two different isometric embeddings with the same Gauss map N , the two parallel isomorphisms F, F˜ : T M → N ⊥ yield a parallel and orthogonal automorphism F −1 ◦ F˜ of T M . Generically, i.e. if the holonomy group is SO(m) with m > 2, such an automorphism must be trivial, thus an isometric embedding with prescribed Gauss map is unique. In the case of minimal surfaces described above, the automorphism is rotation by a constant angle. A similar phenomenon occurs in higher dimensions for pluri-minimal submanifolds, cf. Ref. 7. f This is the famous Sym-Bobenko Formula; see Ref. 6 for details and generalization.

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to extrinsic symmetric spaces (see next section). Using extended matrix computations, Nikolaevskii could show that this last case is completely rigid: the Gauss map image determines the submanifold uniquely. We wish to give a conceptual proof for this theorem (as stated in the introduction). 4. Extrinsic symmetric spaces A symmetric space is a Riemannian manifold M with an isometric point reflection (symmetry) at every point x ∈ M , i.e. there is an isometry sx of M with sx (x) = x and (dsx )x = −I. We are interested in a submanifold version of this notion which is defined as follows. For every submanifold M ⊂ E and any point x ∈ M we let τx be the normal reflection at x, i.e. the affine isometry of E fixing x whose linear part (differential) τx∗ has eigenvalues 1 on Nx and −1 on Tx . A submanifold M ⊂ E is called extrinsic symmetric if it is preserved by all these normal reflections τx , x ∈ M (cf. Ref. 1, Section 3.7). Clearly, an extrinsic symmetric space with its induced Riemannian metric is a symmetric space where the symmetry at x is τx |M . Examples are euclidean spheres Sm ⊂ Em+1 , the orthogonal group O(n) ⊂ Rn×n or the Grassmannians where a k-dimensional linear subspace is replaced with the orthogonal reflection at this subspace and the receiving space E consists of all symmetric n × n-matrices.g In fact, most symmetric spaces (so called symmetric R-spaces 12 ) allow such an embedding. The hermitian symmetric spaces are of particular interest. These are symmetric spaces M with a K¨ahler structure J such that all symmetries are holomorphic. Then at any x ∈ M , the complex structure Jx on Tx M is a skew symmetric derivation of the curvature tensor at x and hence an element of the isotropy Lie algebra gx ⊂ g (see footnote n); the embedding x 7→ Jx : M → g (standard embedding) is extrinsic symmetric (cf. Ref. 8, Ref. 6). Extrinsic symmetric spaces are the most beautiful submanifolds, playing a similar rˆ ole for submanifold geometry as symmetric spaces do for Riemannian geometry. While the latter spaces have parallel curvature tensor, the former ones have parallel second fundamental form which follows from the invariance of ∇α under τx∗ .h The Gauss equations (1) show that α is even more fundamental than R. As symmetric spaces are stable in the sense of “intrinsic pinching” (a Riemannian manifold with ∇R ≈ 0 is already g Cf.

Ref. 3 for details and further examples. = τx∗ ∇α(u, v, w) = ∇α(τx∗ u, τx∗ v, τx∗ w) = −∇α(u, v, w).

h ∇α(u, v, w)

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diffeomorphic to a locally symmetric space10,13 ), extrinsic symmetric submanifolds are stable in the sense of “extrinsic pinching”:17 any submanifold with ∇α ≈ 0 is already close to an extrinsic symmetric one. Now let us consider the Gauss map of an extrinsic symmetric space M ⊂ E. We may assume that M is contained in the unit spherei S ⊂ E. Then we have x ∈ Nx for all x ∈ M and thus τx is linear, τx = τx∗ . Since M is invariant under τx , the same is true for the set of all normal spaces of M . But Gr is a symmetric space whose symmetry at Nx ∈ Gr is precisely the reflection τx , therefore the Gauss image Q = N (M ) is a symmetric subspacej of Gr. Since the kernel of dN = −A would be a parallel distribution on M whose leaves are affine subspaces of E, the kernel must be zero, thus N : M → Q is a covering map (in many cases a diffeomorphism). Moreover it is equivariant with respect to the group K = {A ∈ O(n); A(M ) = M } acting transitively on M , and therefore N is an isometry (up to scaling) on each isotropy irreducible componentk of M . In the case of the sphere M = Sm , the Gauss map N identifies Sm with the Grassmannian of oriented one-dimensional subspaces in Em+1 . 5. Algebra of extrinsic symmetric spaces According to work of Ferus,4,8 each irreducible extrinsic symmetric space (up to congruence and rescaling) is a certain orbit of the isotropy representation of another symmetric space as follows.l Let G be a compact Lie group with two commuting involutions σ and τ . Then we have a i Since α is parallel, the same holds for the mean curvature vector η = trace α and for the corresponding Weingarten map Aη . Hence the eigenspaces of Aη are also parallel and moreover invariant under any Aξ , due to the parallelity of η and the Ricci equation which implies [Aξ , Aη ] = 0. The leaves of each eigenspace distribution are submanifolds in a subspace of E and M decomposes into such submanifolds which are the intersections of M with an orthogonal decomposition of E (extrinsic splitting). Each factor can be considered separately. If it is not an affine subspace, we have Aη = λI for some nonzero λ, and thus M lies in some sphere of radius 1/|λ|. Normalizing we may assume that M is contained in the unit sphere Sn−1 . j If P is a symmetric space with symmetries σ , p ∈ P , and if Q ⊂ M is a connected p subset which is invariant under each σq , q ∈ Q, then Q is a symmetric subspace, i.e. a complete totally geodesic submanifold of P , see Ref. 9 or Ref. 3. k In fact, if M does not split extrinsically as M × M ⊂ En1 × En2 = E with M ⊂ 1 2 i Eni (extrinsic irreducibility), it is also intrinsically isotropy irreducible, i.e. the isotropy representation is irreducible (Ref. 4, Theorem 4). l Recently, J.R. Kim gave a similar characterization for extrinsic symmetric spaces with indefinite inner product, cf. Ref. 11, Ref. 5.

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common decomposition of the Lie algebra g = k− + k+ + p− + p+

(3)

where k = k+ + k− and g+ = k+ + p+ are the Lie subalgebras fixed by σ∗ and τ∗ , respectively. Further we require that τ is inner and of a very simple type: τ∗ = eπad(η) for some η ∈ p+ with ad(η)3 = −ad(η), i.e.  the eigenvalues of I on g+ ad(η) are 0 and ±i. More precisely, since τ∗ = eπad(η) = , −I on g−  0 on g+ ad(η) = (4) J on g− where J is a complex structure on g+ (eigenvalues ±i). Let K ⊂ G be the fixed group of σ. Then M = Ad(K)η ⊂ p =: E is an extrinsic symmetric space whose normal reflection at the point η is τ∗ . The tangent and normal spaces of M at η are p− and p+ . The ambient space E = p is a Lie triple, i.e. [p, [p, p]] ⊂ p. The corresponding trilinear map on p, R(u, v)w = [w, [u, v]]

(5)

is called Lie triple product; it is the curvature tensor of the symmetric space G/K whose tangent space at eK can be identified with p. Both subspaces p+ , p− (being the fixed sets of the involutions τ∗ and σ∗ τ∗ on p) are Lie subtriples, i.e. they are preserved by R. The Lie algebra k consists of the derivationsm of R, and k+ (resp. k− ) contains those elements of k which preserve (resp. reverse) the splitting p = p− + p+ . The same η ∈ p+ ⊂ g+ (cf. (4)) generates yet another extrinsic symˆ = g− and ˆ = Ad(G)η ⊂ g with Tη M metric space: the full adjoint orbit M ˆ = g+ . The complex structure J on g− defined in (4) makes M ˆ hermiNη M ˆ tian symmetric, and M ⊂ g is the standard embedding mentioned above. Thus we see that any extrinsic symmetric space M lies in a standard emˆ ; in fact, M is a real form of M ˆ since bedded hermitian symmetric space M ˆ. it is the fixed set of σ∗ |Mˆ which acts as a complex conjugation on M We can also view the hermitian case as a special case of the general construction of extrinsic symmetric spaces as follows. To avoid confusion with the notation in the general case we change symbols renaming g → h, J → j, η → ζ. The compact Lie group H (previous G) is considered as a symmetric space H = G/K with G = H × H and K = {(h, h); h ∈ H}. m Derivations of R are skew adjoint linear maps A : p → p with A R(x, y)z = R(Ax, y)z + R(x, Ay)z + R(x, y)Az. They form the Lie algebra of the automorphism group K of p.

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Therefore we put g = h ⊕ h and p = {(X, −X); X ∈ h},

k = {(X, X); X ∈ h}.

(6)

The Cartan decomposition h = h+ + h− extends to g = g+ + g− , and we have η = (ζ, −ζ) ∈ p+ and J = (j, −j). 6. The second fundamental form Now let Mo ⊂ E = p be an irreducible extrinsic symmetric space and M ⊂ E any submanifold of the same dimension with Gauss image N (M ) ⊂ Q := N (Mo ). We fix some point x ∈ M and let η ∈ Mo such that Nx M = Nη Mo = p+ and Tx M = Tη Mo = p− . For tangent and normal vector fields v, w and ξ on M we consider the Levi-Civita derivatives ∇v w = (∂v w)T ,

∇v ξ = (∂v ξ)N

(7)

Together they form a covariant derivative on the trivial bundle M × E = T M ⊕ N M . We show first that the constant Lie triple product R is parallel also with respect to this derivative: Lemma 6.1. ∇R = 0. Proof. Let a, b, c, d be ∇-parallel vector fields along any curve x(t) in M ; we may assume that each of them is either tangent or normal. Put d we have to show R(a, b, c, d)′ = R(a, b, c, d) := hR(a, b)c, di. Denoting ′ = dt 0. But R(a, b, c, d)′ = R(a′ , b, c, d) + R(a, b′ , c, d) + R(a, b, c′ , c) + R(a, b, c, d′ ). Since both the tangent and normal space, p− and p+ , are Lie subtriples, we have R(a, b, c, d) = 0 unless R has an even number of entries of each type (tangent vs. normal); e.g. R(p− , p− , p− , p+ ) = 0 since R(p− , p− )p− ⊂ p− ⊥ p+ , and by the curvature identities this remains true with permuted entries. But the derivative of a ∇-parallel tangent (resp. normal) field is normal (resp. tangent), hence each term on the right hand side contains an odd number of entries of each sort, thus it has to be zero. The difference tensor between the two derivatives ∇ and ∂ on M × E, L := ∂ − ∇

(8)

is essentially the second fundamental form; we have Lv w = α(v, w),

Lv ξ = −Aξ v.

(9)

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Since ∂v R = 0 and ∇v R = 0 by the previous lemma, we get Lv R = 0, i.e. Lv is a derivation of R and thereforen Lv ∈ k, acting as ad(Lv ). More precisely, Lv interchanges p− and p+ , hence Lv ∈ k− . Thus we have defined a linear map L : p− → k− , v 7→ Lv = Lv.

(10)

Let Lt : k− → p− be the transposed of L with respect to an Ad(G)-invariant inner product on g extending the given Ad(K)-invariant inner product on p = E. Lemma 6.2. For all ξ ∈ p+ we have Lt ◦ ad(ξ) = −ad(ξ) ◦ L,

t

L ◦ ad(ξ) = −ad(ξ) ◦ L .

(11) (12)

Proof. The first equation follows from the symmetry of α which reads [Lv, w] = [Lw, v]

(13)

for all v, w ∈ p− . In fact, for any ξ ∈ p+ we have h[Lv, w], ξi = hLv, [w, ξ]i = −hv, Lt ad(ξ)wi, h[Lw, v], ξi = hv, [ξ, Lw]i = hv, ad(ξ)Lwi which proves (11). The second equation (12) is obtained by composing (11) from both sides with ad(η). From (4) we see that ad(η) and ad(ξ) commute (since [ad(η), ad(ξ)] = ad([η, ξ]) = 0), and further ad(η)2 = J 2 = −I on g− . Using (11) for η in place of ξ, ad(η)Lt ad(ξ)ad(η) = ad(η)Lt ad(η)ad(ξ) = −ad(η)2 L ad(ξ) = L ad(ξ), −ad(η)ad(ξ)L ad(η) = −ad(ξ)ad(η)L ad(η) = ad(ξ)Lt ad(η)2 = −ad(ξ)Lt . The left hand sides are equal by (11), hence we have proved (12). The second fundamental form L defines a skew symmetric linear map ˆ L : g− → g− interchanging p− and k− :  L on p− ˆ L= (14) −Lt on k− n Any

Lie triple derivation L : p → p extends uniquely to a Lie algebra derivation L : g → g where for every A ∈ k we let L(A) ∈ k with [L(A), x] = L[A, x] − [A, Lx] for any x ∈ p. Since g is semisimple, each derivation is inner, L = ad(l) for some l ∈ g. In fact l ∈ k since L = ad(l) preserves p and k.

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ˆ commutes with the action of ˜g+ = ˜k+ + p+ on g− , where Lemma 6.3. L ˜k+ = [p+ , p+ ] ⊂ k+ . Proof. From Equations (11) and (12) of the previous Lemma we obtain ˆ ◦ ad(ξ) = ad(ξ) ◦ L ˆ L

(15)

ˆ commutes with the action of p+ on g− , and consefor all ξ ∈ p+ . Hence L quently with the action of [p+ , p+ ] = ˜k+ . 7. The group action Lemma 7.1. Let Mo ⊂ p be extrinsic symmetric and irreducible, but Mo 6= Sm . Then [p+ , p+ ] = k+ and hence ˜g+ = g+ . Proof. Note that any A ∈ k+ with A ⊥ [p+ , p+ ] acts trivially on p+ . Hence (˜ g+ , ˜k+ ) is the effective pairo corresponding to the Lie triple p+ . We show by inspection of the tables for Mo (cf. Ref. 14, p. 241 f) that k+ = ˜k+ = [p+ , p+ ]

(16)

in all cases but the one corresponding to Mo = Sm (Case No. 13 for i = 1). In fact, the symmetric pairs (k, k+ ) corresponding to k− are listed on p. 242, first column while the last column contains the pairs (˜g+ , ˜k+ ) corresponding to p+ .p We have ˜k+ = k+ in all cases except No. 13 for i = 1 where p+ = R and the pair (R + so(n − 2), so(n − 2)) is ineffective; the effective one would be (˜ g+ , ˜k+ ) = (R, 0). Lemma 7.2. ˆ = λ ad(η) L

(17)

for some real λ. Proof. The vector space g− has a complex structure J = ad(η)|g− which ˆ (by (15)), i.e. these commutes with the action of g+ (by (4)) and of L actions are complex linear. o We call the pair of Lie algebras (g, k) with k ⊂ g ineffective if there is a nonzero subalgebra l ⊂ k with [l, g] ⊂ k; in other words, the subgroup L ⊂ K corresponding to l acts trivially on G/K. p The table in Ref. 2, p. 735 f is very similar: It displays the pairs (k, k ) on p. 735, third + column and the noncompact duals of (˜ g+ , ˜k+ ) on p. 736, second column

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If Mo belongs to the cases 7–18 in Ref. 14 or Ref. 2, the Lie algebra g is simple and hence the symmetric pair (g, g+ ) is irreducible.q Thus g+ acts ˆ commutes with this action. By Schur’s lemma, irreducibly on g− , and L ˆ = αI for some α ∈ C. But L ˆ is antisymmetric as a real endomorphism, L ˆ = λJ = λ ad(η)|g− . hence α is purely imaginary, α = λi and L The cases 1–6 are those where Mo is hermitian and p itself a Lie algebra. More precisely, there is a simple compact Lie algebra h with Cartan decomposition h = h+ + h− such that g = h ⊕ h and p = {(X, −X); X ∈ h},

k = {(X, X); X ∈ h}.

Now g− = h− ⊕h− has two irreducible factors for the action of g+ = h+ ⊕h+ , and the complex structure J = (j, −j) is given by the Lie bracket with ˆ has (imaginary) eigenvalues λi, µi η = (ζ, −ζ) ∈ p+ . By Schur’s lemma, L on the two irreducible factors, and in particular ˆ L(X, X) = (λjX, −µjX)

ˆ − ) ⊂ p− , hence for all X ∈ h− . But on the other hand L(k ˆ L(X, X) = (Y, −Y )

for some Y ∈ h− . Comparing these equations shows λ = µ. Lemma 7.3. λ = const

(18)

Proof. Locally we may view λ and η as smooth functions on M . Since η is the position vector of Mo ⊂ S, its derivatives ∂v η lie in Tη Mo = Tx M , hence η is a parallel normal vector field on M . By Lemma 1 we have ∇R = 0, and hence ad(η) = R(., η) : p− → k− ⊂ Hom(p+ , p− ) is ∇-parallel.r Codazzi equations show (∇v L)w = (∇w L)v

(19)

for any two tangent vectors v, w ∈ p− . Using (17) and (4) we obtain (∂v λ)Jw = (∂w λ)Jv.

(20)

If v, w are linearly independent we have ∂v λ = 0 = ∂w λ. Thus λ = const. Proof of the Theorem. If λ = 0, then L = 0 and our submanifold M is affine. Otherwise we may assume λ = ±1 (up to scaling), and replacing η q see

Ref. 14, p. 241, third column or the dual in Ref. 2, p. 736, first column that p− , p+ now depend on x ∈ M ; they are subbundles of M × E.

r Remind

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by −η if necessary, we have λ = 1. Since ad(η) : p− → k− ⊂ Hom(p+ , p− ) is a linear isometry up to scaling, being the differential of the (equivariant) Gauss map of Mo , the same holds for L. Thus both Gauss maps N : M → Q and No : Mo → Q are local isometries, and using these maps to identify the abstract Riemannian manifold M with an open subset of Mo , the second ˆ = ad(η)). Now the theorem follows fundamental forms are the same (L from the uniqueness part of the existence and uniqueness theorem for submanifolds. References 1. J. Berndt, S. Console and C. Olmos, Submanifolds and Holonomy (CRC Press 2003). 2. J. Berndt, J.-H. Eschenburg, H. Naitoh and K. Tsukada, Symmetric submanifolds associated with irreducible symmetric R-spaces, Math. Ann. 332 (2005) 721–737. 3. J.-H. Eschenburg, Lecture Notes on Symmetric Spaces, Preprint Augsburg 2000. 4. J.-H. Eschenburg and E. Heintze, Extrinsic symmetric spaces and orbits of srepresentations, manuscripta math. 88 (1995) 517–524; Erratum manuscripta math. 92 (1997) 408. 5. J.-H. Eschenburg and J.R. Kim, Indefinite Extrinsic Symmetric Spaces, in preparation. 6. J.-H. Eschenburg and P. Quast, Pluriharmonic maps into K¨ahler symmetric spaces, Preprint Augsburg 2006. 7. J.-H. Eschenburg and R. Tribuzy, Associated families of pluriharmonic maps and isotropy, manuscripta math. 95 (1998) 295–310. 8. D. Ferus, Symmetric submanifolds of Euclidean space, Math. Ann. 246 (1980) 81–93. 9. S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, 1978). 10. A. Katsuda, A pinching problem for locally homogeneous spaces. J. Math. Soc. Japan 41 (1989) 57–74. 11. J.R. Kim, Indefinite Extrinsic Symmetric Spaces, Thesis Augsburg 2005. 12. S. Kobayashi and T. Nagano, On filtered Lie algebras and geometric structures, I, J. Math. Mech. 13 (1964) 875–907. 13. M. Min-Oo and E.A. Ruh, Comparison theorems for compact symmetric ´ Norm. Sup.(4) 12 (1979) 335–353. spaces. Ann. sc. Ec. 14. H. Naitoh, Symmetric submanifolds of compact symmetric spaces, Tsukuba J. Math. 10 (1986) 215–224. 15. Y.A. Nikolaevskii, Classification of Multidimensional Submanifolds in Euclidian Space with Totally Geodesic Gauss Image, Russian Academy of Sciences Sbornik Mathematics 76 (1993) 225–246. 16. C.-L. Terng (Ed.), Integrable Systems, Geometry, and Topology (AMS/IP Studies in Advanced Mathematics 2006).

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17. P. Quast, A pinching theorem for extrinsically symmetric submanifolds of Euclidean space. Manuscr. Math. 115 (2004) 427–436. 18. E. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970) 569–573.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 133–146

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Ricci flow on almost flat manifolds Galina Guzhvina Mathematisches Institut, Westf¨ alische Wilhelms-Universit¨ at (WWU), 62 Einsteinstrasse, 48149 M¨ unster, Germany E-mail: [email protected] The present work studies how the Ricci flow acts on almost flat manifolds. We show that the Ricci flow exists on any ε-flat Riemannian manifold (M, g) with ε small enough for any t ∈ R≥0 , that limt→∞ |K|(M,gt ) · diam2 (M, gt ) = 0 along the Ricci flow and in the case when the fundamental group of (M, gt ) is (almost) Abelian we obtain the C 0 -convergence of the metric to a flat limit metric. The cases of π1 (M, gt ) Abelian and non-Abelian are handled in two different ways. Keywords: Ricci flow, curvature pinching, nilmanifold. MS classification: 51K10, 58J90.

1. Introduction A compact Riemannian manifold M n is called ε-flat if its curvature is bounded in terms of the diameter as follows: |K| ≤ ε  d(M )−2 , where K denotes the sectional curvature and d(M ) the diameter of M. If one scales an ε−flat metric it remains ε−flat. By almost flat we mean that the manifold carries ε-flat metrics for arbitrary ε > 0. The (unnormalized) Ricci flow is the geometric evolution equation in which one starts with a smooth Riemannian manifold (M n , g0 ) and evolves its metric by the equation: ∂ g = −2ricg , (1) ∂t where ricg denotes the Ricci tensor of the metric g. The present paper studies how the Ricci flow acts on almost flat manifolds. We show that on a sufficiently flat Riemannian manifold (M, g0 ) the

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Ricci flow exists for all t ∈ [0, ∞), limt→∞ |K|g(t) · d(M, g(t))2 = 0 as g(t) evolves along (1), moreover, if π1 (M, g0 ) is Abelian, g(t) converges along the Ricci flow to a flat metric. More precisely, we establish the following result: Theorem 1.1 (Main Th. (Ricci Flow on Almost Flat Manifolds)). In any dimension n there exists an ε(n) > 0 such that for any ε ≤ ε(n) an ε-flat Riemannian manifold (M n , g) has the following properties: (i) the solution g(t) to the Ricci flow (1) ∂g = −2ricg , ∂t

g(0) = g,

exists for all t ∈ [0, ∞), (ii) along the flow (1) one has lim |K|gt  d2 (M, gt ) = 0

t→∞

(iii) g(t) converges to a flat metric along the flow (1), if and only if the fundamental group of M is (almost) Abelian (= Abelian up to a subgroup of finite index). Note that in (iii) convergence is of class C 0 . Actually, a little more can be said: the limit manifold (M, g∞ ), where g∞ is the limit of g(t) along the Ricci flow (1), is isometric to a flat manifold. Fundamental results concerning the algebraic structure of almost flat manifolds were obtained by Gromov at the end of 70’s. In fact, Gromov1 showed that each nilmanifold (= compact quotient of a nilpotent Lie group) is almost flat. It means that almost flat manifolds which do not carry flat metrics exist and occur rather naturally. Moreover, the next theorem asserts that nilmanifolds are, up to finite quotients, the only almost flat manifolds. Theorem 1.2 (Gromov). Let M n be an ε(n)− flat manifold, where ε(n) = exp(− exp(exp n2 )). Then M is finitely covered by a nilmanifold (compact quotient of a nilpotent Lie group). More precisely: (i) The fundamental group π1 (M ) contains a torsion-free nilpotent normal subgroup Γ of rank n, (ii) The quotient G = π1 (M )/Γ has finite order and is isomorphic to a subgroup of O(n), (iii) the finite covering of M with fundamental group Γ and deckgroup G is diffeomorphic to a nilmanifold N/Γ,

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135

(iv) The simply connected nilpotent group N is uniquely determined by π1 (M ). From this theorem it is not clear whether M is diffeomorphic to the quotient of N by a uniform discrete group of isometries for a suitable left invariant metric on N. Ruh7 proved a stronger version of this theorem. He showed that, under strict curvature assumptions, M itself, and not only the finite cover, possesses a locally homogeneous structure. Theorem 1.3 (Ruh). For a compact Riemannian manifold M n and a suitably small number ε = ε(n) the fact that M n is ε−flat implies that M is diffeomorphic to the compact quotient N/Γ, where N is a simply connected nilpotent Lie group and Γ is an extension of a lattice L in N by a finite group H.

2. Almost Flat Manifolds with Non-Abelian Fundamental Group Gromov’s Theorem establishes a close connection between the ε−flat manifolds with ε sufficiently small and nilmanifolds. The following Theorem shows that sequences of universal covers of εk −flat manifolds with εk → 0 subconverge to nilmanifolds thus giving a kind of motivation to consider first the Ricci flow on nilmanifolds. Theorem 2.1. For any sequence of numbers εk → 0 take a sequence of εk −flat Riemannian n−manifolds (Mkn , gk ) such that maxMk kRkk = 1 and for any i ∈ N ∪ {0} there exists a constant c(i, n) such that k∇i Rkk ≤ ˜ n , g˜k ) of the universal coverings c(i, n) for all k. Consider the sequence (M k n of (Mk , gk ) with the covering metrics. ˜ n , g˜k ) subconverges w. r. to Gromov-Hausdorff topology to a Then (M k nilpotent Lie group with a left-invariant metric on it. Interesting observations concerning the behaviour of the Ricci flow on almost flat manifolds were made by J.Lauret (cf. Ref. 4). From Ref. 4 follows (implicitly) the next property of the Ricci flow: Theorem 2.2. On a nilpotent Lie group Ricci flow (1) is a gradient flow of the functional F = trRic2 .

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To obtain the necessary estimates it often makes sense to consider instead of (1) the normalised Ricci flow (2): ∂g = −2ricg − 2kricg k2g g, ∂t

(2)

where kricg k2g = trRic2g and we normalise the scalar curvature sc(g0 ) = −1. The normalised Ricci flow (2) differs from the unnormalised one (1) only by parametrisation and scaling; moreover, the scalar curvature is preserved along (2) (cf. Ref. 4). It is not difficult to see that on a nilpotent Lie group the flow (2) preserves the scalar curvature. There is an analogue of Theorem 2.2 for (2): Theorem 2.3. On a nilpotent Lie group N with the Lie algebra η the normalised Ricci flow (2) is a gradient flow of the functional F = trRic2 restricted to the sphere of Λ2 η ⋆ ⊗ η, corresponding to the normalisation of the scalar curvature: sc = −1. S = {µ ∈ Λ2 η ⋆ ⊗ η : kµk2 = 4}.

(3)

Lauret4 obtained also the following characterisation of the critical points of flow (2): Theorem 2.4 (Ref. 4, 4.2). For a nilpotent Lie group structure µ on S ∩ N ∈ Λ2 η ⋆ ⊗ η, where N can be regarded as the set of nilpotent algebra brackets on η, the following properties are equivalent: (i) (Nµ , < ·, · >) is a Ricci soliton, (ii) µ is a critical point of F restricted to S = µ ∈ Λη ⋆ ⊗ η : kµk = 1, (iii) µ is a critical point of F restricted to S ∩ Gl(η) · µ, where Gl(η) · µ can be identified with the set of all left-invariant metrics on Nµ . Definition 2.1. A steady Ricci soliton is a special solution of the Ricci flow on a Riemannian manifold M which moves along the equation by diffeomorphisms, that is, where the metric g(t) is the pull-back of the initial metric g(0) by a one-parameter group of diffeomorphisms. In general,a Ricci soliton is a special solution of the Ricci flow which moves along the equation by a diffeomorphism and also shrinks or expands by a factor at the same time: if ϕt is a one-parameter group of diffeomorphisms generated by some vector field X ∈ χ(M ), g(t) = ect ϕ⋆t g, g(0) = g0 , for some c ∈ R. The Ricci soliton (M, g) is called expanding if c > 0 and shrinking if c < 0.

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137

It is not difficult to see that in case when the manifold N possesses the structure of a nilpotent Lie group we can consider the one-parameter group of diffeomorphisms ϕt in the definition of the homothetic Ricci soliton as a one-parameter group of automorphisms of N . If ϕt = exp(− 2t D), ∂ D ∈ Der(η) then ∂t |0 ϕ⋆t g = g(−D·, ·). Operators of the type D on nilmanifolds were studied in great detail by J. Heber.3 In particular, he established the following property of their eigenvalues: Theorem 2.5 (Ref. 3, 4.14). For some positive multiple the operator D0 = ξD, has as eigenvalues positive integers with no common divisor. Let λ1 < ... < λm be the eigenvalues of D0 , di , i = 1, ..., m be the corresponding multiplicities. Then we call the tuple (λ1 < ... < λm ; d1 , ..., dm ) the eigenvalue type of D. Corollary 2.1 (Ref. 3, 4.11). In every dimension, only finitely many eigenvalue types can occur. We are now ready to establish the key property of the nilsoliton metrics which serves as a starting point for the proof of the Main Theorem. Theorem 2.6. Every nilsoliton strongly contracts the metric. In other words, there exists a constant λ > 0, such that, if (N, g) is a Ricci nilsoliton, then along the flow (2), for any t ≥ 0, h > 0, holds g(h+t) < e−λh g(t), where g is considered as a symmetric operator on η. Now let U be a neighbourhood of all solitons in N′ := S ∩ N such that on any manifold in U the corresponding left-invariant metric contracts along the flow (2). More precisely, there exists a λ > 0 such that for any λ (Nµ , g) ∈ U, as long as g(t) ∈ U , ∀t > 0, ∀h > 0, holds: g(t + h) < e− 2 h g(t) along the flow (2). Such a neighbourhood exists, as follows from the theory of the continuous dependence of solutions of ODE’s on the initial data. Thus theorem 2.4 permits us to obtain the following important result: Proposition 2.1. Choose a neighbourhood of the critical set as above. There exists a constant C such that for any nilmanifold (Nµ , g) ∈ N′ , along the normalised Ricci flow flow (2) the measure of the set I := {t : (N, g(t)) ∈ / U } is less than or equal to C.

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Recall that up to now we considered nilmanifolds with scalar curvature normalised to −1, or, equivalently, with the norm of the algebraic structure constants normalised to 2. The sectional curvature function K can be expressed as a quadratic function of the structure constants µijk (cf, for example, Ref. 5): X 1 K(e1 , e2 ) = ( µ12k (−µ12k + µ2k1 + µk12 ) 2 k (4) 1 − (µ12k − µ2k1 + µk12 )(µ12k + µ2k1 − µk12 ) − µk11 µk22 ) 4 This means that, since along the flow (2) scg ≡ −1, there exists a constant L(n) > 1 depending only on the dimension such that √ 1 √ ≤ kRk ≤ L L

(5)

for the curvature tensor R. Thus along (2) for any h > 0, t ≥ 0 holds: kRk(t+h) ≤ LkRkt.

(6)

On a compact manifold the flow (2) is equivalent to the flow (1) up to reparametrisation and scaling. More precisely, g(1) (t˜) = ψ(t)g(2) (t), 1 kR(1) kt˜ = kR(2) kt , ψ(t)

(7) (8)

where g(1) , g(2) and R(1) , R(2) are metric and curvature tensors corresponding, accordingly, to the flows (2) and (1), and R

t

ψ(t) = e2 0 kricg k Z t˜ = ψ(t)dt.

2

ds

,

(9) (10)

√ Remark 2.1. Note, that while the norm of the curvature kRk ∈ [ √1L , L] with L defined as above along the normalised Ricci flow (2), 8 and 9 show that kRkt → 0 along the Ricci flow (1). For any t˜ ∈ R≥0 kR(1) kt˜ ≤

kR(2) kt e

2t n2 L

≤

L 2t

e n2 L

kR(1) k0 ,

(11)

R n2 L·lnL hence for t˜ ≥ 0 ψ(t)dt, kR(1) kt˜ ≤ L1 kR(1) k0 . Thus on a nilmanifold the curvature always shrinks along the Ricci flow.

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139

Theorem 2.7. In any dimension n there exist constants c1 (n), c2 (n) ≥ 1 such that for any n-dimensional nilmanifold (N, g) along the Ricci flow ∂g = −2ricg ∂t with g(0) = g holds: if kRk0 ∈ [ 10c12 (n) , 10], then (i) kRkt g(t)
c1 (n), (ii) kRkt g(t) < c2 (n)kRk0 g(0)

(13)

for any t > 0. Note also that the constants obtained in 2.7, are universal for all leftinvariant metrics and a given algebraic structure, since Gl(n) acts transitively on Sym(η)−the set of all symmetric bilinear forms on η. Now we prove the main result on a segment. In the proof we make use of the ideas of Gromov-Hausdorff convergence. The structure of the limit space provides us with information about the behaviour of the spaces “close” to it. Note that under the norm of the tensor R at t (which is denoted as kRkt ) we understand the sup-norm of R on the manifold (M, g(t)). Theorem 2.8. In any dimension n and any T > 0 there exists an ε(n) such that for any ε ≤ ε(n) and any ε - flat Riemannian manifold (M n , g) (i) the solution of the Ricci flow (1) ∂g = −2ricg ∂t exists on M for all t ∈ [0, T ), (ii) parametrise the curvature at t = 0 as kRk0 = 1. Then, for c1 , c2 defined as in 2.7 along the flow (1) kRkt g(t)
0, s. t. the evolution equation (1) ∂(gij (x, t)) = −2ricij ∂t

(21)

Ricci flow on almost flat manifolds

141

on M , gij (x, o) = gij (x)

(22)

for any x on M has a smooth solution gij (x) > 0 on M for a short time 0 ≤ t ≤ T (n, k0 ) and satisfies the following estimates: for any integer m ≥ 0 there exist constants c(m, n), depending only on m and n such that k∇m Rijkl (x, t)k2 ≤

c(m, n) · k0 tm

(23)

for any t ∈ [0, T ] Note that the estimates in theorem 3.1 follow the natural parabolic scaling in which time behaves as distance squared. Note too that the estimates are stated in a form that deteriorates as t → 0. This is the best one can do without making further assumptions on the initial metric. Recall also, that √ the lifetime of a maximal solution is bounded below by C(n) , where C(n) k0 is a universal constant depending only on the dimension. Let now (M, g) be an ε−flat Riemannian manifold with Abelian funda1

ε8 . For ε0 sufficiently mental group. Put maxM |R|0 = 1, ε0 := ε, δ0 = kR 0k small, the Ricci flow a priori exists for t ≤ δ0 and, from 3.1, for t ≤ δ0

maxM |R|t ≤ c(0, n).

(24)

Hence we have the following estimation for kRick on the same segment: kRickt ≤ n2 kRkt ≤ c(0, n)n2 kRk0

(25)

for any t ≤ δ0 . Lemma 3.1. Define δ0 as above and let g(t) is the metric on M evolving along the Ricci flow (1). Then for any t ∈ [0, δ0 ], e−2

R

t 0

krics kg(s) ds

g(0) ≤ g(t) ≤ e2

Rt 0

krics kg(s) ds

g(0).

(26)

this lemma is proved in Ref. 8. Rδ We have the following estimate on 0 0 krict k along the segment [0, δ0 ]: Z δ0 1 krict kg(t) ≤ n2 · c(0, n) · δ0 kR0 k ≤ n2 · c(0, n) · ε08 . (27) 0

Which means that, for ε0 sufficiently small and for any integers i, j ≤ n and any t ∈ [0, δ0 ], 1 gij (0) ≤ gij (t) ≤ 2gij (0) : 2

(28)

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The proof of the point iii) of the Main Theorem uses the following three inequalities: 3/2

kRkδ0 ≤ c1 (n)k∇Rkδ0 , 2

(29)

kRk2t ),

k∇Rkt ≤ c2 (n) · d(M, gt )(k∇ Rkt + c(2, n) k∇2 Rkδ0 ≤ kRk0 , δ0

(30) (31)

where the last one follows from Theorem 3.1. From these three inequalities we have finally 3/2 kRkδ0

≤ 2c1 (n)c2 (n) · d(M, g0 )kRk0 c(0, n)kRk0 +

c(2, n) 1/8

ε0

!

kRk0 ,

thus there exists a constant c(n) such that c(n)3/2 · d(M, g0 ) · kR0 k2

3/2

kRkδ0 ≤

1/8 ε0

3/8

3/2

≤ c(n)3/2 · ε0 kRk0 ,

(32)

1 4

kRkδ0 ≤ c(n) · ε0 kRk0 . So, after the time δ0 , the initial curvature diminishes by absolute value by 1/4 ε0 . We have also the following estimation for the pinching constant ε1 at t = δ0 : 5

ε1 = kRkδ0 · d2 (M, dδ0 ) ≤ 4c(n) · ε04 .

(33)

Define the sequence of points ti on R≥0 such that t0 = 0, ti+1 = ti + δi , 1

ε8

where δi := kRit k , εi = d(M, gti )2 · kRti k. i Note that 1

1

δi =

d(M, gti−1 ) 4 (d(M, gti )2 · kRkti ) 8 ≥ 1 7 7 7 kRkti 32 2 4 c(n) 8 εi−1 · kRkt8i−1 1

≥

8 εi−1 1 4

7 8

=

7 32

2 c(n) εi−1 · kRkti−1

1 1 4

7 8

3 32

,

2 c(n) εi−1 · kRkti−1

It means that δi ≥

δi−1 1 4

7 8

7

,

32 2 c(n) εi−1 · kRkti−1

hence the segments [ti , ti+1 ] cover the whole of R≥0 .

(34)

Ricci flow on almost flat manifolds

R∞ The last inequality and (25) permit us to estimate 0 krickgs ds: Z ti+1 Z ∞ krickgs ds ≤ Σ∞ krickgs ds ≤ Σ∞ i=0 δi · kRickti i=0 ti

0

≤n

2

Σ∞ i=0 δi

· c(0, n)kRkti ≤ n

2

1

8 Σ∞ i=0 εi

143

(35)

· c(0, n).

From the same considerations as in (33) we get that 5

4 , εi ≤ 4c(n) · εi−1

(36)

1

8 so, the series Σ∞ i=0 εi is a geometric progression, therefore, for ε0 small enough, it converges. Hence the integral converges on the real line and curvature along the Ricci flow tends to zero. Note, that the convergence of the metrics along the Ricci flow (1) is of class C 0 , since from (26) we have

kgt − g∞ kgt ≤ −1 + e2

R

∞ t

kricgτ kdτ

→0

as t → ∞. The limit manifold (M, g∞ ) is a Gromov-Hausdorff limit of the family (M, gt ) of Riemannian manifolds, where gt evolves along (1). As t → ∞ we have the convergence of the corresponding sectional curvatures to zero: |Kt |t→∞ → 0. We can also show that the volumes of (M, gt ) remain bounded from below (volt ≥ v > 0) for any t. Indeed, from (26), for any t ∈ R≥0 we have gt ≥ e−2

Rt 0

kricgτ kdτ

g0

(37) Rt

Therefore, for ε0 small enough, vol(M, gt ) ≥ e−n 0 kricgτ kdτ vol(M, g0 ) for any t. Now we can use the argument of Cheeger (cf., for example, Ref. 6): Theorem 3.2 (Cheeger). Given n ≥ 2 and v, k ∈ (0, ∞) and a compact n-manifold (M, g) with |K| ≤ k, volB(p, 1) ≥ v for all p ∈ M, then injM ≥ i0 , where i0 depends only on n, k and v. So, we can conclude that the injectivity radius of (M, gt ) is uniformly bounded from below and the Convergence Theorem of Riemannian Geometry6 can be applied to this family of manifolds. We get that any sequence in this family subconverges in the Gromov-Hausdorff topology to a flat manifold. Now, since the Gromov-Hausdorff limit is unique up to isometries, we can conclude that (M, g∞ ) is isometric to a flat manifold.

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So, we have shown that on any ε−flat Riemannian manifold with the Abelian fundamental group the Ricci flow (1) converges to a flat metric for ε small enough. Of course, this condition on the fundamental group is also necessary. Now a few words about the proofs of the inequalities (29) and (30). Concerning (29) we have the result Theorem 3.3. In any dimension n there exist an ε(n) and a c(n) such that for any ε ≤ ε(n) and for any ε-flat manifold M we have 3/2

kRkM ≤ c(n)k∇RM k.

(38)

Note that for 3.3 we do not need any conditions on the fundamental group. In the proof we make use of a standard compactness argument and theorem 2.1: if we argue by contradiction, the algebraic structure of the limit manifold will prove to be incompatible with the structure of a nilpotent Lie group. More specific is the inequality (30). The following statement holds: Theorem 3.4. In any dimension n there exists an ε(n) such that for any ε ≤ ε(n) and for any ε-flat manifold (M n , g) with (almost) Abelian fundamental group, there exists a constant c = c(n), depending only n such that k∇Rk ≤ c(n) · d(M )(k∇2 Rk + kRk2 ),

(39)

where R is the curvature tensor of (M n , g). To prove it we consider the Taylor expansion of ∇R along a geodesic loop on M in the direction v s.t. |∇v R| := k∇Rk. Recall that to each geodesic loop α at p we associate its holonomy motion m(α) : Tp M → Tp M,

m(α)(x) = r(α)x + t(α),

(40)

where r(α) is a parallel transport around α (rotational part of m(α)) and t(α) = α(0) ˙

(41)

- the translational part of α. This gives an embedding of the fundamental group π1 (M ) into Iso(Tp M ) = Iso(Rn ), whereby the equivalence class of each loop [γ] is mapped into m(γ). One of the important intermediary results of Gromov’s

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Theorem says that, if we put πρ := {γ ∈ π1 (M ) : |γ| ≤ ρ}, the map i : πρ → Iso(Rn ) is injective for ρ and ε sufficiently small. Proposition 3.1 (Commutator estimates1 2.4.1). Let (M, g) be a complete Riemannian manifold, |KM | < Λ2 . Consider α, β at p ∈ M such π . Then that |α| + |β| ≤ 3Λ d([r(α), r(β)], r[β, α]) ≤

5 2 5 Λ ·|t(α)|·|t(β)|+ Λ2 |t[α, β]|(|t(α)|+|t(β)|), (42) 3 6

|t[m(α), m(β)] − t[β, α]| ≤ + |t(β)|) +

10 2 Λ |t(α)||t(β)|(|t(α)| 3

10 |t([β, α])|Λ2 |t(α)||t(β)|(|t(α)| + |t(β)|). 6

(43)

Proposition 3.1 shows that the Gromov product and commutator of geodesic loops are almost compatible with the easily computable product and commutator of the holonomy motions of the loops. The error is curvature controlled. In case when the fundamental group is Abelian, more precise estimations can be obtained: Theorem 3.5. In any dimension n there exists an ε(n) > 0 such that for any ε ≤ ε(n) and for any ε-flat n-dimensional manifold (M, g) with (almost) Abelian fundamental group for a ρ s.t. ρ ≤ c2 (n)d(M ), for c2 (n) depending only on the dimension of M , for any geodesic loop α s.t. |α| < ρ we have that kr(α)k < 10n c2 (n)2 ε.

(44)

In other words, theorem 3.5 states that in its setting the holonomy angles of the geodesic loops are of the order ε, where ε is the pinching constant of M . Thus, the parallel transport of any tensor along such a loop is periodic “modulo” ε. This (informal) consideration concludes the proof of (30) References 1. P. Buser and H. Karcher, Gromov’s almost flat manifolds. In: P´eriodique mensuel de la Soci´et´e Math´ematique de France (1981, Ast´erisque 81). 2. B. Chow and D. Knopf, The Ricci Flow: An Introduction (Mathematical Surveys and Monographs, vol. 110, AMS, 2004). 3. J. Heber, Noncompact homogeneous Einstein spaces, Invent. Math. 133 (2) (1998) 279–352.

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4. J. Lauret, Ricci soliton homogeneous manifolds. Math. Ann. 319 (4) (2001) 715–733. 5. J. Milnor, Curvatures of left-invariant metrics on Lie groups, Advances in Math. 21 (3) (1976) 293–329. 6. P. Petersen, Riemannian Geometry (Springer 171, 1962). 7. E. Ruh, Almost flat manifolds, J. Differential Geom. 17 (1) (1982) 1–14. 8. W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1) (1989) 223–301. 9. E.N. Wilson, Isometry groups on homogeneous manifolds, Geom. Dedicata 12 (3) (1982) 337–346.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 147–156

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Compact minimal CR submanifolds of a complex projective space with positive Ricci curvature Mayuko Kon Department of Mathematics, Hokkaido University, Kita 10 Nishi 8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan E-mail: mayuko [email protected] n-dimensional minimal proper CR submanifold M immersed in a complex projective space CP m with the complex structure J under the assumption that the Ricci curvature of M is equal or greater than n − 1. Moreover, we classify compact n-dimensional minimal CR submanifolds whose Ricci tensor S satisfies S(X, X) ≥ (n − 1)g(X, X) + kg(P X, P X), k = 0, 1, 2, for any vector field X tangent to M , where P X is the tangential part of JX. Keywords: Ricci curvature, CR submanifold, complex projective space. MS classification: 53C40, 53C55.

1. Introduction The purpose of the present paper is to study the pinching problem in terms of Ricci curvatures of minimal CR submanifolds immersed in a complex projective space. Let CP m denote the complex projective space of real dimension 2m (complex dimension m) with constant holomorphic sectional curvature 4 and K¨ ahler structure (J, g). Let M be a real n-dimensional Riemannian manifold isometrically immersed in CP m with induced metric g. If there exist a differentiable holomorphic distribution H : x 7→ Hx ⊂ Tx (M ) and the complementary orthogonal anti-invariant distribution H ⊥ , then M is called a CR submanifold. Especially, when M satisfies JTx (M )⊥ ⊂ Tx (M ) for any point x of M , M is called a generic submanifold. Any real hypersurface is obviously generic. Kon6 proved that if the Ricci tensor S of a compact n-dimensional minimal CR submanifold M of CP m satisfies S(X, X) ≥ (n − 1)g(X, X) + 2g(P X, P X), then M is a real projective space RP n , or a complex pro-

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√ jective√space CP m , or a pseudo-Einstein real hypersurface π(S k (1/ 2) × S k (1/ 2)) (k = (n + 1)/2) of some CP (n+1)/2 in CP m , where S k (r) is a k-dimensional sphere of radius r, π is the Hopf fibration and P X is the tangential part of JX (see also Kon5 ). For a compact minimal real hypersurface M of CP m , Maeda7 studied the pinching problem in terms of Ricci curvatures of M of CP m (m ≥ 3). He proved that if the Ricci tensor S of a minimal real hypersurface satisfies (2m X) ≤ S(X, X) ≤ 2mg(X, X), then it is congruent to √ − 2)g(X, √ π(S m (1/ 2) × S m (1/ 2)). On the other hand, Yamagata-Kon12 proved that if the Ricci tensor S of a compact n-dimensional minimal generic submanifold M of CP m , which is not totally real, satisfies S(X, X) ≥ (n − 1)g(X, X), then M is a real hypersurface of CP m , that is, 2m − n = 1. In this paper, we prove a reduction theorem of the codimension of a compact n-dimensional minimal proper CR submanifold M in CP m . We prove that if the Ricci curvature of M is equal or greater than n−1, then M is a real hypersurface of some CP (n+1)/2 in CP m . Using this result, we classify compact n-dimensional minimal CR submanifold M immersed in CP m whose Ricci tensor S satisfies S(X, X) ≥ (n − 1)g(X, X) + kg(P X, P X), k = 0, 1, 2, for any vector field X tangent to M . 2. Preliminaries Let CP m denote the complex projective space of complex dimension m with constant holomorphic sectional curvature 4. We denote by J the complex structure, and by g the metric of CP m . Let M be a real n-dimensional Riemannian manifold isometrically immersed in CP m . We denote by the same g the Riemannian metric on M induced from g, and by p the codimension of M , that is, p = 2m − n. We denote by Tx (M ) and Tx (M )⊥ the tangent space and the normal space of M at x, respectively. ˜ with complex Definition 2.1. A submanifold M of a K¨ahler manifold M ˜ strucature J is called a CR submanifold of M if there exists a differentiable distribution H : x 7→ Hx ⊂ Tx (M ) on M satisfying the following conditions: (i) H is holomorphic, i.e., JHx = Hx for each x ∈ M , and (ii) the complementary orthogonal distribution H ⊥ : x 7→ Hx⊥ ⊂ Tx (M ) is anti-invariant, i.e. JHx⊥ ⊂ Tx (M )⊥ for each x ∈ M . In the following, we put dimHx =h and dimHx⊥ = q. If q = 0 (resp. h =

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149

0) for any x ∈ M , then the CR submanifold M is a complex submanifold ˜ . If h > 0 and q > 0, then a CR (resp. totally real submanifold) of M submanifold M is said to be proper. ˜ the operator of covariant differentiation in CP m , and We denote by ∇ by ∇ that in M determined by the induced metric. Then the Gauss and Weingarten formulas are given respectively by ˜ X Y = ∇X Y + B(X, Y ), ∇

˜ X V = −AV X + DX V ∇

for any vector fields X and Y tangent to M and any vector field V normal to M , where D denotes the normal connection. We call both A and B the second fundamental form of M and are related by g(B(X, Y ), V ) = g(AV X, Y ). The second fundamental forms A and B are symmetric with respect to X and Y . The mean curvature vector field of M is defined to be the trace of the P second fundamental form B, that is, trB = i B(ei , ei ), {ei } being an orthonormal basis of Tx (M ). If the mean curvature vector field vanishes identically, then M is said to be minimal. The covariant derivative (∇X A)V Y of A is defined by (∇X A)V Y = ∇X (AV Y ) − ADX V Y − AV ∇X Y. If (∇X A)V Y = 0 for any vector fields X and Y tangent to M , then the second fundamental form of M is said to be parallel in the direction of the normal vector V. If the second fundamental form is parallel in any direction, it is said to be parallel. A vector field V normal to M is said to be parallel if DX V = 0 for any vector field X tangent to M . For x ∈ M , the first normal space N1 (x) is the orthogonal complement in Tx (M )⊥ of the set N0 (x) = {V ∈ Tx (M )⊥ : AV = 0}. If DX V ∈ N1 (x) for any vector field V with Vx ∈ N1 (x) and any vector field X of M at x, then the first normal space N1 (x) is said to be parallel with respect to the normal connection. In the sequel, we assume that M is a CR submanifold of CP m . The tangent space Tx (M ) of M is decomposed as Tx (M ) = Hx + Hx⊥ at each point x of M . Similarly, we see that Tx (M )⊥ = JHx⊥ + Nx , where Nx is the orthogonal complement of JHx⊥ in Tx (M )⊥ . For any vector field X tangent to M , we put JX = P X + F X, where P X is the tangential part of JX and F X the normal part of JX. For any vector field V normal to M , we put JV = tV + f V , where tV is the tangential part of JV and f V the normal part of JV . Then we see that F P = 0, f F = 0, tf = 0 and P t = 0.

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We define the covariant derivatives of P , F , t and f by (∇X P )Y = ∇X (P Y ) − P ∇X Y , (∇X F )Y = DX (F Y ) − F ∇X Y , (∇X t)V = ∇X (tV ) − tDX V and (∇X f )V = DX (f V ) − f DX V , respectively. ˜ of a complex projective space CP m The Riemannian curvature tensor R is given by ˜ R(X, Y )Z = g(Y, Z)X − g(X, Z)Y + g(JY, Z)JX − g(JX, Z)JY + 2g(X, JY )JZ

for any vector fields X, Y and Z of CP m . Then the equation of Gauss and the equation of Codazzi are given respectively by R(X, Y )Z = g(Y, Z)X − g(X, Z)Y + g(P Y, Z)P X − g(P X, Z)P Y and

− 2g(P X, Y )P Z + AB(Y,Z) X − AB(X,Z) Y

g((∇X A)V Y, Z) − g((∇Y A)V X, Z)

= g(Y, P Z)g(X, tV ) − g(X, P Z)g(Y, tV ) − 2g(X, P Y )g(Z, tV ),

where R is the Riemannian curvature tensor field of M . We denote by S the Ricci tensor field of M . Then S(X, Y ) = (n − 1)g(X, Y ) + 3g(P X, P Y ) X X + trAa g(Aa X, Y ) − g(A2a X, Y ), a

(1)

a

where Aa is the second fundamental form in the direction of va , {v1 , · · · , vp } being an orthonormal basis of Tx (M )⊥ , and tr denotes the trace of an operator. If the Ricci tensor S satisfies S(X, Y ) = αg(X, Y ) for some constant α, then M is called Einstein. When M is a real hypersurface of CP m with a unit normal vector field V , if the Ricci tensor S satisfies S(X, Y ) = αg(X, Y ) + βg(X, tV )g(Y, tV ) for some constants α and β, then M is said to be pseudo-Einstein. We need the following examples of CR submanifolds in CP m . Example 2.1. An n-dimensional complete totally geodesic submanifold M of CP m is either a complex projective space CP n/2 or a real projective space RP n of constant curvature 1. A real projective space RP n is a totally real submanifold of CP m (see Abe1 ). Example 2.2. Let z 0 , z 1 , · · · , z m be a homogeneous coordinates of CP m . The complex quadric Qm−1 is a complex hypersurface of CP m defined by the equation (z 0 )2 + (z 1 )2 + · · · + (z m )2 = 0.

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Then Qm−1 is a K¨ ahler manifold. Moreover, Qm−1 is an Einstein manifold with Ricci curvature 2(m − 1) (see Smith11 ). Example 2.3. For an integer k and for 0 < r < π/2, we define M (k, r) in S 2m+1 by k X j=0

|zj |2 = cos2 r,

m X

j=k+1

|zj |2 = sin2 r.

M (k, r) is a standard product S 2k+1 (cos r)× S 2l+1 (sin r), l = m− k − 1. We consider the Hopf fibration π : S 2m+1 −→ CP m , where S 2m+1 denotes the unit sphere. Then M c (k, r) = π(M (k, r)) is a real hypersurface in CP m . For an integer 1 ≤ k ≤ m − 2, we see that M c (k, r) is the tube of radius r 3 over CP k (see Cecil and Ryan p ). p When r satisfies cos r = (2k + 1)/(2m) and sin r = (2l + 1)/(2m), M c (k, r) is a minimal real hypersurface of CP m . Moreover, we see that M c (k, r) is a pseudo-Einstein real minimal hypersurface of CP m if and only if k = l = (m − 1)/2 and r = π/4. Then the Ricci tensor S satisfies S(X, Y ) = (2m − 2)(X, Y ) + 2g(P X, P Y ). 3. Reduction of the codimension In this section we prove the following reduction theorem of a codimension. Theorem 3.1. Let M be a compact n-dimensional minimal proper CR submanifold of a complex projective space CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X) for any vector X tangent to M , then M is a real hypersurface of some CP (n+1)/2 in CP m . First of all, using (1) and the formula for the Ricci tensor of the submanifold M (see Yano-Kon;13 p.44), we have Lemma 3.1. Let M be a compact n-dimensional minimal CR submanifold of CP m which is not a complex submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X), then M is a real projective space RP n or q = 1, that is, dimHx⊥ = 1. In the following, we shall prove that the first normal space of M is just F H ⊥ and is of dimension 1 when M is proper and satisfies the condition of Lemma 3.1. To prove this, we prepare some lemmas.

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Lemma 3.2. Let M be a compact n-dimensional minimal proper CR submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X), then the following hold: (a) ∇f = 0. (b) For any X tangent to M and any V ∈ F H ⊥ , we have DX V ∈ F H ⊥ . (c) For any X tangent to M and any U ∈ N , we have DX U ∈ N . Lemma 3.3. Let M be a compact n-dimensional minimal proper CR submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X), then the second fundamental form A satisfies the following: (a) Av P Av = P , where v is a unit vector field in F H ⊥ . (b) |[P, Av ]|2 = 2trA2v − 2(n − 1), where v is a unit vector field in F H ⊥ . (c) AV AU = AU AV for any V ∈ F H ⊥ and U ∈ N . (d) P AU = Af U and P AU + AU P = 0 for any U ∈ N . We use the following theorem directly given by the results in Simons.10 Theorem 3.2. Let M be an n-dimensional minimal submanifold of a complex projective space CP m . Then we have g(∇2 A, A) = (n − 3) +4

X

trA2a + 3

a

trA2f a

a

X X (g(Aa tvb , Ab tva ) − g(Aa tva , Ab tvb )) − 8 trAa Af a P a,b

+3

X

X a

|[P, Aa ]|2 −

X a,b

|[Aa , Ab ]|2 −

X

a

(trAa Ab )2 .

a,b

Using Theorem 3.2 and Lemma 3.3, we next compute the Laplacian for the square of the length of the second fundamental form of the minimal submanifold in CP m whose Ricci tensor satisfies S(X, X) ≥ (n− 1)g(X, X) for any tangent vector field X. Lemma 3.4. Let M be a compact n-dimensional minimal proper CR submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X), then X g(∇2 A, A) = (n + 3)trA2v + (n + 4) trA2f a − 6(n − 1) −

X a,b

2

|[Aa , Ab ]| −

X a,b

a

(trAa Ab )2 .

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From this, we obtain the following Lemma 3.5. Let M be a compact n-dimensional minimal proper CR submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X), then X g((∇2 A)v ej , Av ej ) = (n + 3)trA2v − 6(n − 1) − (trA2v )2 , j

X

g((∇2 A)a ej , Aa ej ) =

X a

a≥2,j

trA2f a −

X a,b

|[Aa , Ab ]|2 −

X

(trAa Ab )2 ,

a,b≥2

where {ei } is an orthonormal basis of Tx (M ). P P Next we give inequalities for a,b |[Aa , Ab ]|2 and a,b≥2 (trAa Ab )2 in the equation in Lemma 3.5. Lemma 3.6. Let M be a compact n-dimensional minimal proper CR submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X), then X X |[Aa , Ab ]|2 ≤ 4 trA2f a , a

a,b

X

(trAa Ab )2 ≤

a,b≥2

1 X ( trA2f a )2 . 2 a

Using Lemma 3.1-Lemma 3.6, we prove the following lemma. Lemma 3.7. Let M be a compact n-dimensional minimal proper CR submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X), then Af a = 0 for all a. From Lemma 3.2 and Lemma 3.7, the first normal space of M is of dimension 1 and parallel. By the similar method in the proof of Theorem 3.6 in Yano-Kon;13 p.227, we see that M is a real hypersurface of some totally geodesic complex projective space CP (n+1)/2 in CP m . 4. Pinching theorems of the Ricci curvature In the following, we take the unit normal vector field v of a real hypersurface M in CP m , and we put ξ = −Jv. Then ξ is the unit tangent vector field of M and P 2 X = −X + g(X, ξ)ξ, P ξ = 0. We also put Av = A to simplify the notation. Then ∇X ξ = P AX for any vector field X tangent to M.

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To prove our theorems, we need a well-known result given by T. E. Cecil and P. J. Ryan: Proposition 4.1 (Ref. 3). Let M be a real hypersurface (with unit normal vector v) of a complex projective space CP m on which ξ is a principal curvature vector with principal curvature α = 2cot2r and the focal map φr has constant rank on M . Then the following hold: (a) M lies on a tube (in the direction η = γ ′ (r), where γ(r) = expx (rv) and x is a base point of the normal vector v) of radius r over a certain K¨ ahler submanifold N in CP m . (b) Let cotθ, 0 < θ < π, be a principal curvature of the second fundamental form Aη at y = γ(r) of the K¨ ahler submanifold N . Then the real hypersurface M has a principal curvature cot(r − θ) at x = γ(0). And we use the following proposition given by S. Maeda: Proposition 4.2 (Ref. 8). Let M be a real hypersurface of a complex projective space CP m . If Aξ = 0, except for the null set on which the focal map φr degenerates, M is locally congruent to one of the following: (a) a homogeneous real hypersurface which lies on a tube of radius π/4 over a totally geodesic CP k (1 ≤ k ≤ m − 1), (b) a nonhomogeneous real hypersurface which lies on a tube of radius π/4 over a K¨ ahler submanifold N with nonzero principal curvatures 6= ±1. From these results and Theorem 3.1, we have the following Theorem 4.1. Let M be a compact n-dimensional minimal CR submanifold of a complex projective space CP m which is not a complex submanifold of CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X) for any vector X tangent to M , then M is congruent to one of the following: (a) a totally geodesic real projective space RP n of CP m , (b) a pseudo-Einstein real hypersurface M c ((n − 1)/4, π/4) of some CP (n+1)/2 in CP m , (c) a real hypersurface of some CP (n+1)/2 in CP m which lies on a tube of radius π/4 over certain K¨ ahler submanifold N with principal curvatures cot θ, 0 < θ ≤ π/12. Remark 4.1. The author does not know examples of certain K¨ahler submanifold N having the properties required in Case (c) in Theorem 4.1. Corollary 4.1. Let M be a compact n-dimensional minimal proper CR submanifold of a complex projective space CP m . If the Ricci tensor S of

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M satisfies S(X, X) ≥ (n − 1)g(X, X), then M is congruent to one of the following: (a) a pseudo-Einstein real hypersurface M c ((n − 1)/4, π/4) of some CP (n+1)/2 in CP m , (b) a real hypersurface of some CP (n+1)/2 in CP m which lies on a tube of radius π/4 over certain K¨ ahler submanifold N with principal curvatures cot θ, 0 < θ ≤ π/12. Using the theorem given by Maeda,7 we have Corollary 4.2. Let M be a compact n-dimensional minimal proper CR submanifold of a complex projective space CP m . If the Ricci tensor S satisfies (n − 1)g(X, X) ≤ S(X, X) ≤ (n + 1)g(X, X), then M is congruent to a pseudo-Einstein real hypersurface M c ((n − 1)/4, π/4) of some CP (n+1)/2 in CP m . Moreover, we obtain the following theorem. Theorem 4.2. Let M be a compact n-dimensional minimal CR submanifold of a complex projective space CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X) + g(P X, P X) for any vector X tangent to M , then M is congruent to one of the following: (a) a totally geodesic real projective space RP n of CP m , (b) a totally geodesic complex projective space CP n/2 of CP m , (c) a complex (n/2) dimensional complex quadric Q(n/2) of some CP n/2+1 of CP m , (d) a pseudo-Einstein real hypersurface M c ((n − 1)/4, π/4) of some CP (n+1)/2 in CP m , (e) a real hypersurface of some CP (n+1)/2 in CP m which lies on a tube of radius π/4 over certain K¨ ahler submanifold N with principal curvatures cot θ, where θ satisfies 0 < sin 2θ ≤ 1/3. Remark 4.2. About the proof above, in 1974, Chen and Ogiue2 proved that if the Ricci curvature of n-dimensional K¨ahler submanifold of CP m is everywhere equal to n/2, then M is locally Qn in some CP n+1 in CP m (see also Ogiue9 ). We suppose that M is a compact n-dimensional minimal CR submanifold of a complex projective space CP m . When the Ricci tensor S of M satisfies S(X, X) ≥ (n − 1)g(X, X) + 2g(P X, P X) for any vector X tangent to M , the cases (c) and (e) in Theorem 4.2 do not occur. Thus we obtain a result by Masahiro Kon:

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Theorem 4.3 (Ref. 6). Let M be a compact n-dimensional minimal CR submanifold of a complex projective space CP m . If the Ricci tensor S of M satisfies S(X, X) ≥ (n−1)g(X, X)+2g(P X, P X) for any vector X tangent to M , then M is equivalent to one of the following: (a) a totally geodesic real projective space RP n of CP m , (b) a totally geodesic complex projective space CP n/2 of CP m , (c) a pseudo-Einstein real hypersurface M c ((n − 1)/4, π/4) of some CP (n+1)/2 in CP m . References 1. K. Abe, Applications of Riccati type differential equation to Riemannian manifolds with totally geodesic distribution, Tˆ ohoku Math. J. 25 (1973) 425– 444. 2. B-Y. Chen and K. Ogiue, A characterization of the complex sphere. Michigan Math. J. 21 (1974) 231–232. 3. T.E. Cecil and P.J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982) 481–499. 4. Masahiro Kon, On some complex submanifolds in Kaehler manifolds, Canad. J. Math. 26 (1974) 1442–1449. 5. Masahiro Kon, Generic minimal submanifolds of a complex projective space, Bull. London Math. Soc. 12 (1980) 355–360. 6. Masahiro Kon, Minimal CR submanifolds immersed in a complex projective space, Geom. Dedicata 31 (1989) 357–368. 7. S. Maeda, Real hypersurfaces of a complex projective space II, Bull. Austral. Math. Soc. 29 (1984) 123–127. 8. S. Maeda, Ricci tensors of real hypersurfaces in a complex projective space, Proc. Amer. Math. Soc. 122 (1994) 1229–1235. 9. K. Ogiue, Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974) 73–114. 10. J. Simons, Minimal varieties in riemannian manifolds, Ann. of Math. 88 (1968) 62–105. 11. B. Smith, Differential geometry of complex hypersurfaces, Ann. of Math. 85 (1967) 246–266. 12. M. Yamagata and Masahiro Kon, Reduction of the codimension of a generic minimal submanifold immersed in a complex projective space, Coll. Math. 74 (1997) 185–190. 13. K. Yano and Masahiro Kon, Structures on manifolds (World Scientific Publishing, Singapore, 1984).

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 157–169

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Unique determination of domains A.P. Kopylov Sobolev Institute of Mathematics, Akademik Koptyug Pr. 4, Novosibirsk, 630090, Russia; Novosibirsk State University, Pirogova street 2, Novosibirsk, 630090, Russia E-mail: [email protected] This paper is an extension of the author’s survey lecture with the same title given at the 10th International Conference on Differential Geometry and its Applications dedicated to celebrations of the 300th anniversary of birth of Leonhard Euler, which was held in Olomouc (Czech Republic) in August 2007. The paper is devoted to two new trends in differential geometry connected with investigation of the classical problems concerning the uniqueness of determination of closed convex surfaces by their intrinsic metrics. Keywords: Unique determination, intrinsic and relative metrics of boundary, relative conformal modulus, conformal-type unique determination. MS classification: 53C24, 53A30.

1. Introduction A classical theorem says (see, e.g., Ref. 1) that: If two bounded closed convex surfaces in Euclidean 3-space are isometric in their intrinsic metrics, then they are equal, i.e., can be superposed by a motion. The problem of unique determination of closed convex surfaces by means of their intrinsic metrics dates back to Cauchy (1813) who proved the unique determination of convex polyhedrons (in Euclidean 3-space R3 ) by their unfoldings. Since then this problem became a subject of investigation by a number of mathematicians for about 150 years, e.g., it was studied by Minkowski, Hilbert, Weyl, Blashke, Cohn-Vossen, Aleksandrov, Pogorelov and other prominent mathematicians (see, for instance, a historical survey in Ref. 1, Chapter 3); finally, its complete solution, which is just the theorem we have cited at the beginning, was obtained by A. V. Pogorelov. Regarding generalizations of Pogorelov’s result to higher dimensions see Ref. 2. In Ref. 3 a new approach to the problem of unique determination of sur-

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faces was suggested which provided an essentially larger framework for that problem. The following model situation fairly well illustrates the essence of that approach. Let U1 and U2 be domains (i.e., open connected sets) in the real Euclidean n-space Rn whose closures cl Uj , where j = 1, 2, are Lipschitz manifolds (such that fr (cl Uj ) = fr Uj 6= ∅ where fr E is the boundary of E(⊂ Rn ) in Rn ). This means that for each point x ∈ cl Uj , there are a neighborhood Vj,x and a bi-Lipschitz mapping hj,x : Vj,x → B(0, 1) of this

neighborhood onto the standard unit ball B(0, 1) = x = (x1 , x2 , . . . , xn ) ∈  n 1/2  P n 2 R |x| = (xs ) < 1 such that either hj,x (Uj ∩ Vj,x ) = B(0, 1) s=1 or hj,x (Uj ∩ Vj,x ) = {x ∈ B(0, 1) x1 ≤ 0}. Assume that the boundaries fr U1 and fr U2 of these domains (which coincide with borders of manifolds cl U1 and cl U2 ) are isometric with respect to their relative metrics ρfr Uj ,Uj (j = 1, 2). These metrics are the restrictions to the boundaries fr Uj of the extensions (by continuity) to cl Uj of the inner metrics of the domains Uj . The natural question arises: Under what additional conditions are the domains U1 and U2 isometric? We must stress that the problem of unique determination of closed convex surfaces mentioned at the beginning of the paper, is the most important particular case of the stated question. To see this, assume that S1 and S2 are closed convex surfaces in R3 , i.e., are boundaries of two bounded convex domains Gj ⊂ R3 (j = 1, 2). Let Uj = R3 \ cl Gj . Then, the intrinsic metrics on the surfaces Sj = fr Uj coincide with the relative metrics ρfr Uj ,Uj on the boundaries of the domains Uj . This clearly shows that the problem of the unique determination of closed convex surfaces via their intrinsic metrics is indeed a particular case of the problem of the unique determination of domains via the relative metrics on their boundaries. The generalization of the problem of the unique determination of surfaces ensuing from a new approach suggested in Ref. 3 manifests itself by the fact that the unique determination of domains by the relative metrics on their boundaries takes place not only when their complements are bounded and convex but also in the following cases. Domain U1 is bounded and convex, while domain U2 is arbitrary (A.P. Kopylov (see Theorems 2.1, 4.1 below)). Domain U1 is strictly convex, U2 is arbitrary (A.D. Aleksandrov (Theorem 3.1)). Domains U1 , U2 are bounded and have smooth boundaries (V.A. Aleksandrov (Theorem 2.2)). Domains U1 , U2 have nonempty bounded complements, while their boundaries are (n − 1)-dimensional con-

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nected C 1 -manifolds without boundaries, n ≥ 3 (V.A. Aleksandrov (Theorem 3.2)); etc. The present paper can be divided into two parts. The first one mainly contains a survey of results obtained since 1984 which pertain to the unique determination of domains by the relative metrics on their boundaries (see Sections 2-6 of the paper). The second (very important) part is represented in Section 7. It contains a one more new approach in studying a unique determination of domains in Euclidean spaces. This approach is closely connected with the approach discussed in the first part and deals with the examination of a conformaltype unique determination problem. The main result in the second part is Theorem 7.1 which treats the unique determination of bounded convex polyhedral domains in terms of relative conformal moduli of their boundary condensers. 2. Theorems of unique determination of bounded domains The first result concerning the problem of the unique determination of domains by the relative metrics of their boundaries was as follows (see Ref. 3). Theorem 2.1. Let n ≥ 2 and U1 ⊂ Rn be a bounded convex domain, and let U2 ⊂ Rn be a bounded domain whose boundary is an (n − 1)dimensional C 0 -manifold without boundary. Suppose also that U2 has the property that the inner metric of U2 may be extended by continuity to cl U2 . Furthermore, assume that the boundaries fr U1 and fr U2 of the domains U1 and U2 are isometric in their relative metrics, i.e., there exists a surjective and isometric in the relative metrics of boundaries, mapping f : fr U1 → fr U2 (ρfr U2 ,U2 (f (x′ ), f (x′′ )) = ρfr U1 ,U1 (x′ , x′′ ) if x′ , x′′ ∈ fr U1 ). Then U1 and U2 are isometric in Euclidean metrics. Remark 2.1. By a C k -manifold (k = 0, 1, 2, . . . , ∞), real analytic manifold and Lipschitz manifold we mean corresponding submanifolds of Rn (regarding definition of Lipschitz n-dimensional manifold see Section 1; Lipschitz manifold of arbitrary dimension is defined analogously). The relative metric ρfr Uj ,Uj (j = 1, 2) of the boundary fr Uj of Uj is the metric being the restriction to fr Uj of the extension (by continuity) to cl Uj of the inner metric of Uj . Remark 2.2. Although Theorem 2.1 may seem more general than Theorem 3 of Ref. 3, they are, in fact, equivalent. Moreover, Theorem 2.1 could

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be proved repeating verbatim the argumentations in Theorem 3 of Ref. 3. We should stress that by virtue of Theorem 2.1, unlike classical Cauchy point of view on problems of unique determination of surfaces, a new approach to these problems (based on notions from Ref. 3; see also Definition 2.1 below) provides a unified and comprehensive examination of them even for n = 2. Next Theorem by V.A. Aleksandrov4 shows that a new approach permits to eliminate such assumptions as convexity of domains and connectivity of their boundaries what turns out to be redundant. Theorem 2.2. Let U1 and U2 be bounded domains in Rn , n ≥ 2, such that each cl Uj , j = 1, 2, is an n-dimensional C 1 -manifold with boundary fr Uj . Suppose that fr U1 and fr U2 are isometric in their relative metrics. Then the domains U1 and U2 are isometric too. Note that Theorem 2.2 could be generalized to domains with piecewise smooth boundaries (see Ref. 4). We will now state the propositions of that Section in a somewhat different form which will serve subsequently as a model pattern. To this end, denote by A = A(n) the class of all bounded domains U in Rn , n ≥ 2, with the following properties. The closure cl U of U is an n-dimensional C 0 manifold with boundary fr U , and the inner metric of U can be extended by continuity to its closure. Next we introduce the following notion which is one of the most important in our paper. Definition 2.1. Let A0 ⊂ A and U ∈ A0 . We will say that a domain U is uniquely determined in the class A0 by the relative metric on its boundary if the following condition is satisfied. Assume that a domain V belongs to the class A0 and there is a surjection f : fr V → fr U preserving the relative metric of the boundary. This means that for any two points x′ , x′′ ∈ fr V , we have ρfr U,U (f (x′ ), f (x′′ )) = ρfr V,V (x′ , x′′ ). Then V can be isometrically mapped onto U . In other words, U could be determined in A0 up to an additional isometric transformation of Rn . Denote by A1 = A1 (n) the class of all bounded domains U in Rn , n ≥ 2, whose closures are n-dimensional C 1 -manifolds with boundary fr U . Using Definition 2.1, we may now restate Theorems 2.1 and 2.2 as follows. Each bounded convex domain in Rn , n ≥ 2, is uniquely determined in the class A(n) by the relative metric on its boundary (Theorem 2.1).

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If a domain U belongs to A1 (n), n ≥ 2, then it is uniquely determined in the class A1 (n) by the relative metric on its boundary (Theorem 2.2). 3. Case of unbounded domains For unbounded domains, the unique determination problem by the relative metrics on their boundaries differs essentially as compared to the case of bounded domains. Indeed, the half-space Rn− = {(x1 , x2 , . . . , xn−1 , xn ) ∈ Rn xn < 0},

n ≥ 2 (and consequently, each n-dimensional open half-space of Rn ), is not uniquely determined by the relative metric on its boundary, even in the class of all domains whose boundaries are real analytic (n − 1)-dimensional manifolds without boundary. To see this, let us consider, e.g., the domain  U = (x1 , x2 , . . . , xn−1 , xn ) ∈ Rn xn < (xn−1 )2 .

The boundary of this domain is isometric to the boundary of Rn− (in the intrinsic metrics on boundaries). It is not difficult to verify that the mapping f : fr U → fr Rn− ,
f : x1 , x2 , . . . , xn−2 , xn−1 , (xn−1 )2 7→ x1 , x2 , . . . , xn−2 , p p

2−1 xn−1 1 + 4(xn−1 )2 + 4−1 ln 2xn−1 + 1 + 4(xn−1 )2 , 0 , (x1 , x2 , . . . , xn−1 ) ∈ Rn−1 ,

is surjective and isometric in the intrinsic metrics of boundaries of U and Rn− . Since Rn \ U is convex, we conclude that f is isometric in the relative metrics on boundaries fr U and fr Rn− . But at the same time these domains are not isometric. Before starting discussion of results in this Section, we must note that as previously, we consider only domains U ⊂ Rn , 6= Rn , admitting extension by continuity of inner metrics ρU to cl U . The following result is due to A. D. Aleksandrov (see Ref. 4). Theorem 3.1. Let U1 be a strictly convex domain (i.e., a convex domain whose boundary does not contain any linear interval ). Assume that U2 is any domain such that fr U1 and fr U2 are isometric in their relative metrics ρfr U1 ,U1 and ρfr U2 ,U2 . Then U1 and U2 are isometric. Let B = B(n) be a class of domains U in Rn (n ≥ 2) with nonempty bounded complements, whose boundaries are connected (n−1)-dimensional C 1 -manifolds without boundaries. The following result is obtained in Ref. 5.

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Theorem 3.2. Let n ≥ 3. Then each domain U ∈ B is uniquely determined in the class B by the relative metric on its boundary. Remark 3.1. The notion of unique determination of unbounded domains is introduced analogously to Definition 2.1. Theorem 3.2 is no longer true for n = 2 what could be seen in the following example. Example 3.1. Let U1 and U2 be unbounded domains in R2 with convex bounded complements R2 \ Uj , satisfying R2 \ cl Uj 6= ∅ (j = 1, 2) and such that their boundaries fr U1 , fr U2 are of the same length. Then fr U1 and fr U2 are isometric in their intrinsic, hence in relative, metrics. But at the same time, the domains U1 and U2 need not be isometric. Next Theorem proved by A. V. Kuz’minykh,6 gives a complete solution to the problem of unique determination of convex domains in the class C = C(n) of all domains U in Rn , n ≥ 2, admitting extensions by continuity of their inner metrics to the closures cl U . Theorem 3.3. Each convex domain in Rn , n ≥ 2, different from Rn and from any open half-space of Rn , is uniquely determined in C(n) by the relative metric on its boundary. Observe that Theorems 2.1, 3.1 are the particular cases of Theorem 3.3. Let ∆(U ) denote the interior of the convex hull of the complement of U in Rn , n ≥ 2: ∆(U ) = int (conv (Rn \ U )). The following proposition holds (see Ref. 7). Theorem 3.4. Assume that the boundary fr U of a domain U ⊂ Rn (n ≥ 2) is a connected (n − 1)-dimensional C 1 -manifold without boundary and satisfies the condition fr U ⊂ ∆(U ). Then U is uniquely determined by the relative metric on its boundary in the class A∗1 = A∗1 (n) of all domains in Rn with smooth boundaries (i.e., domains V whose closures cl V are n-dimensional C 1 -manifolds with boundary fr V ). The following result (see Ref. 7) is valid for domains with analytic boundaries, i.e., domains V ⊂ Rn (n ≥ 2) such that their closures cl V are n-dimensional (real) analytic manifolds with boundary fr V . Theorem 3.5. Each domain U ⊂ Rn (n ≥ 2) with connected analytic boundary satisfying fr U ∩ ∆(U ) 6= ∅, is uniquely determined by the relative metric on its boundary in the class E = E(n) of all domains in Rn with analytic boundaries.

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4. Domains with Hausdorff boundary Domains U we considered above satisfy the condition that the inner metric of U can be extended by continuity onto cl U . But this condition excludes the case of domains whose (Euclidean) boundaries have very simple configuration. For instance, such is an open cube in R3 ”incised” with a half-plane   U1 = x = (x1 , x2 , x3 ) ∈ R3 |xj | < 1, j = 1, 2, 3 \ x ∈ R3 x1 ≥ 0, x3 = 0 . One more example is the domain   3 X 2 3 3 (xj ) ≤ 1, x3 = 0 , U2 = R \ x ∈ R j=1

which is a complement of a disk with respect to R3 . This drawback could be eliminated if we pass from Euclidean boundaries discussed in previous Sections, to what we call Hausdorff boundaries defined as follows. Definition 4.1. Let U ⊂ Rn (n ≥ 2) be a domain and ρU its inner metric. Consider a completion of the metric space (U, ρU ) and let us identify each point x ∈ U with the point corresponding to x in the completion. Then eliminating U from the completion we obtain a metric space (fr H U, ρfr H U,U ). The set fr H U is called the Hausdorff boundary of the domain U , while ρfr H U,U is called the relative metric of its Hausdorff boundary. Remark 4.1. If a domain U admits an extension by continuity of its inner metric ρU onto cl U then the Hausdorff boundary fr H U of U is identified in a natural way with its Euclidean boundary. Definition 4.2. Let U be a class of domains in Rn , n ≥ 2. We say that a domain U ∈ U is uniquely determined in U by the relative metric on its Hausdorff boundary if each domain V ∈ U, whose Hausdorff boundary is isometric in the relative metrics to the Hausdorff boundary of U , is itself isometric to U (in Euclidean metrics). Remark 4.2. The isometry of the Hausdorff boundaries of domains U and V in their relative metrics means that there exists a surjective isometric mapping f : (fr H U, ρfr H U,U ) → (fr H V, ρfr H V,V ) of these boundaries. There are domains U such that U is uniquely determined by the relative metric on its boundary in one class of domains but this is not true in some another, larger class. In this connection, there arises a question about the

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existence of domains uniquely determined by the relative metric on their boundaries in the class of all domains in Rn . One of possible answers to this question is given by the following proposition which generalizes and strengthens Theorem 2.1 in the case n = 2. Theorem 4.1. Each bounded convex domain U ⊂ R2 is uniquely determined in the class of all domains V ⊂ R2 by the relative metric on its Hausdorff boundary (in the sense of Definition 4.2). The next Theorem (see Ref. 8) gives an example of domains in Rn of any dimension n ≥ 2 for which the solution of the unique determination problem appeals necessarily to considering Hausdorff boundaries. Theorem 4.2. Each bounded domain U ⊂ Rn (n ≥ 2) with polyhedral boundary is uniquely determined in the class of all such domains by the relative metric of its Hausdorff boundary. By a polyhedral domain (in Theorem 4.2) we mean a domain whose boundary is a polyhedron, i.e., the union of a finitely many cells (possibly of different dimensions), where a cell is defined to be a bounded set which is the intersection of a finite number of closed half-spaces. Now in conclusion, making use of the notion of Hausdorff boundary, let us give the full statement of A. V. Pogorelov’s Theorem on the unique determination of closed convex domains as follows. Each domain U ⊂ R3 whose complement R3 \ U is a bounded convex set, is uniquely determined in the class of all such domains by the relative metric of its Hausdorff boundary. Remark 4.3. The case when the complement R3 \ U of U is a onedimensional segment was not included in the statement of the Theorem. But since this particular case is trivial, the formulation of Pogorelov’s Theorem stated above may be regarded as complete. In conclusion of the Section, we call reader’s attention to very interesting results of recent M. V. Korobkov’s works.9,10 In these papers M.V. Korobkov obtained the necessary and sufficient conditions for unique determination of a plane domain in the class of all domains by the relative metric on its Hausdorff boundary.

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5. Unique determination of domains (local variant) In previous Sections we dealt with the problem of the unique determination of domains by the relative metrics of their boundaries. In the present Section we give some results permitting to answer the following natural question. Is it sufficient for the isometry of domains in Euclidean spaces to assume that their boundaries (Euclidean, Hausdorff ) are isometric in their relative metrics in, so to say, some local sense? Definition 5.1. Let M be a class of domains in Rn , n ≥ 2. We say that a domain U ∈ M is uniquely determined in M by a condition of local isometry in the relative metrics of Hausdorff boundaries of domains, if the assumptions 1) V ∈ M and 2) fr H V is locally isometric in the relative metrics to fr H U imply the isometry of U and V (in Euclidean metrics). The local isometry in the relative metrics of Hausdorff boundaries fr H U, fr H V of domains U, V means that there exists a bijection f : fr H U → fr H V locally isometric in the relative metrics of fr H U and fr H V . In other words, f is such that for each y ∈ fr H U , there is ε > 0 satisfying the condition: for each two elements a, b belonging to the ε neighborhood Z(y) = {z ∈ fr H U ρfr H U,U (z, y) < ε} of y, the equality ρfr H U,U (a, b) = ρfr H V,V (f (a), f (b)) holds. Remark 5.1. The notion of a unique determination of domains by the condition of local isometry in the relative metrics of Euclidean boundaries is introduced analogously. Here we assume that the domains admit extension by continuity of their inner metrics onto the closures of domains. First results related to the discussed in this Section approach in the study of the unique determination of domains by the condition of local isometry of their boundaries in the relative metrics, were obtained by A. V. Kuz’minykh.6 One of them is the following assertion. Theorem 5.1. Let U ⊂ Rn (n ≥ 2) be a convex domain such that fr U does not contain any strip, i.e., a convex hull of two parallel (n − 2)-dimensional planes. Assume that V ∈ Rn is a domain admitting extension by continuity of its inner metric ρV onto cl V . Let ε > 0 be given and suppose that there exists a bijection f : fr U → fr V such that for each α ∈ ]0, ε[ and for each points x′ , x′′ ∈ fr U, the equality ρfr U,U (x′ , x′′ ) = α holds if and only if ρfr V,V (f (x′ ), f (x′′ )) = α. Then f can be extended to an isometry F : U → V in the Euclidean metrics.

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The next result is obtained by M. K. Borovikova.11 It contains a complete description of the case where a bounded polygonal domain in R2 is uniquely determined in the class of all such domains by the condition of local isometry of Hausdorff boundaries in the relative metrics. Theorem 5.2. In order that a bounded polygonal domain U ⊂ R2 be uniquely determined in the class of all such domains by the condition of local isometry of Hausdorff boundaries in the relative metrics, it is necessary and sufficient that U be convex. Observe that the condition “U is bounded” cannot be rejected (see Ref. 11). We complete this Section with a result by V. A. Aleksandrov,12 which permits to conclude that the situation in R3 differs essentially from the case studied by Borovikova in Theorem 5.2 for two-dimensional polyhedral domains. Slightly coarsening, we can formulate Aleksandrov’s result as follows: If a domain U ⊂ R3 has the compact polyhedral boundary then it is uniquely determined in the class of all domains in R3 by the condition of local isometry in the relative metrics of Hausdorff boundaries of domains. Note that although the boundary of U in the statement of this Theorem is compact, the domain U itself may be unbounded. 6. Unique determination and Riemannian manifolds Suppose that (X, g) is an n-dimensional connected smooth Riemannian manifold without boundary. Let Yj , j = 1, 2, be two n-dimensional compact connected C 0 -submanifolds of X with boundaries ∂Yj (6= ∅) satisfying the following conditions: (1) if x, y ∈ Yj then ρYj (x, y) =

lim inf

x′ →x,y ′ →y;x′ ,y ′ ∈int Yj

{inf [l(γx′ ,y′ ,int Yj )]} < ∞,

where inf [l(γx′ ,y′ ,int Yj )] is the infimum of the lengths l(γx′ ,y′ ,int Yj ) of the smooth paths γx′ ,y′ ,int Yj : [0, 1] → int Yj joining x′ and y ′ in the interior int Yj of Yj , moreover, ρYj is a metric in Yj ; (2) for arbitrary two points a, b ∈ Yj there exist points c, d ∈ ∂Yj which may be joint in Yj by a shortest curve γ : [0, 1] → Yj in the metric ρYj such that a, b ∈ γ. Denote also by ρX (x, y) (where x, y ∈ X) the infimum of the lengths l(γx,y,X ) of the smooth paths γx,y,X : [0, 1] → X joining x and y.

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Theorem 6.1. In addition to the above-mentioned conditions, assume that Y1 is strictly convex in the metric ρY1 , i.e., if α, β ∈ Y1 then, for each shortest curve γ = γα,β,Y1 : [0, 1] → Y1 in the metric ρY1 joining α and β in Y1 , we have γ(t) ∈ int Y1 , where 0 < t < 1. Suppose also that the boundaries of Y1 and Y2 are isometric in the metrics ρYj , j = 1, 2, i.e., there exists a surjection f : ∂Y1 → ∂Y2 which is an isometry in these metrics. Then Y2 is strictly convex with respect to ρY2 . Remark 6.1. Assume that the manifold X in Theorem 6.1 has the following property: ρX (x, y) = ρY (x, y) for any two points x, y in every ndimensional compact connected C 0 -submanifold Y ⊂ X with boundary satisfying the condition (1) (where Yj = Y ) and strictly convex with respect to the metric ρY . Then (by this Theorem) we obtain the following assertion (A) (on the rigidity, i.e., on the unique determination of the boundary of Y1 ): ∂Y1 and ∂Y2 are isometric in the metric ρX of X. This assertion is very close to A. D. Aleksandrov’s Theorem 3.1: Theorem 3.1 is an immediate consequence of assertion (A) (in the case where the closures of domains Uj , j = 1, 2, in Theorem 3.1 are compact C 0 -submanifolds in Rn (with ∂{cl Uj } = fr Uj ) satisfying the condition (1)). Proof. Theorem 6.1 is proved by methods similar to those used in the proof of Theorem 3.1 (see Ref. 4). 7. On unique determination of conformal type In the previous Sections we considered the problem of unique determination of domains in Euclidean spaces which has, as may be termed, an isometric type. In other words, we discussed the problem of whether given domains U, V are isometric provided that their boundaries are isometric in some sense. In the present Section we proceed with further development of that topic discussing the problem of unique determination of conformal type of domains in Rn . Let us start with details concerning the contents of the Section. Definition 7.1. Let U ⊂ Rn be a domain with Lipschitz boundary (i.e., a domain whose boundary fr U is an (n − 1)-dimensional Lipschitz manifold without boundary). By a boundary condenser F = {F1 , F2 } of the domain U we mean a couple of disjoint closed subsets F1 , F2 of fr U , at least one of them being bounded. The sets F1 , F2 are called the components of the condenser F (or “plates” of F ).

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Definition 7.2. A relative conformal modulus M U (F ) of a boundary condenser F of a domain U is an n-modulus Mn (ΓF1 ,F2 ,U ) of the family ΓF1 ,F2 ,U of all continuous paths γ : [0, 1] → cl U connecting the components F1 and F2 of the condenser F in the domain U (i.e., we mean paths γ such that γ(0) ∈ F1 , γ(1) ∈ F2 , and γ(t) ∈ U for t ∈ ]0, 1[). Remark 7.1. As to the notion of n-modulus Mn (Γ) of a family Γ of continuous curves (paths) in Rn , see, e.g., Ref. 13. Let L0 = L0 (n) be a subclass of the class L = L(n) of all domains U ⊂ R , n ≥ 3, the closure of each of them being an n-dimensional Lipschitz manifold (with boundary fr U 6= ∅). n

Definition 7.3. We say that a domain U ∈ L0 is uniquely determined by the relative conformal moduli of their boundary condensers in the class L0 if the following condition is satisfied. Assume that V ∈ L0 and there exists a homeomorphism f : fr V → fr U of fr V onto fr U preserving the relative conformal moduli of boundary condensers: M V (F ) = M U (f (F )) = M U ({f (F1 ), f (F2 )}) for each boundary condenser F of V . Then the domain V can be conformally mapped onto U (in other words, U can be determined in the class L0 up to possibly additional M¨obius transformation). In 2006 (see Refs. 14,15), Kopylov proved the following assertion (about unique determination of convex polyhedral domains by relative conformal moduli of their boundary condensers). Theorem 7.1. Assume n ≥ 4. Then each bounded convex polyhedral domain U ⊂ Rn ( i.e., a nonempty bounded intersection of finitely many open n-dimensional half-spaces) is uniquely determined by the relative conformal moduli of their boundary condensers in the class P of all bounded convex polyhedral domains V ⊂ Rn . Moreover, U can be determined in the class P up to possibly additional affine conformal mapping F : Rn → Rn . It should be observed that Theorem 7.1 (as well as the rest results of our paper) may be considered as one of important applications of notions and ideas of differential geometry in the theory of conformal and (in prospect) quasiconformal mappings for study of their boundary values. Acknowledgments This research was carried out with the partial support of the Russian Foundation for Basic Researches (project no. 05–01–00482), the Programm

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of Support of Leading Scientific School (project no. NSh-8526.2006.1), the Exchange Program between the Russian and Polish Academies of Sciences (2005-2007, project ”Stability and regularity of solutions to PDE’s system and related problems of the theories of quasiconformal mappings and harmonic fields”), and the Interdisciplinary integration project 2007 of the Siberian Division, Russian Academy of Sciences. References 1. A.V. Pogorelov, Extrinsic Geometry of Convex Surfaces (American Mathematical Society, Providence, 1973). 2. E.P. Sen’kin, Rigidity of convex hypersurfaces, In: Ukr. geometr. sb., vyp. 12 (Izd-vo KhGU, Kharkov, 1972) 131–152. 3. A.P. Kopylov, On boundary values of mappings which are close to isometric mappings, Siberian Math. J. 25 (1985) 438–447. 4. V.A. Aleksandrov, Isometry of domains in Rn and relative isometry of their boundaries, Siberian Math. J. 25 (1985) 339–347. 5. V.A. Aleksandrov, Isometry of domains in Rn and relative isometry of their boundaries. II, Siberian Math. J. 26 (1986) 783–787. 6. A.V. Kuz’minykh, On isometry of domains whose boundaries are isometric in the relative metrics, Siberian Math. J. 26 (1986) 380–387. 7. V.A. Aleksandrov, On domains which are uniquely determined by the relative metrics of their boundaries, In: Trudy Inst. Mat. Akad. Nauk SSSR, Sibirsk. Otd., 7: Research in Geometry and Mathematical Analysis (Sobolev Institute of Mathematics 1987) 5–19, in Russian. 8. V.A. Aleksandrov, Unique determination of domains with non-Jordan boundaries, Siberian Math. J. 30 (1989) 1–8. 9. M.V. Korobkov, On necessary and sufficient conditions for unique determination of plane domains, Preprint/RAS. Sib. Div., Sobolev Institute of Mathematics, Novosibirsk, 2006, no. 174, pp. 27. 10. M.V. Korobkov, On necessary and sufficient conditions for unique determination of plane domains, to appear in Siberian Math. J. 11. M.K. Borovikova, On isometry of polygonal domains with boundaries locally isometric in relative metrics, Siberian Math. J. 33 (1993) 571–580. 12. V.A. Aleksandrov, On isometry of polyhedral domains with boundaries locally isometric in relative metrics, Siberian Math. J. 33 (1992) 177–182. 13. J. V¨ ais¨ al¨ a, Lectures on n-Dimensional Quasiconformal Mappings (SpringerVerlag, Berlin-Heidelberg-New York, 1971). 14. A.P. Kopylov, Unique determination of convex polyhedral domains by relative conformal moduli of boundary condensers, Doklady Mathematics 74 (2006) 637–639. 15. A.P. Kopylov, Unique determination of domains in Euclidean spaces, Sovremennaya matematika. Fundamental’nye napravleniya 22 (2007) 139–167, in Russian.

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Invariance of g-natural metrics on tangent bundles Oldˇrich Kowalski Faculty of Mathematics and Physics, Charles University, Sokolovsk´ a 83, 186 75 Praha 8,Czech Republic E-mail: [email protected]ff.cuni.cz Masami Sekizawa Department of Mathematics, Tokyo Gakugei University, Koganei-shi Nukuikita-machi 4-1-1, Tokyo 184-8501, Japan E-mail: [email protected] In this paper we prove that each g-natural metric on a tangent bundle T M over a Riemannian manifold (M, g) is invariant with respect to the induced map of a (local) isometry of the base manifold. Then we define natural metrics on unit tangent sphere bundles and prove their homogeneity in case that (M, g) is a two-point homogeneous space. As a corollary, we re-prove a result from a paper by Musso and Tricerri obtained by another method. Keywords: Riemannian manifold, tangent bundle, tangent sphere bundle, g-natural metrics, homogeneity. MS classification: 53C07, 53C20, 53C21, 53C40.

1. Introduction There are well-known classical examples of “lifted metrics” on the tangent bundle T M over a Riemannian manifold (M, g). Namely, these are the Sasaki metric, the horizontal lift and the vertical lift. As one can see, the classical constructions are examples of “natural transformations of second order”. In Ref. 14, the present authors fully classified all (possibly degenerate) naturally lifted metrics “of second order” on T M . They proved that the complete family of such natural metrics (for a fixed base metric) is a module over real functions generated by some generalizations of known classical lifts. Our idea of naturality is closely related to that of A. Nijenhuis, D. B. A. Epstein, P. Stredder and others (see Ref. 13 for the full references). We have used for our purposes the concepts and methods developed by D. Krupka15,16 and D. Krupka and V. Mikol´aˇsov´a.17 See also

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I. Kol´ aˇr, P. W. Michor and J. Slov´ak,13 pp. 227–280, and D. Krupka and J. Janyˇska,18 pp. 160–166 for other presentations of our study and for the concept of naturality in general. We shall use further the name “g-natural metrics” for our natural metrics on T M as it was proposed by M. T. K. Abbassi in Ref. 1. One of the properties of g-natural lifts of metrics to be expected is the “invariance property” saying that the lifts of (local) isometries are again (local) isometries. After a short survey about our classification we present it in the convenient setting due to M. T. K. Abbassi and M. Sarih.4 Then we prove that the invariance property really holds. Next we pass over to the tangent sphere bundles. We define g-natural metrics on tangent sphere bundles Tr M of constant radius r > 0 as restrictions of g-natural metrics on T M to the hypersurfaces Tr M of T M , and we present an explicit formula belonging to M. T. K. Abbassi. Then we prove that each tangent sphere ˜ over a two-point homobundle Tr M equipped with a g-natural metric G geneous space is a homogeneous space. The local version of this statement follows automatically. As a corollary, we re-prove the result by E. Musso and F. Tricerri in Ref. 19 saying that the homogeneity holds for the Sasaki metric. Another consequence is that, over a space (M, g) of constant curvature, a tangent sphere bundle Tr M with a g-natural metric is locally homogeneous. 2. Lifts of vectors The tangent bundle T M over a smooth manifold M consists of all pairs (x, u), where x is a point of M and u is a vector from the tangent space Mx of M at x. We denote by p the natural projection of T M to M defined by p(x, u) = x. Let g be a Riemannian metric on the manifold M and ∇ its Levi-Civita connection. Then the tangent space (T M )(x,u) of T M at (x, u) ∈ T M splits into the horizontal and vertical subspace H(x,u) and V(x,u) with respect to ∇: (T M )(x,u) = H(x,u) ⊕ V(x,u) . If a point (x, u) ∈ T M and a vector X ∈ Mx are given, then there exists a unique vector X h ∈ H(x,u) such that p∗ (X h ) = X. We call X h the horizontal lift of X to T M at (x, u). The vertical lift of X to (x, u) is a unique vector X v ∈ V(x,u) such that X v (df ) = Xf for all smooth functions f on M . Here we consider a one-form df on M as a function on T M defined by (df )(x, u) = uf for all (x, u) ∈ T M . The map X 7−→ X h

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is an isomorphism between Mx and H(x,u) , and the map X 7−→ X v is an isomorphism between Mx and V(x,u) . In an obvious way we can define horizontal and vertical lifts of vector fields on M . These are uniquely defined vector fields on T M . 3. g-natural metrics We say that a bundle morphism of the form ζ : T M ⊕T M ⊕T M −→ M × R is an F-metric on M if it is linear in the second and the third argument (and smooth in the first argument). We also say that ζ is symmetric or skew-symmetric if it is symmetric or skew-symmetric with respect to the second and third argument, respectively. (We use here the letter “F” to recall the Finsler geometry.) Any Riemannian metric g on M is a symmetric F-metric which is independent on u. In our special case, letting g be a given Riemannian metric on M , we speak about natural F-metrics derived from g which are F-metrics ζ, for a fixed u ∈ T M , whose components ζ(u)ij = ζ(u, ∂/∂xi , ∂/∂xj ) with respect to a system of local coordinates (x1 , x2 , . . . , xn ) in M are solutions of the system of differential equations 2

n X

gap

a=1

∂ζij ∂ζij − uq p = ζip δjq + ζpj δiq , ∂gaq ∂u

i, j, p, q = 1, 2, . . . , n.

We obtain Theorem 3.1 (Ref. 14). Let (M, g) be an n-dimensional oriented Riemannian manifold. Then all natural F-metrics ζ on M derived from g are given as follows: (1) For n = 2, all symmetric natural F-metrics are of the form ζ(u; X, Y ) = α(kuk2 )g(X, Y ) + β(kuk2 )g(X, u)g(Y, u) + γ(kuk2 ){g(X, u)g(Y, Ju) + g(X, Ju)g(Y, u)}, and all skew-symmetric natural F-metrics are of the form ζ(u; X, Y ) = δ(kuk2 ){g(X, u)g(Y, Ju) − g(X, Ju)g(Y, u)}, where α, β, γ and δ are arbitrary smooth functions of kuk2 = g(u, u) and J is one of the two canonical almost complex structures of (M, g) (for which (M, g, J) is a K¨ ahler manifold). (2) For n = 3, all symmetric natural F-metrics are of the form ζ(u; X, Y ) = α(kuk2 )g(X, Y ) + β(kuk2 )g(X, u)g(Y, u)

(1)

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and all skew-symmetric natural F-metrics are of the form ζ(u; X, Y ) = ϕ(kuk2 )g(X × Y, u),

where α, β and ϕ are arbitrary smooth functions of kuk2 = g(u, u), and X × Y is the usual vector product of X and Y . (3) For n > 3, all natural F-metrics are symmetric and of the form (1). I. Kol´ aˇr, P. W. Michor and J. Slov´ak have given in Ref. 13, Proposition 33.22 a new and elegant proof of the classification of natural F-metrics on non-oriented Riemannian manifolds. Theorem 3.2 (Ref. 13). Let (M, g) be an n-dimensional non-oriented Riemannian manifold, n ≥ 1. Then all natural F-metrics ζ on M derived from g are symmetric and given by (1), where α and β are arbitrary smooth functions defined on the interval (0, ∞). In particular, β = 0 if n = 1. M. T. K. Abbassi has proved in Ref. 1 explicitly that all basic functions from Theorems 3.1 and 3.2 can be prolonged to smooth functions on the set of all non-negative real numbers. This result found many applications in the techniques used for the thorough investigation of g-natural metrics by M. T. K. Abbassi. For a given Riemannian metric g on M , we define the classical lifts of F-metrics from M to T M , with respect to g, as follows: (a) The Sasaki lift ζ s,g of a symmetric F-metric ζ with respect to g is defined by s,g (X h , Y h ) = ζx (u; X, Y ), ζ(x,u)

s,g (X h , Y v ) = 0, ζ(x,u)

s,g (X v , Y h ) = 0, ζ(x,u)

s,g (X v , Y v ) = ζx (u; X, Y ) ζ(x,u)

for all X, Y ∈ Mx . (b) The horizontal lift ζ h,g of an arbitrary F-metric ζ with respect to g is defined by h,g ζ(x,u) (X h , Y h ) = 0,

h,g ζ(x,u) (X h , Y v ) = ζx (u; Y, X),

h,g ζ(x,u) (X v , Y h ) = ζx (u; X, Y ),

h,g ζ(x,u) (X v , Y v ) = 0

for all X, Y ∈ Mx . (c) The vertical lift ζ v of a symmetric F-metric ζ with respect to g is defined by v (X h , Y h ) = ζx (u; X, Y ), ζ(x,u)

v (X h , Y v ) = 0, ζ(x,u)

v ζ(x,u) (X v , Y h ) = 0,

v (X v , Y v ) = 0 ζ(x,u)

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for all X, Y ∈ Mx . Obviously, the vertical lift does not depend on the choice of g. We note that ζ s,g , ζ h,g and ζ v are (not necessarily regular) pseudo-Riemannian metrics on T M . If we take ζ = g, then ζ s,g , ζ h,g and ζ v are just the classical lifts g s , g h and g v , respectively. Then we can present all metrics on T M which come from a second order natural transformation of a given Riemannian manifold (M, g) in the following form: Theorem 3.3 (Ref. 14). Let g be a Riemannian metric on an n-dimensional (oriented or non-oriented ) smooth manifold M , n ≥ 2, and let G be a metric (possibly degenerate) pseudo-Riemannian metric on the tangent bundle T M which comes from a second order natural transformation of g. Then there are natural F-metrics ζ1 , ζ2 and ζ3 derived from g, where ζ1 and ζ3 are symmetric, such that G = ζ1 s,g + ζ2 h,g + ζ3 v . Moreover, all natural F-metrics derived from g are given by Theorem 3.1. If n = 2, then the family of all natural metrics G on T M depends on 10 arbitrary functions of one variable, for n = 3 it depends on seven arbitrary functions of one variable, and for n > 3 on six arbitrary functions of one variable. M. T. K. Abbassi in Ref. 1 started to call natural metrics on tangent bundles as “g-natural metrics”. For the purpose of this paper, we shall use the alternative formulas belonging to Abbassi and Sarih, with a little modification. Theorem 3.4 (Refs. 1,4). Let (M, g) be an n-dimensional (nonoriented ) Riemannian manifold and G be a g-natural metric on the tangent bundle T M . Then there are real valued functions αi and βi , i = 1, 2, 3, defined on [0, ∞) such that G(x,u) (X h , Y h ) = (α1 + α3 )(kuk2 )gx (X, Y )

+ (β1 + β3 )(kuk2 )gx (X, u)gx (Y, u), G(x,u) (X h , Y v ) = α2 (kuk2 )gx (X, Y ) + β2 (kuk2 )gx (X, u)gx (Y, u), v

h

2

(2)

2

G(x,u) (X , Y ) = α2 (kuk )gx (X, Y ) + β2 (kuk )gx (X, u)gx (Y, u), G(x,u) (X v , Y v ) = α1 (kuk2 )gx (X, Y ) + β1 (kuk2 )gx (X, u)gx (Y, u) hold at each point (x, u) ∈ T M for all u, X, Y ∈ Mx , where k·k denotes the norm of the vector. For n = 1, the same holds with βi = 0, i = 1, 2, 3.

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4. Invariance of g-natural metrics Let φ be a (local) transformation of a manifold M . Then we define a transformation Φ of T M by Φ(x, u) = (φx, φ∗x u) for all (x, u) ∈ T M . Proposition 4.1. Let φ be a (local ) affine transformation of a manifold M with an affine connection ∇ and let Φ be the lift of φ to T M defined as above. Then we have Φ∗ (X h ) = (φ∗ X)h ,

Φ∗ (X v ) = (φ∗ X)v

for all X ∈ X(M ). Proof. We use the formula p ◦ Φ = φ ◦ p. For all functions f on M we calculate: h (p∗Φ(x,u) (Φ∗(x,u) (X(x,u) )))f h h (f ◦ φ ◦ p) (f ◦ p ◦ Φ) = X(x,u) = X(x,u) h ))(f ◦ φ) = Xx (f ◦ φ) = (p∗(x,u) (X(x,u)

= (φ∗x Xx )f. h Since Φ preserves the horizontal distribution, we have Φ∗(x,u) (X(x,u) ) = h (φ∗x Xx )Φ(x,u) by the definition of the horizontal lift. Next, we have (df ) ◦ Φ = d(f ◦ φ). In fact, since, by the definition, (df )(x, u) = uf at (x, u) ∈ T M holds for all functions f on M , we calculate at every point (x, u) ∈ T M as

((df ) ◦ Φ)(x, u) = (df )(φx, φ∗x u) = (φ∗x u)f = u(f ◦ φ) = (d(f ◦ φ))(x, u). Hence we obtain v (Φ∗(x,u) X(x,u) )(df ) v v (d(f ◦ φ)) ((df ) ◦ Φ) = X(x,u) = X(x,u)

= Xx (f ◦ φ) = (φ∗x Xx )(f ) = (φ∗x Xx )v(φx,φ∗x u) (df ) v Since Φ preserves the vertical distribution, we have Φ∗(x,u) (X(x,u) ) = v (φ∗x Xx )Φ(x,u) by the definition of the vertical lift.

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Theorem 4.1. Let φ be a (local ) isometry of a Riemannian manifold (M, g). Then every g-natural metric G on the tangent bundle T M over (M, g) is invariant by the lift Φ of φ. In other words, Φ is a local isometry of (T M, G) whose projection on (M, g) is φ. Proof. Let X, Y be vectors from Mx . Then, by Proposition 4.1 and the first formula of (2), we have at any point (x, u) ∈ T M that h h ) , Y(x,u) (Φ∗ G)(x,u) (X(x,u)

h h = GΦ(x,u) (Φ∗(x,u) X(x,u) , Φ∗(x,u) Y(x,u) )

= G(φx,φ∗x u) ((φ∗x X)h(φx,φ∗x u) , (φ∗x Y )h(φx,φ∗x u) ) = (α1 + α3 )(kφ∗x uk2 )gφx (φ∗x X, φ∗x Y ) + (β1 + β3 )(kφ∗x uk2 )gφx (φ∗x X, φ∗x u)gφx (φ∗x Y, φ∗x u). Now, since φ is an isometry of (M, g), the right-hand side of this formula h h is just G(x,u) (X(x,u) , Y(x,u) ). Thus we have h h h h (Φ∗ G)(x,u) (X(x,u) , Y(x,u) ) = G(x,u) (X(x,u) , Y(x,u) )

for all X, Y ∈ Mx . The rest of the assertion is proved by similar calculations. Let us remark that the natural projection p of (T M, G) onto (M, g) is not a Riemannian submersion, in general. 5. Tangent sphere bundles Let r be a positive number. Then the tangent sphere bundle of radius r over a Riemannian manifold (M, g) is a smooth hypersurface Tr M = {(x, u) ∈ T M | gx (u, u) = r2 } of the tangent bundle T M . For any vector field X tangent to M , the horizontal lift X h is always tangent to Tr M at each point (x, u) ∈ Tr M . Yet, in general, the vertical lift X v is not tangent to Tr M at (x, u). Proposition 5.1. Let (M, g) be a Riemannian manifold. Then the tangent space of Tr M at a point (x, u) is given by v h | X ∈ Mx , Y ∈ {u}⊥ ⊂ Mx }, + Y(x,u) (Tr M )(x,u) = {X(x,u)

where {u}⊥ is the orthogonal complement of the subspace spanned by u in Mx .

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Proof. Let α be a curve of Tr M . Then it is written in the form α : t 7−→ α(t) = (x(t), u(t)). A vertical vector tangent to Tr M at (x, u0 ) ∈ Tr M is a vector tangent to some curve α : t 7−→ α(t) = (x, u(t)) such that u(0) = u0 . The velocity of α at t = 0 is α′ (0) = (0, u′ (0)). Since g(u(t), u(t)) = r2 is constant, we have gx (u′ (0), u(0)) = 0, that is, u′ (0) ∈ Mx is orthogonal to u0 = u(0). Now v the vertical space V(x,u0 ) of T M is spanned by the all vertical lifts X(x,u 0) v of X ∈ Mx , and hence it is linearly isomorphic to Mx . Moreover, X(x,u0 ) is tangent to Tr M if and only if X is orthogonal to u0 in Mx . Hence the assertion follows. Definition 5.1. Consider a smooth Riemannian manifold (M, g). A g-nat˜ ural metric on the tangent sphere bundle Tr M over (M, g) is restriction G of any g-natural metric G given by (2) on the tangent bundle T M to the hypersurface Tr M of T M . All g-natural metrics on the tangent sphere bundle Tr M over a Riemannian manifold (M, g) has been characterized by M. T. K. Abbassi and O. Kowalski (preprint, unpublished). We reproduce it here (with the kind agreement of our friend M. T. K. Abbassi) also the short but detailed proof. Proposition 5.2. Let r > 0 and (M, g) be a Riemannian manifold. For ˜ on Tr M induced by a Riemannian g-natural G every Riemannian metric G on T M , there exist four constants a, b, c and d, with a > 0, a(a+c)−b2 > 0 and a(a + c + dr2 ) − b2 > 0, such that ˜ = a g˜s + b g˜h + c g˜v + d k˜v , G

(3)

where k is the natural F-metric on M defined by k(u; X, Y ) = g(X, u)g(Y, u) for all (u, X, Y ) ∈ T M ⊕ T M ⊕ T M , and g˜s , g˜h , g˜v and k˜v are considered here as the metrics on Tr M induced by the lifts g s , g h , g v and k v , respectively. Proof. Let us fix (x, u) ∈ Tr M (i.e., kuk = r). Then, in the formulas (2) we see that each right-hand side reduces to only one term whenever one of the vectors X, Y is orthogonal to u. Due to Proposition 5.1, the induced

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˜ (x,u) is completely characterized by the identities metric G ˜ (x,u) (X h , X h ) = (α1 + α3 )(r2 )gx (X1 , X2 ) G 1 2 + (β1 + β3 )(r2 )gx (X1 , u)gx (X2 , u), ˜ (x,u) (X1h , Y1v ) = α2 (r2 )gx (X1 , Y1 ), G ˜ (x,u) (Y1v , Y2v ) = α1 (r2 )gx (Y1 , Y2 ) G

(4)

˜ (x,u) depends on for all X1 , X2 ∈ Mx and Y1 , Y2 ∈ {u}⊥. In other words, G 2 2 2 2 (α1 + α3 )(r ), (β1 + β3 )(r ), α2 (r ) and α1 (r ). But, for any element of ˜ (x,u) depends (x, u) ∈ Tr M , the norm of u is a constant equal to r, then G 2 2 on four constants. We put a = α1 (r ), b = α2 (r ), c = α3 (r2 ) and d = (β1 +β3 )(r2 ). Then a, b, c and d are constants, and the formula (3) follows at once from the formulas (4). Now, the fact that G is Riemannian is equivalent to the inequalities in the Proposition. Hence the Proposition follows. It is well-known that if (M, g) is a two-point homogeneous space, then each tangent sphere bundle Tr M , r > 0, is a homogeneous space with respect to the induced maps of the isometries of (M, g). From Theorem 4.1 we obtain immediately Proposition 5.3. Let (M, g) be a two-point homogeneous space, and let ˜ over the tangent sphere bundle Tr M be equipped with a g-natural metric G ˜ is homogeneous as Riemannian manifold. (M, g). Then (Tr M, G) As a corollary, we have reproved the special result by E. Musso and F. Tricerri in Ref. 19 saying that, for the induced Sasaki metric g˜s , the unit tangent sphere bundle (T1 M, g˜s ) is a homogeneous space. We can formulate also the local version of Proposition 5.3: Proposition 5.4. Let (M, g) be locally isometric to a two-point homogeneous space, and let the unit tangent sphere bundle Tr M be equipped with a ˜ over (M, g). Then the space (Tr M, G) ˜ is locally homog-natural metric G geneous. Hence we obtain Corollary 5.1. Let (M, g) be a space of constant sectional curvature, and ˜ let the tangent sphere bundle Tr M be equipped with a g-natural metric G ˜ over (M, g). Then the space (Tr M, G) is locally homogeneous.

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Acknowledgments ˇ 201/05/2707 and The first author was supported by the grant GA CR by the project MSM 0021620839. References 1. M.T.K. Abbassi, Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifolds (M, g), Comment. Math. Univ. Carolinae, 45 (4) (2004) 591–596. 2. M.T.K. Abbassi and G. Calvaruso, g-natural contact metrics on unit tangent sphere bundles, Monatsh. Math. 151 (2) (2007) 89–109. 3. M.T.K. Abbassi and O. Kowalski, On g-natural metrics with constant scalar curvature on unit tangent sphere bundles, In: Topics in almost Hermitian geometry and related fields (World Sci. Publ., Hackensack, NJ, 2005) 1–29. 4. M.T.K. Abbassi and M. Sarih, On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds, Diff. Geom. Appl. 22 (2005) 19–47. 5. M.T.K. Abbassi and M. Sarih, On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. (Brno) 41 (2005) 71–92. 6. M.T.K. Abbassi and M. Sarih, On Riemannian g-natural metrics of the form a.g s + b.g h + c.g v on the tangent bundle of a Riemannian manifold (M, g), Mediter. J. Math. 2 (2005) 19–43. 7. A.L. Besse, Einstein manifolds (Springer-Verlag, Berlin-Heidelberg-New York, 1987). 8. D. Blair, When is the tangent sphere bundle locally symmetric?, In: Geometry and topology (World Sci. Publishing, Singapore, 1989) 15–30. 9. E. Boeckx and L. Vanhecke, Geometry of the tangent sphere bundle, In: Proceedings of the Workshop on Recent Topics in Differential Geometry ((L.A. Cordero and E. Garc´ıa-R´ıo, Eds.) Santiago de Compostela, Spain, 1997) 5– 17. 10. E. Boeckx and L. Vanhecke, Curvature homogeneous unit tangent sphere bundles, Publ. Math. Debrecen 35 (1998) 389–413. 11. E. Boeckx and L. Vanhecke, Unit tangent sphere bundles and two-point homogeneous spaces, Periodica Math. Hung. 36 (1998) 79–95. 12. P. Dombrowski, On the geometry of the tangent bundles, J. Reine Angew. Math. 210 (1962) 73–88. 13. I. Kol´ aˇr, P.W. Michor and J. Slov´ ak, Natural Operations in Differential Geometry (Springer-Verlag, Berlin-Heidelberg-New York, 1993). 14. O. Kowalski and M. Sekizawa, Natural transformations of Riemannian metrics on manifolds to metrics on tangent bundles—A classification—, Bull. Tokyo Gakugei Univ. 40 (4) (1988) 1–29; http://ir.u-gakugei.ac.jp/handle /2309/36051. 15. D. Krupka, Elementary theory of differential invariants, Arch. Math.(Brno) 4 (1978) 207–214. 16. D. Krupka, Differential invariants (Lecture Notes, Faculty of Science, Purkynˇe University, Brno, 1979).

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17. D. Krupka and V. Mikol´ aˇsov´ a, On the uniqueness of some differential invariants: d, [ , ], ∇, Czechoslovak Math. J. 34 (1984) 588–597. 18. D. Krupka and J. Janyˇska, Lectures on Differential Invariants (University J. E. Purkynˇe in Brno, 1990). 19. E. Musso and F. Tricerri, Riemannian metrics on tangent bundles, Ann. Mat. Pura Appl. (4) 150 (1988) 1–20. 20. S. Sasaki, On the differential geometry of tangent bundles, Tˆ ohoku Math. J. 10 (1958) 338–354.

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Some variational problems in geometry of submanifolds Haizhong Li Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China E-mail: [email protected] In this paper, we present a survey of some variational problems in geometry of submanifolds, which includes our recent research results in geometry of rminimal submanifolds, geometry of Willmore submanifolds and variations of some parametric elliptic functional. We also propose some open problems at the end of paper. Keywords: r-minimal submanifold, Willmore submanifold, r-anisotropic mean curvature, Wulff shape. MS classification: 53C42, 53A10, 49Q10.

1. Introduction This paper is to present our recent research results about variational problems in geometry of submanifolds and divides into six sections. Now we describe in more detail some of the contents of this paper, where we refer to the papers for the explanation of notations and undefined terms. The section 1 is the introduction. The section 2 is preliminaries and is devoted to some formulas and notations of submanifolds in geometry of submanifolds. In section 3 we present the first and second variational formula of the volume of n-dimensional submanifolds in an (n + p)-dimensional manifold N n+p . Besides, we give two important results of J. Simons. In section 4, we will recall our recent works in Ref. 4. We introduce the functional Z Fr (S0 , S2 , · · · , Sr )dvg Jr = M

where function Fr is a suitable function on M , and introduce the concepts of r-minimal submanifolds and stability. We study the stability of compact r-minimal submanifold in the unit sphere S n+p . In section 5 we give some facts (see Refs. 10,16,17) about variational problems of Willmore functional on submanifolds. In particular, we recall the Euler-Lagrange equa-

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tion of W (x) for an n-dimensional submanifold in an (n + p)-dimensional Riemannian manifold N n+p , give some typical examples of Willmore submanifolds and give integral inequality of Simons’ type for n-dimensional closed Willmore submanifolds in S n+p . In section 6 we consider a variation problem concerning certain parametric elliptic functional and collect some results of B. Palmer and He-Li. We state integral formula of Minkowski’s type8 and new characterizations of the Wulff shape.9,22 2. Preliminaries Let(N n+p , h) be an n + p-dimensional oriented smooth Riemannian manifold, and x : M n → N n+p be an n-dimensional submanifold of (N n+p , h). We will agree on the following index convention: 1 ≤ i, j, k, · · · ≤ n;

n + 1 ≤ α, β, γ, · · · ≤ n + p;

1 ≤ A, B, C, · · · ≤ n + p

Let{eA } be a local orthonormal basis for T N n+p with dual basis {θA } such that when restricted to M n , {ei } is a local orthonormal basis for T M and {eα } is a local orthonormal basis for the normal bundle of x : M n → N n+p . Let {ωAB } be the connection forms of (N n+p , h), they are characterized by the following structure equations dωA =

X B

ωAB ∧ ωB ,

ωAB = ωAC ∧ ωCB −

ωAB + ωBA = 0

(1)

1X¯ RABCD ωC ∧ ωD 2

(2)

C,D

¯ ABCD are the components of the Riemannian curvature tensor of whereR n+p (N , h). Now we restrict to a neighborhood of x : M n → N n+p . Let θA , θAB be the restrictions of ωA , ωAB to M n , then we have θα = 0

(3)

Taking its exterior derivative and by (1) we get X θαi ∧ θi = 0

(4)

α hα ij = hji .

(5)

i

By Cartan’s lemma we have X hα θiα = ij θj , j

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We can define the second fundamental form Bij and the mean curvature vector H of x as follows X 1X α 1X α hii := H eα (6) Bij := hα H= ij eα , n i,α n α α Let S = |B|2 =

P

2 (hα ij ) and H = |H| be the norm square of B and

i,j,α

the mean curvature of x : M n → N n+p , respectively. If we denote by Rijkl , Rij , R the Riemannian curvature tensor, Ricci curvature and scalar curvature of M , respectively. We have Gauss equations, Ricci equations and Codazzi equations as followings (see Refs. 3,10) ¯ ijkl + Rijkl = R

X α

¯ αβij + Rαβij = R

X α

α ¯ hα ijk − hikj = Rαikj ,

α α α (hα ik hjl − hil hjk ) β α β (hα ik hkj − hjk hki ),

(7) (8) (9)

α where hα ijk is the covariant derivative of hij . In particular, when the ambient space N n+p is a space form Rn+p (c), the Gauss equation, Ricci equation and Codazzi equation are (also see Refs. 14, 15) X α α α Rijkl = c(δik δji − δil δjk ) + (hα (10) ik hjl − hil hjk ),

Rαβij = hα ijk =

X

α

β (hα ik hkj

α hα ikj .

−

β hα jk hki ),

(11)

(12)

3. Minimal submanifolds in a Riemannian manifold Let x0 : M → N n+p be an n-dimensional submanifolds in an (n + p)dimensional manifold N n+p , we assume, without loss of generality, that M is compact with (possibly empty) boundary. If otherwise, we will consider the variation with compact support. Let x : M ×R → N n+p be a smooth variation of x0 such that x(·, t) = x0 and dxt (T M ) = dx0 (T M ) on ∂M for each small t, where xt (p) = x(p, t). Let V (t) be the volume functional of xt (M ), i.e. Z θ1 ∧ · · · ∧ θn . (13) V (t) = M

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We have (see Refs. 3,21) V ′ (t) = −n where

P

Z

M

X α

H α aα θ1 ∧ · · · ∧ θn ,

(14)

aα eα is the normal variation vector field.

α

Definition 3.1. Let x0 : M → N n+p be an n-dimensional submanifold in an (n + p)-dimensional manifold N n+p , if X ~ = 1 H H α eα ≡ 0, (15) n α we call M be a minimal submanifold.

Proposition 3.1 (Refs. 3,27). Let x0 : M → N n+p be an n-dimensional submanifold in an (n + p)-dimensional manifold N n+p , then we have Z X n+p n X X X ′′ ⊥ ˜ αiβj aβ )θ1 ∧ · · · ∧ θn , V (0) = − aα (∆ aα + σαβ aβ + R M

α

β

i=1 β=n+1

(16) ˜ where R is the components of Riemannian curvature tensor, σαβ = P α β αiβj hij hij , and ∆⊥ aα is the Laplacian of aα in the normal bundle. i,j

Definition 3.2. Let x : M → N n+p be an n-dimensional submanifold in P N n+p . If for any normal variational vector W = aα eα , we have V ′′ (0) ≥ 0, we call M is stable.

α

Theorem 3.1 (J. Simons, Ref. 27). There exists no any n-dimensional closed stable minimal submanifolds in an (n + 1)-dimensional unit sphere S n+p . Example 3.1 (see Refs. 1,13). Clifford tori: r r m n−m m n−m Cm,n−m = S ( )×S ( ) , n n

1≤m≤n−1

are minimal hypersurfaces in S n+1 . Theorem 3.2 (Refs. 1,13,27). Let M be an n-dimensional (n ≥ 2) closed minimal submanifold in (n + p)-dimensional unit sphere S n+p . Then we have   Z n − S dv ≤ 0. (17) S 2 − 1/p M

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In particular, if 0≤S≤

n , 2 − 1/p

(18)

n . In the latter 2 − 1/p case, either p = 1 and M is a Clifford torus Cm,n−m ; or n = 2, p = 2 and M is the Veronese surface. then either S ≡ 0 and M is totally geodesic, or S ≡

4. r-minimal submanifolds in a space form We recall the definition of the generalized Kronecker symbols. If r i1 , · · · , ir and j1 , · · · , jr are integers between 1 and n, then δji11 ···i ···jr is +1 ′ ′ or −1 according as the i s are distinct and the j s are an even or odd permutation of the i′ s, and is 0 in all other cases. Let (M, g) be an n-dimensional submanifold in an (n + p)-dimensional space form Rn+p (c), and B be the second fundamental form of M . Suppose {ei } is a local orthonormal basis for T M with dual basis {θi } and {eα } is a local orthonormal basis for the normal bundle of x : M n → Rn+p (c), (Bij ) is the matrix with respect to the frame {e1 , · · · , en } on M . Then for any even integer r ∈ {0, 1, · · · , n − 1}, we introduce r-th mean curvature ~r+1 as follows (see function Sr and (r + 1)-th mean curvature vector field S Refs. 4,5,25,26: Sr =

1 X i1 ···ir δ hBi1 j1 , Bi2 j2 i · · · hBir−1 jr−1 , Bir jr i r! i ···ir j1 ···jr 1 j1 ···jr

1X T α = r i,j,α r−1 ~r+1 = S

1 (r + 1)!

α ij hij ,

X

i1 ···ir+1 j1 ···jr+1

i ···i

δj11 ···jr+1 hBi1 j1 , Bi2 j2 i · · · hBir−1 jr−1 , Bir jr iBir+1 jr+1 r+1

1 X i α T h eα , r + 1 i,j,α rj ij     n n ~r+1 = ~ r+1 , Sr = Hr , S H r r+1 =


where nr being the binomial coefficient. Besides, we define the following (0, 2)-tensor Tr for r ∈ {1, · · · , n − 1}:

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• If r is even, we set Tr =

1 X i1 ···ir i δ hBi1 j1 , Bi2 j2 i · · · hBir−1 jr−1 , Bir jr iθi ⊗ θj r! i ···ir i j1 ···jr j 1 j1 ···jr j

=

X i,j

Trji θi ⊗ θj .

• If r − 1 is odd, we set Tr−1 =

1 (r − 1)!

X

i1 ···ir−1 i j1 ···jr−1 j

i ···i

i

hBi1 j1 , Bi2 j2 i · · · hBir−3 jr−3 , Bir−2 jr−2 i δj11 ···jr−1 r−1 j

·Bir−1 jr−1 θi ⊗ θj X α Tr−1 = ij θi ⊗ θj eα , i,j,α

By convention, we put H0 = S0 = 1, T0 =identity. Let x : M → Rn+p (c) be an n-dimensional compact submanifold in Rn+p (c). Assume that r is even and r ∈ {0, 1, · · · , n − 1}, , in Ref. 4 the authors introduce a curvature integral Jr for r even and r ∈ {0, 1, · · · , n−1} Z Fr (S0 , S2 , · · · , Sr )dv Jr = M

where the function Fr are defined inductively by ( F0 = 1 Fr = Sr + (n−r+1)c Fr−2 , for 2 ≤ r ≤ n − 1. r−1 Theorem 4.1 (Ref. 4, the first variational formula). Let M be an ndimensional compact, possibly with boundary, submanifold in an (n + p)dimensional space form Rn+p (c) . Assume that r is even and r ∈ {0, 1, · · · n − 1}, then we have Z ~r+1 , V idvgt hS Jr ′ (t) = −(r + 1) Mt

where V is the variational vector field. Definition 4.1 (Ref. 4). Let M be an n-dimensional submanifold in an (n + p)-dimensional space form Rn+p (c) . Assume that r is even and r ∈ {0, 1, · · · , n − 1}, we call x to be r-minimal if its (r + 1)-th mean curvature ~r+1 vanishes on M . vector S

Some variational problems in geometry of submanifolds

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Theorem 4.2 (Ref. 4, the second variational formula). Let M be an n-dimensional compact r-minimal submanifold in Rn+p (c) for some even integer r ∈ {0, 1, · · · , n − 1}, we have Z X X 1 ir−1 ir i β ′′ α J (0) = − Vα { Tr−2jr−1 jr j hir−1 jr−1 hir jr Vβ,ij r − 1 M α i ir i +

X i,j

r−1 jr−1 jr j β

Trji Vα,ij + c · (n − r)(Sr Vα +

− (r + 1)

X

i,j,β

Tr+1 αij hβij Vβ }dv

X

Tr−1 αij hβij Vβ )

i,j,β

Definition 4.2 (Ref. 4). Assume that r is even and r ∈ {0, 1, · · · , n − 1}, and let M be an n-dimensional compact r-minimal submanifold in an (n + ′′ p)-dimensional space form Rn+p (c). If Jr (0) ≥ 0 for arbitrary variations, we call M to be stable. Remark 4.1. From definition 4.1, we know that concept of 0-minimal submanifolds is the concept of minimal submanifolds. Theorem 4.3 (Ref. 4). Assume that r is even and r ∈ {0, 1, · · · , n−1}. If Sr is positive, then there exists no any closed stable r-minimal submanifold in the unit sphere S n+p . Remark 4.2. Noting when r = 0, i.e. S0 = 1, it is obvious to see that our Theorem 4.3 reduces to J. Simons’ Theorem 3.2. Theorem 4.4 (Ref. 4). Assume that r is even and r ∈ {1, · · · , n − 1}. If Sr ≥ 0, then any closed stable r-minimal hypersurface in unit sphere S n+1 must be a geodesic sphere. 5. Willmore submanifolds in a Riemannian manifold Let N n+p be an (n + p)-dimensional Riemannian manifold and x : M → N an isometric immersion of an n-dimensional Riemannian manifold M . Willmore functional is the following non-negative functional:(see Refs. 2,24 or Ref. 28) Z Z n n W (x) := ρ dv = (S − nH 2 ) 2 dv. n+p

where S =

P

α,i,j

M

M

2 (hα ij ) and H are respectively the norm square of the second

fundamental form and the mean curvature of the immersion x, dv is the

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H. Li

volume element of M . An immersion x : M → N n+p is called Willmore if it is an extremal submanifold of the Willmore functional. The famous Willmore conjecture can be stated in a equivalent way as that W (x) ≥ 4π 2

holds for all immersed tori x : M → S 3 (see Ref. 30). It is well known (see Refs. 2,24 or Ref. 28) that Willmore functional is invariant under conformal transformations of N n+p . In Refs. 16,24,28, the authors calculated the first variation formula of Euler-Lagrangian equation of W (x) for an n-dimensional submanifold in Rn+p (c). In Ref. 6, the authors calculated the second variation formula of W (x) for Willmore submanifolds x : M n → Sn+p without umbilic points in terms of M¨obius geometry and gave many examples of Willmore submanifolds. In Ref. 10 the authors calculated the Euler-Lagrangian equation for the critical points of W (x) for the most general case. Theorem 5.1 (Ref. 10). The variation of the Willmore functional depends only on the normal component of the variation vector field. A submanifold x : M → N n+p is a Willmore submanifold if and only if h X X X ˜ βiαj hβ − ρn−2 hβij hβik hα R H β hβij hα kj + ij ij −

X i,β

i,j,k,β

i,j,β

˜ βiαi − ρ H H R 2

β

o

α

i

i,j,β

Xn

n−2 hα + 2 ρn−2 i hα ijj + ρ i,j ij i,j

α n−2 + ρn−2 hα ) − ρn−2 ∆⊥ H α − 2 ijij − H ∆(ρ

n + 1 ≤ α ≤ n + p.

X i

ρ


n−2

i

(19)

H,iα = 0,

˜ ABCD are the components of the Riemannian curvature tensor of where R n+p (N , h). In particular, when N n+p = Rn+p (c), we have Theorem 5.2 (Refs. 16,24). A submanifold x : M n → Rn+p (c) is a Willmore submanifold if and only if h X i X 2 α ρn−2 hβij hβik hα H β hβij hα kj − ij − ρ H i,j,k,β

α

+ (n − 1)H ∆ ρ −

X i,j

ρ

n−2




i,j

n−2




α

i,j,β

+ 2(n − 1)

nH δij −

hα ij




X i

= 0,


ρn−2 i H,iα + (n − 1)ρn−2 ∆⊥ H α (20) n + 1 ≤ α ≤ n + p.

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191

Example 5.1 (Ref. 29). Every minimal surface in Rn+p (c) is Willmore. We note that there are much more abundance of non-minimal Willmore surfaces in R2+p (c), see, e.g. Refs. 20,23, among many others. Example 5.2 (Refs. 6,24). Every n-dimensional (≥ 3) minimal and Einstein submanifold in N n+p (c) is Willmore. Example 5.3 (Refs. 6,16). Wn1 ,··· ,np+1 = S n1 (a1 ) × · · · × S np+1 (ap+1 ) is an n-dimensional Willmore submanifold in S n+p (1), where n = n1 + · · · + np+1 and ai are defined by r n − ni , i = 1, · · · , p + 1. ai = np Furthermore, Wn1 ,··· ,np+1 is a minimal submanifold in S n+p (1) if and only if it is Einstein with r 1 n n1 = · · · = np+1 = , ai = . p+1 p+1 Example 5.4 (Refs. 6,16,17). Willmore tori: r r n−m m n−m m )×S ( ), Wm,n−m = S ( n n

1≤m≤n−1

are Willmore hypersurfaces in S n+1 (1). Example 5.5 (Ref. 11). Define the Lagrangian sphere Ψ : Sn → Cn by Ψ(x1 , · · · , xn , x√ n+1 )

n−1

=

2 2n eiβ(xn+1 ) √ 2n √ n−1 · (x1 , · · · , xn ), √ 2n [(1 + xn+1 ) n−1 + (1 − xn+1 ) n−1 ] 2n

β(xn+1 ) =

q

2(n−1) n

arctan(

(1+xn+1 ) (1+xn+1 )

√ n √ n 2(n−1) −(1−xn+1 ) 2(n−1) √ n ). √ n 2(n−1)

+(1−xn+1 )

2(n−1)

Then Ψ is a Lagrangian Willmore submanifold. We call Ψ as the Lagrangian Willmore sphere. We note that when n = 2, Ψ is Whitney sphere Theorem 5.3 (Refs. 16,17). Let M be an n-dimensional (n ≥ 2) closed Willmore submanifold in (n + p)-dimensional unit sphere S n+p . Then we have   Z n n 2 (21) ρ − ρ dv ≤ 0. 2 − 1/p M

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H. Li

In particular, if 0 ≤ ρ2 ≤

n , 2 − 1/p

(22)

n then either ρ2 ≡ 0 and M is totally umbilic, or ρ2 ≡ 2−1/p . In the latter case, either p = 1 and M is a Willmore torus Wm,n−m ; or n = 2, p = 2 and M is the Veronese surface.

For n = 2, the following result was proved by Li18 (also see Li-Simon19 ) Theorem 5.4 (Refs. 18,19). Let M be a closed Willmore surface in an (2 + p)-dimensional unit sphere S 2+p . Then we have Z 3 (23) ρ2 (2 − ρ2 )dv ≤ 0. 2 M In particular, if 0 ≤ ρ2 ≤

4 , 3

(24)

then either ρ2 = 0 and M is totally umbilic, or ρ2 = 43 . In the latter case, p = 2 and M is the Veronese surface. Remark 5.1. In Ref. 21, the authors give some examples of 3-dimensional Lagrangian Willmore submanifolds in S 6 . In Ref. 7, the authors study the R variational problem of the functional F (x) = M (S − nH 2 )dv for submanifold x : M n → S n+p and get an integral inequality of Simons’ type. 6. Variations of Some Parametric Elliptic Functional Let F : S n → R+ be a smooth function which satisfies the following convexity condition: (D2 F + F 1)x > 0,

∀x ∈ S n ,

(25)

where D2 F denotes the intrinsic Hessian of F on S n and 1 denotes the identity on Tx S n , > 0 means that the matrix is positive definite. We consider the map φ : S n → Rn+1

x → F (x)x + (gradS n F )x ,

its image WF = φ(S n ) is a smooth, convex hypersurface in Rn+1 called the Wulff shape of F (see Refs. 8,9,12,22).

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193

Now let X : M → Rn+1 be a smooth immersion of a compact, orientable hypersurface without boundary. Let ν : M → S n denotes its Gauss map, that is, ν is an unit inner normal vector of M . Let AF = D2 F + F 1, SF = −AF ◦ dν. SF is called the F -Weingarten operator, and the eigenvalues of SF are called anisotropic principal curvatures. Let σr be the elementary symmetric functions of the anisotropic principal curvatures λ1 , λ2 , · · · , λn : X σr = λi1 · · · λir (1 ≤ r ≤ n). i1 c˜, then the submanifold is not minimal and hence a generalized Veronese manifold is not a Veronese manifold, in general. It is well-known that generalized Veronese manifolds are the only examples of non totally umbilic but parallel immersions of space forms into space forms. These submanifolds have various nice geometric properties. For example, for each geodesic γ on the submanifold S n (nc/(2n + 2)) the curve f ◦ γ is fn+p (˜ a circle in q the ambient space M c). Moreover, the immersion f given
by (1) is ( 2nc/(n + 1) − c˜ -)isotropic (for the definition of “isotropic”, see section 2). The notion of isotropic immersions plays a key role in this paper. In this paper, we study Veronese manifolds and generalized Veronese manifolds in terms of isotropic immersions. The authors would like to express their hearty thanks to Professor Oldrich Kowalski for his valuable suggestion and encouragement during the preparation of this paper. 2. Isotropic immersions f of an n-dimensional Riemannian An isometric immersion f : M → M manifold into an (n + p)-dimensional Riemannian manifold is said to be (λ(x)-)isotropic at x ∈ M if kσ(X, X)k/kXk2(= λ(x)) does not depend on the choice of X(6= 0) ∈ Tx M , where σ is the second fundamental form of the immersion f . If the immersion is (λ(x)-)isotropic at every point x of M , then the immersion is said to be (λ-) isotropic (see Ref. 11). When λ = 0 on the submanifold M , this submanifold is totally geodesic in the ambient manifold f. Note that a totally umbilic immersion is isotropic, but not vice versa M (for details, see Refs. 2,6,14). The notion of isotropic immersions gives us

Generalized Veronese manifolds and isotropic immersions

199

valuable information in some cases. For example, we consider each parallel submanifold (M n , f ) of a standard sphere S n+p (˜ c). Then our manifold M n is a compact Riemannian symmetric space. Moreover, by the classification theorem of parallel submanifolds in S n+p (˜ c) we can see that M n is of rank one if and only if the isometric immersion f : M n → S n+p (˜ c) is isotropic (see Refs. 4,13). That is, in this case the notion of isotropic immersions distinguishes symmetric spaces of rank one from symmetric spaces of higher rank. 3. Characterizations of Veronese manifolds We first review the following fundamental lemma (Ref. 11): Lemma 3.1. Let f be a λ(> 0)-isotropic immersion of a Riemannian fn+p . The discriminant manifold M n (n ≧ 2) into a Riemannian manifold M e ∆x (X, Y ) at x ∈ M is defined by ∆x (X, Y ) = K(X, Y ) − K(X, Y ) for a e pair (X, Y ) of linearly independent vectors, where K(X, Y )(resp. K(X, Y )) denotes the sectional curvature of the plane spanned by X, Y ∈ Tx M for M f). Suppose that the discriminant ∆x at x ∈ M is constant for each (resp. M pair of vectors (X, Y ). Then the following inequalities hold at x : n+2 λ(x)2 ≦ ∆x ≦ λ(x)2 . − 2(n − 1) Moreover,

(i) ∆x = λ(x)2 ⇐⇒ f is umbilic at x ⇐⇒ dim Nx1 = 1, (ii) ∆x = −{(n + 2)/2(n − 1)}λ(x)2 ⇐⇒ f is minimal at x ⇐⇒ dim Nx1 = (n(n + 1)/2) − 1, (iii) −{(n + 2)/2(n − 1)}λ(x)2 < ∆x < λ(x)2 ⇐⇒ dim Nx1 = n(n + 1)/2. Here, we denote by Nx1 the first normal space at x, namely Nx1 = SpanR {σ(X, Y ) : X, Y ∈ Tx M }. Using Lemma 3.1, we establish the following local theorem which characterizes Veronese manifolds (cf. Refs. 1,8). Theorem 3.1. Let f be a λ-isotropic immersion of an n(≧ 2)-dimensional Riemannian manifold M n of constant sectional curvature k into an (n+ p)˜ fn+p of constant sectional curvature k. dimensional Riemannian manifold M 2 ˜ Suppose that p ≦ (n + n − 2)/2 and k < k. Then f is locally equivalent to a ˜ where k˜ = 2(n+1)k/n (parallel) minimal immersion of S n (k) into S n+p (k), and p = (n2 + n − 2)/2. Hence the submanifold (M n , f ) is locally congruent to a Veronese manifold.

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We here recall the following global theorem which characterizes Veronese manifolds. This theorem can be proved by using a well-known differential equation of the second fundamental form of the immersion (Ref. 12 and Ref. 5). Theorem 3.2. Let M n (n ≧ 2) be a connected compact oriented minimal but not totally geodesic submanifold of S n+p (˜ c) through a full isometric immersion f . Suppose that every sectional curvature K of M n satisfies K ≧ n˜ c/(2n + 2). Then K = n˜ c/(2n + 2), p = (n2 + n − 2)/2 and f is a n parallel immersion. Hence (M , f ) is a Veronese manifold. 4. Characterizations of generalized Veronese manifolds Motivated by Theorem 3.2, we obtained the following theorem (cf. Ref. 10). Theorem 4.1. Let M n be an n(≧ 3)-dimensional connected compact oriented isotropic submanifold whose mean curvature vector is parallel with respect to the normal connection in an (n + p)-dimensional space form fn+p (˜ M c) of constant sectional curvature c˜ through an isometric immersion f . Suppose that every sectional curvature K of M n satisfies K ≧ (n/2(n + 1))(˜ c + H 2 ), where H is the mean curvature of the immersion f . Then the immersion f has parallel second fundamental form and the submanifold (M n , f ) is congruent to one of the following: (1) M n is a congruent to S n (K) of constant sectional curvature K = c˜ + H 2 and f is a totally umbilic embedding; (2) (M n , f ) is a generalized Veronese manifold. That is, M n is congruent to S n (K) of constant sectional curvature K = (n/2(n + 1))(˜ c + H 2 ) and f is decomposed as: 2 f1 f2 fn+p (˜ f = f2 ◦ f1 : S n (K) −→ S n+(n +n−2)/2 (2(n + 1)K/n) −→ M c), where f1 is a minimal (parallel) immersion and f2 is a totally umbilic embedding. When n = 2, Theorem 4.1 also holds without the assumption that M n fn+p (˜ is isotropic in M c) (see Ref. 10).

Theorem 4.2. Let M 2 be a 2-dimensional connected compact oriented submanifold whose mean curvature vector is parallel with respect to the normal f2+p (˜ connection in a (2 + p)-dimensional space form M c) of constant sectional curvature c˜ through an isometric immersion f . Suppose that every sectional curvature K of M 2 satisfies K ≧ (1/3)(˜ c + H 2 ), where H is the

Generalized Veronese manifolds and isotropic immersions

201

mean curvature of f . Then the immersion f has parallel second fundamental form and the submanifold (M 2 , f ) is congruent to one of the following: (1) M 2 is congruent to S 2 (K) of constant sectional curvature K = c˜ + H 2 and f is a totally umbilic embedding; (2) (M 2 , f ) is a generalized Veronese surface. That is, M 2 is congruent to S 2 (K) of constant sectional curvature K = (1/3)(˜ c + H 2 ) and f is decomposed as: f1 f2 f2+p (˜ f = f2 ◦ f1 : S 2 (K) −→ S 4 (3K) −→ M c), where f1 is a minimal (parallel) immersion and f2 is a totally umbilic embedding. Remark 4.1. (a) In the assumption of Theorem 4.1, if we replace the condition that K ≧ (n/2(n+ 1))(˜ c + H 2 ) by a stronger condition that (n/2(n+ 1))(˜ c + H 2) ≦ 2 K < c˜ + H , then we obtain a corollary which characterizes just an n(≧ 3)-dimensional generalized Veronese manifold. Also in the assumption of Theorem 4.2, replacing “K ≧ (1/3)(˜ c + H 2 )” by “(1/3)(˜ c + H 2) ≦ 2 K < c˜ + H ”, we get a corollary which characterizes just a generalized Veronese surface. (b) The following proposition shows that Theorems 4.1 and 4.2 are no longer true if we replace the condition that “the mean curvature vector is parallel with respect to the normal connection” by a weaker condition that “the mean curvature is constant” in assumptions of these theorems (Ref. 9). Proposition 4.1. For each n(≧ 2) there exists at least one Riemannian submanifold M n of S n+p (˜ c) satisfying the following five conditions. (1) (2) (3) (4)

M n is a connected compact oriented Riemannian manifold. M n is isotropic in S n+p (˜ c). The mean curvature H of M n in S n+p (˜ c) is nonzero constant. Every sectional curvature K of M n satisfies K≧

 n c˜ + H 2  > (˜ c + H 2) . 2 2n + 2

(5) The mean curvature vector of M n in S n+p (˜ c) is not parallel with respect to the normal connection, so that in particular M n is not a parallel submanifold of S n+p (˜ c). Hence M n is neither totally umbilic in S n+p (˜ c) nor a generalized Veronese manifold.

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At the end of this paper, in consideration of Theorems 3.2, 4.1 and 4.2 we pose the following open problem: Problem. If we remove the assumption that the immersion is isotropic in Theorem 4.1, does this theorem hold? References 1.

2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13.

14.

N. Boumuki, Isotropic immersions with low codimension of space forms into space forms, Mem. Fac. Sci. Eng. Shimane Univ. Ser. B: Math. Sci. 37 (2004) 1–4. R. Bryant, Conformal and minimal immersions of compact surfaces into the 4-sphere, J. Differential Geom. 17 (1982) 455–473. M. Do Carmo and N. Wallach, Minimal immersions of spheres into spheres, Ann. Math. 93 (1971) 43–62. D. Ferus, Immersions with parallel second fundamental form, Math. Z. 140 (1974) 87–92. T. Itoh, Addendum to my paper “On Veronese manifolds” J. Math. Soc. Japan 30 (1978) 73–74. T. Itoh and K. Ogiue, Isotropic immersions, J. Differential Geometry 8 (1973) 305–316. T. Itoh and K. Ogiue, Isotropic immersions and Veronese manifolds, Trans. Amer. Math. Soc. 209 (1975) 109–117. S. Maeda, Isotropic immersions with parallel second fundamental form, Canad. Math. Bull. 26 (1983) 291–296. S. Maeda, On some isotropic submanifolds in spheres, Proc. Japan Acad. Ser. A 77 (2001) 173–175. S. Maeda and S. Udagawa, Characterization of parallel isometric immersions of space forms into space forms in the class of isotropic immersions, to appear in Canad. J. Math. B. O’Neill, Isotropic and Kaehler immersions, Canadian J. Math. 17 (1965) 905–915. J. Simons, Minimal varieties in riemannian manifolds, Ann. Math. 88 (1968) 62–105. M. Takeuchi, Parallel submanifolds of space forms, In: Manifolds and Lie groups (Notre Dame, Ind., 1980, in honor of Y. Matsushima, Birkh¨ auser, Boston, Mass. Progr. Math. 14, 1981) 429–447. K. Tsukada, Isotropic minimal immersions of spheres into spheres, J. Math. Soc. Japan 35 (1983) 355–379.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 203–214

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Integral formulae for foliations on Riemannian manifolds V. Rovenski Mathematical Department, University of Haifa, Haifa, 31905, Israel E-mail: [email protected] http://math.haifa.ac.il/ROVENSKI/rovenski.html P. Walczak Katedra Geometrii, Uniwersytet L´ odzki, ul. Banacha 22 L´ od´ z, Poland E-mail: [email protected] http://math.uni.lodz.pl/˜ pawelwal/ We generalize the integral formulae for foliations on space forms by Brito– Langevin–Rosenberg (1991) and by Brito–Naveira (2000). Our integral formulae concern foliations on complete Riemannian manifolds of a finite volume and involve the co-nullity operator, the mixed curvature operator, and their products. To prove the integral formulae for foliations of arbitrary codimension we introduce and study the tangent sphere bundle of a foliation. This is a submanifold of the tangent sphere bundle with the Sasaki metric studied by Borisenko–Yampol’skii (1991), Kowalski–Sekizawa (2000), and others. Keywords: Riemannian manifold, foliation, higher mean curvatures, mixed curvature, Sasaki metric. MS classification: 53C12.

1. Introduction Recent years brought increasing interest in the dynamics and extrinsic geometry of foliations, 13,18 etc. In what follows (M, g) denotes a compact (or a finite volume) Riemannian manifold with the Levi-Civita connection ∇, and the curvature tensor R; F = {L} a transversely oriented foliation on M , T F (T F ⊥ ) its tangent (orthogonal) distribution, Rξmix (y) = R(y, ξ) ξ (ξ ∈ T F , y ∈ T F ⊥ ), the mixed curvature operator, K mix the mixed sectional curvature (of planes containing some ξ ∈ T F , and y ∈ T F ⊥ ), Cξ y = (∇y ξ)⊥ the co-nullity operator, Aξ Weingarten operator (for the unit ξ), σi (Aξ ) the mean curvatures

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P of Aξ , i.e. the coefficients of the polynomial det(Id + tAξ ) = i σi (Aξ ) ti . For the codimension 1 foliations case, ξ denotes the unit normal vector field. The first known integral formula for codimension one foliations reads 12 Z H dVM = 0, where H = σ1 (Aξ )/ dim F − the mean curvature. (1) M

A general integral formula for pairs of complementary orthogonal plane fields on compact manifolds 15 is Z
K(F , F ⊥ ) + |A2 |2 − |H2 |2 − |T2 |2 + |A2 |2 − |H2 |2 − |T2 |2 dVM = 0 (2) M

P where K(F , F ⊥ ) = i,α g(R(ei , eα )ei , eα ), H1 , T1 , A1 and H2 , T2 , A2 are the mean curvature vector, the integrability tensor, and the 2-nd fundamental form of F and F ⊥ . In codimension one foliation case Eq. (2) reduces to Z 1 (3) [σ2 (Aξ ) − Ric(ξ)] dVM = 0. 2 M These formulae are of some interest, they can be useful for the problems: prescribing higher mean curvatures σi of a foliation; minimizing functions like volume and energy defined for plane fields on Riemannian manifolds; existence of foliations with all the leaves enjoying a given geometric property such as being totally geodesic, totally umbilical, minimal, etc. (see, among the others,4,11,16,17 the survey13 and the bibliography there) Brito, Langevin and Rosenberg 3 have shown that the integrals of σi (Aξ ) of a codimension-one foliation F on a compact space form M n+1 (c) do not depend on F : they depend on n, i, c and the volume of M only. Theorem 1.1 (Brito–Langevin–Rosenberg3 ). Let F be a codimension one foliation on a compact space form M n+1 (c). Then the integrals of σk (Aξ ) (ξ is a unit normal to the leaves) do not depend on F :  ( k Z c 2 n/2 k/2 vol(M ), n, k even; (4) σk (Aξ ) dM = M 0, n or k odd. For i = 1 and 2, these integral formulae have known generalizations, Eqs. (1), (3), to arbitrary Riemannian manifolds. As far as totally geodesic foliations of codimension n on compact space 5 forms M n+p (c) concern, Brito and Naveira have shown that total 2kR ⊥ dimensional curvatures, γ2k (F ) = M γ2k (Fx⊥ ) d V do not depend on a foliation, they depend on k, p, n, c and the volume of M only.

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Definition 1.1. Given x ∈ M take orthonormal bases {ei }1≤i≤p of T Fx i and {eα }1≤α≤n of T Fx⊥ , and set Cα,β = g(Cei eα , eβ ). The 2k-dimensional ⊥ curvature of Fx is defined by  Xp 1 X (Cαi11 ,β1 Cαi12 ,β2− Cαi11 ,β2 Cαi12 ,β1 )× εα,β γ2k (Fx⊥ ) = k i1 =1 α, β (2k)! 2 Xp  (5) (Cαik2k−1 ,β2k−1 Cαik2k ,β2k − Cαik2k−1 ,β2k Cαik2k ,β2k−1 ) ... × ik =1

where εα,β = εα1 ,...α2k ;β1 ,...β2k is +1 (−1) if α = (α1 , . . . α2k ) are 2k distinct integers and β = (β1 , . . . β2k ) are even (odd) permutations of α1 , . . . α2k ; otherwise it is 0. The sum is taken over all αi , βi between 1 and n.

Theorem 1.2 (Brito–Naveira5 ). Let F p be a totally geodesic foliation on a compact space form M n+p (c) with c ≥ 0. Then the total 2kdimensional curvatures do not depend on F :
   p+2k−1  n/2 2k 
ck vol(M ) if n even, p odd;  (p+2k−1)/2  k k    γ2k (F ⊥ ) = (6) 22k (k!)2 p/2+k−1 n/2  ck vol(M ) if n, p are even;  (2k)! k k   0 if n odd. Examples of geometrically distinct totally geodesic foliations on space forms are abundant, see discussion in Ref. 5. A compact space form M (c) with c < 0 does not admit a totally geodesic foliation of dimension 6= 0, dim M .19

Brito 4 has shown also relations between some differential forms (involving connection and curvature forms) on arbitrary foliated Riemannian manifolds. His results suggest the existence of similar to Eqs. (1), (3) formulae for σi ’s with i > 2. Here, we provide such formulae (see Sec. 3): they generalize Eqs. (4), (6) (the constancy of curvature condition is dropped) and relate the integrals of σi ’s with those involving some algebraic invariants obtained from the co-nullity operator , the mixed curvature operator and their products. We write several such formulae explicitly, on locally symmetric spaces as well as on arbitrary Riemannian manifolds where they involve also covariant derivatives of the mixed curvature operator. The structure of the present work is as follows: Sec. 2 contains algebraic preliminaries. Sec. 3 presents the main results (Theorems 1, 2, and their corollaries). Secs. 4, 5 contain proofs. 2. Algebraic preliminaries We define and study the invariants σλ (A1 , . . . , Am ) of a set of matrices that generalize the elementary symmetric functions of a symmetric ma-

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trix A. 2.1. Invariants of a set of square matrices Given arbitrary quadratic matrices A1 , . . . Am of order n and a unit matrix In one can consider the determinant det(In + t1 A1 + . . . + tm Am ) and express it as a polynomial of real variables t = (t1 , . . . tm ). Given λ = (λ1 , . . . λm ), a sequence of nonnegative integers with |λ| := λ1 + . . . + λm ≤ n, we shall denote by σλ (A1 , . . . , Am ) its coefficient at tλ = tλ1 1 · . . . tλmm : X det(In + t1 A1 + . . . + tm Am ) = σλ (A1 , . . . Am ) tλ . (7) |λ| ≤n

Certainly, σi (A) coincides with the i-th elementary symmetric polynomial of the eigenvalues {kj } of a single symmetric matrix A. The next lemma collects properties of these invariants which will be used in the sequel. Lemma 2.1 (Ref. 14). For any λ = (λ1 , . . . λm ), a, b ∈ R and any n × n matrices Ai , A and B one has (I) σλ (0, A2 , . . . Am ) = σ0,λˆ (A1 , A2 , . . . Am ) = σλˆ (A2 , . . . Am ), where ˆ λ = (λ2 , . . . λm ), (II) σλ (As(1) , . . . As(m) ) = σλ◦s (A1 , . . . Am ), where s ∈ Sm is a permutation of m elements and λ ◦ s = (λs(1) , . . . λs(m) ), ˆ
λ| σλˆ (A2 , . . . Am ), (III) σλ (In , A2 , . . . Am ) = n−| λ1
λ1 +λ2 (IV) σλ (A, A, A3 , . . . Am ) = λ1 σ(λ1 +λ2 ,λ3 ,...λm ) (A, A3 , . . . Am ), (V) σ1,λˆ (A + B, A2 , . . . Am ) = σ1,λˆ (A, . . . Am ) + σ1,λˆ (B, . . . Am ) and σλ (aA1 , A2 , . . . Am ) = aλ1 σλ (A1 , . . . Am ). The invariants defined above can be used in calculation of the deterP∞ i minant of a matrix B(t) expressed as a power series B(t) = i=0 t Bi . j The coefficient at t in the series of det(B(t)) depends only on the part P i i≤j t Bi of B(t). P i Lemma 2.2. If B(t) is n × n matrix given by B(t) = ∞ i=0 t Bi , then  X∞  X det(B(t)) = 1 + σλ (B1 , . . . Bk ) tk , (8) k=1

λ,kλk=k

where kλk = λ1 + 2λ2 + . . . + kλk for λ = (λ1 , . . . λk ).

By the First Fundamental Theorem of Matrix Invariants,7 all σλ are expressed in terms of the traces of the matrices involved and their products. The following Lemma provides a tool for writing down the explicit formulae for all σλ ’s in terms of traces (of given matrices and their products).

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Lemma 2.3 (Ref. 14). For any n × n-matrices Ai , λ = (λ1 , . . . λm ), k one has σ(λ,k) (A1 , . . . Am , Am+1 ) = σλ (A1 , . . . Am ) σk (Am+1 ) X (9) − σµ ( A1 , . . . Am , Am+1 , A1 Am+1 , . . . Am Am+1 ), µ

where the sum ranges over all the sequences µ = (λ1 − j1 , . . . λm − jm , k − j1 − . . . − jm , j1 , . . . jm ) with j1 ≤ λ1 , . . . jm ≤ λm , 0 < j1 + . . . + jm ≤ k. For example, σk,1 (B, C) = σk (B) Tr(C) − σk−1,1 (B, BC), etc.

2.2. Integral relations for σλ Let T : (Rp )h × Rn → Rn be a multilinear map. For each ξ ∈ Rp define a linear operator Tξ := T (ξ, . . . ξ; ·) : Rn → Rn . In coordinate form ξ = P P ei1 , . . . e¯ip ; ·). ¯i we have Tξ = i Ti1 ,...ip ξi1 · . . . ξip , where Ti1 ,...ip = T (¯ i ξi e R Lemma 2.4. If h and m are odd, then kξk=1 σm (Tξ ) d ωp−1 = 0.

Proof. By Lemma 2.1 (V ), σm (T−ξ ) = σm ((−1)h Tξ ) = (−1)h m σm (Tξ ) . R Q λi 10 that Set Iλ := kξk=1 ξ λ d ωp−1 where ξ λ = i≤p ξi . It is known
λi Q 1+(−1) 1+λi 2 Iλ = Γ p + 1 P λ Γ 2 , where Γ is Gamma function. 2 ( 2 2 i i ) i≤p p−1 Γ(1/2+λ ) p/2 1 2π = Vol(S1p−1 ), I2λ1 ,0,...0 = 2 π 2 Γ(p/2+λ , etc. For example, I0,...0 = Γ(p/2) 1) R P Lemma 2.5. If h = 1 then kξk=1 σ2k−1 ( i≤p ξi Ti ) d ωp−1 = 0 and Z X X σ2k ( ξi Ti ) d ωp−1 = I2λ σ2λ (T1 , . . . Tp ). (10) kξk=1

i≤p

|λ|= k

P Proof. Using Eq. (7) one obtains for Tξ = i≤p ξi Ti X X X X (t ξi ) Ti )= tj σj (Tξ ) tj = det(In + j

j

i≤p

Hence

σj (Tξ ) =

X

|λ|= j

|λ|= j

σλ (T1 , . . . Tp ) ξ λ .

σλ (T1 , . . . Tp ) ξ λ .

(11)

From this and definition of I2λ follows Eq. (10). Example 2.1. The cases k = 1, 2, . . . of Eq. (10) read Z X σ2 (Tξ ) d ωp−1 = I2,0,...0 σ2 (Ti ), i≤p kξk=1 Z X X σ4 (Tξ ) d ωp−1 = I4,0,...0 σ4 (Ti ) + I2,2,0,...0 kξk=1

i≤p

i,j≤p

σ2,2 (Ti , Tj ),

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p/2

π 3π and so on, where I2,0,...0 = Γ(p/2+1) , I4,0...0 = 2Γ(p/2+2) , I2,2,0...0 = P p/2 π i,j ξi ξj Tij (h = 2), then similarly to Eq. (11) one has 2Γ(p/2+2) . If Tξ = X Y σk (Tξ ) = σ(λ11 ,...λij ,...λpp ) (T11 , . . . Tij , . . . Tpp ) (ξi ξj )λij , (12) |λ|=k i,j≤p P and so on. The cases k = 1, 2 of Eq. (12) read σ1 (Tξ ) = i,j σ1 (Tij ) ξi ξj , P P σ2 (Tξ ) = i,j σ2 (Tij ) ξi ξj + i,j,s,l σ1,1 (Tij , Ts l ) ξi ξRj ξs ξl . These integrals can we written similarly to Eq. (10). For instance, kξk=1 σ1 (Tξ ) dωp−1 = P π p/2 i≤p σ1 (T ii ). Γ(p/2+1)

3. Main Results

In this section we generalize theorems by Brito–Langevin–Rosenberg3 and by Brito–Naveira,5 mentioned in Introduction. Namely, we drop the restriction that the ambient space has a constant sectional curvature. Theorem 3.1. Let F be a codimension one foliation (with the unit normal ξ) on a complete locally symmetric space M n+1 of a finite volume. Assume that (M, F ) has bounded geometry, i.e. sup kRξ k < ∞, M

Then for any m > 0 Z X M

where B2k =

(−1)k (2k)!

kλk=m

sup kAξ k < ∞.

σλ (B1 , . . . , Bm ) dVM = 0,

Rξk , B2k+1 =

(13)

M

(−1)k (2k+1)!

(14)

Rξk Aξ .

Similar formulae can be derived on arbitrary Riemannian manifolds. They are more complicated since they contain terms which depend on covariant derivatives of Rξ . More precisely, Eq. (14) contains just terms of the (k) (1) (2) form Rξ with k ≤ m − 2, where Rξ = ∇ξ Rξ , Rξ = ∇ξ ∇ξ Rξ and so on. For few initial values of m = 1, . . . 5, Eq. (14) give Eq. (1), Eq. (3) and
1 1 (15) σ3 (Aξ ) − Ric(ξ) Tr(Aξ ) + Tr(Rξ Aξ ) dVM = 0, 2 3 ZM 1 1 1 1 (σ4 (Aξ )+ σ2 (Rξ )− σ1,1 (Aξ , Rξ Aξ )+ σ1 (Rξ2 )− σ2,1 (Aξ , Rξ ))dVM =0, 4 6 24 2 ZM  1 1 1 σ5 (Ax ) + σ1,2 (Ax , Rξ )− σ2,1 (Ax , Rξ Ax )+ σ1,1 (Rξ , Rξ Ax ) 4 6 12 M  1 1 1 + σ1,1 (Ax , Rξ2 ) − σ3,1 (Ax , Rξ ) + Tr(Rξ2 Ax ) dVM = 0. (16) 24 2 5! Z

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209

Corollary 3.1. Let F be a codimension 1 foliation on a complete Riemannian manifold M n+1 of a finite volume with K mix = c and supM kAξ k < ∞. Then, for any k > 0, Eqs. (4) are valid. Remark 3.1. (a) From (14) one may get obstructions for existence of codimension 1 totally geodesic/umbilical foliations.14 (b) We apply Gray’s tube formulae 6 and work also with foliations of codimension 1 (or vector fields) which admit ”good” (in a sense) singularities, see details in Ref. 14. (c) By Corollary 3.1, there are no codimension 1 foliations with K mix = const 6= 0 on compact even-dimensional M . Theorem 3.2. Let F be a p-dimensional totally geodesic foliation on a complete Riemannian manifold M p+n of a finite volume with the condition ∇ξ Rξmix = 0 Then for any m > 0 one has Z Z X M

where Bξ,2k =

kξk=1

kλk=m

1 mix k (2k)! (−Rξ ) ,

(ξ ∈ T F ).

  σλ (Bξ,1 , . . . Bξ,m ) d ω dVM = 0,

Bξ,2k+1 =

(17)

(18)

1 mix k (2k+1)! (−Rξ ) Cξ .

The expressions for Bξ,i (i > 2) in Theorem 3.2 are more complicated without assumption of Eq. (17), since they also contain terms with covariant derivatives of the mixed curvature operator. For example, 1 1 mix 2 2 mix mix Cξ +∇ξ R mix Bξ,3 =− (R mix ξ ), Bξ,4 = ((Rξ ) −∇ξ R ξ −2(∇ξ R ξ ) Cξ ). 6 ξ 4! R Note that kλk = odd ⇒ kξk=1 σλ (Bξ,1 (x), . . . Bξ,m (x)) d ωp−1 = 0. For few initial values of m, m = 2, 4, Eqs. (18) read as follows: Z Z
1 σ2 (Cξ ) − Tr Rξmix d ω) dVM = 0, ( (19) 2 kξk=1 M Z Z 1 1 1 Tr((Rξmix )2 ) − Tr(∇2ξ Rξmix ) σ4 (Cξ ) + σ2 (Rξmix ) + ( 4 24 24 M kξk=1
1 1 − Tr((∇ξ Rξmix )Cξ ) − σ1,1 (Cξ , Rξmix Cξ ) d ω) dVM = 0. (20) 12 6

One may reduce the integrals to ones over M , see Sec. 2.2. The Eq. (20) is new. The Eq. (19) reduces to Eq. (2). 3.1. The
total k-th mean curvature is the integral σk (F ⊥ ) = RDefinition R ⊥ M kξk=1 σk (Cξ ) d ω dVM . Note that σ2k+1 (F ) = 0.

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Corollary 3.2. Let F p be a totally geodesic foliation on a complete M p+n of a finite volume with K mix = c. Then, for any k > 0,  p/2 ( n/2 2π k Γ(p/2) c vol(M ), n even, k σ2k (F ⊥ ) = (21) 0, n odd. Remark 3.2. (a) From K mix = c follows that Rξmix y = c y, and hence Eq. (17) holds. (b) For p = 1, Eq. (21) is equivalent to Eq. (6). For p > 1 one may show that γ2k (Fx⊥ ) 6= σ2k (Fx⊥ ). For p = 1 the equality γ2k (Fx⊥ ) = σ2k (Fx⊥ ) holds, moreover, Eq. (21) and Eq. (6) coincide when p = 1. (c) By Corollary 3.2, there are no totally geodesic foliations F p (p > 0) with K mix = const 6= 0 on a complete M p+n of a finite volume for n odd. 4. Proof of Theorem 3.1 Let ξ be a smooth unit vector field on a complete oriented Riemannian manifold (M n+1 , g) of class C ∞ with bounded geometry due (13). For simplicity, one may assume that ξ is orthogonal to a codimension-1 foliation F , dim F = n. We call expx , as usual, the exponential map at x: expx : Tx M → M of M . Let γx : t 7→ expx (t ξ) be the unique geodesic in M with γx (0) = x and γx′ (0) = ξ. Choose a positively oriented orthonormal frame (e1 , . . . en ) of ξx⊥ (= Tx F ) and extend it by parallel translation to the frame (Ex1 , . . . Exn ) of vector fields along γx . Denote also by Exn+1 the parallel field along γx with Exn+1 (0) = ξx . For any i ≤ n, denote by Yxi (t) the Jacobi field along γx satisfying Yxi (0) = ei and Yxi ′ (0) = Ax (ei ), where Ax is the Weingarten operator (of a leaf at x) relative to ξx . Denote by Rξ the curvature operator X 7→ R(X, ξ)ξ in ξ ⊥ (= T F ) and by Rx (t) the matrix with entries hR(Exi (t), Exn+1 (t))Exn+1 (t), Exi (t)i (“Jacobi operator”). Denote also by Yx (t) the n × n matrix consisting of the scalar products hYxi (t), Exj (t)i (“Jacobi tensor”). Then Yx (0) = In and Yx′ (0) = Ax . Consider the maps {φt : M → M, t ∈ (−ε, ε)} defined by φt (x) = expx (t ξ). It is known (see, for instance,8 Lemma 3.1.17) that |dφt (x)| = det Yx (t).

(22)

In view of (13), there exists ε > 0 such that for all x ∈ M there are no focal points of a leaf Lx along γx (t), t ∈ [0, ε). Hence, {φt : M → M }t∈(−ε,ε) are diffeomorphisms. Assume that (M, g) is locally symmetric (∇R = 0) and consider the Jacobian |dφt (x)| of φt at a point x of M . For short, write Rξ := Rx (0). The Jacobi equation Yx′′ = −Rx (t)Yx implies that (2m) (2m+1) Yx (0) = (−Rξ )m , Yx (0) = (−Rξ )m Ax , m = 0, 1, 2, . . . Hence,

Integral formulae for foliations on Riemannian manifolds

our Jacobi tensor Yx =

P∞

m=0

(m)

Yx

211

m

(0) tm! has the form

t3 t4 t2 Rξ − Rξ Ax + Rξ2 + . . . (23) 2! 3! 4! Certainly, the radius of convergence of the series in Eq. (23) is uniformly bounded from below on M (by 1/kRk > 0). Therefore – by Lebesque Dominated Convergence Theorem – its integration together with Exchange Variable Theorem yield the equality for arbitrary t small enough Z
t3 t4 t2 det In + tAx − Rξ − Rξ Ax + Rξ2 + . . . dVM . (24) vol(M ) = 2! 3! 4! M Yx (t) = In + tAx −

Formula (24) together with Lemma 2.2 imply (14). 5. Proof of Theorem 3.2

First, we recall the basic facts about tangent bundles.2,9 Let T M be the tangent bundle of M with the projection π : T M → M . Split T T M (the tangent bundle of T M ) into the sum T T M = V ⊕ H of vertical and horizontal subbundles. For any ξ ∈ (T M )x the subspaces V(x,ξ) , H(x,ξ) can be identified with (T M )x via π∗ : T T M → T M and the connection map K : T T M → T M . The linear map K is determined by the condition K(Z∗ ξ) = ∇ξ Z

(for all ξ ∈ T M and any vector field Z on M ).

A Sasaki metric g¯ on T M is the unique Riemannian structure for which V and H are orthogonal and all the maps π∗|H : H(x,ξ) → (T M )x and K|V : V(x,ξ) → (T M )x are isometries. The unit tangent sphere bundle over (M, g) is T1 M = {(x, ξ) ∈ T M : |ξ| = 1} (the hypersurface of T M ). Let F = {L} be a transversely oriented foliation on M . We call T F = {(x, ξ) ∈ T M : x ∈ M, ξ ∈ (T L)x } the tangent bundle of F , and F¯ = {T L} the lift foliation on T F . The metric g¯ in T F is the restriction of the Sasaki metric of T M . Definition 5.1. We call T1 F = {(x, ξ) ∈ T F : |ξ| = 1} the tangent sphere bundle of radius 1 of F , a hypersurface in T F consisting of all pairs (x, ξ), where x ∈ M and ξ ∈ T L is a vector of the length 1. We call Fe = {T1 L} the lift foliation on T1 F .

The canonical vertical vector field ξ¯ is outward normal to T1 F in T F at each point (x, ξ) ∈ T1 F . The tangential lift of y is a vector field y˜t tangent ¯ ξ. ¯ The hypersurface T1 F ⊂ T F to T1 F and defined by y˜t = y¯V − g¯(¯ y V , ξ) is endowed with the Riemannian metric g˜ ¯ g¯(¯ ¯ g˜(˜ z H, y˜H ) = g¯(¯ z H, y¯H ), g˜(˜ z H, y˜t ) = 0, g˜(˜ z t, y˜t ) = g¯(¯ z V, y¯V )−¯ g (¯ z V, ξ) y V, ξ).

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The natural projection π ˜ : T1 F → M is the Riemannian submersion with totally geodesic fibers S1p−1 , the spheres of a radius 1. Hence the volume V ol(T1 F ) is the product of V ol(S1p−1 ) and V ol(M ), see Ref. 1. Lemma 5.1. Let F = {Lp } be a totally geodesic foliation on M . Then F˜ = {T1 L} (the (2p − 1)-dimensional lift foliation on T1 F ) is totally geodesic. Proof. follows from the fact that T L is totally geodesic in T M if and only if L is a totally geodesic submanifold in M , see Ref. 2. The tangent bundle to T1 F is orthogonally decomposed into sum of hore In view of isomorphism izontal and vertical subbundles: T (T1 F ) = Ve ⊕ H. e → T M , the almost product structure T M = T F ⊕ T F ⊥ induces the π ˜∗ : H e into sum of subbundles: H e =H e⊤⊕H e ⊥. orthogonal decomposition of H

e ⊤ (velocity of the Definition 5.2. The characteristic vector field N ⊂ H geodesic flow on T1 F ) at a point x e = (x, ξ) is defined by π e∗ (N ) = ξ. The characteristic Weingarten and Jacobi operators on T1 F are defined by e y˜N − h∇ e y˜N, N iN and R ˜ N : y˜ → R(˜ ˜ y , N )N . A˜N y˜ = ∇

For totally geodesic F , the restriction of N on any leaf T1 L generates the e ⊥ is ∇-parallel e geodesic flow on this leaf, and the distribution H along N .

Lemma 5.2. Let F be a totally geodesic foliation on (M, g), y˜ a vector field on T1 F and y = π∗ (˜ y ) the projection on M . Then e y˜N ⊂ Ve ⊕ H e⊤ A˜N y˜ = ∇

e ⊤ ), (˜ y ⊂ Ve ⊕ H

^ ⊥ e y˜ N = (∇ e e ), A˜N y˜ = ∇ (˜ y⊂H y ξ) ⊂ H ⊤ ˜ N y˜ ⊂ Ve ⊕ H e e ⊤ ), R (˜ y ⊂ Ve ⊕ H

^ mix y ⊂ H ˜ N y˜ = R e⊥ R ξ

⊥

e ⊥, (˜ y⊂H

⊥

ξ=π e∗ (N )).

(25) (26) (27) (28)

Proof of Theorem 3.2. By Lemma 5.2, the Weingarten and Jacobi op˜ N , and hence the Jacobi operator Y˜N have 2 × 2 block view: erators, A˜N , R  A˜L ∗    Y L (t) ∗   ˜L N N ˜ ˜ N = RN 0 A˜N = , R mix , YN (t) = 0 Rξ 0 Yξ⊥ (t) 0 Cξ

˜L ˜ ˜ where A˜L N and RN are the restrictions of AN and RN on T1 L. By Schur identity, we have det Y˜N (t) = det YNL (t) det Yξ⊥ (t). Recall that N generates the geodesic flow on each T1 L and det YNL (t) presents the volume element in L. Hence, by Liouville’s theorem, det YNL (t) = 1 for all t. Finally, we have det Y˜N (t) = det Yξ⊥ (t) for all t. Given ξ ∈ (T L)x , choose a positive

Integral formulae for foliations on Riemannian manifolds

213

oriented orthonormal frame (e1 , . . . en ) of (T L⊥ )x and extend it by parallel translation to the frame (E1 , . . . En ) of vector fields along the geodesic γ : t 7→ expx (t ξ). Hence for the Jacobi tensor Yξ⊥ (t) we get 2

3

mix − t3! (R mix Yξ⊥ (t) = In + tCξ − t2! R mix ξ ξ Cξ + ∇ξ R ξ ) (2) mix t4 mix 2 mix + 4! ((R ξ ) − ∇ξ R ξ − 2∇ξ R ξ Cξ ) + . . .

(29)

˜ (x)) of T1 F are defined for all The diffeomorphisms φ t (x) = expx (t N t ∈ R (since the leaves of F are complete). The Jacobian of φ t satisfies 14 |dφ t (x)| = det Y˜N˜ (t). The rest of proof is similar to codimension 1 case. Proof of Corollary 3.2. By the proof of Theorem 3.2 we have X z m σm (F ⊥ ) = vol(T1 F ) (1 ± |c| z 2 )n/2 . m≤n

For n odd we obtain c = 0 (the case c 6= 0 leads to a contradiction). References 1. M. Berger, A Panoramic View of Riemannian Geometry (Springer, 2002). 2. A. Borisenko and A. Yampolskii, Riemannian geometry of fiber bundles, Russian Math. Surveys 46 (6) (1991) 55–106. 3. F. Brito and R. Langevin and H. Rosenberg, Integrales de courbure sur des varietes feuilletees, J. Diff. Geom. 16 (1981) 19–50. 4. F. Brito, Une obstruction geometrique a l’existence de feuilletages de codimension 1 totalement geodesiques, J. Diff. Geom. 16 (1981) 675–684. 5. F. Brito and A. Naveira, Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom. 18 (2000) 371–383. 6. A. Gray, Tubes (Birkhauser, Boston, 2nd edition, 2004). 7. G.G. Gurevich, Foundations of the Theory of Algebraic Invariants (Noordhof, Groningen, 1964). 8. W. Klingenberg, Riemannian Geometry (Walter de Gruyter, Berlin, 1995). 9. O. Kowalskii and M. Sekizawa, On Tangent Sphere Bundles with Small or Large Constant Radius, Global Anal. Geom. 18 (2000) 207–219. 10. A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series (Vol. 3, Gordon & Breach Sci. Publ., New York, 1990). 11. A. Ranjan, Structural equations and an integral formula for foliated manifolds, Geom. Dedicata 20 (1986) 85–91. 12. G. Reeb, Sur la courbure moyenne des vari´et´es int´egrales d’une ´equation de Pfaff ω = 0, C. R. Acad. Sci. Paris 231 (1950) 101–102. 13. V. Rovenski, Foliations on Riemannian Manifolds and Submanifolds (Birkhauser, Boston, 1998). 14. V. Rovenski and P. Walczak, Integral formulae on foliated symmetric spaces, Preprint, University of Lodz, Fac. Math. Comp. Sci. 2007/13 (2007) 1–27.

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15. P. Walczak, An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Colloq. Math. 58 (1990) 243–252. 16. P. Walczak and F. Brito, On the energy of unit vector fields with isolated singularities, Ann. Pol. Math. 73 (3) (2000) 269–274. 17. P. Walczak and P. Schweitzer, Prescribing mean curvature vectors for foliations, Illinois J. Math. 48 (1) (2004) 21–35. 18. P. Walczak, Dynamics of foliations, groups and pseudogroups (Birkhauser, Basel, 2004). 19. A. Zeghib, Feuilletages g´eod´esiques des vari´et´es localement sym´etriques, Topology 36 (1997) 805–828.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 217–222

217

The prequantization of Tk1 Rn Adara M. Blaga Department of Mathematics and Computer Sciences, West University from Timi¸soara, Timi¸soara, Romˆ ania E-mail: [email protected] www.uvt.ro The aim of this paper is to give a prequantization of the manifold Tk1 Rn . Keywords: Prequantization, k-symplectic manifold. MS classification: 53D05, 53D50.

1. Introduction Prequantization is the first step in the geometric quantization procedure. Two different ways of performing the prequantization of a symplectic manifold (M, ω) were described by Kostant1 and respectively, Souriau.2 The first one consists in finding a complex line bundle with a connection such that its curvature is i ω and the second one refferes to determine a ~ principal S 1 -bundle π : Y −→ M with an S 1 -invariant 1-form α such that R ∗ dα = π ω and f ibre α = 2π~ (where ~ stands for the Plank’s constant). Using the prequantization of the standard k-symplectic manifold ((Tk1 )∗ Rn , ωi , V )1≤i≤k (see Ref. 3), we will define a prequantization of the k-symplectic manifold (Tk1 Rn , (ωL )i , VL )1≤i≤k associated to a regular Lagrangian L : Tk1 Rn −→ R and point out an irreducible representation of it. 2. Hamiltonian vector fields on a k-symplectic manifold Let M be an n + nk dimensional differential manifold. Definition 2.1 (Ref. 4). A k-symplectic structure on M is a family (ωi , V )1≤i≤k where ωi , 1 ≤ i ≤ k are closed 2-forms on M and V is an nk dimensional integrable distribution on M , such that ωi |V ×V = 0, 1 ≤ i ≤ k.

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We call (M, ωi , V )1≤i≤k a k-symplectic manifold. The standard example of k-symplectic manifold is provided by the Whitney sum of k copies of T ∗ Rn , i.e. (Tk1 )∗ Rn , together with the 2-forms ωi , 1 ≤ i ≤ k and the distribution V , given in a system of local coordinates (q α , piα ), 1 ≤ α ≤ n, 1 ≤ i ≤ k, by: ωi = dq α ∧ dpiα ; V =h

∂ i ∂piα

(1)

(the Einstein convention is used whenever a lower and and an equal upper indices appear). Recall that the manifold of one jets of maps from an n dimensional manifold Q to Rk with target at 0 ∈ Rk , J 1 (Q, Rk )0 , can be canonically identified with the Whitney sum (Tk1 )∗ Q of k copies of T ∗ Q, that is J 1 (Q, Rk )0 −→ T ∗ Q ⊕ · · · ⊕ T ∗ Q, 1 jq,0 σ ≡ (α1 (q), . . . , αk (q)) where αi (q) = d(πi ◦ σ)(q) and πi : Rk −→ R are the canonical projections, 1≤i≤k . The Darboux’s theorem states the following: Theorem 2.1. Around any point of a k-symplectic manifold (M, ωi , V )1≤i≤k , there exists a system of local coordinates (q α , piα ), 1 ≤ α ≤ n, 1 ≤ i ≤ k, such that: ωi = dq α ∧ dpiα ; V =h

∂ i. ∂piα

For H ∈ C ∞ (M, R) a Hamiltonian on M , we shall obtain the Hamiltonian vector fields i i i XH = (X1H , ..., XkH ), 1 ≤ i ≤ k,

as the solutions of the Hamilton’s equations (see Ref. 5): k X j=1

i ωj = dH, 1 ≤ i ≤ k, iXjH

(2)

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219

of the form: i XjH =

∂H ∂pjα

·

∂H ∂ ∂ − δji α · j , 1 ≤ i, j ≤ k. α ∂q ∂q ∂pα

(3)

Consider (q α , viα )1≤α≤n,1≤i≤k a system of local coordinates on Tk1 Rn , where Tk1 Rn is the Whitney sum of k copies of T Rn and let L ∈ C ∞ (Tk1 Rn , R) be a hyperregular Lagrangian on Tk1 Rn . Then the Legendre transformation T L : Tk1 Rn −→ (Tk1 )∗ Rn defined by (T L(v1q , ..., vkq ))i (wq ) :=

d |s=0 L(v1q , ..., viq+swq , ..., vkq ), ∀1 ≤ i ≤ k, (4) ds

is a diffeomorphism. Using this transformation, we can transfer the ksymplectic structure (ωi , V )1≤i≤k from (Tk1 )∗ Rn on Tk1 Rn , defining (see Ref. 6): (1) (ωL )i = (T L)∗ ωi (2) VL = (T L)−1 ∗ V We saw that in a weaker case [i.e. when the Lagrangian is only regular (see Ref. 6)] the family (Tk1 Rn , (ωL )i , VL )1≤i≤k ,

(5)

is a k-symplectic manifold. Therefore, it happens in our case, too. 3. Poisson structures on the k-symplectic manifold (Tk1 Rn , (ωL )i , VL )1≤i≤k associated to the Lagrangian L Let f, g ∈ C ∞ ((Tk1 )∗ Rn , R) and denote by: i i ), 1 ≤ i ≤ k, , ..., Xkf Xfi = (X1f

and i i Xgi = (X1g , ..., Xkg ), 1 ≤ i ≤ k,

the corresponding Hamiltonian vector fields. Definition 3.1 (Ref. 3). The Poisson bracket of f and g is the smooth, real-valued function defined on (Tk1 )∗ Rn : {f, g} =

k X i=1

i i ωi (Xif , Xig ).

(6)

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Using the Legendre transformation associated to the hyperregular Lagrangian L, we can now define a Poisson structure on Tk1 Rn , as follows: {f, g}L = (T L)∗ {(T L)∗

−1

f, (T L)∗

−1

g},

(7)

for any f, g ∈ C ∞ (Tk1 Rn , R). We will verify only the Jacobi identity: Proposition 3.1. The Poisson bracket 7 satisfies the Jacobi identity: {f, {g, h}L}L + {g, {h, f }L}L + {h, {f, g}L}L = 0.

Proof. Indeed, let f, g, h ∈ C ∞ (Tk1 Rn , R). Then: {f, {g, h}L}L = {f, (T L)∗ ({(T L)∗ ∗ −1

∗

= (T L) ({(T L)

= (T L)∗ ({(T L)∗ ∗ −1

= {(T L)

−1

∗ −1

f, (T L)

f, {(T L)∗ ∗ −1

f, {(T L)

−1

g, (T L)∗

−1

h})}L

∗ −1

∗

((T L) ({(T L)

−1

g, (T L)∗ ∗ −1

g, (T L)

−1

g, (T L)∗ −1 h}))}L )

h}}L)

h}}L ◦ T L.

Analogous, we will obtain: {g, {h, f }L}L = {(T L)∗ −1 g, {(T L)∗−1 h, (T L)∗ −1 f }}L ◦ T L and {h, {f, g}L}L = {(T L)∗ −1 h, {(T L)∗−1 f, (T L)∗ −1 g}}L ◦ T L. Adding the three relations and taking into account that the Poisson bracket {, } satisfies the Jacobi identity, we obtain that {, }L satisfies the same identity. The proposition above allows us to state: Proposition 3.2. (C ∞ (Tk1 Rn , R), {, }L ) is an infinite dimensional Lie algebra. Proof. The axioms of a Lie algebra can now be easily checked.

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221

4. The prequantization of the k-symplectic manifold (Tk1 Rn , (ωL )i , VL )1≤i≤k associated to the Lagrangian L Let ((Tk1 )∗ Rn , ωi , V )1≤i≤k be the standard k-symplectic manifold, the k-symplectic structure being given by: ωi = dq α ∧ dpiα ; V =h

∂ i ∂piα

for 1 ≤ α ≤ n, 1 ≤ i ≤ k. Definition 4.1 (Ref. 3). A prequantization of the standard k-symplectic manifold ((Tk1 )∗ Rn , ωi , V )1≤i≤k is a correspondence which assigns to each function f ∈ C ∞ ((Tk1 )∗ Rn , R) a self-adjoint operator δf on a Hilbert space H, such that the Dirac conditions are satisfied: (1) (D1 ) δf +g = δf + δg ; (2) (D2 ) δλf = λδf ; (3) (D3 ) δid(T 1 )∗ Rn = idH ; k

(4) (D4 ) [δf , δg ] = i~δ{f,g} , for any f, g ∈ C ∞ ((Tk1 )∗ Rn , R). Proposition 4.1 (Ref. 3). The pair (H, δ), where: (1) H = L2 ((Tk1 )∗ Rn , C); (2) δ : f ∈ C ∞ ((Tk1 )∗ Rn , R) 7−→ δf and Pk j • δf = −i~Xf − j=1 θj (Xjf ) + f; j α • θj = pα dq , for 1 ≤ j ≤ k, is a prequantization of the k-symplectic manifold ((Tk1 )∗ Rn , ωi , V )1≤i≤k . Using the Legendre transformation associated to the hyperregular Lagrangian L, we can now define a prequantization of the manifold 5: Proposition 4.2. The pair (H, δ), where: (1) H = L2 (Tk1 Rn , C); (2) δ : f ∈ C ∞ (Tk1 Rn , R) 7−→ δf and P j • δf = −i~Xf − kj=1 (θL )j (Xjf ) + f; ∂L α • (θL )j = dq ∂vjα

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for 1 ≤ j ≤ k, is a prequantization of the k-symplectic manifold 5. References 1. B. Kostant, Quantization and unitary reprezentations (Lectures in modern analysis and applications III, LNM 170, Springer Verlag, 1970). 2. J.M. Souriau, Structure des Syst` emes dynamiques (Dunod, Paris, 1970). 3. M. Puta, S. Chirici and E. Merino, On the prequantization of (T1k )∗ Rn Bull. Math. Soc. Sc. Math. Roumanie 44 (2001) 277–284. 4. A. Awane, k-symplectic structures, Math. Phys. 33 (1992) 4046–4052. 5. M. Puta, Some remarks on k-symplectic manifolds, Tensor 47 (1988) 109– 115. 6. F. Munteanu, A. Rey and M. Salgado, The G¨ unther’s formalism in classical field theory: momentum map and reduction, Math. Phys. 45 (2004) 1730– 1751.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 223–238

223

Extension of connections Miroslav Doupovec Department of Mathematics, Brno University of Technology, FSI VUT Brno, Technick´ a 2, 616 69 Brno, Czech Republic E-mail: [email protected] Wlodzimierz M. Mikulski Institute of Mathematics, Jagiellonian University, Reymonta 4, Krak´ ow, Poland E-mail: [email protected] We generalize the well known Ehresmann prolongation of a connection Γ : Y → J 1 Y in the following way: Given a projectable classical linear connection on Y , we introduce an extension of Γ into an r-th order holonomic connection Y → J r Y . Then we describe all second and third order holonomic extensions of Γ. We also study symmetrizations of semiholonomic and nonholonomic jets. Keywords: Ehresmann prolongation, holonomic jets, higher order connections. MS classification: 58A05, 58A20, 58A32.

1. Introduction In general, an r-th order holonomic connection on a fibered manifold Y → M in the sense of C. Ehresmann is a smooth section Y → J r Y of the r-th holonomic jet prolongation of Y . If we replace J r Y by semiholonomic r prolongation J Y or by nonholonomic prolongation Jer Y , then we obtain the concept of semiholonomic or nonholonomic connection, respectively. Clearly, for r = 1 we have a general connection Γ : Y → J 1 Y , which can be also interpreted as the lifting map Y ×M T M → T Y . Our starting point is the classical Ehresmann prolongation, which transforms Γ into an r-th order semiholonomic connection. However, the fundamental role in mathematical physics and in differential geometry is played by classical holonomic jets and connections. That is why we study extensions of Γ into higher order holonomic connections. Up till now, the only This paper is in the final form and no version of it will be published elsewhere.

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known geometric construction of this type is of the form 2

C (2) (Γ ∗ Γ) : Y → J 2 Y,

(1)

where Γ ∗ Γ : Y → J Y is the second order Ehresmann prolongation of 2 Γ and C (2) : J Y → J 2 Y is the well known symmetrization of second order semiholonomic jets. This simple geometric construction leads us to the idea that the holonomic extension of connections is closely related with the problem of symmetrization of jets. The aim of this paper is to study both these problems in the systematic way. Our research is based on the results from Ref. 2, where we introduced an r-th order holonomic extension of Γ and also symmetrizations r J Y → J r Y and Jer Y → J r Y for any r. We also clarified that for r > 2, such geometric constructions depend in an essential way on an auxiliary projectable classical linear connection on Y . The structure of the paper is as follows. In Section 2 we recall some concepts which we need in the sequel. Section 3 contains the survey of some of our recent results from Ref. 2. In Section 4 we describe all second and third order holonomic extensions of a connection Γ. Finally, in Section 5 we present some applications of higher order connections. In particular, we introduce new geometric constructions of connections on higher order principal prolongations. It is well known that the classical theory of higher order jets and connections was established by C. Ehresmann.3,4 Next, an important contribution in this research has been made by P. Libermann,13 J. Pradines,15 I. Kol´aˇr7,8 and G. Virsik.16 Other recent results can be found e.g. in Refs. 1,10,12. Moreover, the basic concepts from the theory of connections can be formulated by means of jets. Using such a point of view, the monograph Ref. 9 can serve as an introduction to the systematic theory of jets and connections. In what follows our geometric constructions will be reflected as natural transformations or natural differential operators in the sense of the book Ref. 9. We denote by F Mm,n the category of fibered manifolds and fiber respecting mappings with m-dimensional bases, n-dimensional fibres and local fibered diffeomorphisms and by PBm (G) the category of principal G-bundles with m-dimensional bases and their local principal G-bundle isomorphisms with the identity isomorphisms of G. All manifolds and maps are assumed to be infinitely differentiable. 2. Ehresmann prolongation First we recall the definition of higher order jet prolongations of a fibered manifold Y → M . The classical r-th holonomic prolongation J r Y is the

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225

bundle of all r-jets of local sections of Y . Using induction, we can define the r-th nonholonomic prolongation Jer Y by Je1 Y = J 1 Y,

Jer Y = J 1 (Jer−1 Y → M ).

Then we have the canonical inclusion J r Y ⊂ Jer Y determined by jxr s 7→ jx1 (u 7→ jur−1 s) for every local section s of Y , u ∈ M . Further, write 1 r−1 J Y = J 1 Y and assume we have defined J Y ⊂ Jer−1 Y such that the r−1 r−2 r−1 r−2 e e Y into J Y , where restriction of βJer−2 Y : J Y →J Y maps J βY : J 1 Y → Y means the projection. Then we can define the r-th semiholonomic prolongation of Y by r

J Y = {U ∈ J 1 J

r−1

Y ; βJ r−1 Y (U ) = J 1 βJer−2 Y (U ) ∈ J

r−1

Y }.

r

Clearly, for r > 2 we have J r Y ⊂ J Y ⊂ Jer Y . Moreover, we write J

r,r−1

r

Y := {U ∈ J Y ; βJ r−1 Y (U ) ∈ J r−1 Y }

(2)

r

for the subspace of J Y such that the underlying (r − 1)-jets are holonomic. Denoting by (xi , y p ) the local coordinates on Y → M , the induced coordir r,r−1 p , . . . , yip1 ...ir ). Then J Y is characterized nates on J Y are (xi , y p , yip , yij by the full symmetry of yip1 ...is for all 2 ≤ s ≤ r − 1 and J r Y by the full symmetry in all subscripts. The product of two connections Γ1 : Y → Jer Y and Γ2 : Y → Jes Y is an (r + s)-th order nonholonomic connection Γ1 ∗ Γ2 : Y → Jer+s Y defined by Γ1 ∗ Γ2 := Jes Γ1 ◦ Γ2 .

Definition 2.1. Given a connection Γ : Y → J 1 Y , its Ehresmann prolongation is the r-th order connection Γ(r−1) : Y → Jer Y defined by Γ(1) := Γ ∗ Γ,

Γ(r−1) := Γ(r−2) ∗ Γ.

By Ref. 5, Γ(r−1) is semiholonomic and G. Virsik16 proved that Γ(r−1) is holonomic if and only if Γ is curvature free. One evaluates directly that if yip = Γpi (x, y) is the coordinate expression of Γ, then its Ehresmann 2 prolongation Γ ∗ Γ : Y → J Y has equations yip = Γpi ,

p yij =

∂Γpi ∂Γpi q + Γ . ∂xj ∂y q j

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3. Holonomic extension of connections and symmetrization of jets (a) Holonomic extension of connections. Definition 3.1. An r-th order holonomic connection Γ : Y → J r Y is called a holonomic extension of Γ : Y → J 1 Y , if π1r ◦Γ = Γ, where πsr : J r Y → J s Y the jet projection. r

Example 3.1. Consider Ehresmann prolongation Γ(r−1) : Y → J Y of Γ : r Y → J 1 Y . Applying an arbitrary natural transformation C : J Y → J r Y , we obtain a holonomic connection C ◦ Γ(r−1) : Y → J r Y,

(3)

which is the holonomic extension of Γ. Clearly, (1) is a special case of (3) for the well known symmetrization C = C (2) of second order semi2 holonomic jets. Obviously, C (2) : J Y → J 2 Y has the coordinate form p p p ) 7→ (xi , y p , yip , yij + yji ). We remark that all natural transfor(xi , y p , yip , yij r r mations C : J Y → J Y will be described in Theorem 3.2 below. Theorem 3.1 (Ref. 2). All F Mm,n -natural operators A transforming connections Γ : Y → J 1 Y into r-th order holonomic connections A(Γ) : Y → J r Y are of the form (1) (2) (3) (4)

r

For m = 1 and arbitrary natural r, A(Γ) = Γ(r−1) : Y → J Y = J r Y . For r = 1 and m ≥ 2, A(Γ) = Γ : Y → J 1 Y . For r = 2 and m ≥ 2, A(Γ) = C (2) (Γ ∗ Γ) : Y → J 2 Y . For r ≥ 3 and m ≥ 2 there is no F Mm,n -natural operator Γ 7→ A(Γ) in question.

This means that to construct a holonomic extension of Γ for r ≥ 3 and m ≥ 2, it is unavoidable to use some additional geometric object. That is why we construct an r-th order holonomic extension of Γ for any r by means of an auxiliary classical linear connection on Y . We recall that a classical linear connection ∇ : T Y → J 1 (T Y → Y ) on p : Y → M is called projectable, if there is an underlying connection ∇ on M such that ∇ ◦ T p(y) = Jx1 T p ◦ ∇(y) for every y ∈ Yx . Let Γ be a connection on Y → M and let ∇ be a projectable classical linear connection on Y with the underlying connection ∇ on M . Let y ∈ Yx , x ∈ M . Lemma 3.1 (Ref. 2). (1) There is a normal coordinate system (U, Φ) of ∇ with centre y covering a normal coordinate system (U , Φ) of ∇ with centre x such that J 1 Φ(Γ(y)) = j01 (0).

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227

(2) If (U, Ψ) is another normal coordinate system of ∇ with centre y covering a normal coordinate system (U , Ψ) of ∇ with centre x such that J 1 Ψ(Γ(y)) = j01 (0), then there exists A ∈ GL(Rm ) and B ∈ GL(Rn ) such that Ψ = (A × B) ◦ Φ on some neighbourhood of y. Example 3.2. Given a general connection Γ on Y → M and a projectable classical linear connection ∇ on Y , we define an r-th order holonomic connection Ar (Γ, ∇) : Y → J r Y by Ar (Γ, ∇)(y) := J r (Φ−1 )(j0r (0))

(4)

where Φ is as in Lemma 3.1(1). By Lemma 3.1(2), the definition of Ar (Γ, ∇)(y) is independent of the choice of Φ. Clearly, Ar (Γ, ∇) is an rth order holonomic extension of Γ by means of ∇ and the correspondence Ar : (Γ, ∇) 7→ Ar (Γ, ∇) is an F Mm,n -natural operator. One can show easily that Ar (Γ, ∇) can be also constructed by the compor sition (3) from Ehresmann prolongation Γ(r−1) : Y → J Y of Γ. Indeed, if sr (∇) : Jer Y → J r Y is the natural transformation from Example 3.5, then Ar (Γ, ∇) = sr (∇) ◦ Γ(r−1) .

Remark 3.1. From Theorem 3.1 it follows that for m ≥ 2 and r ≥ 3, the connection Ar (Γ, ∇) given by (4) depends on ∇ in an essential way. Moreover, Theorem 3.1 also yields that to define an r-th order holonomic extension of Γ : Y → J 1 Y for m ≥ 2 and r ≥ 3, the use of a projectable classical linear connection ∇ on Y is unavoidable. Now we introduce an extension of s-th order holonomic connections into r-th order ones, where s ≤ r. Example 3.3. Let Θ : Y → J s Y be a holonomic s-th order connection on Y → M and let ∇ be a projectable classical linear connection on Y with the underlying connection ∇ on M . Let y ∈ Yx , x ∈ M and denote by Γ := π1s ◦ Θ : Y → J 1 Y the first order underlying connection. We define an r-th order holonomic connection Bsr (Θ, ∇) : Y → J r Y by Bsr (Θ, ∇)(y) := J r (Φ−1 )(j0r (σ)) where Φ is as in Lemma 3.1(1) and σ : Rm → Rn is the unique polynomial of degree ≤ s such that J s Φ(Θ(y)) = j0s σ. Using Lemma 3.1(2), one evaluates easily that the definition of Bsr (Θ, ∇)(y) is independent of the choice of Φ. Clearly, we have πsr ◦ Bsr (Θ, ∇) = Θ, where πsr : J r Y → J s Y is the jet projection. Because of the canonical character of the construction, the

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correspondence Bsr : (Θ, ∇) 7→ Bsr (Θ, ∇) is an F Mm,n -natural operator. Obviously, we have B1r = Ar , where Ar is the operator from Example 3.2. (b) Symmetrization of jets By Example 3.1, symmetrization of semiholonomic jets can be used to construct holonomic extension of connections. However, the general idea of symmetrization plays an important role in the theory of jets and also in many areas of mathematical physics. That is why we pay an attention to the problems of symmetrization of semiholonomic and nonholonomic jets. First we recall that I. Kol´ aˇr6 defined a symmetrization C (r) : J

r,r−1

Y → JrY

2,1

2

of special semiholonomic jets (2). Clearly, for r = 2 we have J Y = J Y , so that C (r) generalizes the well known symmetrization C (2) from Example 3.1. In Ref. 2 we have presented another geometric construction of C (r) and we have proved Proposition 3.1. Any F Mm,n -natural transformation A : J J r Y is C (r) .

r,r−1

Y →

Obviously, C (r) has the coordinate expression p (xi , y p , yip , . . . yip1 ...ir−1 , yip1 ...ir ) 7→ (xi , y p , yip , . . . yip1 ...ir−1 , y(i ). 1 ...ir ) r

Theorem 3.2 (Ref. 2). All F Mm,n -natural transformations C : J Y → J r Y are of the form (1) (2) (3) (4)

r

For m = 1 and arbitrary natural r, C = id : J Y = J r Y → J r Y . 1 For r = 1 and m ≥ 2, C = id : J Y = J 1 Y → J 1 Y . 2 For r = 2 and m ≥ 2, C = C (2) : J Y → J 2 Y . For r ≥ 3 and m ≥ 2 there is no F Mm,n -natural transformation C in question.

Proposition 3.2 (Ref. 2). For r ≥ 2 and m ≥ 2 there is no natural transformation Jer Y → J r Y . r

In Ref. 2 we defined symmetrizations c(r) (∇) : J Y → J r Y and sr (∇) : Jer Y → J r Y for any r depending on a projectable classical linear connection ∇ on Y .

Example 3.4. Let ∇ be a projectable classical linear connection on Y covr ering ∇ on M . We define a natural transformation c(r) (∇) : J Y → J r Y r as follows. Take σ ∈ (J Y )y , y ∈ Yx , x ∈ M and let σ ∈ (J 1 Y )y be the

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underlying element of σ. Similarly to Lemma 3.1, denote by Φ a ∇-normal coordinate system on Y with centre y covering a ∇-normal coordinate system Φ such that J 1 Φ(σ) = j01 (0). We define r

c(r) (∇)(σ) := J r Φ−1 (I −1 (⊕rk=1 Symk (I(J Φ(σ))))),

(5)

r

where I : (J Rm,n )(0,0) → ⊕rk=1 ⊗k T0∗ Rm ⊗ V(0,0) Rm,n and I : (J r Rm,n )(0,0) → ⊕rk=1 S k T0∗ Rm ⊗ V(0,0) Rm,n are the standard identifications and Symk : ⊗k T0∗ Rm ⊗ V(0,0) Rm,n → S k T0∗ Rm ⊗ V(0,0) Rm,n are given by the symmetrizations. If Φ1 is another ∇-normal coordinate system with centre y satisfying J 1 Φ1 (σ) = j01 (0), then near y we have Φ1 = A ◦ Φ for some A ∈ GL(m) × GL(n) (see also Lemma 3.1(2)). Clearly, I and I are GL(m) × GL(n)-invariant. One can verify easily that the right hand sides of (5) for Φ and Φ1 coincide. Then the definition of c(r) (∇)(σ) is correct (it is independent of the choice of Φ). Example 3.5. Given a projectable classical linear connection ∇ on Y covering ∇ on M , we define a natural transformation sr (∇) : Jer Y → J r Y as follows. Let σ ∈ (Jer Y )y , y ∈ Yx , x ∈ M and let σ ∈ (J 1 Y )y be the underlying element of σ. Taking the same Φ as in Example 3.4, we put sr (∇)(σ) := J r Φ−1 (j0r (0)) .

(6)

Quite analogously to Example 3.4 we show that the definition of sr (∇)(σ) is correct. Remark 3.2. By Theorem 3.2 and Proposition 3.2, to define symmetrizar tions J Y → J r Y or Jer Y → J r Y for m ≥ 2 and r ≥ 3, the use of an auxiliary projectable classical linear connection ∇ on Y is unavoidable. 4. Classification of holonomic extensions (a) Classification of second order holonomic extensions From Theorem 3.1 it follows directly that for m ≥ 2, the operator Γ 7→ C (2) (Γ ∗ Γ) from Example 3.1 is the unique F Mm,n -natural operator transforming general connections Γ : Y → J 1 Y into second order holonomic connections Y → J 2 Y . It is well known that J r Y → J r−1 Y is an affine bundle with the associated vector bundle S r T ∗ M ⊗ V Y . Thus, we have an F Mm,n -natural operator Q sending general connections Γ on Y → M and projectable classical linear connections ∇ on Y into sections Q(Γ, ∇) := A2 (Γ, ∇) − C (2) (Γ ∗ Γ) : Y → S 2 T ∗ M ⊗ V Y,

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where A2 is the operator from Example 3.2 for r = 2. In Remark 4.1 below we show another (more classical) interpretation of the operator Q(Γ, ∇). Theorem 4.1 (Ref. 2). All F Mm,n -natural operators A transforming general connections Γ : Y → J 1 Y and projectable torsion free classical linear connections ∇ on Y into second order holonomic connections A(Γ, ∇) : Y → J 2 Y are of the form A(Γ, ∇) = C (2) (Γ ∗ Γ) + tQ(Γ, ∇),

t ∈ R.

(7)

(b) Classification of third order holonomic extensions From now on we denote the connection (7) by Ahti (Γ, ∇), i.e. Ahti (Γ, ∇) := C (2) (Γ ∗ Γ) + tQ(Γ, ∇).

Let B23 be the operator from Example 3.3 transforming second order holonomic connections Θ : Y → J 2 Y and projectable classical linear connections ∇ on Y into third order holonomic connections B23 (Θ, ∇) : Y → J 3 Y . Then for Θ = Ahti (Γ, ∇) we have third order holonomic connections B hti (Γ, ∇) := B23 (Ahti (Γ, ∇), ∇) : Y → J 3 Y,

t ∈ R,

which project onto Ahti (Γ, ∇) via the jet projection π23 : J 3 Y → J 2 Y . Theorem 4.2. All F Mm,n -natural operators A transforming general connections Γ : Y → J 1 Y and projectable torsion free classical linear connections ∇ on Y into third order holonomic connections A(Γ, ∇) : Y → J 3 Y are of the form A(Γ, ∇) = B hti (Γ, ∇) + ∆(Γ, ∇) for some t ∈ R and some section ∆(Γ, ∇) : Y → S 3 T ∗ M ⊗ V Y . Proof. Any such A(Γ, ∇) : Y → J 3 Y projects onto π23 ◦ B(Γ, ∇) : Y → J 2 Y , which is exactly Ahti (Γ, ∇) for some t ∈ R because of Theorem 4.1. Then our assertion follows directly from the fact that J 3 Y → J 2 Y is an affine bundle with the associated vector bundle S 3 T ∗ M ⊗ V Y . In the rest of this section we describe explicitly all ∆’s in question. We show that the vector space of all such ∆’s is of dimension 3 for n ≥ 2 and of dimension 2 for n = 1. Roughly speaking, we prove that all F Mm,n -natural operators A transforming general connections Γ : Y → J 1 Y and projectable torsion free classical linear connections ∇ on Y into third order holonomic connections A(Γ, ∇) : Y → J 3 Y form the one parameter family of three(resp. two-)dimensional affine spaces if n ≥ 2 (resp. if n = 1). So it suffices

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to describe all F Mm,n -natural operators ∆ transforming connections Γ and ∇ as above into sections ∆(Γ, ∇) : Y → S 3 T ∗ M ⊗ V Y . We have the following general example of such operators ∆. Example 4.1. Let H : J 2 (J 1 (Rm,n → Rm ) → Rm,n )j01 (0) → S 3 T0∗ Rm ⊗ V(0,0) Rm,n (the fibre over j01 (0) with respect to the projection π02 : J 2 Y → Y ) be a GL(m) × GL(n)-equivariant map (we note that GL(m) × GL(n) preserve j01 (0) and then the invariance has sense). Let Γ be a general connection on Y → M and ∇ be a projectable classical linear connection on Y with the underlying classical linear connection ∇ on M . Define ∆H (Γ, ∇) : Y → S 3 T ∗ M ⊗ V Y as follows 2 (Φ∗ Γ))) ∈ S 3 Tx∗ M ⊗ Vy Y, (8) ∆H (Γ, ∇)(y) := S 3 T ∗ Φ−1 ⊗ V Φ−1 (H(j(0,0)

y ∈ Yx , x ∈ M where Φ is a normal fiber coordinate system of ∇ with centre y with the underlying normal coordinate system Φ with centre x such that J 1 Φ(Γ(y)) = j01 (0). If Φ1 is another such normal coordinate system, then Φ1 = D ◦ Φ for some D ∈ GL(m) × GL(n). So, because of the GL(m) × GL(n)-invariance of H we see that (8) does not depend on the choice of Φ in question. The correspondence ∆H : (Γ, ∇) → ∆H (Γ, ∇) is an F Mm,n -natural operator. We prove (see also Proposition 4.3) below Proposition 4.1. Any F Mm,n -natural operator ∆ transforming general connections Γ : Y → J 1 Y and torsion free projectable classical linear connections ∇ on Y into sections ∆(Γ, ∇) : Y → S 3 T ∗ M ⊗ V Y is of the form ∆(Γ, ∇) = ∆H (Γ, ∇) for some GL(m) × GL(n)-equivariant map H from Example 4.1. Proof. Clearly, ∆ is determined by the values h(∆(Γ, ∇)(0, 0), u ⊙ u ⊙ u ⊗ ωi ∈ R

(9)

for all connections Γ : Rm,n → J 1 (Rm,n → Rm ), all torsion free projectable classical linear connections ∇ on Rm,n , all u ∈ T0 Rm and all ω ∈ T0∗ Rn . Using respective normal coordinates with centre zero we can assume additionally that Γ(0, 0) = j01 (0), the Christoffel symbols of ∇ are ∇kij (0) = 0, ∂ and ω = d0 y 1 . Because of non linear Peetre theorem (see u = uo = ∂x 1 |0

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Corollary 19.8 in Ref. 9) we can assume that the coefficients of Γ are polynomials in x, y of (fixed) degree ≤ K for arbitrary large K ∈ N, Γ=

m X i=1

m

dxi ⊗

n

XX ∂ + i ∂x i=1 j=1

X

|α|+|β|≤K

Γjiαβ xα y β dxi ⊗

∂ . ∂y j

Applying the invariance of ∆ with respect to the homotheties (tx1 , . . . , txm , ty 1 , . . . , ty n ) and using homogeneous function theorem from Ref. 9 we deduce that (9) is the linear combination of ∇lij;s (0, 0) and of the polynomial in the derivatives of Γ of order ≤ 2. So ∆ is determined by the (well-defined) values 2 H(j(0,0) Γ) := ∆(∇o , Γ)(0, 0) m,n

1

m,n

(10) j01 (0)

for all connections Γ : R →J R with Γ(0, 0) = (here ∇o is the usual flat connection on Rm,n ) together with the values (9) with the trivial connection Γo on Rm,n → Rm instead of Γ. The values (9) with the trivial connection Γo instead of Γ under the above assumptions are linear combinations of ∇kij;s (0). Then using the invariance of ∆ with respect to the homotheties (t1 x1 , . . . , tm xm , τ1 y 1 , . . . , τn y n ) we see that the last values ∂ are proportional to ∇m+1 11;1 (0) (we have the assumptions u = uo = ∂x1 |0 and

ω = d0 y 1 ), and then are zero, as ∇m+1 11;1 (0, 0) = 0 in normal coordinates (see Lemma 4.1 below). By the GL(m) × GL(n)-invariance of ∆, the map H : J 2 (J 1 (Rm,n → Rm ) → Rm,n )j01 (0) → S 3 T0∗ Rm ⊗ V(0,0) Rm,n given by (10) is GL(m) × GL(n)-equivariant. Clearly ∆ = ∆H because both ∆ and ∆H determine the same values (10). Lemma 4.1. Let ∇ be a classical linear connection on Rk such that the identity map idRk is its normal coordinate system with centre zero. Then ∇l11;1 (0) = 0 for l = 1, . . . , k. Proof. Obviously, the straight lines passing through 0 are geodesics of ∇. Using the well known equations for geodesics we get k X

i,j=1

∇lij (x)xi xj = 0,

l = 1, . . . , k.

Then our assertion follows easily from the last equality. Namely, we pass to formal Taylor series at 0 of booth sides of the above equality (for l = 1, . . . , k) and then consider the coefficients at (x1 )3 . So it remains to describe all GL(m) × GL(n)-equivariant maps

H : J 2 (J 1 (Rm,n → Rn ) → Rm,n )j01 (0) → S 3 T0∗ Rm ⊗ V(0,0) Rm,n .

(11)

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233

Obviously, any element τ ∈ J 2 (J 1 (Rm,n → Rm ) → Rm,n )j01 (0) is of the 2 form τ = j(0,0) Γ for a general connection Γ=

m X i=1

+

dxi ⊗

m X n X

j=1 k,l=1

+

k=1

bkl,j y l dxj ⊗

m X n X

n m X X ∂ ∂ ckl1 ,l2 ,j y l1 y l2 dxj ⊗ k + ∂y k ∂y j=1

eki,l,j xi y l dxj ⊗

i,j=1 k,l=1 m,n

n m X X ∂ ∂ aki,j xi dxj ⊗ k + ∂xi i,j=1 ∂y

l1 ,l2 =1 m X

∂ + ∂y k i

1 ,i2 ,j=1

n X

k=1

fik1 ,i2 ,j xi1 xi2 dxj ⊗

(12) ∂ ∂y k

on R for some (uniquely determined by τ ) real numbers aki,j , bkj,l , ckl1 ,l2 ,j , eki,l,j , fik1 ,i2 ,j with ckl1 ,l2 ,j = ckl2 ,l1 ,j and fik1 ,i2 ,j = fik2 ,i1 ,j . That is why, we have J 2 (J 1 (Rm,n → Rn ) → Rm,n )j01 (0) = W1 ⊕ W2 ⊕ W3 ⊕ W4 ⊕ W5 modulo the obvious isomorphism of GL(m) × GL(n)-modules, where W1 = Rm∗ ⊗ Rm∗ ⊗ Rn , W2 = Rn∗ ⊗ Rm∗ ⊗ Rn , W3 = S 2 Rn∗ ⊗ Rm∗ ⊗ Rn , W4 = Rm∗ ⊗ Rn∗ ⊗ Rm∗ ⊗ Rn and W5 = S 2 Rm∗ ⊗ Rm∗ ⊗ Rn . Quite similarly, S 3 T0∗ Rm ⊗ V(0,0) Rm,n = S 3 Rm∗ ⊗ Rn modulo the obvious isomorphism of GL(m) × GL(n)-modules. Thus it remains to describe all GL(m) × GL(n)-equivariant maps H : W1 ⊕ W2 ⊕ W3 ⊕ W4 ⊕ W5 → S 3 Rm∗ ⊗ Rn .

(13)

We have the following examples of GL(m) × GL(n)-maps (13). Example 4.2. Let τ = (τ1 , ..., τ5 ) ∈ W1 ⊕ ... ⊕ W5 . We define Hi (τ ) as follows: (a) We symmetrize τ5 H1 (τ ) := Sym(τ5 ) ∈ S 3 Rm∗ ⊗ Rn .

(b) Consider τ1 ⊗ τ2 ∈ Rm∗ ⊗ Rm∗ ⊗ Rn ⊗ Rn∗ ⊗ Rm∗ ⊗ Rn . Applying ∗ two obvious contractions C43 and C46 , we obtain C43 (τ1 ⊗ τ2 ) ∈ Rm ⊗ Rm∗ ⊗ Rm∗ ⊗Rn and C46 (τ1 ⊗τ2 ) ∈ Rm∗ ⊗Rm∗ ⊗Rn ⊗Rm∗ = Rm∗ ⊗Rm∗ ⊗Rm∗ ⊗ Rn (modulo obvious isomorphism of GL(m) × GL(n)-modules). Then the symmetrization yields H2 (τ ) := Sym(C43 (τ1 ⊗ τ2 )) ∈ S 3 Rm∗ ⊗ Rn and H3 (τ ) := Sym(C46 (τ1 ⊗ τ2 )) ∈ S 3 Rm∗ ⊗ Rn .

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Proposition 4.2. (a) If n ≥ 2, then any GL(m) × GL(n)-map H : W1 ⊕ · · · ⊕ W5 → S 3 Rm∗ ⊗ Rn is a linear combination of H1 , H2 and H3 (described in Example 4.2) with real coefficients. (b) If n = 1, then any GL(m) × GL(n)-map H : W1 ⊕ · · · ⊕ W5 → S 3 Rm∗ ⊗ Rn is a linear combination of H1 and H2 with real coefficients. Proof. It is easy to see that if n ≥ 2, then H1 , H2 and H3 are linearly independent in the vector space of all GL(m) × GL(n)-maps (13). If n = 1, then H1 and H2 are also linearly independent. So by the dimension argument and our identifications, it suffices to show that the vector space of all GL(m) × GL(n)-maps (11) is of dimension ≤ 3 for n ≥ 2 and of dimension ≤ 2 for n = 1. Consider a GL(m) × GL(n)-map C as in (11). Because of the GL(m) × GL(n)-invariance, H is determined by the values E D 2 (14) H(j(0,0) Γ), uo ⊙ uo ⊙ uo ⊗ d0 y 1 ∈ R for all Γ as in (12), where uo = to

∂ ∂x1 |0 .

Using equivariance of H with respect

(x1 , tx2 , . . . , txm , y 1 , . . . , y n ) ∈ GL(m) × GL(n) for t 6= 0 and putting t → 0, we see that (14) does not depend on many coordinates of Γ. Consequently we see that H is determined by the values (14) for Γ of the form Γ=

m X i=1

+ +

dxi ⊗

n X

n n X X ∂ ∂ ∂ k 1 1 a x dx ⊗ bkl y l dx1 ⊗ k + + ∂xi ∂y k ∂y

ckl1 ,l2 y l1 y l2 dx1 ⊗

l1 ,l2 ,k=1 n X k 1 2 k=1

k=1

f (x ) dx1 ⊗

k,l=1

∂ + ∂y k

n X

l,k=1

ekl y l x1 dx1 ⊗

∂ ∂y k

(15)

∂ ∂y k

for all real numbers ak , bkl , ckl1 ,l2 , ekl , f k with ckl1 ,l2 = ckl2 ,l1 . Using equivariance of H with respect to the homotheties tidRm,n ∈ GL(m) × GL(n) for t 6= 0 and homogeneous function theorem we see that the left hand side of (14) for Γ as in (15) is the linear combination of ckl1 ,l2 , ekl , f k and of the combination of monomials in ak , bkl of degree 2. Next, applying equivariance of H with respect to ϕt,τ = 1t idRm × τ idRn ∈ GL(m) × GL(n) for t, τ 6= 0

Extension of connections

235

we get the homogeneous condition E D 2 ((ϕt,τ )∗ Γ)), uo ⊙ uo ⊙ uo ⊗ d0 y 1 H(j(0,0) D E 2 = t3 τ H(j(0,0) Γ), uo ⊙ uo ⊙ uo ⊗ do y 1 ,

where Γ is as in (15). This yields n n E X D X 2 αk f k + H(j(0,0) Γ), uo ⊙ uo ⊙ uo ⊗ d0 y 1 = k=1

βkl21 ,k2 ak1 bkl22 (16)

k1 ,k2 ,l2 =1

for Γ as in (15) for some (determined by H) real numbers αk , βkl21 ,k2 . Using invariance of H with respect to (x1 , . . . , xm , y 1 , ty 2 , . . . , ty n ) ∈ GL(m) × GL(n) for t 6= 0 we deduce easily n n E D X X 2 Γ), uo ⊙ uo ⊙ uo ⊗ d0 y 1 =αf 1+βa1 b11+ β k ak b1k + γ k a1 bkk (17) H(j(0,0 k=2

k=2

k

k

for some (determined by H) real numbers α, β, β , γ , where Γ is as in (15). Therefore if n = 1, then the vector space of all H as in (11) is of dimension ≤ 2. Assume now n ≥ 2. Using equivariance of H with respect to permutations of coordinates y 2 , . . . , y n (they belong to GL(m) × GL(n)) we deduce easily β 2 = · · · = β n , γ 2 = · · · = γ n . From (17) we have E D 2 (18) Γ), uo ⊙ uo ⊙ uo ⊗ d0 y 1 = 0 H(j(0,0) Pm ∂ ∂ ∂ 1 1 2 1 for Γ = i=1 dxi ⊗ ∂x i + x dx ⊗ ∂y 1 + y dx ⊗ ∂y 1 . Applying equivariance 1 m 1 2 1 3 of H with respect to (x , . . . , x , y , y + y , y , . . . , y n ) ∈ GL(m) × GL(n) Pm ∂ ∂ ∂ 1 1 2 1 1 we have (18) for Γ = i=1 dxi ⊗ ∂x i + x dx ⊗ ( ∂y 1 + ∂y 2 ) + (y − y )dx ⊗ ( ∂y∂ 1 + ∂y∂ 2 ). But from(16) it follows that the left hand side of (18) is equal to (−β + ∗), where ∗ is a linear combination of β 2 and γ 2 . Then (−β + ∗) = 0. That is why the vector space of all H as in (11) is of dimension ≤ 3. Remark 4.1. Let (Γ, ∇) be as in Theorem 4.1 or 4.2. We have the tangent valued 1-form Γ : Y → T ∗ Y ⊗ V Y (the horizontal projection of Γ). Its covariant derivative is a tensor field ∇Γ : Y → ⊗2 T ∗ Y ⊗ V Y , where ∇Γ(U, W ) = (∇U Γ)(W ) for vector fields U, W . Composing symmetrization of ∇Γ with the horizontal lifting map h : Y → T ∗ M ⊗T Y of Γ, we have tene ∇) : Y → S 2 T ∗ M ⊗ V Y . Because of the dimension argument, sor field Q(Γ, e the operator Q from Theorem 4.1 is proportional to Q. 2 3 ∗ Write ∇ Γ = ∇(∇Γ) : Y → ⊗ T Y ⊗ V Y for the second covariant derivative. Moreover, we have tensor fields ∇Γ(∇Γ, .) : Y → ⊗3 T ∗ Y ⊗ V Y

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and C11 (∇Γ) ⊗ ∇Γ : Y → ⊗3 T ∗ Y ⊗ V Y , where the first tensor is the substitution of ∇Γ into the first position of ∇Γ and C11 denotes the obvious contraction. Composing symmetrization of these tensor fields with the horizontal lifting map h, we define tensor fields Ei (Γ, ∇) : Y → S 3 T ∗ M ⊗ V Y , i = 1, 2, 3. Thus we have the corresponding F Mm,n -natural operators Ei transforming Γ and ∇ into tensor fields Ei (Γ, ∇) : Y → S 3 T ∗ M ⊗ V Y . One can standardly show that the operators E1 , E2 , E3 are linearly independent for n ≥ 2. If n = 1, then E1 and E2 are linearly independent (E2 and E3 are proportional). By Propositions 4.1, 4.2 and the dimension argument we have Proposition 4.3. If n ≥ 2, then any F Mm,n -natural operator ∆ of the type as in Proposition 4.1 is a linear combination of E1 , E2 , E3 with real coefficients. If n = 1, then such operators are linear combinations of E1 and E2 . 5. Some applications of higher order connections Let P → M be a principal bundle with structure group G. Then we have a right action of G on J r P defined by (jxr s(y))g = jxr (s(y)g), where s : M → P is a local section, x, y ∈ M , g ∈ G. This enables us to define Ginvariant r-th order holonomic connections Γ : P → J r P . Clearly, for r = 1 we obtain the well known concept of a principal connection on P → M . We r recall that the r-th principal prolongation Wm P of P is defined as the space r of all r-jets j(0,e) ϕ of local trivializations ϕ : Rm × G → P , where e ∈ G r is the unit. By Ref. 9, Wm P → M is a principal bundle with the structure r r group Wm G := J(0,e) (Rm × G, Rm × G)(0,−) and the fibered manifold r r P = P r M ×M J r P , P → M coincides with the fibered product Wm Wm r r m where P M := invJ0 (R , M ) is the r-th order frame bundle of M . It is r plays a fundamental role in the well known that the bundle functor Wm theory of gauge natural operators and in mathematical physics. Example 5.1. Let Γ be a principal connection on P → M and Λ be a classical linear connection on M . We have a reduction of principal bundles 1 µΓ : P 1 M ×M P → P 1 M ×M J 1 P = Wm P,

µΓ (l, p) = (l, Γ(p))

(19)

1 with the obvious group monomorphism GL(m) × G → Wm G, cf. Ref. 9. Using right translations, the product connection Λ × Γ on P 1 M ×M P can 1 be extended into a principal connection p(Γ, Λ) on Wm P → M . By 17.2. and 16.6. of Ref. 9, reduction µΓ can be used in the coordinate description

Extension of connections

237

of geometric object fields. Moreover, connection p(Γ, Λ) plays an important role in the geometric constructions on principal prolongations of principal bundles, see Ref. 10. Example 5.2. Suppose we have a G-invariant r-th order holonomic connection D(Γ, Λ) : P → J r P depending on a principal connection Γ : P → J 1 P and a classical linear connection Λ on M . Then we can generalize (19) to the reduction r µD : P r M ×M P → P r M ×M J r P = Wm P, µD (l, p) = (l, D(Γ, Λ)(p)), (20)

which can be used for the geometric description of higher order geometric object fields. Moreover, if Λ is a torsion free classical linear connection on M , then we can define its exponential prolongation Λexpr : T M → J r T M , which can be identified with a principal connection on P r M , see e.g. Ref. 14. On P r M ×M P we have a product connection Λexpr × Γ, which can be r extended via right translations of Wm G to a principal connection r r P → J 1 (Wm P → M ), pD (Γ, Λ) : Wm

(21)

which generalizes p(Γ, Λ) from Example 5.1. Clearly, for r = 1 and D(Γ, Λ) = Γ, (21) is exactly p(Γ, Λ). Example 5.3. We recall that a torsion free classical linear connection Λ on M and a principal connection Γ on P → M induce a G-invariant projectable classical linear connection NP (Γ, Λ) on P , see Ref. 9. Using our operator Ar from Example 3.2, we can write D(Γ, Λ) = Ar (Γ, NP (Γ, Λ)). Then (21) r P → M , which generalizes p(Γ, Λ). yields a connection pr (Γ, Λ) on Wm Example 5.4. One can replace D(Γ, Λ) in Example 5.3 by an arbitrary e transforming couples (Γ, Λ) consisting PBm (G)-gauge natural operator D of principal connections Γ on P → M and classical linear connections Λ on e M into G-invariant r-th order holonomic connections D(Γ, Λ) : P → J r P . e can be constructed In the case r = 2 or r = 3, many examples of such D from operators A classified in Theorems 4.1 and 4.2. In fact, if Y = P → M is a principal G-bundle, Γ is a principal connection on P → M and Λ is a torsion free classical linear connection on M , then all operators A from Thee orem 4.1 and 4.2 give G-invariant connections D(Γ, Λ) := A(Γ, NP (Γ, Λ)). e Connections D(Γ, Λ) are in fact G-invariant, because the right translations of G on P are in particular F Mm,n -maps, such translations preserve Γ (as Γ are principal), they preserve ∇ = NP (Γ, Λ) (as NP (Γ, Λ) are G-invariant) and they also preserve A (as A are F Mm,n -natural). This yields other gen2 3 eralizations of p(Γ, Λ) to Wm P → M and to Wm P → M.

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Acknowledgments ˇ The first author was supported by a grant of the GA CR No 201/05/0523. References 1. A. Cabras and I. Kol´ aˇr, Second order connections on some functional bundles, Arch. Math. (Brno) 35 (1999) 347–365. 2. M. Doupovec and W.M. Mikulski, Holonomic extension of connections and symmetrization of jets, Rep. Math. Phys. 60 (2) (2007) 299–316. 3. C. Ehresmann, Extension du calcul des jets aux jets non holonomes, CRAS Paris 239 (1954) 1762–1764. 4. C. Ehresmann, Sur les connexions d’ ordre sup´erieur, In: Atti del V. Cong. del’ Unione Mat. Italiana (1955, Roma Cremonese, 1956) 344–346. 5. I. Kol´ aˇr, On the torsion of spaces with connection, Czech. Math. J. 21 (1971) 124–136. 6. I. Kol´ aˇr, The contact of spaces with connection, J. Diff. Geom. 7 (1972) 563–570. 7. I. Kol´ aˇr, Higher order absolute differentiation with respect to generalized connections, Differential Geometry, Banach Center Publications 12 (1984) 153–162. 8. I. Kol´ aˇr, A general point of view to nonholonomic jet bundles, Cahiers Topol. G´eom. Diff. 44 (2003) 149–160. 9. I. Kol´ aˇr, P.W. Michor and J. Slov´ ak, Natural Operations in Differential Geometry (Springer–Verlag, 1993). 10. I. Kol´ aˇr, G. Virsik, Connections in first principal prolongations, Rend. Circ. Mat. Palermo, Serie II, Suppl. 43 (1996) 163–171. 11. D. Krupka, J. Janyˇska, Lectures on Differential Invariants (Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis, Mathematica, University J. E. Purkynˇe, Brno, 1990). 12. M. Kureˇs, On the symmetrisation of nonholonomic jets, Math. Proc. of the Royal Irish Academy 105A (2005) 93–106. 13. P. Libermann, Introduction to the theory of semi-holonomic jets, Arch. Math. (Brno) 33 (1997) 173–189. 14. W.M. Mikulski, Higher order linear connections from first order ones, Arch. Math. (Brno) 43 (4) (2007) 285–288. 15. J. Pradines, Fibr´es vectoriels doubles symm´etriques et jets holonomes d’ ordre 2, CRAS Paris 278 (1974) 1557–1560. 16. G. Virsik, On the holonomity of higher order connections, Cahiers Topol. G´eom. Diff. 12 (1971) 197–212.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 239–251

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Conformally-projectively flat statistical structures on tangent bundles over statistical manifolds Izumi Hasegawa Department of Mathematics, Hokkaido University of Education, Sapporo, 002-8502, Japan E-mail: [email protected] Kazunari Yamauchi Department of Mathematics, Asahikawa Medical College, Asahikawa, 078-8510, Japan E-mail: [email protected] Let (M, h, ∇) be a statistical manifold and T M the tangent bundle over M . If T M with “horizontal lift” statistical structure (hH , C H ) is conformallyprojectively flat, then (M, h, ∇) is of constant curvature. In particular, if ∇ is non-flat, then ∇ is the Levi-Civita connection of h. Keywords: Statistical structure, tangent bundle, horizontal lift, conformallyprojectively flat. MS classification: 53A15, 53B05.

1. Introduction Let M be a differentiable manifold, h a semi-Riemannian metric on M , and ∇ a symmetric affine connection on M . If the covariant derivative ∇h is symmetric, then ∇ is said to be compatible to h. A pair (h, ∇) of semiRiemannian metric with compatible affine connection is called a statistical structure on M . A manifold M together with a statistical structure is called a statistical manifold. On a statistical manifold (M, h, ∇), the symmetric tensor field C := ∇h is called the cubic form of M ; C = (Skji ) = Skji dxk ⊗ dxj ⊗ dxi , where Skji := ∇k hji and (xh ) is the local coordinate system of M . The tensor field S of type (1, 2) is defined by h(S(X, Y ), Z) := C(X, Y, Z) = (∇X h)(Y, Z) for any X, Y, Z. S is called the skewness operator of (M, h, ∇); S = (Sji h ) = Sji h dxj ⊗ dxi ⊗ ∂x∂ h , where Sji h := Sjia hah and

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(hji ) := (hji )−1 . Then we have ∇ = ∇0 − 12 S, i.e., Γji h = (Γ )jih − 21 Sji h , 0
where ∇0 denotes the Levi-Civita
connection of h and Γji h resp. (Γ )jih are the coefficients of ∇ resp. ∇0 . For every statistical manifold (M, h, ∇), there exists a naturally associated symmetric trilinear form C called the cubic form. Conversely let (M, h, C) be a semi-Riemannian manifold with symmetric trilinear form C. Define a tensor field S of type (1, 2) by h(S(X, Y ), Z) := C(X, Y, Z), and an affine connection ∇ by ∇ := ∇0 − 12 S. Then ∇ is symmetric and satisfies ∇h = C. Hence the triplet (M, h, ∇) becomes a statistical manifold. We call this symmetric affine connection a compatible connection with respect to (h, C). Thus equipping a statistical structure (h, ∇) is equivalent to equipping a pair (h, C) consisting of a semi-Riemannian metric h and a symmetric trilinear form C. Therefore (h, C) is also called a statistical structure. Let (M, h, ∇) be an n-dimensional statistical manifold. The curvature tensor K of (M, h, ∇) is defined by the curvature tensor of ∇, i.e., Kkjih := ∂k Γji h − ∂j Γki h + Γji a Γkah − Γki a Γjah . The curvature tensor satisfies Kkjih = −Kjkih ,

Kkjih + Kjikh + Kikj h = 0, Kkjih + Kjhik + Khkij = 0, Kkjia hah .

where Kkjih := The Ricci tensor of (M, h, ∇) is defined by Rji := Kajia . In statistical geometry, the Ricci tensor is not necessarily to be symmetric. The scalar curvature of (M, h, ∇) is defined by ρ := Rba hba . (M, h, ∇) is called an Einstein statistical manifold if ρ is constant and Rji = nρ hji . (M, h, ∇) is ρ (δkh hji − said to be of constant curvature if ρ is constant and Kkjih = n(n−1) h δj hki ). (M, h, ∇) is said to be locally flat if K = 0. In Riemannian geometry we have the following:

Theorem 1.1 (Ref. 11). Let (M, h) be a semi-Riemannian manifold of dimension n(≥ 2). Then (T M, hC ) is conformally flat if and only if (M, h) is of constant curvature. J. Inoguchi9 suggested to generalize this theorem from the view point of statistical geometry. We can construct many statistical structure on the tangent bundle T M over statistical manifold (M, h, ∇). For example, the

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pair (hC , ∇C ) of complete lift of h and complete lift of ∇ is a statistical structure on T M . In our previous paper4 we proved the following: Theorem 1.2. Let (M, h, ∇) be a statistical manifold of dimension n(≥ 2). Then (T M, hC , ∇C ) is conformally-projectively flat if and only if (M, h, ∇) is of constant curvature. In particular, if (T M, hC , ∇C ) is ±1-conformally flat, then (M, h, ∇) and (T M, hC , ∇C ) are locally flat. In this paper we prove the following: Theorem 1.3. Let (M, h, ∇) be a statistical manifold of dimension n(≥ 2). If (T M, hH , C H ) is conformally-projectively flat, then (M, h, ∇) is of constant curvature. In particular, if M is non-flat, then (h, ∇) is semi-Riemannian structure (i.e., ∇ is the Levi-Civita connection of h ). Therefore, if ∇ is not the Levi-Civita connection of h, then (M, h, ∇) is locally flat. Corollary 1.1. Let (M, h, ∇) be a statistical manifold of dimension n(≥ 2). If (T M, hH , C H ) is λ(6= 0)-conformally flat, then (M, h, ∇) and (T M, hH , C H ) are locally flat. The following corollary coincides with Theorem 1.1 since the complete lift hC is coincides with horizontal lift hH with respect to Levi-Civita connection of h. Corollary 1.2. Let (M, h) be a semi-Riemannian manifold of dimension n(≥ 2). Then (T M, hH ) is conformally flat if and only if (M, h) is of constant curvature. 2. Conformal-projective equivalence e be two statistical structures on M . We recall the Let (h, ∇) and (e h, ∇) notion of λ-conformality formulated by Kurose.6

Definition 2.1. For any fixed real number λ, two statistical structures e are said to be λ-conformally equivalent if there exists a (h, ∇) and (e h, ∇) function ϕ on M satisfying: and

e hji = exp(ϕ)hji

1−λ h 1+λ hji ϕh + (δj ϕi + δih ϕj ), Γeji h = Γji h − 2 2

(1)

(2)

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i. e., Sekji = exp(ϕ) {Skji + λ(hkj ϕi + hji ϕk + hik ϕj )} ,

where ϕi := ∂i ϕ and ϕh := hha ϕa . A statistical manifold (M, h, ∇) is said to be λ-conformally flat if (M, h, ∇) is λ-conformally equivalent to a flat statistical manifold in a neighbourhood of an arbitrary point of M . Remark 2.1 (Remarks of Definition 2.1). e are conformally equivalent in statis(1) If λ = 0, then (h, ∇) and (e h, ∇) tical geometry. In case of Riemannian geometry, (2) comes from (1). e are projectively equivalent. (2) If λ = −1, then ∇ and ∇ e are said to be dual-projectively equiv(3) If λ = 1, then (h, ∇) and (e h, ∇) alent (cf. Ref. 5).

H. Matsuzoe8 established the notion of conformal-projective equivalence relation and gave a necessary and sufficient condition for a conformallyprojectively flat statistical manifold to be realized by a nondegenerate centroaffine immersion of codimension 2. e on M are said Definition 2.2. Two statistical structures (h, ∇) and (e h, ∇) to be conformally-projectively equivalent if there exist two functions ϕ and ψ on M satisfying: (3)

and

e hji = exp(ϕ + ψ)hji

(4)

i. e.,

Γeji h = Γji h − hji ϕh + δjh ψi + δih ψj ,

Sekji = exp(ϕ + ψ) {Skji + hkj (ϕi − ψi ) + hji (ϕk − ψk ) + hik (ϕj − ψj )} .

A statistical manifold (M, h, ∇) is said to be conformally-projectively flat if (M, h, ∇) is conformally-projectively equivalent to a flat statistical manifold in a neighbourhood of an arbitrary point of M . Remark 2.2 (Remarks of Definition 2.2). e are conformally equivalent in statis(1) If ϕ = ψ, then (h, ∇) and (e h, ∇) tical geometry. e are projectively equivalent. (2) If ϕ is constant, then ∇ and ∇ e are dual-projectively equiva(3) If ψ is constant, then (h, ∇) and (e h, ∇) lent.

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e on M are λLemma 2.1. If two statistical structures (h, ∇) and (e h, ∇) e are conformally-projectively conformally equivalent, then (h, ∇) and (e h, ∇) equivalent. 3. Lift metrics and connections on T M (cf. Ref. 12) Let M be a differentiable manifold with an affine connection ∇, and Γji h the coefficients of ∇, i.e., Γji a ∂a := ∇∂j ∂i , where ∂h = ∂x∂ h and (xh ) is the local coordinates of M . We define a local frame {Ei , E¯i } of T M as follows: h

h

Ei := ∂i − y b Γib a ∂a¯

and E¯i := ∂¯i ,

(5) ∂ ∂y i .

where (x , y ) is the induced coordinates of T M and ∂¯i := This frame {Ei , E¯i } is called the adapted frame of T M with respect to ∇. Then {dxh , δy h } is the dual frame of {Ei , E¯i }, where δy h := dy h + y b Γabh dxa . Then, by straightforward calculation, we have the following Lemma 3.1. The Lie brackets of the adapted frame of T M satisfy the following identities: (1) [Ej , Ei ] = y b Kijb a Ea¯ , (2) [Ej , E¯i ] = Γji a Ea¯ , (3) [E¯j , E¯i ] = 0, where K = (Kkji h ) denotes the curvature tensor of (M, ∇) defined by Kkji h := ∂k Γji h − ∂j Γki h + Γji a Γkah − Γki a Γja h . Let h be a semi-Riemannian metric with the components hji on (M, ∇), i.e., h = hji dxj ⊗ dxi . The horizontal lift metric hH of h with respect to ∇ is defined as follows: a

hH = hji (δy j ⊗ dxi + dxj ⊗ δy i ).

(6)

Let X = X ∂a be a vector field on M . The vertical lift X zontal lift X H of X are defined as follows: X H := X a Ea

V

and the hori-

and X V := X a Ea¯ .

(7)

Let (M, h, ∇) be a statistical manifold. We have the following lift tensors on T M : C H = Skji (δy k ⊗ dxj ⊗ dxi + dxk ⊗ δy j ⊗ dxi + dxk ⊗ dxj ⊗ δy i ). (8) S H = Sji h (δy j ⊗ dxi ⊗ Eh¯ + dxj ⊗ δy i ⊗ Eh¯ + dxj ⊗ dxi ⊗ Eh ).

Then we have the following: e Ye , Z) e hH (S H (X)

e Ye , Z) e = C H (X, e k¯ e j

ei

(9)

(10) e k e ¯j

ei

ek ej

e¯i

= Skji (X Y Z + X Y Z + X Y Z ),

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e =X e a ∂a +X e a¯ ∂a¯ , Ye = Ye a ∂a +Ye a¯ ∂a¯ , Z e=Z ea ∂a +Z ea¯ ∂a¯ ∈ T1 (T M ). for any X 0 H H Therefore (h , C ) is a statistical structure on T M . Lemma 3.2. Let ∇H be the compatible connection with respect to (hH , C H ) on T M . Then we have 1 ∇HEj Ei = (Γji a + Sji a )Ea − y b K aibj Ea¯ , ∇HE¯j E¯i = 0, 2 1 a 1 a H ∇ Ej E¯i = (Γji − Sji )Ea¯ , ∇HE¯j Ei = − Sji a Ea¯ . 2 2

(11)

We call this structure (hH , ∇H ) the horizontal lift statistical structure. Remark 3.1. This connection ∇H is different from the usual horizontal lift connection of ∇. This ∇H is symmetric connection on T M and coincides with the usual horizontal lift connection if and only if S = 0 and K = 0.

4. Conformal-projective flatness of T M with (hH , C H ) Theorem 1.3. Let (M, h, ∇) be a statistical manifold of dimension n(≥ 2). If (T M, hH , C H ) is conformally-projectively flat, then (M, h, ∇) is of constant curvature. In particular, if M is non-flat, then (h, ∇) is a semi-Riemannian structure (i.e., ∇ is the Levi-Civita connection of h). e on T M which is Proof. There exists a flat statistical structure (e h, ∇) H H conformally-projectively equivalent to (h , C ). Then, using Lemma 3.2, we have e Ej Ei = (Γji a + 1 Sji a + δja ψei + δia ψej )Ea − y b K aibj Ea¯ , ∇ 2 1 a e e + δja ψe¯i )Ea + (Γji a − Sji a − hji ϕ ∇Ej E¯i = (−hji ϕ ea¯ + δia ψej )Ea¯ , 2 (12) 1 a a ¯ ae a ae e e + δj ψi )Ea¯ , e + δi ψ¯j )Ea + (− Sji − hji ϕ ∇E¯j Ei = (−hji ϕ 2
e E¯ E¯i = δ a ψe¯i + δ a ψe¯j Ea¯ . ∇ i j j

Conformally-projectively flat statistical structures on tangent bundles

245

e be the curvature tensor of ∇. e Using (12), we obtain (13), · · · , (18). Let K e k , Ej )Ei 0 = K(E n 1 1 = Kkjia + (∇k Sji a − ∇j Skia ) + (Sji b Skba − Skib Sjba ) 2 o 4 n b a a a + δj αki − δk αji Ea + y (∇j K ibk − ∇k K aibj )

(13)

1 − y b (Sji c K acbk − Skic K acbj + Sjca K cibk 2 o − Skca K cibj + Sic a Kkjbc ) Ea¯ .

e k , Ej )E¯i 0 = K(E n o = δja βki − δka βji − hji Bka + hki Bj a + (ψeA ϕ eA )(δja hki − δka hji ) Ea n (14) + Kkjia − hji Aka + hki Aja o 1 1 − (∇k Sji a − ∇j Skia ) + (Sji b Skba − Skib Sjba ) Ea¯ . 2 4 e k , E¯j )Ei 0 = K(E n o = − δka βij − hji Bka − (ψeA ϕ eA )δka hji Ea (15) o n 1 1 + K aijk − ∇k Sji a + (Sji b Skba +Skib Sjba )+δja αki −hji Aka Ea¯ . 2 4 e k¯ , E¯j )Ei 0 = K(E n o = hki (∂¯j ϕ ea − ϕ e¯j ϕ ea ) − hji (∂k¯ ϕ ea − ϕ ek¯ ϕ ea ) Ea (16) n o + δja βik − δka βij − hji Cka + hki Cj a + (ψeA ϕ eA )(δja hki − δka hji ) Ea¯ . e k , E¯j )E¯i 0 = K(E n o = hki (∂¯j ϕ ea − ϕ e¯j ϕ ea ) − δka (∂¯j ψe¯i − ψe¯j ψe¯i ) Ea o n + δja βki + hki Cj a + (ψeA ϕ eA )δja hki Ea¯ .

n o e k¯ , E¯j )E¯i = δja (∂k¯ ψe¯i − ψek¯ ψe¯i ) − δka (∂¯j ψe¯i − ψe¯j ψe¯i ) Ea¯ . 0 = K(E

(17)

(18)

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Here we put as follows: 1 αji :=Ej ψei − ψej ψei − Γji c ψec − Sji c ψec + y b K cibj ψec¯. 2 1 βji :=Ej ψe¯i − ψej ψe¯i − Γji c ψec¯ + Sji c ψec¯ 2 1 =∂¯i ψej − ψej ψe¯i + Sji c ψec¯. 2 1 ¯ ¯ eh − ϕ ej ϕ eh + Γjc h ϕ ec¯ − Sjch ϕ ec¯ − y b K hcbj ϕ ec . Ajh :=Ej ϕ 2 1 ec + Sjch ϕ ec . Bj h :=Ej ϕ eh − ϕ ej ϕ eh + Γjc h ϕ 2 1 ¯ ¯ Cj h :=∂¯j ϕ eh − ϕ e¯j ϕ eh − Sjch ϕ ec . 2 From the definition of αji , we have
αji − αij = Ej ψei − Ei ψej − y b K ajbi ψea¯ − K aibj ψea¯
= y b K a ψea¯ − K a ψea¯ + K a ψea¯ ijb

jbi

ibj

(19)

(20)

= 0.

We put Aji := Aja hai . Then we have also


Aji − Aij = y b K aibj − K ajbi − Kjib a ϕ ea¯ = 0.

From (15), we have
1 1 Khijk = ∇k Sjih − 3Sji a Skha + Skia Sjha + Akh hji − αki hjh , 2 4 from which we have 0 = Khijk + Kihjk
= ∇k Sjih − Sji a Skha + Skia Sjha

+ Akh − αkh hji + Aki − αki hjh .

(21)

(22)

(23)

Applying the first Bianchi identity to (22), we get

0 = Khijk + Kijhk + Kjhik


= Akh − αkh hji + Aki − αki hjh + Akj − αkj hih
3 + ∇k Sjih − Sji a Skha + Sjha Skia + Siha Skja . 2 Comparing (23) with (24), we have
1 ∇k Sjih = Sji a Skha − Akh − αkh hji 2
= Sjha Skia − Aki − αki hjh
= Siha Skja − Akj − αkj hih ,

(24)

(25)

Conformally-projectively flat statistical structures on tangent bundles

247

from which we get (n − 1)(Aji − αji ) = Sji a Sabb − Sjba Siab

(26)

Aji − αji = (Aaa − αaa )hji − (Sji a Sabb − Sjba Siab ).

(27)

and

Using (26) and (27), we have n(Aji − αji ) = (Aaa − αaa )hji

(28)

and Sji a Sabb − Sjba Siab =

n−1 (Aaa − αaa )hji . n

(29)

From (13) we have
1
1 ∇k Sjih − ∇j Skih + Sji a Skha − Skia Sjha 2 4 + αji hkh − αki hjh .

Kkjih = −

(30)

On the other hand, using (21) and (22), we have Kkjih = Kkhij − Kjhik
1
1 = − ∇k Sjih − ∇j Skih + Sji a Skha − Skia Sjha 2 4 + αkh hji − αjh hki ,

(31)

Comparing (30) with (31), we get αji hkh − αki hjh = αkh hji − αjh hki , from which we have αji =

1 a α hji . n a

(32)

Substituting (32) into (30), we get
1
1 Kkjih = αaa hkh hji − hki hjh − ∇k Sjih − ∇j Skih n 2
1 + Sji a Skha − Skia Sjha . 4

(33)

Substituting (33) into (28), we obtain Aji =

1 a A hji . n a

(34)

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Next, from (14), we have 1 Kkjih = Akh hji − Ajh hki + (∇k Sjih − ∇j Skih ) 2 (35) 3 a a − (Sji Skha − Ski Sjha ). 4 Substituting (34) into (35), we get
1
1 Kkjih = Aaa hkh hji − hki hjh + ∇k Sjih − ∇j Skih n 2 (36)
3 a a − Sji Skha − Ski Sjha . 4 From (34) and (36), we obtain

1
1 Aaa + αaa hkh hji − hki hjh − Sji a Skha − Skia Sjha , (37) 2Kkjih = n 2 from which Kkjih = −Kkjhi .

(38)

Using (33), (36) and (38), we obtain 0 = Kkjih + Kkjhi = −∇k Sjih + ∇j Skih

1 = A a − αaa hkh hji − hki hjh − Sji a Skha + Skia Sjha , n a from which

1 Sji a Skha − Skia Sjha = Aaa − αaa hkh hji − hki hjh . n Substituting (39) into (37), we obtain

1 Kkjih = Aaa + 3αaa hkh hji − hki hjh , 4n from which
ρ Kkjih = hkh hji − hki hjh n(n − 1)

(39)

(40)

and

n−1 (Aaa + 3αaa ), 4 where ρ denotes the scalar curvature of K. Lastly we prove that ρ is constant. From (13) we have ρ=

(41)

∇k K hilj − ∇j K hilk (42) 1 = − (Sji a K halk − Skia K halj + Sjah K ailk − Skah K ailj + Siah Kkjla ). 2

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Substituting (40) into (42) and contracting k and h, we have (n − 2)(∇j ρ)hli + (∇l ρ)hji = Transvecting hji to (43), we get


1 Sla a hji + Siaa hlj − (n + 1)Slji . 2 ∇l ρ = 0.

(43)

(44)

Therefore ρ is constant. Transvecting hli to (43) and using (44), we get ρSjaa = −2(n − 1)∇j ρ = 0,

(45)

Substituting (44) and (45) into (43), we obtain ρSkji = 0.

(46)

Therefore, if ρ 6= 0 (i.e., ∇ is non-flat connection), then ∇ = ∇0 . Corollary 1.1. Let (M, h, ∇) be a statistical manifold of dimension n(≥ 2). If (T M, hH , C H ) is λ(6= 0)-conformally flat, then (M, h, ∇) and (T M, hH , C H ) are locally flat. Proof. First, we prove (M, h, ∇) is locally flat. By virtue of Theorem 1.3, it is good that we consider only the case C = 0. If λ 6= ±1, 0, then, using (16) and (18), there exists a scalar function ϕ on M such that αji =

1−λ 1−λ (∇j ϕi − ϕj ϕi ) 2 2

(47)

Aji =

1+λ 1+λ (∇j ϕi − ϕj ϕi ). 2 2

(48)

and

From (26), (47) and (48) we have 0 = Aji − αji = λ(∇j ϕi − ϕj ϕi ), from which 0 = ∇k ∇j ϕi − ∇j ∇k ϕi = −Kkjia ϕa = − i.e., ρϕj = 0.

ρ (ϕk hji − ϕj hki ), n(n − 1) (49)

3(n−1) a If λ = ±1 then, using (39) and (41), ρ = n−1 αaa ) = 0 4 Aa (resp., = 4 H H according as (T M, h , C ) is 1-conformally flat (resp., (-1)-conformally flat). Therefore (M, h, ∇) is locally flat.

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A e be the curvature tensor of ∇H and K e Next, let K DCB the components e with respect to the adapted frame. Then we have of K
e ¯ h¯ = −K e ¯ h¯ = K hijk − 1 ∇k Sji h + 1 Sji a Skah + Skia Sjah . K kji jki 2 4
1
1 ¯ h h h h e Kkj¯i = Kkji − ∇k Sji − ∇j Ski + Sji a Skah − Skia Sjah . 2 4
1
1 h h h h e (50) Kkji = Kkji + ∇k Sji − ∇j Ski + Sji a Skah − Skia Sjah . 2 4 n e kjih¯ = y b (∇j K hibk − ∇k K hibj ) K o 1 − (Sji c K hcbk − Skic K hcbj + Sjch K cibk − Skch K cibj + Sic h Kkjbc ) . 2 e vanish. On the other hand, from (25), we have Other components of K

∇k Sji h = Sji a Skah = Skia Sjah = Skja Siah ,

from which ∇k Sji h = and


1 S a S h + Skia Sjah 2 ji ka

∇k Sji h − ∇j Skih = e = 0. Therefore we have K


1 Sji a Skah − Skia Sjah = 0. 2

(51)

(52)

Corollary 1.2. Let (M, h) be a semi-Riemannian manifold of dimension n(≥ 2). Then (T M, hH ) is conformally flat if and only if (M, h, ∇) is of constant curvature. e the RiemannProof. Assume that (M, h) is of constant curvature. Let K H A e e with respect ian curvature tensor of h and KDCB the components of K to the adapted frame. Then we have
e h¯ = y a ∇k K h − ∇j K h (53) K aji kji aki

and

e ¯ h¯ = K h . e ¯h¯ = K e ¯ h¯ = K e h=K K kji kji kji kj i kji

(54)

e varnish. Then the Ricci tensor R e has the following Other components of K components: eji = 2Rji , R

e¯ji = R ej¯i = R e¯j¯i = 0. R

(55)

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The scalar curvature ρe of (T M, hH ) vanishes. Therefore, if M is of constant curvature, then the Weyl conformal curvature tensor of (T M, hH ) vanishes, i.e., (T M, hH ) is conformally flat. References 1. S. Amari and H. Nagaoka, Methods of Information Geometry (Amer. Math. Soc., Oxford Univ. Press, 2000). 2. H. Furuhata, H. Matsuzoe and H. Urakawa, Open problems in affine differential geometry and related topics, Interdisciplinary Information Sciences 4 (2) (1998) 125–127. 3. I. Hasegawa and K. Yamauchi, λ-conformal flatness of tangent bundle with complete lift statistical structure, Journal of Hokkaido University of Education (Natural Sciences) 58 (1) (2007) 1–14. 4. I. Hasegawa and K. Yamauchi, Conformal-projective flatness of tangent bundle with complete lift statistical structure, preprint. 5. S. Ivanov, On dual-projectively flat affine connections, Journal of Geometry 53 (1995) 89–99. 6. T. Kurose, On the realization of statistical manifolds in affine space, preprint (1991). 7. S. Lauritzen, Statistical manifolds, Differential Geometry in Statistical Inference, IMS Lecture Notes Monographs, Hayward California, Series 10 (1987) 96–163. 8. H. Matsuzoe, On realization of conformally-projectively flat statistical manifolds and the divergences, Hokkaido Math. J. 27 (1998) 409–421. 9. H. Matsuzoe and J. Inoguchi, Statistical structures on Tangent bundles, Applied Sciences 5 (1) (2003) 55–75. 10. C. Murathan and I. G¨ uney, Vertical and complete lifts on statistical manifolds and on the univariate Gaussian manifolds, Comm. Fac. Sci. Univ. Ankara Ser. A1 Math. Statist. 42 (1993) 69–76. 11. T.J. Willmore, Riemann extensions and affine differential geometry, Results in Math. 13 (1988) 403–408. 12. K. Yano and S. Ishihara, Tangent and Cotangent Bundles (Marcel Dekker, 1973). 13. K. Yano and S. Kobayashi, Prolongations of tensor fields and connections to tangent bundles I, II, III, J. Math. Soc. Japan 18 (1966) 194–210; 236–246; 19 (1967) 486–488.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 253–261

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Morphisms of almost product projective geometries Jaroslav Hrdina and Jan Slov´ ak Institute of Mathematics and Statistics, Faculty of Science, Masaryk University, Jan´ aˇ ckovo n´ am. 2a, 662 95 Brno, Czech Republic E-mail: [email protected], [email protected] We discuss almost product projective geometry and the relations to a distinguished class of curves. Our approach is based on an observation that well known general techniques2,5,8 apply, and our goal is to illustrate the power of the general parabolic geometry theory on a quite explicit example. Therefore, some rudiments of the general theory are mentioned on the way, too. Keywords: Planar curves, almost product projective geometry, parabolic geometry, generalized geodetics. MS classification: 53C10, 53C22, 53A20.

1. An almost product projective structure Let M be a smooth manifold of dimension 2m. An almost product structure on M is a smooth trace–free affinor J in Γ(T ⋆ M ⊗ T M ) satisfying J 2 = idT M . For better understanding, we describe an almost product structure at each tangent space in a fixed basis, i.e. with the help of real matrices:   I 0 J := m . 0 −Im The eigenvalues of J have to be ±1 and Tx M = TxL M ⊕ TxR M, where the subspaces are of the form       c 0 L R m m T M := J+ = , T M := J− = . |c∈R |c∈R 0 c Thus, we can equivalently define an almost product structure J on M as a reduction of the linear frame bundle P 1 M to the appropriate structure group, i.e. as a G–structure with the structure group L of all automorphisms

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preserving the affinor J:    A 0 L := |A, B ∈ GL(m, R) ∼ = GL(m, R) × GL(m, R) ⊂ GL(2m, R). 0 B These G–structures are of infinite type, however each choice of a linear connection ∇ compatible with the affinor J, i.e. ∇J = 0, determines a finite type geometry similar to products of projective structures, which we shall study below. The difference Υ between two projectively equivalent connections is a smooth one–form given by ˆ ξ η = ∇ξ η + Υ(ξ)η + Υ(η)ξ. ∇ Instead, we shall consider a class of connections parameterized also by all smooth one–forms Υ, but with the transformation rule ˆ ξL +ξR (η L + η R ) = ∇ξL +ξR (η L + η R ) + ΥL (ξ L )η L + ΥL (η L )ξ L ∇ + ΥR (ξ R )η R + ΥR (η R )ξ R ,

(1)

where the indices at the forms and fields indicate the components in the subbundles T LM and T R M , respectively. Clearly such a transformed connection will make J parallel again. Definition 1.1. Let M be a smooth manifold of dimension 2m. An almost product projective structure on M is a couple (J, [∇]), where J is an almost product structure, ∇ is a linear connection preserving J, and [∇] is the class of connections obtained from ∇ by the transformations given by all smooth one–forms Υ as in (1). The standard tool for the study of G–structures is the classical prolongation theory. The first step usually provides a class of distinguished connections with minimized (or preferred) torsions. In our case, the torsion T (ξ, η) = ∇ξ η − ∇η ξ − [ξ, η] of ∇ will always involve the obstructions against the integrability of T L M and T R M , i.e. the appropriate projections of the antisymmetric term given by the Lie bracket of vector fields. Our next goal is to identify a class of almost product projective geometries which fit into a wider class of the so called normal parabolic geometries. We shall see, that the appropriate requirement on the torsion of ∇ will be that the integrability obstructions are the only non–zero components.

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Example 1.1 (The homogeneous model). Let us consider the homogeneous space M = G/P given as the product of two projective spaces GL /PL × GR /PR , i.e. GL = GR = SL(n + 1, R) while P = PL × PR , where PR = PL is the usual parabolic subgroup corresponding to the block upper triangular matrices of the block sizes (1, n). Clearly, any product connection on M built of the linear connections ∇L and ∇R from the two projective classes on the product components provides a homogeneous example of an almost product projective structure. At the same time, the Maurer–Cartan form on G = GL × GR provides the homogeneous model of the |1|-graded parabolic geometries of type (G, P ). The Lie algebra of P × P is a parabolic subalgebra of the real form sl(n + 1, R) ⊕ sl(n + 1, R) of the complex algebra sl(n + 1, C) ⊕ sl(n + 1, C). In matrix form, we can illustrate the grading from our example as: 

gL 0  gL −1 g=  0 0

gL 1 gL 0 0 0

0 0 gR 0 gR −1

 0 0   gR 1 gR 0

2. Parabolic geometries and Weyl connections The classical prolongation procedure for G–structures starts with finding a minimal available torsion for connections belonging to the structure on the given manifold M . Our normalization will come from the general theory of parabolic geometries. As a rule, the |1|–graded parabolic geometries are completely given by certain classical G–structures on the underlying manifolds.1,2 In our case, however, both components of the semisimple Lie algebra belong to the series of exceptions and only the choice of an appropriate class of connections defines the Cartan geometry completely.1,2 The normalization of the Cartan geometries is based on cohomological interpretation of the curvature. More explicitly, the normal geometries enjoy co–closed curvature.1,2 In our case, the appropriate cohomology is easily computed by the K¨ unneth formula from the classical Kostant’s formulae and the computation10 provides all six irreducible components of the curvature, only two of which are of torsion type. Of course, the integrability obstructions of the bundles T LM and T R M are just those two. Therefore, a normal almost product projective structure (M, J, [∇]) has this minimal torsion.

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Now, the general theory provides for each normal almost product projective structure (M, J, [∇]) the construction of the unique principal bundle G → M with structure group P , equipped by the normal Cartan connection. Furthermore, there is the distinguished class of the so called Weyl connections corresponding to all choices of reductions of the parabolic structure group to its reductive subgroup G0 . All Weyl connections are parametrized just by smooth one–forms and they all share the torsion of the Cartan connection. The transformation formulae for the Weyl connections are generally given by the Lie bracket in the algebra in question. Of course, this is just the formula (1) we used for the definition of the almost product projective structures. Let us express (1) as:2 ˆ X Y = ∇X Y + [[X, Υ], Y ] ∇

(2)

where we use the frame forms X, Y : G → g−1 of vector fields, and similarly for Υ : G → g1 . Consequently, [Υ, X] is a frame form of an affinor valued in g0 and the bracket with Y expresses the action of such an affinor on the differentiated field. According to the general theory, this transformation rule works for all covariant derivatives ∇ with respect to Weyl connections.2 The general theory of the |1|–graded geometries also provides a formula for the unique normal Cartan connection ω in terms of any chosen Weyl connection and its curvature. Technically, this formula computes the difference between the two connections as the so called Schouten Rho tensor P = −⊓ ⊔−1 ∂ ∗ R, where R is the curvature of the Weyl connection, ∂ ∗ is the Kostant’s codifferential, and ⊓ ⊔−1 is the inverse of the Kostant’s Laplacian.2 This observation shows that our normal almost product projective geometries form a category equivalent to normal Cartan connections of type (G, P ) with homogeneous model discussed in Example 1.1. 3. J–planar curves Let us remind the notion of planarity with respect to affinors:8 Definition 3.1. Let (M, J) be an almost product structure. A smooth curve c : R → M is called J–planar with respect to a linear connection ∇ if ∇c˙ c˙ ∈ hc, ˙ J(c)i, ˙ where c˙ means the tangent velocity field along the curve c and the brackets indicate the linear hulls of the two vectors in the individual tangent spaces. Next, we observe that there is a nice link between J–planar curves and connections from the class defining an almost product projective structure:

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Theorem 3.1. Let (M, J, [∇]) be a smooth almost product projective structure on a manifold M . A curve c is J–planar with respect to at least one ¯ on M if and only if there is a parametrization of c Weyl connection ∇ which is a geodesic trajectory of some Weyl connection ∇. Moreover, this happens if and only if c is J–planar with respect to all Weyl connections. Proof. First, let us compute the bracket [[c, ˙ Υ], c] ˙ appearing in (2). We L R write c˙ = c + c and similarly for Υ: !    L    0 0 0 0 0 cL 0 0

0 0 0 0

0 0 0 cR

0 0 0 0

,

0 0 0 0

Υ 0 0 0

0 0 0 0

0 0 ΥR 0

,

0 cL 0 0

0 0 0 0

0 0 0 cR

0 0 0 0

=

cL ΥL (cL ) 0 0 0 0 0 0 0 R R R 0 0 Υ (c )c 0

.

We may write shortly [[c, ˙ Υ], c] ˙ = cL ΥL (cL ) + ΥR (cR )cR . Now, suppose c : R → M is a geodetics with respect to a connection ∇ and compute: ˆ c˙ c˙ = ∇c˙ c˙ + [[c, ∇ ˙ Υ], c] ˙ = [[c, ˙ Υ], c] ˙ = cL ΥL (cL ) + ΥR (cR )cR =

= (ΥL (cL ) + ΥR (cR ))c˙ + (ΥL (cL ) − ΥR (cR ))J(c) ˙ ∈ hc, ˙ J(c)i. ˙

ˆ i.e. with Thus, the geodetics c is J–planar with respect to connection ∇, respect to all Weyl connections. On the other hand, let us suppose that c : R → M is J–planar with ¯ i.e. ∇ ¯ c˙ c˙ = a(c) respect to ∇, ˙ c˙ + b(c)J( ˙ c) ˙ for some functions a(c) ˙ and b(c) ˙ along the curve. We have to find a one form Υ = ΥL + ΥR such that the formula for the transformed connection kills all the necessary terms along the curve c. Since there are many such forms Υ, there is a Weyl connection ∇ such that ∇c˙ c˙ = 0. 4. Generalized planar curves and mappings Definition 4.1 (Ref. 5). Let M be a smooth manifold of dimension n. Let A be a smooth l–rank (l < n) vector subbundle in T ∗ M ⊗ T M , such that the identity affinor E = idT M restricted to Tx M belongs to Ax ⊂ T ∗ M ⊗ T M at each point x ∈ M . We say that M is equipped by ℓ– dimensional A–structure. An almost product projective structure (M, J, [∇]) carries the A– structure with A = hE, Ji. Let us remind, that the A–planarity does not depend on the choice of the Weyl connection ∇ in the class in view of Theorem 3.1.

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For any tangent vector X ∈ Tx M , we shall write A(X) for the vector subspace A(X) = {F (X)| F ∈ Ax M } ⊂ Tx M, and we call A(X) the A–hull of the vector X. In order to work out relations between morphisms of our geometries and planarity, we shall follow our earlier work.5 We start by quoting a few definitions and results: Definition 4.2. Let (M, A) be a smooth manifold M equipped with an ℓ–dimensional A–structure. We say that A–structure has • generic rank ℓ if for each x ∈ M the subset of vectors (X, Y ) ∈ Tx M ⊕ Tx M , such that the A–hulls A(X) and A(Y ) generate a vector subspace A(X) ⊕ A(Y ) of dimension 2ℓ is open and dense. • weak generic rank ℓ if for each x ∈ M the subset of vectors V := {X ∈ Tx M | dim A(X) = ℓ} is open and dense in Tx M . Lemma 4.1. Every almost product structure (M, J) on a manifold M , dim M ≥ 2, has weak generic rank 2. Proof. Let as consider that X ∈ / V, this fact implies that ∃F ∈ A : F (X) = 0, i.e. the vector X has to be an eigenvector of J, i.e. X has to belong to m–dimensional subspace T L M or T R M of T M . Finally, the complement V is open and dense. Theorem 4.1 (Ref. 4). Let (M, A) be a smooth manifold of dimension n with ℓ–dimensional A–structure, such that ℓ ≧ 2 dim M . • If Ax is an algebra (i.e. for all f, g ∈ Ax , f g := f ◦ g ∈ Ax ) for all x ∈ M and A has weak generic rank ℓ, then the structure has generic rank ℓ. • If Ax ⊂ Tx⋆ M ⊗ Tx M is an algebra with inversion then A has generic rank ℓ. Each almost product structure on a smooth manifold M has generic rank 2 because of lemma and theorem above. Definition 4.3 (Refs. 5,8). Let M be a smooth manifold equipped with an A–structure and a linear connection ∇.

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• A smooth curve C is told to be A–planar if there is trajectory c : R → M such that ∇c˙ c˙ ∈ A(c). ˙ ¯ be another manifold with a linear connection ∇ ¯ and B– • Let M ¯ is called (A, B)–planar f structure. A diffeomorphism f : M → M each A–planar curve C on M is mapped onto the B–planar curve f⋆ C on M . In the special case, where A is the trivial structure given by hEi, we talk about B–planar maps. Theorem 4.2 (Ref. 5). Let M be a manifold with a linear connection ∇, ¯ let N be a manifold of the same dimension with a linear connection ∇ and with A-structure of generic rank ℓ, and suppose dimM ≧ 2ℓ. Then a diffeomorphism f : M → N is a A-planar if and only if ¯ − ∇) ∈ f ∗ (A(1) ) Sym(f ∗ ∇

(3)

where Sym denotes the symmetrization of the difference of the two connection. Theorem 4.3 (Ref. 5). Let M be a manifold with linear connection ∇ and an A-structure, N be a manifold of the same dimension with a linear ¯ and B-structure with generic rank ℓ. Then a diffeomorphism connection ∇ f : M → N is (A, B)-planar if and only if f is B-planar and A(X) ⊂ (f ⋆ (B))(X) for all X ∈ T M . Theorem 4.4 (Refs. 4,5). Let (M, A), (M ′ , A′ ) be smooth manifolds of dimension m equipped with A–structure and A′ –structure of the same generic rank ℓ ≤ 2m and assume that the A–structure satisfies the property ∀X ∈ Tx M, ∀F ∈ A, ∃cX | c˙X = X, ∇c˙X c˙X = β(X)F (X),

(4)

where β(X) 6= 0. If f : M → M ′ is an (A, A′ )–planar mapping, then f is a morphism of the A–structures, i.e f ∗ A′ = A. Finally, we can apply the above concepts and theorems to our situation: Theorem 4.5. Let f : M → M ′ be a diffeomorphism between two almost product projective manifolds of dimension at least four. Then f is a morphism of the almost product structures if and only if it preserves the class of unparameterized geodesics of all Weyl connections on M and M ′ . Proof. In view of the series of theorems above and the fact that f ∗ A = A implies f ∗ J = ±J (i.e. f ∗ J preserves the eigenspaces of J), we only have to prove that an almost product structure (M, J) has the property (4).

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Consider F = aE + bJ ∈ hE, F i and X ∈ T M . First we may solve the system of equations: a(X) + b(X) = 2ΥL (X L ) a(X) − b(X) = 2ΥR (X R ) ˆ where: with respect to Υ. Second, we may to define a new connection ∇, ˆ Y Z = ∇Y Z − [[Y, Υ], Z] ∇

ˆ such that c˙ = X. Finally, we recognize and we may find a geodetics c of ∇, that c is the requested curve from (4) because: ¯ X X − [[X, Υ], X] = [[X, Υ], X] ∇X X = ∇

= (ΥL (X L ) + ΥR (X R ))X = a(X)X + b(X)J(X) = F (X).

5. Almost complex projective structure Another possibility of a Lie algebra whose complexification is sl(n, C) ⊕ sl(n, C) is the complex algebra sl(n, C) viewed as a real algebra. The corresponding geometry analogous to the almost product projective geometry is the almost complex projective geometry. Definition 5.1. Let M be a smooth manifold of dimension 2m. An almost complex projective structure on M is defined by a smooth affinor I on M satisfying I 2 = − idT M and by a choice of a linear connection ∇ with ∇I = 0. In this case, the minimal torsion equals the Nijenhuis tensor obstructing the integrability of I and we may use the same technique as above to verify that this is the only component of the torsion of the normal almost complex projective geometry. There has been a lot of interest on such geometries recently. All the above approach works equally well and we shall come to further discussion on these questions in the context of existing literature elsewhere. Acknowledgments Research supported by grants GACR 201/05/H005 and 201/05/2117. References ˇ 1. A. Cap and J. Slov´ ak, Parabolic geometries, Mathematical Surveys and Monographs, AMS Publishing House, to appear (2008).

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ˇ and J. Slov´ 2. A. Cap ak, Weyl structures for parabolic geometries, Math. Scand. 93 (2003) 53–90. 3. J. Hrdina, H-planar curves, In: Diff. Geom. and its Appl. (Proc. Conf. Prague, Czech Republic, 2004) 265–272. 4. J. Hrdina , Generalized planar curves and quaternionic geometry, Ph.D. thesis, Faculty of science MU Brno, 2007. 5. J. Hrdina and J. Slov´ ak, Generalized planar curves and quaternionic geometry, Global analysis and geometry 29 (2006) 349–360. 6. S. Kobayashi, Transformation groups in differential geometry (Springer, 1972). 7. I. Kol´ aˇr, P.W. Michor and J. Slov´ ak, Natural operations in differential geometry (Springer, 1993); http://www.emis.de/monographs/KSM. 8. J. Mikeˇs and N.S. Sinyukov, On quasiplanar mappings of spaces of affine connection, Sov. Math. 27 (1983) (1) 63–70. 9. K. Nomizu and S. Kobayashi, Foundations of Differential Geometry, vol. 1-2 (John Wiley and Sons, 1963). ˇ 10. J. Silhan, Algorithmic computations of Lie algebras cohomologies, In: Proceedings of the 22nd Winter School “Geometry and Physics” (Srn´ı, 2002, Rend. Circ. Mat. Palermo (2) Suppl. No. 71, 2003) 191–197; www.math.muni.cz/∼silhan.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 263–277

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Flows of Spin(7)-structures Spiro Karigiannis Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK E-mail: [email protected] www.maths.ox.ac.uk/˜karigiannis We consider flows of Spin(7)-structures. We use local coordinates to describe the torsion tensor of a Spin(7)-structure and derive the evolution equations for a general flow of a Spin(7)-structure Φ on an 8-manifold M . Specifically, we compute the evolution of the metric and the torsion tensor. We also give an explicit description of the decomposition of the space of forms on a manifold with Spin(7)-structure, and derive an analogue of the second Bianchi identity in Spin(7)-geometry. This identity yields an explicit formula for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. Keywords: Spin(7)-structures, geometric flows. MS classification: 53C44, 53C10.

1. Introduction This paper discusses general flows of Spin(7)-structures in a manner similar to the author’s analogous results for flows of G2 -structures, which were studied in Ref. 7. Many of the calculations are similar in spirit, although more involved, so we often omit proofs. The reader is advised to familiarize themselves with Ref. 7 first. A general evolution of a Spin(7)-structure is described by a symmetric tensor h and a skew-symmetric tensor X satisfying some further algebraic condition, and it is only h which affects the evolution of the associated Riemannian metric. However, the evolution of the torsion tensor is determined by both h and X. In Section 2, we review Spin(7)-structures, the decomposition of the space of forms, and the torsion tensor of a Spin(7)-structure. In Section 3 we compute the evolution equations for the metric and the torsion tensor for a general flow of Spin(7)-structures. In Section 4, we apply our evolution equations to derive a Bianchi-type identity in Spin(7)-geometry. This leads

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to an explicit formula for the Ricci tensor of a general Spin(7)-structure in terms of the torsion. An Appendix collects various identities in Spin(7)geometry. The notation used in this paper is identical to that of Ref. 7. Throughout this paper, M is a (not necessarily compact) smooth manifold of dimension 8 which admits a Spin(7)-structure. 2. Manifolds with Spin(7)-structure In this section we review the concept of a Spin(7)-structure on a manifold M and the associated decompositions of the space of forms. More details about Spin(7)-structures can be found, for example, in Refs. 1,3–5. We also describe explicitly the torsion tensor associated to a Spin(7)-structure. Consider an 8-manifold M with a Spin(7) structure Φ. The existence of such a structure is a topological condition. The space of 4-forms Φ on M which determine a Spin(7)-structure is a subbundle A of the bundle Ω4 of 4-forms on M , called the bundle of admissible 4-forms. This is not a vector subbundle, and unlike the G2 case, it is not even an open subbundle. A Spin(7)-structure Φ determines a Riemannian metric gΦ and an orientation in a non-linear fashion. Details can be found in Ref. 5, although that paper uses a different orientation convention (see also Ref. 6.) We will not have need for the explicit formula for the metric. The metric gΦ and orientation (determined by the volume form) determine a Hodge star operator ∗, and the 4-form Φ is self-dual. That is, ∗Φ = Φ. The metric also determines the Levi-Civita connection ∇, and the manifold (M, Φ) is called a Spin(7) manifold if ∇Φ = 0. This is a nonlinear partial differential equation for Φ, since ∇ depends on g which depends non-linearly on Φ. Such manifolds (where Φ is parallel) have Riemannian holonomy Holg (M ) contained in the group Spin(7) ⊂ SO(8). A parallel Spin(7)-structure is also called torsion-free. 2.1. Decomposition of the space of forms The existence of a Spin(7)-structure Φ on M (with no condition on ∇Φ) determines a decomposition of the spaces of differential forms on M into irreducible Spin(7) representations. We will see explicitly that the spaces Ω2 , Ω3 , and Ω4 decompose as Ω2 = Ω27 ⊕ Ω221

Ω3 = Ω38 ⊕ Ω348

Ω4 = Ω41 ⊕ Ω47 ⊕ Ω427 ⊕ Ω435

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265

where Ωkl has (pointwise) dimension l and this decomposition is orthogonal with respect to the metric g. For k = 2 and k = 3, the explicit descriptions are as follows: Ω27 = {β ∈ Ω2 ; ∗(Φ ∧ β) = −3β}

Ω38 = {X Φ; X ∈ Γ(T M )}

Ω221 = {β ∈ Ω2 ; ∗(Φ ∧ β) = β}(1) Ω348 = {γ ∈ Ω3 ; γ ∧ Φ = 0}

(2)

For k > 4, we have Ωkl = ∗Ωl8−k . We need these decompositions in local coordinates. The following proposition is easy to verify. Proposition 2.1. Let βij be a 2-form, γijk a 3-form, and X k a vector field. Then βij ∈ Ω27 ⇔ βab g ap g bq Φpqij = −6 βij βij ∈ Ω221 ⇔ βab g ap g bq Φpqij = 2 βij γijk ∈ Ω38 ⇔ γijk = X l Φijkl

γijk ∈ Ω348 ⇔ γijk g ia g jb g kc Φabcd = 0

and the projection operators π7 and π21 on Ω2 are given by 1 1 βij − βab g ap g bq Φpqij 4 8 3 1 = βij + βab g ap g bq Φpqij 4 8

π7 (β)ij = π21 (β)ij

(3) (4)

Remark 2.1. One can show using Proposition 2.1 and Lemma A.1 that if βij ∈ Ω221 , βab g bl Φlpqr = βpi g ij Φjqra + βqi g ij Φjrpa + βri g ij Φjpqa which can then be used to show that Ω221 is a Lie algebra with respect to the commutator of matrices: [β, µ]ij = βil g lm µmj − µil g lm βmj

In fact, Ω221 ∼ = so(7), the Lie algebra of Spin(7).

The decomposition of the space Ω4 of 4-forms can be understood by considering the infinitesimal action of GL(8, R) on Φ. Let A = Ail ∈ gl(8, R). Hence eAt ∈ GL(8, R), and we have eAt · Φ =

1 Φijkl (eAt dxi ) ∧ (eAt dxj ) ∧ (eAt dxk ) ∧ (eAt dxl ) 24

Differentiating with respect to t and setting t = 0, we obtain:   d ∂ At m i Φ (e · Φ) = Ai dx ∧ dt t=0 ∂xm

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mj Now let Am Aij , and decompose Aij = Sij + Cij into symmetric and i = g skew-symmetric parts, where Sij = 12 (Aij + Aji ) and Cij = 12 (Aij − Aji ). We have a map

D : gl(8, R) → Ω4   d ∂ At jm i Φ D : A 7→ (e · Φ) = Aij g dx ∧ dt ∂xm t=0

Proposition 2.2. The kernel of D is isomorphic to the subspace Ω221 . It is also isomorphic to the Lie algebra so(7) of the Lie group Spin(7) which is the subgroup of GL(8, R) which preserves Φ. Proof. Since we are defining Spin(7) to be the group preserving Φ, the kernel of D is isomorphic to so(7) by definition. To show explicitly that this is isomorphic to Ω221 , suppose that Cij is in Ω221 . Then D(C) is
1 Cim Φmjkl + Cjm Φimkl + Ckm Φijml + Clm Φijkm dxi ∧ dxj ∧ dxk ∧ dxl 24

From Proposition 2.1, we have Cij = 12 Cab g ap g bq Φpqij . Using this together with the final equation of Lemma A.1, one can compute that Cim Φmjkl + Cjm Φimkl + Ckm Φijml + Clm Φijkm = − 3 Cim Φmjkl + Cjm Φimkl + Ckm Φijml + Clm Φijkm




and hence D(C) = 0. Thus Ω221 is in the kernel of D. We show below that D restricted to Ω27 or to S 2 (T ) is injective. This completes the proof. By counting dimensions, we must have Ω47 = D(Ω27 ) and also Ω41 ⊕Ω435 = D(S 2 ). We now proceed to establish these explicitly. The proofs of the next two propositions are very similar to analogous results in Ref. 7 and are left to the reader. Proposition 2.3. Suppose that Aij is a tensor. Consider the 4-form D(A) given by   ∂ jm i Φ D(A) = Aij g dx ∧ ∂xm or equivalently D(A)ijkl = Aim g mn Φnjkl + Ajm g mn Φinkl + Akm g mn Φijnl + Alm g mn Φijkn

(5)

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Then the Hodge star of D(A) is ¯ = A¯ij g jm dxi ∧ ∗D(A) = D(A) where A¯ij =

1 4



∂ ∂xm

Φ



Trg (A)gij − Aji . That is, as a matrix, A¯ =

1 4

Trg (A)I − AT .

Proposition 2.4. Suppose Aij and Bij are two tensors. Let D(A) and D(B) be their corresponding forms in Ω4 . We write Aij = 18 Trg (A)gij + (A0 )ij + (A7 )ij , where A0 is the symmetric traceless component of A, and A7 is the component in Ω27 . Similarly for B. (We can assume they have no Ω221 component since that is in the kernel of D.) Then we have 7 Trg (A) Trg (B) + 4 Trg (A0 B0 ) − 16 Trg (A7 B7 ) 2 where A0 B0 and A7 B7 mean matrix multiplication. gΦ (D(A), D(B)) =

Corollary 2.1. The map D : gl(8, R) → Ω4 is injective on S 2 ⊕ Ω27 . It is therefore an isomorphism onto its image, Ω41 ⊕ Ω47 ⊕ Ω435 . Proof. This follows immediately from Proposition 2.4, since if D(A) = 0 and A is pure trace, traceless symmetric, or in Ω27 , we see that A = 0. We still need to understand the space Ω427 . To do this we give another characterization of the space of 4-forms using the Spin(7)-structure, which may be well-known to experts but has apparently not appeared in print before. Definition 2.1. We define a Spin(7)-equivariant linear operator ΛΦ on Ω4 as follows. Let σ ∈ Ω4 . Use the notation (σ · Φ)ijkl to denote σijmn g mp g nq Φpqkl . Then ΛΦ (σ) ∈ Ω4 is given by (ΛΦ (σ))ijkl = (σ·Φ)ijkl +(σ·Φ)iklj +(σ·Φ)iljk +(σ·Φ)jkil +(σ·Φ)jlki +(σ·Φ)klij We now explain the motivation for introducing this operator ΛΦ . If σ ∈ Ω427 , then g(σ, D(A)) = 0 for all A ∈ gl(8, R) since D(A) ∈ Ω41 ⊕ Ω47 ⊕ Ω435 and the splitting is orthogonal. Writing this in coordinates using (5) gives σ ∈ Ω427

⇔

σabcd Φijkl g jb g kc g ld = 0 for all a, i = 1, . . . , 8

Taking the above expression and contracting it with Φ, and using Lemma A.1, after some laborious calculation one can show that σ ∈ Ω427

⇔

σijkl =

1 (ΛΦ (σ))ijkl 4

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which says that Ω427 is an eigenspace of ΛΦ with eigenvalue +4. Suppose now that σ = D(A) ∈ Ω41 ⊕ Ω47 ⊕ Ω435 . Then another brute force calculation using Definition 2.1 and Lemma A.1 shows ΛΦ (D(A)) = D(6AT − 6A − 3 Trg (A)I) and from the above relation it is a simple matter to verify the following characterization of Ω4 . Proposition 2.5. The spaces Ω41 , Ω47 , Ω427 , and Ω435 are all eigenspaces of ΛΦ with distinct eigenvalues. Specifically, Ω41 = {σ ∈ Ω4 ; ΛΦ (σ) = −24 σ}

Ω47 = {σ ∈ Ω4 ; ΛΦ (σ) = −12 σ}

Ω427 = {σ ∈ Ω4 ; ΛΦ (σ) = +4 σ}

Ω435 = {σ ∈ Ω4 ; ΛΦ (σ) = 0}

In addition, we have Ω41 = {D(λg); λ ∈ R}

Ω47 = {D(A7 ); A7 ∈ Ω27 }

Ω435 = {D(A0 ); A0 ∈ S02 }

where S02 is the space of symmetric traceless tensors. Also, Proposition 2.3 shows that Ω4+ = {σ ∈ Ω4 ; ∗σ = σ} = Ω41 ⊕ Ω47 ⊕ Ω427

Ω4− = {σ ∈ Ω4 ; ∗σ = −σ} = Ω435

is the decomposition into self-dual and anti-self dual 4-forms. Finally, we have the following result, which is also proved using Lemma A.1. Proposition 2.6. Let σ ∈ Ω4 . Then if we act on σ by ΛΦ twice, we have ΛΦ (ΛΦ (σ))ijkl = 24 σijkl − 16 ΛΦ (σ)ijkl + 2 Φijmn g mp g nq σpqrs g ra g sb Φabkl + 2 Φikmn g mp g nq σpqrs g ra g sb Φablj + 2 Φilmn g mp g nq σpqrs g ra g sb Φabjk

We will need Propositions 2.5 and 2.6 in Section 2.2 to study the torsion of a Spin(7)-structure. 2.2. The torsion tensor of a Spin(7)-structure In order to define the torsion tensor T of a Spin(7)-structure Φ, we need to first study the decomposition of ∇X Φ into its components in Ω4 . Lemma 2.1. For any vector field X, the 4-form ∇X Φ lies in the subspace Ω47 of Ω4 . Hence ∇Φ lies in the space Ω18 ⊗ Ω47 , a 56-dimensional space (pointwise).

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Proof. Let X = ∂x∂m , and consider the 4-form ∇m Φ. Then the second equation of Proposition A.1 tells us that ∇m Φ is orthogonal to S 2 ∼ = Ω41 ⊕ 4 Ω35 , exactly as in the G2 case as discussed in Ref. 7. However, we need to work harder to show that there is no Ω427 component. The essential reason that ∇X Φ ∈ Ω47 is because of the way that the 4-form Φ determines the metric g. From Ref. 5 (which uses a different orientation convention), we have (u v Φ) ∧ (w y Φ) ∧ Φ = −6 g(u ∧ v, w ∧ y)vol + 7 Φ(u, v, w, y)vol (6) Taking ∇X of this identity gives (u v ∇X Φ) ∧ (w y Φ) ∧ Φ + (u v Φ) ∧ (w y ∇X Φ) ∧ Φ

+ (u v Φ) ∧ (w y Φ) ∧ ∇X Φ = 7 ∇X Φ(u, v, w, y)vol

Now since ∗Φ = Φ and ∗(∇X Φ) = ∇X (∗Φ) = ∇X Φ, this can be written as g((u v ∇X Φ) ∧ (w y Φ), Φ) + g((u v Φ) ∧ (w y ∇X Φ), Φ)

+ g(u v Φ) ∧ (w y Φ), ∇X Φ) = 7 ∇X Φ(u, v, w, y)

We write this expression in coordinates, use Lemma A.1 to simplify the contractions of Φ with itself, and skew-symmetrize the result to obtain (∇X Φ)ijkl = + + − −


1 Φijmn g mp g nq (∇X Φ)pqrs g ra g sb Φabkl 12
1 Φikmn g mp g nq (∇X Φ)pqrs g ra g sb Φablj 12
1 Φilmn g mp g nq (∇X Φ)pqrs g ra g sb Φabjk 12 1 ((∇X Φ · Φ)ijkl + (∇X Φ · Φ)iklj + (∇X Φ · Φ)iljk ) 3 1 ((∇X Φ · Φ)jkil + (∇X Φ · Φ)jlki + (∇X Φ · Φ)klij ) 3

using the notation of Definition 2.1. In fact the above expression can also be directly verified using the identities of Lemma A.1 and Proposition A.1. Now using Definition 2.1 and Proposition 2.6 this becomes 1 1 (ΛΦ (ΛΦ (∇X Φ)) − 24 (∇X Φ) + 16 ΛΦ (∇X Φ)) (∇X Φ) = − ΛΦ (∇X Φ) + 3 24 Upon simplification, we have finally succeeded in showing that the basic relation (6) between the metric and the Spin(7)-structure Φ implies ΛΦ (ΛΦ (∇X Φ)) + 8 ΛΦ (∇X Φ) − 48 (∇X Φ) = 0

for any X

(7)

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Let ∇X Φ = σ1 +σ7 +σ27 +σ35 be its decomposition into components, where σk ∈ Ω4k . Using Proposition 2.5, equation (7) says (336) σ1 + (0) σ7 + (240) σ27 − (48) σ35 = 0 which, by linear independence, shows σ1 = σ27 = σ35 = 0. Therefore ∇X Φ ∈ Ω47 . Remark 2.2. The above result was first proved in Ref. 2 by Fern`andez, using different methods. Definition 2.2. Lemma 2.1 says that ∇Φ can be written as ∇m Φijkl = D(Tm )ijkl

= Tm;ip g pq Φqjkl + Tm;jp g pq Φiqkl + Tm;kp g pq Φijql + Tm;lp g pq Φijkq

where for each fixed m, Tm;ab is in Ω27 . This defines the torsion tensor T of the Spin(7)-structure, which is an element of Ω18 ⊗ Ω27 . The following lemma gives an explicit formula for Tm;ab in terms of ∇Φ. This will be used in Section 3.1 to derive the evolution equation for the torsion tensor. Lemma 2.2. The torsion tensor Tm;αβ is equal to Tm;αβ =

1 (∇m Φαjkl )Φβbcd g jb g kc g ld 96

(8)

Proof. This is a simple computation using Definition 2.2 and the identities in Lemma A.1. We close this section with some remarks about the decomposition of T into irreducible components. One can show that Ω18 ⊗ Ω27 ∼ = Ω3 = Ω38 ⊕ Ω348 . Therefore the torsion tensor T is actually a 3-form, with two irreducible components. In fact under this isomorphism T is essentially δΦ, which is the content of the following result. Theorem 2.1 (Fern´ andez, 1986). The Spin(7)-structure corresponding to Φ is torsion-free if and only if dΦ = 0. Since ∗Φ = Φ, this is equivalent to δΦ = 0. Suppose M is simply-connected, as it must be to admit a metric with holonomy exactly equal to Spin(7) in the compact case (see Ref. 4.) Then as in the G2 case, which is described in Ref. 7, the component of the torsion in Ω38 can always be conformally scaled away, once we have made the Ω348

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component vanish, without changing that other component. Therefore in principle we can restrict our attention to trying to make the Ω348 component of the torsion vanish, although it is not clear if this is really a simplification. We will not pursue this here. 3. General flows of Spin(7)-structures ∂ Φ In this section we derive the evolution equations for a general flow ∂t of a Spin(7)-structure Φ. Let Aij = hij + Xij , where hij ∈ S 2 and Xij ∈ Ω27 . Then from the discussion in Section 2.1, a general variation of Φ can be ∂ Φ = D(A). In coordinates, using (5), this is written as ∂t

∂ Φijkl =D(A)ijkl ∂t (9) =Aim g mn Φnjkl +Ajm g mn Φinkl +Akm g mn Φijnl +Alm g mn Φijkn The first thing we need to do is to derive the evolution equations for the metric g and objects related to the metric, specifically the volume form vol and the Christoffel symbols Γkij . We do this using a much simpler argument than that presented in Ref. 7 for the G2 case. This method works for that case as well. Proposition 3.1. The evolution of the metric gij under the flow (9) is given by ∂ gij = 2 hij ∂t

(10)

Proof. We want to know what the first order variation of the metric gΦ is, given a first order variation D(A) of the Spin(7)-structure Φ. It suffices ∂ Φ(t) = to consider any path Φ(t) of Spin(7)-structures that satisfies ∂t t=0 D(A). We take Φ(t) = eAt · Φ =

1 Φijkl (eAt dxi ) ∧ (eAt dxj ) ∧ (eAt dxk ) ∧ (eAt dxl ) 24

Then if g = gij dxi dxj is the metric of Φ = Φ(0), it is easy to see that the metric g(t) of Φ(t) is g(t) = gij (eAt dxi )(eAt dxj ) Now we differentiate ∂ g(t) = gij (Aik dxk )dxj + gij dxi (Ajl dxl ) = Akj dxk dxj + Ali dxi dxl ∂t t=0 = (Aij + Aji ) dxi dxj = 2 hij dxi dxj

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since h is the symmetric part of A. This completes the proof. Corollary 3.1. The evolution of the inverse g ij of the metric, the volume form vol, and the Christoffel symbols Γkij , under the flow (9), are given by ∂ ij g = −2 hij ∂t

∂ vol = Trg (h) vol ∂t

∂ k Γ = g kl (∇i hjl + ∇j hil − ∇l hij ) ∂t ij

Proof. This is a standard result. 3.1. Evolution of the torsion tensor In this section we derive the evolution equation for the torsion tensor T of Φ under the general flow (9). We begin with the evolution of ∇m Φijkl . Lemma 3.1. The evolution of ∇m Φijkl under the flow (9) is given by ∂ (∇m Φijkl ) = Aip g pq (∇m Φqjkl ) + Ajp g pq (∇m Φiqkl ) ∂t + Akp g pq (∇m Φijql ) + Alp g pq (∇m Φijkq ) + (∇p him )g pq Φqjkl + (∇p hjm )g pq Φiqkl + (∇p hkm )g pq Φijql + (∇p hlm )g pq Φijkq − (∇i hpm )g pq Φqjkl − (∇j hpm )g pq Φiqkl − (∇k hpm )g pq Φijql

− (∇l hpm )g pq Φijkq + (∇m Xip )g pq Φqjkl + (∇m Xjp )g pq Φiqkl

+ (∇m Xkp )g pq Φijql + (∇m Xlp )g pq Φijkq where Aij = hij + Xij ∈ S 2 ⊕ Ω27 . Proof. Recall that ∇m Φijkl =

∂ Φijkl − Γnmi Φnjkl − Γnmj Φinkl − Γnmk Φijnl − Γnml Φijkn ∂xm

so if we differentiate this equation with respect to t and simplify, we obtain     ∂ ∂ ∂ n (∇m Φijkl ) = ∇m Φijkl − Γmi Φnjkl ∂t ∂t ∂t       ∂ n ∂ n ∂ n Φinkl − Φijnl − Φijkn Γ Γ Γ − ∂t mj ∂t mk ∂t ml Now we substitute (9) and use Corollary 3.1. After we use the product rule on the first term, all the terms involving ∇m h cancel in pairs. The result now follows.

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Theorem 3.1. The evolution of the torsion tensor Tm;αβ under the flow (9) is given by ∂ Tm;αβ = Aαp g pq Tm;qβ−Aβp g pq Tm;qα +π7 (∇β hαm−∇α hβm+∇m Xαβ ) (11) ∂t where Aij = hij + Xij is the element of S 2 ⊕ Ω27 corresponding to the flow of Φ, and π7 denotes the projection onto Ω27 of the tensor skew-symmetric in α, β for fixed m. Proof. This is a long computation, but is similar in spirit to the analogous result for G2 -structures in Ref. 7. We will describe the main steps, and leave the details to the reader. Begin with Lemma 2.2, and differentiate to obtain:   ∂ 1 ∂ Tm;αβ = ∇m Φαjkl Φβbcd g jb g kc g ld ∂t 96 ∂t  (12)  1 6 ∂ jb kc ld jb kc ld + (∇m Φαjkl ) Φβbcd g g g − (∇m Φαjkl )Φβbcd h g g 96 ∂t 96 ∂ ij where we have used ∂t g = −2 hij from Corollary 3.1. Recall that for a tensor Bij we defined

D(B)ijkl = Bip g pq Φqjkl + Bjp g pq Φiqkl + Bkp g pq Φijql + Blp g pq Φijkq Let us define a similar shorthand notation Dm (B)ijkl to denote Bip g pq ∇m Φqjkl + Bjp g pq ∇m Φiqkl + Bkp g pq ∇m Φijql + Blp g pq ∇m Φijkq Then Lemma 3.1 says that ∂ ∇m Φijkl = Dm (A) + D(B) ∂t

(13)

Bαβ = ∇β hαm − ∇α hβm + ∇m Xαβ

(14)

where we define

We also have

∂ ∂t Φ

= D(A). Therefore (12) becomes

∂ 1 1 Tm;αβ = Dm (A)αjkl Φβbcd g jb g kc g ld + D(B)αjkl Φβbcd g jb g kc g ld ∂t 96 96 (15) 6 1 jb kc ld jb kc ld + (∇m Φαjkl )D(A)βbcd g g g − (∇m Φαjkl )Φβbcd h g g 96 96

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We will break up the computation into several manageable pieces. First, we need the following identity. If A = h + X ∈ S 2 ⊕ Ω27 , then: D(A)ijkl Φabcd g kc g ld = 4 hia gjb − 4 hib gja + 4 hjb gia − 4 hja gib

+ 2 Trg (h)(gia gjb − gib gja − Φijab ) + 16 Xia gjb − 16 Xib gja + 16 Xjb gia − 16 Xja gib − 2 Aip g pq Φqjab + 2 Ajp g pq Φqiab

(16)

+ 2 Apa g pq Φqbij − 2 Apb g pq Φqaij

which can be proved using Lemma A.1. Also, it is easy to check that if X and Y are both in Ω27 , then Xip g pq Yqj g ia g jb Φabkl = Xkp g pq Yql − Ykp g pq Xql

(17)

which essentially says that the Lie bracket of two elements of Ω27 is always in Ω221 . Now using the identities (16) and (17), and Lemma A.1 again, along with some patience, one can establish the following four expressions: D(A)αjkl Φβbcd g jb g kc g ld = 24 hαβ + 18 Trg (h) gαβ + 96 Xαβ Dm (A)αjkl Φβbcd g jb g kc g ld = 48 (hαp g pq Tm;qβ + hβp g pq Tm;qα ) jb kc ld

(∇m Φαjkl )D(A)βbcd g g g

+ 48 (Trg (h) Tm;αβ − g(X, Tm )gαβ )

= 48 (−hαp g pq Tm;qβ − hβp g pq Tm;qα )

+ 48 (Trg (h) Tm;αβ + g(X, Tm )gαβ )

+ 96 (Xαp g pq Tm;qβ − Xβp g pq Tm;qα )

(∇m Φαjkl )Φβbcd hjb g kc g ld = 16 (−hαp g pq Tm;qβ − hβp g pq Tm;qα ) + 16 (Trg (h) Tm;αβ )

Now we use the above four expressions to simplify equation (15). We need to substitute B as defined in (14) for A when we use the first of these expressions. After much cancellation and collecting like terms, we are left with exactly (11). We remark that, just as in the G2 case, the terms with ∇h and with ∇X play quite different roles in the evolution of the torsion tensor in equation (11). One hopes that it is possible to choose h and X in terms of T and possibly also ∇T so that the evolution equations have nice properties. In particular we would like the equation to be parabolic transverse to the action of the diffeomorphism group, for short-time existence. Ideally such a flow exists where the L2 -norm ||T || of the torsion decreases. These are questions for future research.

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4. Bianchi-type identity and curvature formulas In this section, we apply the evolution equation (11) to derive a Bianchitype identity for manifolds with Spin(7)-structure. This yields explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion tensor. As the calculations here are extremely similar to those in Ref. 7, we will be brief. Proposition 4.1. The diffeomorphism invariance of the metric g as a function of the 4-form Φ is equivalent to the vanishing of the Ω41 ⊕ Ω435 component of ∇Y Φ for any vector field Y . This is the fact which was proved earlier in Lemma 2.1. Proof. The proof is identical to the G2 case. In both cases it is due to the fact that the evolution of the metric g depends only on the symmetric part h of A = h + X. Notice that in the Spin(7) case, there is a stronger result that the Ω427 component of ∇Y Φ also vanishes, Lemma 2.1, which does not follow from here. Theorem 4.1. The diffeomorphism invariance of the torsion tensor T as a function of the 4-form Φ is equivalent to the following identity: 1 1 Rpqαβ − Rpqij g ia g jb Φijαβ 4 8 + 2 Tq;αm g mn Tp;nβ − 2 Tp;αm g mn Tq;nβ

∇q Tp;αβ − ∇p Tq;αβ =

(18)

Proof. The proof is very similar to the analogous result for G2 -structures described in Ref. 7, and is left to the reader. The identity (18) can also be established directly by using (8), Lemma A.1, and the Ricci identities. We now examine some consequences of Theorem 4.1. For i and j fixed, the Riemann curvature tensor Rijkl is skew-symmetric in k and l. Hence we can use the decomposition of Ω2 to write it as Rijkl = (π7 (Riem))ijkl + (π21 (Riem))ijkl where by equation (3), we have 1 1 Rijkl − Rijab g ap g bq Φpqkl 4 8 Therefore the identity (18) says that (π7 (Riem))ijkl =

(π7 (Riem))pqαβ = ∇q Tp;αβ − ∇p Tq;αβ

+ 2 (Tp;αm g mn Tq;nβ − Tq;αm g mn Tp;nβ )

(19)

(20)

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Corollary 4.1. If Φ is torsion-free, then the Riemann curvature tensor Rijkl ∈ S 2 (Ω2 ) actually takes values in S 2 (Ω221 ), where Ω221 ∼ = so(7), the Lie algebra of Spin(7). Proof. Setting T = 0 in (20) shows the for fixed i, j, we have Rijkl ∈ Ω221 as a skew-symmetric tensor in k, l. The result now follows from the symmetry Rijkl = Rklij . Remark 4.1. This result is well-known. When T = 0, the Riemannian holonomy of the metric gΦ is contained in the group Spin(7). By the Ambrose-Singer holonomy theorem, the Riemann curvature tensor of the metric is thus an element of S 2 (so(7)). Lemma 4.1. Let Qijkl = Rijab g ap g bq Φpqkl . Then we have Qijkl g il = 0. Proof. This is identical to the G2 case proved in Ref. 7. From Lemma 4.1 and equation (19), we see that the Ricci tensor Rjk can be expressed as Rjk = Rijkl g il = 4 (π7 (Riem))ijkl g il

(21)

Proposition 4.2. Given a Spin(7)-structure Φ with torsion tensor Tm;αβ , its associated metric g has Ricci curvature Rjk given by Rjk = 4 g il ∇i Tj;lk − 4 ∇j (g il Ti;lk ) + 8 Tj;mk Ti;nl g mn g il − 8 Tj;ml Ti;nk g mn g il Proof. This follows immediately from equations (20) and (21). This explicit expression for the Ricci tensor Rjk in terms of the torsion tensor appears to be new. Corollary 4.2. The metric of a torsion-free Spin(7)-structure is necessarily Ricci-flat. Remark 4.2. This is classical, originally proved by Bonan. Here we see a direct proof of this fact. Appendix A. 5. Identities in Spin(7)-geometry In this appendix we collect several identities involving the 4-form Φ of a Spin(7)-structure. They are derived by methods analogous to those for the G2 case as explained in Ref. 7, so we omit the proofs.

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In local coordinates x1 , x2 , . . . , x8 , the 4-form Φ is 1 Φijkl dxi ∧ dxj ∧ dxk ∧ dxl 24 is totally skew-symmetric. The metric is given by gij = Φ=

where Φijkl g( ∂x∂ i , ∂x∂ j ).

Lemma A.1. The following identities hold: Φijkl Φabcd g ia g jb g kc g ld = 336 Φijkl Φabcd g jb g kc g ld = 42gia Φijkl Φabcd g kc g ld = 6gia gjb − 6gib gja − 4Φijab

Φijkl Φabcd g ld = gia gjb gkc + gib gjc gka + gic gja gkb − gia gjc gkb − gib gja gkc − gic gjb gka

− gia Φjkbc − gja Φkibc − gka Φijbc

− gib Φjkca − gjb Φkica − gkb Φijca

− gic Φjkab − gjc Φkiab − gkc Φijab

Proposition A.1. The following identities hold: (∇m Φijkl )Φabcd g ia g jb g kc g ld = 0 (∇m Φijkl )Φabcd g jb g kc g ld = −Φijkl (∇m Φabcd )g jb g kc g ld

(∇m Φijkl )Φabcd g kc g ld = −Φijkl (∇m Φabcd )g kc g ld − 4 ∇m Φijab

References 1. R.L. Bryant and S.M. Salamon, On the Construction of Some Complete Metrics with Exceptional Holonomy, Duke Math. J. 58 (1989) 829–850. 2. M. Fern´ andez, A Classification of Riemannian Manifolds with Structure Group Spin(7), Ann. Mat. Pura Appl. (IV) 143 (1986) 101–122. 3. D.D. Joyce, Compact 8-Manifolds with Holonomy Spin(7), Invent. Math. 123 (1996) 507–552. 4. D.D. Joyce, Compact Manifolds with Special Holonomy (Oxford University Press, 2000). 5. S. Karigiannis, Deformations of G2 and Spin(7)-structures on Manifolds, Canad. J. Math. 57 (2005) 1012–1055. 6. S. Karigiannis, Some Notes on G2 and Spin(7) Geometry; arXiv:math.DG/ 0608618. 7. S. Karigiannis, Geometric Flows on Manifolds with G2 -structure, I., submitted for publication; arXiv:math/0702077v2.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 279–291

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Connections on principal prolongations of principal bundles Ivan Kol´ aˇr Institute of Mathematics and Statistics, Faculty of Science, Masaryk University, Jan´ aˇ ckovo n´ am 2a, 602 00 Brno, Czech Republic E-mail: [email protected] We study the principal connections of the r-th principal prolongation W r P of a principal bundle P (M, G) by using the related Lie algebroids. We deduce that both basic approaches to the concept of torsion are naturally equivalent. We prove that the torsion-free connections on W r P are in bijection with the reductions of W r+1 P to the group G1m × G. Special attention is paid to the flow prolongation of connections. Keywords: Principal prolongation of principal bundle, gauge theories, connection, torsion, Lie algebroid. MS classification: 53A05, 58A20, 58A32.

Consider a principal bundle P (M, G), dim M = m. Its r-th order prinr cipal prolongation W r P is the bundle of all r-jets j(0,e) ϕ of local principal bundle isomorphisms ϕ : Rm × G → P ,

0 ∈ Rm , e = the unit of G.

r G := W0r (Rm × This is a principal bundle over M with structure group Wm r 11 G), whose action on W P is given by the jet composition. If G = {e} is the one-element group, then M × {e} is identified with M and W r (M × {e}) = P r M is the r-th order frame bundle of M . The r-th principal prolongation W r P is a fundamental structure for both the general theory of geometric object fields,11 and the gauge theories of mathematical physics.3 Our main subject are the principal connections on W r P . So we omit the adjective “principal” as a rule. At a few places (in particular at the beginning of Section 4), where we mention arbitrary connections on an arbitrary fibered manifold Y , we call them explicitly “general connections”. It has been clarified recently that the connections Λ on P r M are in

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bijection with the r-th order linear connections λ : T M → J r T M on T M . In Section 2 we point out that in the case of W r P the role of T M is replaced by the Lie algebroid LP = T P/G of P . Using the flow prolongation of right -invariant vector fields, we identify J r (LP → M ) with the Lie algebroid LW r P and prove that the connections ∆ on W r P are in bijection with the linear splittings δ : T M → J r LP . The torsion of ∆ is defined as the covariant exterior differential of the canonical one-form of W r P , while the torsion of δ is introduced by means of the bracket of LP . In Section 3 we prove that the torsions of ∆ and δ are naturally equivalent. A connection Γ on P and a connection Λ on P r M determine a connection W r (Γ, Λ) on W r P by means of the flow prolongation of vector fields. To demonstrate the applicability of the algebroid approach, we express explicitly the torsion of W 1 (Γ, Λ) in terms of the torsion of Λ and the curvature of Γ in Section 4. For arbitrary r, we then deduce that W r (Γ, Λ) is torsion-free, if and only if Λ is torsion-free and Γ is curvature-free. In Section 5 we prove that, analogously to the case of P r M , the torsion-free connections on W r P are identified with certain reductions of W r+1 P . From a general point of view, J r is a fiber product preserving bundle functor. In Section 6 we study an arbitrary functor F of this type and we deduce the algebroid formula for the flow prolongation of the abovementioned pair of connections Γ and Λ. All manifolds and maps are assumed to be infinitely differentiable. Unless otherwise specified, we use the terminology and notations from the book.11 1. The torsion of connections on W r P r G = We write Grm for the r-th jet group in dimension m and Tm r r r We have W P = P M ×M J P and

J0r (Rm , G).

r r W0r (Rm × G) = Wm G = Grm ⋊ Tm G

is the group semidirect product with the group composition
(g1 , C1 )(g2 , C2 ) = g1 ◦ g2 , (C1 ◦ g2 ) • C2 ,

(1)

(2)

r where • denotes the induced group composition in Tm G.11 The first product projection W r P → P r M is a principal bundle morphism with the associr ated group homomorphism Wm G → Grm determined by (1). Write PBm (G) for the category of principal G-bundles with m-dimensional bases and prin¯ , G) cipal G-morphisms with local diffeomorphisms as base maps. Let P¯ (M

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be another object of PBm (G). For every PBm (G)-morphism f : P → P¯ ¯ , we define with base map f : M → M ¯ ×M¯ J r P¯ . W r f = P r f ×f J r f : P r M ×M J r P → P r M

(3)

r−1 P u e = T(0,er−1 ) W r−1 ϕ : Rm × wr−1 m G → Tu1 W

(4)

r Then W r is a functor from the category PBm (G) into PBm (Wm G).11 r r m On P M , we have the canonical one-form ϕr : T P M → R × gr−1 m . On r W P , we introduce analogously a canonical one-form Θr : T W r P → Rm × r−1 r−1 wr−1 (Rm × G), er−1 = the unit of Wm G. Consider m G = T(0,er−1 ) W r r r r−1 r u = j(0,e) ϕ ∈ W P and write u1 = πr−1 (u) ∈ W P , where πr−1 is the jet projection. The tangent map

is a linear isomorphism depending on u only. For every Z ∈ Tu W r P , we define
r (Z) . Θr (Z) = u e−1 T πr−1

Clearly, the following diagram commutes Θr

T W rP

/ / Rm × wr−1 G m

 T P rM

ϕr

(5)

 / / Rm × gr−1 m

r

For a connection Λ on P M , Yuen defined its torsion to be the covariant exterior differential DΛ ϕr . Analogously, in Ref. 13 we introduced Definition 1.1. The torsion of a connection ∆ on W r P is the covariant exterior differential D∆ Θr . 2. Another approach to connections on W r P A linear splitting T M → J r T M is said to be a linear r-th order connection on T M . Since P r is an r-th order bundle functor from the category Mfm of m-dimensional manifolds and local diffeomorphisms into PBm (Grm ), the flow prolongation P r X of every vector field X on M is a right-invariant vector field on P r M .11 The restriction P r X | Pxr M depends on jxr X only. This defines an identification r IM : J r T M → LP r M ,

(6)

where LP r M is the Lie algebroid of P r M .15 Clearly, for every linear splitting λ : T M → J r T M , the rule
r λ(Z) , Z ∈ TM (7) Λ(Z) = IM

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defines a connection Λ on P r M . This establishes a bijection λ → Λ between linear r-th order connections on T M and connections on P r M .8 Such a bijection can be generalized to the case of W r P . The role of T M is replaced by the Lie algebroid LP = T P/G of P . Every section σ : M → LP is identified with a right-invariant vector field σ ¯: P → TP r r such that σ = σ ¯ /G. Since W is a functor from PBm (G) into PBm (Wm G), r r the flow prolongation W (¯ σ ) is a right-invariant vector field on W P . The rule
 r G (8) σ ) | Wxr P Wm IPr (jxr σ) = W r (¯

defines a bijection IPr : J r (LP → M ) → LW r P . This is the principal bundle form of an identification that was established in the Lie algebroid form in Ref. 14. In the same way as in (7), we obtain Proposition 2.1. (8) identifies connections ∆ on W r P with linear splittings δ : T M → J r LP .

(9)

We say that δ is the algebroid form of ∆. 3. The torsions on W r P and J r LP The (r − 1)-jet at x ∈ M of the bracket [X1 , X2 ] of two vector fields X1 and X2 on M depends on the r-jets jxr X1 and jxr X2 . This defines a bilinear morphism [ , ]r : J r T M ×M J r T M → J r−1 T M . For a linear r-th order connection λ : T M → J r T M , one defines its torsion τ λ : T M ×M T M → J r−1 T M by   (Z1 , Z2 ) ∈ T M ×M T M . (10) (τ λ)(Z1 , Z2 ) = λ(Z1 ), λ(Z2 ) r ,

Our result from Ref. 8 reads: If Λ is the principal bundle form of λ, then the torsion DΛ ϕr is naturally identified with the torsion τ λ. We are going to deduce the same result for the case of W r P . Since the bracket [[ , ]] of LP is a first order operator, it determines a bilinear morphism [[ , ]]r : J r LP ×M J r LP → J r−1 LP analogously to [ , ]r .

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Definition 3.1. The torsion of a connection in the algebroid form δ : T M → J r LP is the morphism τ δ : T M ×M T M → J r−1 LP defined by (τ δ)(Z1 , Z2 ) = [[δ(Z1 ), δ(Z2 )]]r ,

(Z1 , Z2 ) ∈ T M ×M T M .

Clearly, τ δ can be viewed as a section of J r−1 LP ⊗ Λ2 T ∗ M . r−1 Write Um = J0r−1 L(Rm × G). Since J r−1 L is an r-th order gauge r natural bundle, every u = j(0,e) ϕ ∈ W r P can be interpreted as a map r−1 J r−1 L(u) := (J r−1 L)(ϕ) |0 : Um → Jxr−1 LP .

(11)

Our identification IPr−1 : J r−1 LP → LW r−1 P is a natural equivalence of functors J r−1 L and LW r−1 . Write Vmr−1 = L0 W r−1 (Rm ×G) = Rm ×wr−1 m . Analogously to (11), we construct LW r−1 (u) := LW r−1 (ϕ) |0 : Vmr−1 → Lx W r−1 P .

(12)

IRr−1 m ×G

r−1 The restriction of over 0 ∈ Rm yields a bijection ε : Um → Vmr−1 . By naturality, the following diagram commutes r−1 Um

J r−1 L(u)

x

r−1 (IP )x

ε

 Vmr−1

/ / J r−1 LP

LW r−1 (u)

(13)

 / / L W r−1 P x

r (u), then LW r−1 (u) If we identify Lx W r−1 P with Tu1 W r−1 P , u1 = πr−1 r−1 2 ∗ is identified with u e from (4). Since J LP ⊗ Λ T M is a fiber bundle r−1 associated to W r P with standard fiber Um ⊗ Λ2 Rm∗ and τ δ is a section, we can consider its frame form11 r−1 {τ δ} : W r P → Um ⊗ Λ2 Rm∗ .

(14)

On the other hand, D∆ Θr is a horizontal 2-form, so that it can be interpreted as a map {D∆ Θr } : W r P → Vmr−1 ⊗ Λ2 Rm∗ . Further we can construct r−1 ε ⊗ id Λ2 Rm∗ : Um ⊗ Λ2 Rm∗ → Vmr−1 ⊗ Λ2 Rm∗ .

(15)

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Proposition 3.1. Under the identifications (14) and (15),
1 {D∆ Θr } = ε ⊗ id Λ2 Rm∗ ◦ {τ δ} . 2

(16)

Proof. If η2 is a vector field on W r P , then Θr (η2 ) is an (Rm × wr−1 m G)r r valued function on W P . Thus, if η1 is another vector field on W P , we can consider the derivative η1 Θr (η2 ) : W r P → Rm × wr−1 m G of Θr (η2 ) in the direction of η1 . First we deduce that for every sections σ1 , σ2 : M → LP we have

¯ rσ ¯1 , W r σ ¯2 ] . (17) σ2 ) = Θr [W W r (¯ σ1 ) Θr W r (¯

Indeed, the rule σ ¯ 7→ Θr (W r σ ¯ ) is a gauge-natural operator. Hence it commutes with the Lie differentiation.11 But the Lie derivative of σ ¯2 with respect to σ ¯1 is the bracket [¯ σ1 , σ ¯2 ]. Consider now u ∈ Wxr P and Z1 , Z2 ∈ Tx M , δ(Zi ) = jxr σi , i = 1, 2. Write u0 : Rm → Tx M for the underlying map of u. If we interpret {D∆ Θr } as a map W r P × Rm × Rm → Vmr−1 , we have

−1 r {D∆ Θr } u, u−1 ¯1 (u), W r σ ¯2 (u) . 0 (Z1 ), u0 (Z2 ) = dΘr W σ Applying the classical formula for dΘr and (17), we obtain

2dΘr (W r σ ¯1 , W r σ ¯2 ) = (W r σ ¯1 )Θr (W r σ ¯2 ) − (W r σ ¯2 )Θr (W r σ ¯1 )

r r − Θr [W σ ¯1 , W σ ¯2 ] = Θr [W σ ¯1 , W σ ¯2 ] .

By the commutativity of (13), the last expression corresponds to [[δ(Z1 ), δ(Z2 )]]r . 4. The flow prolongation of principal connections

First we recall a general result on the flow prolongation of connections on an arbitrary fibered manifold Y → M . Let Σ be a general connection on Y considered in the lifting form Σ : Y ×M T M → T Y . Write F Mm,n for the category of fibered manifolds with m-dimensional bases and n-dimensional fibers and their local isomorphisms. Let F be a bundle functor on F Mm,n of the base order r.11 For every vector field X on M we first construct the Σ-lift ΣX : Y → T Y . The flow prolongation F (ΣX) depends on the r-jets of X. This defines a map F Σ : F Y ×M J r T M → T F Y .

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Let Λ be a principal connection on P r M and λ : T M → J r T M be the corresponding splitting. Then F (Σ, Λ) := F Σ ◦ (id F Y ×M λ) : F Y ×M T M → T F Y is a general connection on F Y , that is called the flow prolongation of Σ with respect to Λ.11 If we consider a principal connection Γ on a principal bundle P (M, G) in the role of Σ, then W r (Γ, Λ) is a principal connection on W r P . The algebroid form γ : T M → LP of Γ is a fibered morphism over id M . Its r-th jet prolongation is a map J r γ : J r T M → J r LP . The following assertion will be proved in Section 6 in a more general setting. Proposition 4.1. The algebroid form of W r (Γ, Λ) is J r γ ◦ λ : T M → J r LP .

(18)

The first application of (18) is the following assertion, that we deduced in a quite different way in Ref. 13. However, we find remarkable that the algebroid approach reduces the proof to a simple direct evaluation. Proposition 4.2. W 1 (Γ, Λ) is torsion-free, iff λ is torsion-free and Γ is curvature-free. Proof. By locality, it suffices to discuss the case P = Rm × G, so that LP = T Rm × g. The sections of LP are pairs (X, ̺) of a vector field X on Rm and a map ̺ : Rm → g with the bracket
[[(X1 , ̺1 ), (X2 , ̺2 )]] = [X1 , X2 ], X1 ̺2 − X2 ̺1 + [̺1 , ̺2 ]g , (19)

where [X1 , X2 ] is the bracket of vector fields and [ , ]g is the bracket in g. Consider the canonical coordinates xi on Rm , the induced coordinates i y on T Rm and some linear coordinates z p on g. Let yji , zip be the induced coordinates on J 1 LP . The map [[ , ]]1 has the coordinate expression
p p i i y1j y2j (20) , y1i z2i − y2i z1i + cpqr z1q z2r , − y2j y1j

where cpqr are the structure constants of G. Let δ : T M → J 1 LP be a connection of the form z p = ∆pi (x)y i ,

yji = ∆ikj (x)y k ,

zip = ∆pji (x)y j .

(21)

Then the coordinate expression of τ δ is
1 k ∆ , ∆p + cpqr ∆qi ∆rj dxi ∧ dxj . 2 ij ij

(22)

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On the other hand, let γ and λ be of the form z p = Γpi (x)y i ,

yji = Λikj (x)y k .

(23)

Then the coordinate expression of J 1 γ is zip =

∂Γpj j y + Γpj yij . ∂xi

(24)

Hence J 1 γ ◦ λ is of the form yji = Λikj y k and zip =

 ∂Γp j ∂xi

 + Γpk Λkij y j .

(25)

By (22), the torsion of J 1 γ ◦ λ is p  1  k ∂Γj p k p q r i j Λij , + Γ Λ + c Γ Γ (26) qr i j dx ∧ dx . k ij 2 ∂xi The first term in (26) is the torsion of Λ. If it vanishes, the second term coincides with the algebroid expression p  1  ∂Γj p q r + c Γ Γ dxi ∧ dxj (27) qr j i 2 ∂xi of the curvature of Γ. Now it is easy to prove the general result. Proposition 4.3. W r (Γ, Λ) is torsion-free, iff Λ is torsion-free and Γ is curvature-free. Proof. If W r (Γ, Λ) is torsion-free, then W 1 (Γ, Λ) is also torsion-free, so that Γ is integrable. Hence there is a local trivialization of P such that Γpi = 0 identically. Then all non-trivial coefficients of J r γ are also zero and the coordinate expression of [[ , ]]r reduces to the case of Λ. So the coordinate expressions of τ (J r γ ◦ λ) and τ λ coincide and our assertion follows from Proposition 4.2 in Ref. 8. 5. Torsion-free connections as reductions Every a ∈ G1m is a matrix that defines a linear map l(a) : Rm → Rm . This yields an injection G1m ֒→ Grm ,

a 7→ j0r l(a) .

In Ref. 5, we deduced that the torsion-free connections on P r M are in bijection with the reductions of P r+1 M to the subgroup G1m ⊂ Gr+1 m . The

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r-jets j0r gb, g ∈ G, of the constant maps b g : Rm → G, x 7→ g, define an r 1 r injection G → Tm G. Then Gm × G is a subgroup of Wm G. In Ref. 13, we 1 proved that the torsion-free connections on W P are in bijection with the reductions of W 2 P to G1m × G. We are going to deduce such a result for an arbitrary order r. For every fibered manifold Y → M , the r-th contact morphism is a map ψr : T J r Y → V J r−1 Y ≈ J r−1 (V Y → M ). In the case of a principal bundle P (M, G), we have V P = P × g. Then J r−1 V P = J r−1 P ×M J r−1 (M, g). r−1 Every frame of Pxr−1 M identities Jxr−1 (M, g) with the Lie algebra tm G of r−1 r r−1 ¯ Tm G. If we modify ψr in this way, we obtain a map ψr : T W P → tm G. r−1 r−1 On the other hand, wr−1 m G = gm × tm G, so that we have the product m r−1 r−1 projection π : R × wm G → tm G. One verifies directly that ψ¯r = π ◦ Θr .

(28)

We have J 1 (W r P ) = J 1 P r M ×M J 1 J r P . In Ref. 5, we described an injection P r+1 M ֒→ J 1 P r M . On the other hand, we have the classical inclusion J r+1 P ֒→ J 1 J r P . This defines an injection ir : W r+1 P → J 1 (W r P ) .

(29)

Let Γ : P → J 1 P be a connection on P = W 0 P . The rule
̺(Γ)(u, v) = u, Γ(v) , (u, v) ∈ P 1 M ×M P ,

defines a reduction ̺(Γ)(P 1 M ×M P ) ⊂ W 1 P to G1m ×G.13 For a connection ∆ : W r P → J 1 W r P , we proceed by the following induction. Let ∆ be such that the underlying connection ∆1 on W r−1 P is torsion-free. Hence ∆1 defines a reduction ̺(∆1 ) : P 1 M ×M P → W r P by the induction hypothesis. Proposition 5.1. ∆ is torsion-free, iff the values of ∆ ◦ ̺(∆1 ) lie in ir (W r+1 P ). Then we define 1 r+1 ̺(∆) = i−1 P. r ◦ ∆ ◦ ̺(∆1 ) : P M ×M P → W

Proof. Every ̺(∆)(u, v), (u, v) ∈ P 1 M ×M P , represents a linear mdimensional subspace S in T W r P , which is identified with a pair of mdimensional linear subspaces S1 ⊂ T P r M and S2 ⊂ T J r P .
By (5) and (28), dΘr | S can be considered as the pair dϕr | S1 , dψ¯r | S2 . By Ref. 5,

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dϕr | S1 = 0 if and only if S1 corresponds to an element of P r+1 M . Analogously to Ref. 6, dψ¯r | S2 = 0 if and only if S2 corresponds to an element of J r+1 P . Proposition 5.2. Proposition 5.1 establishes a bijection between the torsion-free connections on W r P and the reductions of W r+1 P to the subr+1 group G1m × G ⊂ Wm G. Proof. On one hand, one verifies directly that ̺(∆) is a reduction to the subgroup G1m × G. On the other hand, let Q : P 1 M ×M P → W r+1 P be a reduction to the subgroup G1m × G. Write Q1 = πrr+1 ◦ Q : P 1 M ×M P → W r P . For every Q1 (u, v) ∈ W r P , (u, v) ∈ P 1 M ×M P , Q(u, v) represents an m-dimensional horizontal subspace of T W r P . Since our maps are (G1m ×G)equivariant, these subspaces are canonically extended into a connection on W r P . By the proof of Proposition 5.1, this connection is torsion-free. 6. The case of W F P The r-th jet prolongation of fibered manifolds is a fiber product preserving bundle functor J r on the category F Mm of fibered manifolds with m-dimensional bases and fibered morphisms with local diffeomorphisms as base maps. In Ref. 12 we characterized all these functors in terms of Weil algebras, see also Ref. 9. Every such functor F has finite order. If the base order of F is r, then we have an identification F = (A, H, t), where A is a Weil algebra, H : Grm → Aut A is a group homomorphism and t : Drm → A is an equivariant algebra homomorphism, provided Aut A means the group of all algebra automorphisms of A and Drm is the Weil algebra J0r (Rm , R). In the case of J r , we have A = Drm , so that Aut Drm = Grm , H = id Grm and t = id Drm . Analogously to the case of J r , every F = (A, H, t) determines a bundle functor W F on PBm (G)2 W F P = P r M ×M F P ,

W F f = P r f ×f F f .

Similarly to the case of W r , W F (Rm × G) is a Lie group A WH G = Grm ⋊ T A G

(30)

with the group composition
(g1 , C1 )(g2 , C2 ) = g1 ◦ g2 , HG (g2−1 )(C1 )•C2 ,

(31)

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where • denotes the induced group composition in T A G. A Further, W F P is a principal bundle over M with structure group WH G. F A The values of W are in the category PBm (WH G). For every fibered manifold Y → M , t induces a map tY : J r Y → F Y , tY (jxr s) = (F s)(x) ,

x∈M,

(32)

where s is a local section of Y , which is interpreted as a fibered morphism from the trivial fibered manifold M → M into Y , so that F s : M → F Y . In particular, we have tT M : J r T M → F T M . On the other hand, the anchor map q : LP → T M induces F LP → F T M . In Ref. 10 we deduced, by using the theory of semi-direct products, that the Lie algebroid of W F P is LW F P = J r T M ×F T M F LP .

(33)

For every section σ : M → LP , the vector field σ ¯ on P induces the flow prolongation W F (¯ σ ), which is a right-invariant vector field on W F P . To found its algebroid form, we use our general idea of the flow natural transformation of F . According to Ref. 7, see also Ref. 9, for every Y → M there exists a map ψYF : J r T M ×F T M F (T Y → M ) → T (F Y ) with the property that for every projectable vector field η on Y over ξ on M , the flow prolongation F η satisfies F η = ψYF ◦ (j r ξ ×F ξ F η) , provided η is considered as a fibered morphism of T Y → M into T M → M . In particular, this yields Proposition 6.1. For every section σ : M → LP over X = q ◦ σ : M → T M , the flow prolongation W F (¯ σ ) corresponds to the section j r X ×F X F σ : M → LW F P ,

jr X : M → J r T M ,

F σ : M → F (LP → M ) .

Let Γ be a connection on P and Λ a connection on P r M . Hence the flow prolongation W F (Γ, Λ) is a connection on W F P . The algebroid form γ : T M → LP of Γ is a base preserving morphism, so that we can construct F γ : F T M → F LP . Further, we have λ : T M → J r T M . By the very definition of W F (Γ, Λ), we deduce Proposition 6.2. The algebroid form of W F (Γ, Λ) is (λ, F γ ◦ tT M ◦ λ) : T M → LW F P .

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In the case F = J r , we have tT M = id J r T M , so that Proposition 4.1 is a special case of Proposition 6.2. Remark 6.1. There is a natural question whether one can define the torsion of connections on an arbitrary principal bundle W F P . The definition of the canonical form on W r P is essentially based on the fact that W r−1 is the underlying functor of W r of the order r − 1. However, Doupovec clarified that the general concept of underlying functors of arbitrary F is rather sophisticated.1 So it seems to be reasonable to restrict ourselves to the subfunctors E ⊂ J 1 F with the property that the jet projection EY → F Y is surjective. Then Proposition 2 of Ref. 4 implies that there is a canonical form on W F P with good properties and the procedures of the present paper can be applied. Acknowledgments The author was supported by the Ministry of Education of the Czech Republic under the project MSM 0021622409 and the grant GACR No. 201/05/0523 References 1. M. Doupovec, On the underlying lower order bundle functors, Czechoslovak Math. J. 55 (2005) 901–916. 2. M. Doupovec and I. Kol´ aˇr, Iteration of Fiber Product Preserving Bundle Functors, Monatsh. Math. 134 (2001) 39–50. 3. L. Fatibene and M. Francaviglia, Natural and Gauge Natural Formalism for Classical Fields Theories (Kluwer, 2003). 4. I. Kol´ aˇr, Generalized G-structures and G-structures of higher order, Boll. Un. Math. Ital., Suppl. fasc. 3 (1975) 249–256. 5. I. Kol´ aˇr, Torsion-free connections on higher order frame bundles, In: New Developments in Differential Geometry (Proceedings, Kluwer, 1996) 233– 241. 6. I. Kol´ aˇr, On holonomicity criteria in second order geometry, Beitr¨ age zur Algebra und Geometrie 39 (1998) 283–290. 7. I. Kol´ aˇr, On the geometry of fiber product preserving bundle functors, In: Diff. Geom. and Its Applications (Proceedings, Silesian University of Opava, 2002) 85–92. 8. I. Kol´ aˇr, On the torsion of linear higher order connections, Central European Journal of Mathematics 3 (2003) 360–366. 9. I. Kol´ aˇr, Weil Bundles as Generalized Jet Spaces, In: Handbook of Global Analysis (Elsevier, 2007) 625–665. 10. I. Kol´ aˇr and A. Cabras, On the functorial prolongations of principal bundles, Comment. Math. Univ. Carolin. 47 (2006) 719–731.

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11. I. Kol´ aˇr, P.W. Michor and J. Slov´ ak, Natural Operations in Differential Geometry (Springer Verlag, 1993). 12. I. Kol´ aˇr and W. Mikulski, On the fiber product preserving bundle functors, Differential Geometry and Its Applications 11 (1999) 105–115. 13. I. Kol´ aˇr and G. Virsik, Connections in first principal prolongations, Suppl. ai Rendiconti del Circolo Matematico di Palermo, Serie II 43 (1996) 163–171. 14. A. Kumpera and D. Spencer, Lie equations I (Princeton University Press, 1972). 15. K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids (Cambridge University Press, Cambridge 2005).

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The (κ, µ, ν)-contact metric manifolds and their classification in the 3-dimensional case Th. Koufogiorgos Department of Mathematics, University of Ioannina, Ioannina 45100, Greece E-mail: [email protected] M. Markellos and V.J. Papantoniou Department of Mathematics, University of Patras, Rion, GR-26500, Greece E-mail: [email protected], [email protected] The (κ, µ, ν)-contact metric manifolds have been recently introduced by the authors (Ref. 11, 2007). In this aspect, we locally classify three dimensional (κ, µ, ν)-contact metric manifolds M (η, ξ, φ, g) which satisfy the condition ∇ξ τ = 2aτ φ, where a is smooth function on M with ξ(a) = 0 and τ = Lξ g. Moreover, we consider the same condition with a a constant function. Keywords: Contact metric manifolds, (κ, µ, ν)-contact metric manifolds. MS classification: 53C15, 53C25, 53C30.

1. Introduction Let M (η, ξ, φ, g) be a contact metric manifold. Chern and Hamilton7 introduced the torsion τ = Lξ g, where Lξ is the Lie derivative of g with respect to the characteristic vector field ξ, in their study of compact metric three-manifolds. The classification of conformally flat contact metric manifolds is another problem which has been investigated by many researchers. At one hand, in many cases conformally flat contact metric manifolds must have constant sectional curvature.6,8,15 So, it is natural to ask if there are any conformally flat contact metric structures which are not of constant curvature. To this direction, D. E. Blair (Ref. 1, page 108) constructed examples of conformally flat contact metric three-manifolds which don’t have constant sectional curvature. G. Calvaruso4 proved that Blair’s examples

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satisfy the condition ∇ξ τ = 2aτ φ,

(1)

where a is a smooth function with ξ(a) = 0. Here, the composition (τ φ)(X, Y ) has to be interpreted as τ (φX, Y ). In Ref. 11 the authors proved the existence of a new class of contact metric manifolds: the so called (κ, µ, ν)-contact metric manifolds. Such a manifold M is defined through the condition R(X, Y )ξ = κ(η(Y )X − η(X)Y ) + µ(η(Y )hX − η(X)hY ) + ν(η(Y )φhX − η(X)φhY )

(2)

for every X, Y ∈ X (M ) and κ, µ, ν are smooth functions on M . Furthermore, it is shown in Ref. 11 that if dim M > 3, then κ, µ are constants and ν is the zero function. In the same paper, the authors gave a nice geometric interpretation of three dimensional (κ, µ, ν)-contact metric manifolds in terms of harmonic vector fields. More precisely, they proved that the characteristic vector field ξ of a 3-dimensional contact metric manifold M is a harmonic vector field if and only if M is a (κ, µ, ν)-contact metric manifold on an everywhere open and dense subset of M , generalizing a similar result of Perrone.14 The paper is organized in the following way. Section 2 contains the presentation of some basic notions about contact manifolds and (κ, µ)-contact metric manifolds. In Section 3 we set the question of the existence of (κ, µ, ν)-contact metric manifolds where κ, µ, ν are smooth functions independent of the choice of the vector fields X, Y . We state that the answer to the above question is negative for dimension greater than three and positive for dimension equal to three. More precisely, for dimensions greater than three (κ, µ, ν)-contact metric manifolds are reduced to (κ, µ)-contact metric manifolds manifolds or to Sasakian manifolds. In Section 4 we give a partial classification of 3-dimensional (κ, µ, ν)contact metric manifolds. Especially, we assume that these manifolds satisfy the condition (1) where a is a smooth function or a constant. In the case which a is a constant, we give a geometrical meaning of (1). 2. Preliminaries We start by collecting some fundamental notions about contact Riemannian geometry. We refer to Ref. 1 for further details. All manifolds in the present paper are assumed to be connected and smooth.

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A (2n + 1)-dimensional manifold is called contact manifold if it admits a global 1-form η(contact form) such that η ∧ (dη)n 6= 0 everywhere on M . Given η, there exists a unique vector field ξ, called the characteristic vector field or the Reeb vector field, such that η(ξ) = 1 and dη(ξ, X) = 0 for every vector field X on M . It is well known that there also exists a Riemannian metric g and a tensor field φ of type (1, 1) such that φ(ξ) = 0,

φ2 = −Id + η ⊗ ξ,

η◦φ=0

g(φX, φY ) = g(X, Y ) − η(X)η(Y )

(3) (4)

for all vector fields X, Y on M . Moreover, the quadruple (η, ξ, φ, g) can be chosen so that dη(X, Y ) = g(X, φY ). The manifold M together with the structure tensors (η, ξ, φ, g) is called a contact metric manifold and is denoted by M (η, ξ, φ, g). We denote by ∇ the Levi - Civita connection, and by R the corresponding Riemannian curvature tensor field given by R(X, Y ) = [∇X , ∇Y ] − ∇[X,Y ] for all vector fields X, Y on M . Given a contact Riemannian manifold M , we define on M the operators h and τ by hX = 12 (Lξ φ)X, τ (X, Y ) = (Lξ g)(X, Y ) where Lξ is the Lie differentiation in the direction of ξ. The tensor field h of type (1,1) is self-adjoint and satisfies hξ = 0, trh = trhφ = 0, hφ = −φh.

(5)

If X is an eigenvector of h corresponding to the eigenvalue λ, then φX is also an eigenvector of h corresponding to the eigenvalue −λ, since h anticommutes with φ. We also have the following formulas for a contact metric manifold: τ = 2g(φ·, h·)

(6)

∇ξ τ = 2g(φ·, ∇ξ h·)

(7)

Formulas (6) and (7) occur also in Ref. 12. A contact metric manifold for ξ being a Killing vector field is called a K−contact manifold. It is well known that a contact metric manifold is K−contact if and only if h = 0. A contact Riemannian manifold M (η, ξ, φ, g) is called a Sasakian manifold if and only if R(X, Y )ξ = η(Y )X − η(X)Y

(8)

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for every X, Y ∈ X (M ). Every Sasakian manifold is K−contact, but the converse is true only in the three dimensional case. The (κ, µ)−nullity distribution of a contact metric manifold M (η, ξ, φ, g) for the pair (κ, µ) ∈ R2 is the distribution N (κ, µ) : p → Np (κ, µ) = {Z ∈ Tp M |R(X, Y )Z = κ(g(Y, Z)X − g(X, Z)Y ) + µ(g(Y, Z)hX − g(X, Z)hY )}.

for every X, Y ∈ Tp M . So, if the characteristic vector field ξ belongs to the (κ, µ)-nullity distribution, then R(X, Y )ξ = κ(η(Y )X − η(X)Y ) + µ(η(Y )hX − η(X)hY )

(9)

and the manifold M is called (κ, µ)-contact metric manifold.2 If κ, µ are non - constant smooth functions on M, the manifold M is called generalized (κ, µ)-contact metric manifold. It is shown in Ref. 9 that if dim M > 3, then κ and µ are necessarily constants. On the contrary, if dim M = 3 then such generalized manifolds exist. We mention that the class of (κ, µ)-contact metric manifolds extends the class of Sasakian manifolds (κ = 1 and relation (8)). 3. (κ, µ, ν)-contact metric manifolds The following question comes up naturally. Do there exist contact metric manifolds satisfying (2) with κ, µ, ν non-constant smooth functions, independent of the choice of vector fields X, Y ? The answer is positive for the 3-dimensional case and in the following we give an example.11 Example 3.1. Consider the 3-dimensional manifold M = {(x, y, z) ∈ R3 ; x > 0, y > 0, z > 0}, where (x, y, z) are the cartesian coordinates in R3 . We define the following vector fields on M : e1 =

∂ , ∂x

e2 =

∂ , ∂y

∂ 4 ∂ G ∂ e3 = − eG Gy +β +e2 z ∂x ∂y ∂z

where G = G(y, z) < 0 for every (y, z), is a solution of the partial differential equation 2Gyy + G2y = −ze−G, and the function β = β(x, y, z) is a solution of the following system of partial differential equations 4 βx = 2 e G zx

G

and

Gz e 2 1 G 4eG Gy βy = e2 − − . 2z 2 xz

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The vector fields e1 , e2 , e3 are linearly independent at each point of M . We define a Riemannian metric g on M such that g(ei , ej ) = δij , i, j = 1, 2, 3. Let η be the 1-form defined by η(W ) = g(W, e1 ) for every W ∈ X (M ). Then η is a contact form since η ∧ dη 6= 0 everywhere on M . Let φ be the tensor field of type (1,1), defined by φe1 = 0, φe2 = e3 ,φe3 = −e2 . Using the linearity of φ, dη and g, we easily obtain that η(e1 ) = 1, dη(Z, W ) = g(φZ, W ) and g(φZ, φW ) = g(Z, W ) − η(Z)η(W ) for every vector fields Z, W on M . Hence M (η, e1 , φ, g) is a contact metric manifold. Computing the Lie brackets of e1 , e2 , e3 and using the Koszul’s formula, we easily prove that M (η, e1 , φ, g) is a (κ, µ, ν)-contact metric manifold with 2G 2eG 2 κ = 1 − 4e z 2 x4 , µ = 2(1 + zx2 ), ν = − x . Let M (η, ξ, φ, g) be a (2n+1)-dimensional (κ, µ, ν)-contact metric manifold and B = {p ∈ M |κ(p) = 1}. Then, the set N = M \ B is an open subset of M and hence it inherits the contact structure of M i.e. N (η, ξ, φ, g) is a contact metric manifold which satisfies (2) with κ < 1 everywhere. On the contrary, for dimensions greater than three the following Theorem is valid. Theorem 3.1. Every (κ, µ, ν)-contact metric manifold M (η, ξ, φ, g) of dimension greater than 3 is either a Sasakian manifold or a (κ, µ)-contact metric manifold, i.e. the functions κ, µ are constants and ν is the zero function. The proof of this Theorem is given in Ref. 11 and depends largely on the following lemmas. Lemma 3.1. On every (2n+1)-dimensional (κ, µ, ν)-contact metric manifold M (η, ξ, φ, g) the following relations are satisfied h2 = (κ − 1)φ2 , κ ≤ 1

(10)

∇ξ h = µhφ + νh

(11)

ξ(κ) = 2ν(κ − 1)

(12)

Lemma 3.2. On every (2n+1)-dimensional (κ, µ, ν)-contact metric manifold M (η, ξ, φ, g) the following differential equation is satisfied: ξ(κ)[η(Y )X − η(X)Y ] + ξ(µ)[η(Y )hX − η(X)hY ]

+ ξ(ν)[η(Y )φhX − η(X)φhY ] − X(κ)φ2 Y + Y (κ)φ2 X

+ X(µ)hY − Y (µ)hX + X(ν)φhY − Y (ν)φhX = 0

(13)

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Lemma 3.3 (Ref. 9, Lemma 3.4). For every p ∈ N there exists an open neighborhood W of p and orthonormal local vector fields Xi , φXi , ξ, i = 1, 2, . . . n, defined on W , such that hXi = λXi , hφXi = −λφXi , hξ = 0, i = 1, 2, . . . n √ where λ = 1 − κ.

(14)

4. Classification of 3-dimensional (κ, µ, ν)-contact metric manifolds with ∇ξ τ = 2aτ φ The existence of (κ, µ, ν)-contact metric manifolds in dimension 3 as described in Section 3, raises the question of their classification. To this direction, we give some partial answers assuming additionally that they satisfy the condition (1). We mention that for the cases a = 0 or a = 1, we have a complete classification of 3-dimensional (κ, µ, ν)-contact metric manifolds. For the sake of completeness, we give the corresponding classifications.11 Theorem 4.1. Let M (η, ξ, φ, g) be a complete 3-dimensional (κ, µ, ν)contact metric manifold with ∇ξ τ = 0. Then M is a (κ, 0)-contact metric manifold with κ = constant. In particular, M is either a Sasakian manifold (if κ = 1) or locally isometric to one of the following Lie groups, equipped with a left invariant metric: SU (2) if 0 < κ < 1, SL(2, R) if κ < 0 and E(2) if κ = 0. Theorem 4.2. Let M (η, ξ, φ, g) be a complete 3-dimensional (κ, µ, ν)contact metric manifold for which ∇ξ τ = 2τ φ. Then M is a (κ, 2)-contact metric manifold where κ is a constant. Moreover, M is either a Sasakian manifold (if κ = 1) or locally isometric to SL(2, R) if κ 6= 1, equipped with a left invariant metric. In the second case, any two (κ, 2)-contact metric manifolds are D-invariant under a specific D-homothetic transformation. A contact metric manifold M (η, ξ, φ, g) is said to be homogeneous if there exists a connected Lie group of isometries acting transitively on M and leaving η invariant. It is said to be locally homogeneous if the pseudogroup of local isometries acts transitively on M and leaves η invariant. For more details about homogeneous Riemannian manifolds see Ref. 16. In Ref. 13 , D. Perrone studied three-dimensional manifolds admitting a homogeneous contact metric structure. He showed that these manifolds are locally isometric to a Lie group G with a left-invariant contact metric structure (η, ξ, φ, g). Moreover, all such manifolds satisfy the condition (1) with a

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constant (Ball-homogeneous is defined as a Riemannian manifold which has the property that the volume of sufficiently small geodesic spheres or balls depends only on their radius and not on their center). Moreover, G. Calvaruso and D. Perrone5 proved that a three-dimensional contact metric manifold is locally homogeneous if and only if it is ball-homogeneous and satisfies the condition (1) with a constant. In the sequence, we completely classify 3-dimensional (κ, µ, ν)-contact metric manifolds which additionally satisfy the condition (1) with a constant. Theorem 4.3. Let M (η, ξ, φ, g) be a complete 3-dimensional (κ, µ, ν)contact metric manifold. If M satisfies (1) with a = constant, then M is a (κ, µ)-contact metric manifold. Particularly, M is either a Sasakian manifold (κ = 1) or locally isometric to one of the following Lie groups with a left invariant metric: SU (2) (or SO(3)), SL(2, R) (or O(1, 2)), E(2) (the group of rigid motions of the Euclidean 2-space), E(1, 1) (the group of rigid motions of the Minkowski 2-space). Proof. Using (1), (3), (4), (6) and (7), we get (∇ξ h)φX = −2ahX for every X ∈ X (M ). Combining (11) and the last relation, we easily obtain −2ahX = (∇ξ h)φX

= µhφ2 X + νhφX

or (2a − µ)hX + νhφX = 0

(15)

for every X ∈ X (M ). Let p ∈ N , then Lemma 3.3 implies the existence of a local orthonormal basis {e, φe, ξ} on W which satisfies the relations (14). Applying (15) on e we get 0 = (2a − µ)he + νhφe = (2a − µ)λe − λνφe which gives µ = 2a and ν = 0 on W . On the other hand, substituting X = e and Y = φe in (13), we get that e(κ) − λe(µ) − λφe(ν) = 0

−φe(κ) − λφe(µ) + λe(ν) = 0.

(16)

Since the function µ is constant on W and the function ν is the zero function on W , the relations (12) and (16) lead to the constancy of the continuous function κ on every connected component of W . If M \ N 6= ∅, then due

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to the continuity of the function κ we have that κ = 1 everywhere on M i.e. M is a Sasakian manifold. If M \ N = ∅ i.e M = N , then we have µ = 2a, ν = 0 and κ = c 6= 1 everywhere on M , where c is a constant i.e. M is a (κ, µ)-contact metric manifold. From the classification of 3-dimensional (κ, µ)-contact metric manifolds in Ref. 2 it follows that in the non-Sasakian case M is locally isometric to the above Lie groups equipped with a left invariant metric. Remark 4.1. • i) In Ref. 3 , E. Boeckx proved that (κ, µ)-contact metric manifolds are locally homogeneous. As a consequence, 3-dimensional (κ, µ, ν)-contact metric manifolds which satisfy (1) with a constant, are locally homogeneous. • ii) In Example 3.1, the eigenvalues of h are non-constant smooth funcG 2eG tions. In fact, they are 2e zx2 , − zx2 and 0. This shows that the (κ, µ, ν)contact metric manifolds are not necessarily locally homogeneous contact Riemannian manifolds13 or, more generally, curvature homogeneous. Generalizing Theorem 4.3, we obtain Theorem 4.4. Let M (η, ξ, φ, g) be a 3-dimensional (κ, µ, ν)-contact metric manifold for which ∇ξ τ = 2aτ φ where a is a smooth function on M . Then, M is either a Sasakian manifold (κ = 1) or a generalized (κ, µ)-contact metric manifold with κ < 1. Proof. Using (1), (6) and (7) we obtain that (∇ξ h)φX = −2ahX for every X ∈ X (M ). Combining (11) and the last relation, we easily deduce (2a − µ)hX + νhφX = 0

(17)

for every X ∈ X (M ). Let p ∈ N , then Lemma 3.3 implies the existence of a local orthonormal basis {e, φe, ξ} on W which satisfies the relations (14). Applying (17) on e we get µ = 2a and ν = 0 on W . If M \ N 6= ∅, then due to the continuity of the function κ we have that κ = 1 everywhere on M i.e. M is a Sasakian manifold. If M \ N = ∅ i.e M = N , then we have µ = 2a, ν = 0 and κ < 1 everywhere on M . But, in this case κ, µ are smooth functions on M i.e. M is a generalized (κ, µ)-contact metric manifold with κ < 1.

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Now, we give an example of a generalized (κ, µ)-contact metric manifold which satisfy the condition (1) with a smooth function. Example 4.1. We consider the 3-dimensional manifold M = {(x1 , x2 , x3 ) ∈ R3 |x3 6= 0}, where (x1 , x2 , x3 ) are the standard coordinates in R3 . The vector fields ∂ 2x1 ∂ 1 ∂ 1 ∂ ∂ , e2 = −2x2 x3 + 3 − 2 , e3 = e1 = ∂x1 ∂x1 x3 ∂x2 x3 ∂x3 x3 ∂x2 are linearly independent at each point of M . Let g be the Riemannian metric defined by g(ei , ej ) = δij , i, j = 1, 2, 3 and η the dual 1-form to the vector field e1 . We define the tensor field φ of type (1, 1) by φe1 = 0, φe2 = e3 , φe3 = −e2 . Following Ref. 9 , we have that M (η, e1 , φ, g) is a x4 −1 generalized (κ, µ)-contact metric manifold with κ = 3x4 and µ = 2(1− x12 ). 3 3 By a straightforward calculation, we deduce that M satisfies the condition: 1 ∇ξ τ = 2(1 − 2 )τ φ x3 Remark 4.2. We assume that the condition (1) is satisfied on a 3dimensional (κ, µ, ν)-contact metric manifold M (η, ξ, φ, g). Then, applying Theorem 4.4 and Theorem 1.1 of Ref. 14 , we deduce that the characteristic vector field ξ : (M, g) 7→ (T1 M, gS ) determines an harmonic map, where (T1 M, gS ) is the unit tangent sphere bundle equipped with the Sasaki metric gS . Proposition 4.1. Let M (η, ξ, φ, g) be a non-Sasakian 3-dimensional (κ, µ, ν)-contact metric manifold. Suppose that ∇ξ τ = 2aτ φ where a is a smooth function which is constant along the geodesic foliation generated by ξ. Then, M is a generalized (κ, µ)-contact metric manifold with ξ(µ) = 0. We denote by {ξ, X, φX} a local orthonormal frame of eigenvectors of h √ such that hX = λX, λ = 1 − κ > 0. Furthermore, for every p ∈ M there exists a chart (U, (x, y, z)), z < 1, such that the function κ depends only of √ √ the variable z and µ = 2(1 − 1 − κ) or µ = 2(1 + 1 − κ). In the first case, the following are valid, ξ=

∂ ∂x ,

φX =

∂ ∂y ,

∂ ∂ X = α ∂x + b ∂y +

∂ ∂z .

In the second case, the following are valid, ξ=

∂ ∂x ,

X=

∂ ∂y ,

∂ ∂ φX = α1 ∂x + b1 ∂y +

∂ ∂z ,

where α(x, y, z) = −2y + f (z), α1 (x, y, z) = 2y + f (z), b(x, y, z) = p λ′ (z) y + h(z), λ(z) = 1 − κ(z) > 0 and f, h are b1 (x, y, z) = 2λ(z)x − 2λ(z) smooth functions of z.

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Proof. By Theorem 4.4, we get that M is a generalized (κ, µ)-contact metric manifold with µ = 2a and κ < 1. Since ξ(a) = 0, we easily obtain that ξ(µ) = 0. The remaining part of the proposition follows immediately from Ref. 10.

Acknowledgments The second author was partially supported by the Greek State Scholarships Foundation (I.K.Y.) and by the C. Carathe´oodory grant no.C.161 2007 - 10, University of Patras.

References 1. D.E. Blair, Riemannian geometry of contact and symplectic manifolds (Vol. 203, Progress in Math., Birkhauser, Boston, 2002). 2. D.E. Blair, Th. Koufogiorgos and B. Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. Math. 91 (1995) 189–214. 3. E. Boeckx, A class of locally φ-symmetric contact metric spaces, Arch. Math. 72 (1999) 466–472. 4. G. Calvaruso, Einstein-like and conformally flat contact metric threemanifolds, Balkan J. Geom. Appl. 5 (2000) 17–36. 5. G. Calvaruso and D. Perrone, Torsion and homogeneity on contact metric three-manifolds, Ann. Mat. Pura Appl. 178 (2000) 271–285. 6. G. Calvaruso, D. Perrone and L. Vanhecke, Homogeneity on threedimensional contact metric manifolds, Israel J. Math. 114 (1999) 301–321. 7. S.S. Chern and R.S. Hamilton On Riemannian metrics adapted to threedimensional contact manifolds, In: Proc. Meet. Max-Planck-Inst. Math. (Vol.1111, Lect. Notes in Math., Springer-Verlag, Berlin/New York, 1985) 279-308. 8. F. Gouli-Andreou and Ph.J. Xenos, Two classes of conformally flat contact metric 3-manifolds, J.Geom. 64 (1999) 80–88. 9. Th. Koufogiorgos and C. Tsichlias, On the existence of a new class of contact metric manifolds, Canad. Math. Bull. 43 (2000) 440–447. 10. Th. Koufogiorgos and C. Tsichlias, Generalized (κ, µ)-manifolds with ξµ = 0, to be appeared in Tokyo J. Math. 11. T. Koufogiorgos, M. Markellos and V.J. Papantoniou, The harmonicity of the Reeb vector field on contact metric 3-manifolds, Pacific J. Math. 234 (2008) 325–344. 12. D. Perrone, Torsion and critical metrics on contact three-manifolds, Kodai Math. J. 13 (1990) 88–100. 13. D. Perrone, Homogeneous contact Riemannian three-manifolds, Illinois J. Math. 42 (1998) 243–256. 14. D. Perrone, Harmonic characteristic vector fields on contact metric manifolds, Bull. Austral. Math. 67 (2003) 305–315.

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15. S. Tanno, Locally symmetric K-contact Riemannian manifolds, Proc. Japan Acad. 43 (1967) 581–583. 16. F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds (Vol. 83, Lect. Note Series, London Math. Soc., Cambridge University Press, 1989).

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Invariants and submanifolds in almost complex geometry Boris Kruglikov Institute of Mathematics and Statistics, University of Tromsø, Tromsø 90-37, Norway E-mail: [email protected] In this paper we describe the algebra of differential invariants for GL(n, C)structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of higherdimensional pseudoholomorphic submanifolds. Keywords: Almost complex structure, equivalence, differential invariant, Nijenhuis tensor, pseudoholomorphic submanifold. MS classification: 32Q60, 53C15, 53A55.

1. Introduction Let (M, J) be an almost complex manifold, J 2 = −1. In this paper we discuss only local aspects and so suppose n = 12 dim M > 1. In this case the Nijenhuis tensor NJ (ξ, η) = [Jξ, Jη] − J[ξ, Jη] − J[Jξ, η] − [ξ, η] (which is a skew-symmetric (2,1)-tensor) is generically non-zero. Vanishing of NJ is equivalent to local integrability of J.11 It is known that all differential invariants can be expressed via the jet of the Nijenhuis tensor.5 In the first part of the paper we describe how this can be used to solve the equivalence problem of GL(n, C)-structures. This problem is void for n = 1 and was solved in Ref. 7 for n = 2, but we present here a uniform approach via differential invariants suitable for all n. The differential invariants of an almost complex structure also occur in the problem of establishing pseudoholomorphic (PH) submanifolds. They played the crucial role in the proof of non-existence of PH-submanifolds for generic almost complex (M, J).6 In dimension 2n ≥ 8 existence of a single higher-dimensional submanifold already imposes restrictions on the Nijenhuis tensor, so for their existence NJ should be degenerate (though can stay far from being zero). On the other hand existence of 4-dimensional PH-submanifolds for 6-

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dimensional (M, J) does not impose identities-restrictions on the tensor NJ (there are open subsets of admissible tensors in the space of all Nijenhuis tensors). At the second half of the paper we discuss when (M, J) can have PHfoliations and when their number is bounded in non-integrable case.

2. First order invariants of almost complex structures For n = 1 the almost complex structures J are complex and possess no local invariants. So this case will not be considered in what follows. For n = 2 there are non-integrable structures, but there are no first order differential invariants. To explain this let us note that all such invariants must be derived from the Nijenhuis tensor NJ . In dimension 4 the linear Nijenhuis tensor (a purely tensorial object at a point, i.e. an element of Λ2 T ∗ ⊗C¯ T with T = Tx M , see Ref. 5) is special and the GL(2, C) ⊂ GL(4) orbit space consists of two points: zero and non-zero tensor NJ . For non-zero tensor we can talk of the image Π2 = Im(NJ ) which, if we vary the point x, is a two-distribution in M 4 , called Nijenhuis tensor characteristic distribution.7 Provided J is generic, Π2 is generic as well. In particular, there’s the derived rank 3 distribution Π3 = ∂Π2 . This leads to the fact that there’s no second order invariants as well. However, we can associate the second order e-structure {ξi }4i=1 , ξi ∈ DM , to J as follows: ξ1 ∈ C ∞ (Π2 ), ξ3 ∈ C ∞ (Π3 ), NJ (ξ1 , ξ3 ) = ξ1 , ξ2 = Jξ1 , [ξ1 , ξ2 ] = ξ3 , ξ4 = Jξ3 .

This defines the pair ξ1 , ξ2 canonically up to ±1 and the pair ξ3 , ξ4 absolutely canonically.7 When n = 3 there are moduli in the space of linear Nijenhuis tensors.6 This is clearly seen from Theorem 7 loc.sit. Indeed the statement means that the space of differential invariants of order 1 is two-dimensional and the constants of the normal forms provide the invariants. In Ref. 2 Bryant arrived independently to the result about dimension 2, observing that codimension of generic orbits w.r.t. the GL(3, C) ⊂ GL(6) action on the space of Nijenhuis tensors Λ2 T ∗ ⊗C¯ T is 2 (where T = Tx M is a model tangent space of dimension 6), because the stabilizer is two-dimensional. Moreover in Ref. 2 some invariants of almost complex 6-dimensional manifolds were introduced. All of them are expressed via a (1,1)-form ω, which is given in coordinates via the components of the Nijenhuis tensor as

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follows: ωij =

l ¯k l ¯k Nik Njl − Njk Nil √ . −1

Here complex coordinates adapted at the point to J are used (in fact, in Ref. 2 non-holonomic, i.e. frames). Note that this is a real-valued form and it can be written in invariant terms as follows (now we assume all tensors real): ω(ξ, η) = Tr[NJ (ξ, JNJ (η, ·)) − NJ (η, JNJ (ξ, ·))]. In particular, ω(ξ, Jξ) = 2 Tr[NJ (ξ, NJ (ξ, ·))] is not identically zero. The form ω is J-compatible: ω(Jξ, Jη) = ω(ξ, η) and we can associate the quadric q(ξ, η) = ω(ξ, Jη), which equals q(ξ, η) = Tr[NJ (ξ, NJ (η, ·)) + NJ (η, NJ (ξ, ·))]. Indeed, these both 2-tensors are skew-symmetric and symmetric parts l ¯k of the form Tij = Nik Njl and the pair (S, ω) forms a Hermitian metric provided q (or equivalently ω) is non-degenerate (it can be indefinite). Let us investigate ω via the normal forms of NJ .6 Proposition 2.1. The (1,1)-form ω is degenerate precisely in the following cases in terms of differential invariants from classification theorem 7 of Ref. 6: NDG1 : λ = ±1, ϕ = 0, π. NDG2 : ϕ = 0, π. NDG3 : ψ = ± π4 ± π2 = ±ϕ ± π. NDG4 , DG1 (4): Never. DG1 (1-3,5), DG2 (1-2): Always. This is a straightforward tedious calculation. It shows generic nondegeneracy of the 2-form ω. In Refs. 2,12 global implications of nondegeneracy are discussed. The above local aspects show which open strata of Ref. 6 contain non-degenerate forms ω, and this yields topological restrictions on global realization of such ω. The canonical G2 -invariant almost complex structure J on S 6 corresponds to NDG.3 ϕ = 0, ψ = π2 , so in this case ω is non-degenerate. Also note that when the form ω is degenerate, then M possesses a canonical distribution (kernel), which can be used to construct classification in the case of non-general position.

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In dimension n > 3 the orbit space of GL(n, C)-action on Λ2 T ∗ ⊗C¯ T is quite complicated. And indeed the space of invariants is pretty big, as will be discussed below.

3. General background on differential invariants The equivalence problem of geometric structures on M is usually solved either via differential invariants algebra or by constructing a canonical estructure. In the first case the algebra can be represented either via some basic invariants and invariant differentiations or via some more differential invariants and Tresse derivatives. However in the case of geometric structures the number of required differential invariants is smaller and equals n = dim M , provided the restrictions of them to the structure are functionally independent (this is generically so). Indeed, let π be a bundle of geometric structures (associated with a tensorial bundle over M ) and E a section of it (i.e. a geometric structure of the specified type), which can be represented as the image of a section j : M → Eπ . Let ρ be the induced action of the pseudogroup Diff loc (M ) on π and I be a differential invariant. Its restriction to E is the function ∞ IE = j ∗ I ∈ Cloc (M ). Given n functionally independent invariants I 1 , . . . , I n we assume their restrictions IE1 , . . . , IEn are functionally independent (here and in what follows one can assume local treatment), so that they can be considered as local coordinates. Then one gets local frames ∂i = ∂I∂ i and coframes ω i = dIEi . E

...jt Any tensorial field T can be expressed as T = Tij11...i ∂j1 ⊗ . . . ∂jt ⊗ ω i1 ⊗ s js . . . ω and the coefficients are scalar differential invariants. Being expressed via IEi they form the complete set of invariant relations for equivalence problem. This is the principle of n-invariants.1 Two remarks are of order. First: It is clear in this case that canonical frame field ω i gives e-structure; otherwise around is also true, so that e-structures approach4 is equivalent to one with differential invariants. Second: Lifts of the derivations ∂i are invariant differentiations and coefficients DJ i are exactly Tresse derivatives of a differential invariant of dJ|E = DI i ω E

J by the basis I i (see Refs. 9,10). Thus all the discussed approaches are equivalent. Let us apply this to classification of almost complex structures of general position. This means that π is the bundle of almost complex structures over

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M: def π −1 (x) = {J ∈ GL(Tx M ) : J 2 = −1} ≃ GL(2n, R)/ GL(n, C) = J (2n). The pseudogroup G = Diff loc (M ) acts on π. The groupoid of its jets is denoted by Gl ⊂ J l (M, M ), with natural projections being denoted by ρl,k : Gl → Gk and ρl : Gl → M . Denote Glx the fiber over the point (x, x) ∈ G0 = M ×M , which is also a sub-groupoid of G called the differential group of order l (we will sometimes omit reference to the point x). Then Glx acts on the fiber of πk : J k π → M over point x. Moreover denoting Gl+1 = Ker[ρl+1,l : Gl+1 → Glx ] we obtain action of this normal x x subgroup on the fiber of the bundle πl,l−1 . Since for l > 0 the group Gl+1 −1 is abelian, the orbits in Fl = πl,l−1 (xl−1 ) are affine and so the differential invariants can be chosen affine in derivatives of order l. Usually they are non-linear in lower-order derivatives. 4. Equivalence problem for almost complex structures Let (M, J) be an almost complex manifold. If it is in general position, then as we have noticed above, it is enough to find 2n = dim M differential invariants for local classification. Solution of the equivalence problem depends on n (which we can assume to be > 1). Theorem 4.1. The basic scalar differential invariants of J solving the equivalence problem via the described methods can be specified as follows. n = 2 : There are no differential invariants of order ≤ 2, but in order 3 there are (no less than) 4 differential invariants; n = 3 : There are precisely 2 differential invariants of order 1 and 4 invariants of order 2; n > 3 : There are at least n2 (n − 3) > 2n differential invariants of order 1. Proof. Consider the cases. n = 2. It is clear from the description in Section 2 that J has no differential invariants of order 1 or 2. To get invariants of order 3 one proceeds as follows: the Maurer-Cartan coefficients ckij for the described canonical e-structure ξs (defined by the 2-jet of J) are the invariants of order 3: [ξi , ξj ] = ckij ξk . Since ck12 = δ3k and [ξ2 , ξ4 ] can be expressed via other brackets from NJ (ξ1 , ξ3 ) = ξ3 , the number of such differential invariants is 16. Note that the invariants of Ref. 5 §6.1 (canonical 1- and 2-forms on Π2 ) can be expressed via ckij .

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Let us notice that the result can be obtained via pure dimensional count. Indeed, rank of ρ1,0 is 16 and that of π is 8. The action of G1 on F0 = π −1 (x) is transitive (8-dimensional stabilizer). Rank of ρ2,1 is 40 and that of π1,0 is 32. Again the action of G2 on F1 is transitive (8-dimensional stabilizer). Next the rank of ρ3,2 is 80 and that of π2,1 is 80 as well, the corresponding action is transitive. Finally rank of ρ4,3 is 140 and that of π3,2 is 160. The action of G4 cannot be transitive. Moreover if we consider the action of G4 on π3−1 (x), it cannot be transitive as well, because even though the action of G3 has 8 + 8 = 16-dimensional stabilizer, the difference in dimension is 160 − 140 − 16 = 4. Thus there are at least 4 differential invariants. Note though that there are more (as we explained above), so that the action of G4 has a large stabilizer. n = 3. We do the dimensional count. Rank of ρ1,0 is 36 and that of π is 18. The action of G1 on F0 is transitive (18-dimensional stabilizer). Rank of ρ2,1 is 126 and that of π1,0 is 108. It seems that the stabilizer should be 18-dimensional, but as we explained in Section 2 the action of G2 has orbits of codimension 2, so the dimension of stabilizer is by two bigger than can be expected. Next rank of ρ3,2 is 336 and that of π2,1 is 378, so that the pure difference of dimensions gives at least 378 − 336 − 18 · 2 − 2 = 4 differential invariants of order 3. All these invariants can be expressed via the normal forms of Ref. 6. n > 3. Here the dimensional count can be misleading, so we better calculate codimension of orbits of GL(2n)-action on the space of linear Nijenhuis tensors. The stabilizer of a linear complex structure J0 on T is GL(n, C). Since dim GL(n, C) = 2n2 and dim Λ2 T ∗ ⊗C¯ T = n2 (n−1) the largest orbits have codimension n2 (n − 3) + dim St, where St is a stabilizer of a generic point. The result follows. This solves the equivalence problem for almost complex structures. 5. Existence of almost complex submanifolds In a private communication M. Gromov asked the following question: how many higher-dimensional PH-submanifolds can an almost complex manifold possess? According to Refs. 3,6 generically there are none. On the other end, for integrable J there’re plenty. What happens in between? This question is quite difficult if PH-submanifolds are isolated, so we treat the case when they come in families, regularly fashioned, namely as PH-foliations (this allows investigation via linearization8 ).

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We will consider in details 6-dimensional situation, the general case allows certain generalizations. Let us start with some examples. Example 5.1. Let M = C3 with almost complex structure J being given in 2 × 2 block form J = diag(A1 , A2 , A3 ), where the coefficients of Ai ∈ C ∞ (M, J (2)) do depend on all 6 coordinates (x1 , . . . , x6 ) ∈ M in a generic way. Then the Nijenhuis tensor NJ is non-degenerate. Indeed, we have for i, j odd: NJ (∂i , ∂j ) ∈ Ch∂i i ⊕ Ch∂j i, whence existence of kernel ξ = P αi ∂i (αi ∈ C and multiplication means α · η = Re α · η + Im α · Jη) i odd implies that some of the vectors NJ (∂i , ∂j ) have zero components in the above C2 decomposition. In other words if we denote the above splitting as M = V1 ⊕ V2 ⊕ V3 , then the genericity condition is NJ (Vi , Vj ) 6⊂ Vi for all i 6= j. Thus we see that it is possible to have 3 transversal PH-foliation of (M 6 , J) with J being maximally non-degenerate at each point. Generalization to dimension 2n is straightforward. Example 5.2. The above example can be modified as follows. Let Vij = Vi ⊕ Vj and let the almost complex structure have a block form J = diag(A, B) in the splitting M = V1 ⊕ V23 , where A ∈ C ∞ (M, J (2)) and B ∈ C ∞ (M, J (4)). While A-block is allowed to be arbitrary, the B-block is assumed symmetric in V1 -direction, i.e. independent of (x1 , x2 )-coordinates. Then any PH-curve C 2 ⊂ V23 lifts to the 4D PH-submanifold V1 × C 2 ⊂ M . Thus we have an infinite-dimensional family of PH-submanifolds C×C 2 , all of which intersect by a leaf of the 2D PH-foliation V1 . Note that this family will persist if we allow the structure to have the form   AB J= 0 D in the splitting M = V1 ⊕ V23 with the block D projectable along V1 . Definition 5.1. A family of 4D PH-submanifolds Φα intersecting by a PH curve C is a pencil if there exists a 2D PH-foliation in a neighborhood of C such that the projection along it is a PH-map and each Φα is projected to a PH-curve. In other words in a neighborhood of the curve J is represented by the above upper-triangular block form. Then for such a pencil the tensor NJ is degenerate.

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Let us recall basics about degenerations of linear Nijenhuis tensors.6 Such a tensor can be considered as a C-antilinear map NJ : Λ2C T → T of vector spaces of dimC = 3. So if NJ 6= 0 the following situations are possible: NDG: dimC Im NJ = 3 (non-degenerate); DG1 : dimC Im NJ = 2 (weakly degenerate); DG2 : dimC Im NJ = 1, there is a kernel V ∈ GrC 1 (T ), NJ (V, ·) = 0. Generically a pencil belongs to DG1 case. However DG2 can be obtained in the two following cases: 1. A is projectable along V23 and B = 0 or A is constant in the above splitting and B is projectable along V1 . Then if the tensor NJ 6= 0, its kernel coincides with V1 . 2. Almost complex structure D on V23 is integrable. Then if the tensor NJ 6= 0, its kernel is transversal to V1 . Proposition 5.1. Let Φα be a family of 4-dimensional PH-foliations of 6-dimensional (M, J), intersecting by a common foliation V by PH-curves, such that shifts along V is a symmetry of the family as foliations. Let cardinality of indices α be at least 4 and at almost every point x there be 4 leaves Φαi with Tx Φαi /Tx V of general position in Tx M/TxV . Then the family Φα is a pencil: There exists a PH-submersion π : ˜ with V -fibers. (M 6 , J) → (W 4 , J) Before proving this let us discuss the problem how an almost complex structure J is characterized by its PH-submanifolds. This question is nonvoid even in dimension 4, on which we concentrate. In this case the problem can be reformulated as a PH-analog of plane webs. Lemma 5.1. Let Ψa be a PH 4-web of almost complex (W 4 , J), i.e. there are foliations Ψa , 1 ≤ a ≤ 4, by PH-curves, none two of them being tangent anywhere. Then J is determined by Ψa up to sign. Proof. We will prove a more general statement: Let Ψa be a 4-webs of surfaces in W 4 with the same condition of general position at each point. Then there are at most two almost complex structures ±J making Ψa into PH-web. Indeed, this is the question of linear algebra. We have Tx W = Π1 ⊕ Π2 , where Πa = Tx Ψa are 2-dimensional subspaces. Complex structures on Π1 and Π2 determine that on Tx W .

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Since Π3 is a complex subspace it is a graph of a complex linear map F : Π1 → Π2 . This map is nondegenerate and the complex structure on Π1 determines that of Π2 . Now using Π4 , which is also a graph, we get a complex automorphism L : Π1 → Π1 , not proportional to identity. So no two different (up to sign) complex structures can commute with it. This proves uniqueness. Let us discuss existence. It is equivalent to the claim the the spectrum of L is purely complex. Necessity is obvious: if Sp(L) is real simple or L is a Jordan box, no rotation can commute with it. On the other hand, if Sp(L) = { λ±i β }, then J = βL − λI is a complex structure on Π1 and this gives the complex structure on Tx W . The above problem is equivalent to the following: Given a family of PH-foliations Ψa on (W 2n , J) and a diffeomorphism f : W → W , mapping them to PH-foliations, how large should be the index set {a} to ensure that f is a PH-map or anti-PH: f ∗ J = ±J. Imposing general position of leaves, making it into PH-web, the modification of the above proof gives the answer n + 2. Proof of Proposition 5.1. Shift along transversal V maps transversal foliations Φα /V into themselves. Since they are complex PH-lines in T M/T V , Lemma 5.1 implies that the complex structure in quotient tangent spaces is preserved. Thus shifts along V preserve the almost complex structure J in normal direction. Thus the complex structure becomes of the uppertriangular block form and the result follows. Let us call pencils from this Proposition 4-pencils, because there are 4 foliations in it (but then it extends to a continuous family). Remark 5.1. This proposition has certain generalizations to dimensions 2n > 6, but then one should make more specifications (dimension of PHfoliations in the pencil, their number etc), so we do not discuss it. 6. Criteria of integrability We present several approaches basing on existence of many PHsubmanifolds. It was shown in Ref. 6 that whenever through every point x ∈ (M 2n , J) and every complex [dimC = k]-dimensional subspace in Tx M passes a PH-submanifold of dimension 2k (or PH 2-jet), then J is integrable (k is fixed).

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But with this we require an infinite number of PH-submanifolds to ensure integrability. This requirement can be much weakened with the same conclusion. We will specify as above to the case n = 3. 1. We can use the pencils of Proposition 5.1 to get another criterion as follows. Consider 5 foliations Vi of M 6 by PH-curves (these always exist locally), 1 ≤ i ≤ 5, none two of which are tangent and none three have complex dependent tangents at almost any point. Theorem 6.1. Assume that (M, J) admits 10 PH-foliations Φij , such that Φij contain both Vi and Vj and are symmetric with respect to shifts along them. Then the structure J is integrable. Proof. Indeed in this case (M, J) admits 5 pencils of PH-foliations of dimension 4. Proposition 5.1 applies. In fact in this case the pencils become just as in Example 5.2 (without upper-triangular modification) because for any family Aa = {Φak }k6=a there is a transversal PH-foliation Φij , i, j 6= a. Thus the tensor NJ is degenerate. Two pencils can have weak degeneracy along the same complex 2-plane from Gr2 (T M, C), but then the next two show another weak degeneracy, so that there is a kernel. The last pencil gives a weak degeneracy of NJ , independent of this kernel, whence the Nijenhuis tensor vanishes. The hypotheses of the theorem can be modified to have four 4-pencils, each having 3 common PH-foliations with the other pencils, leading to the same conclusion. However this provides the same total amount 10 of PHfoliations. 2. We can skip organizing PH-foliations in pencils and get the same claim, but then the number of foliations should grow. Let us call family Φα of PH-foliations of dimension 4 quadratically non-degenerate if at almost every point x ∈ M the tangents Tx Φα ∈ Gr2 (Tx M, C) ≃ CP 2 do not belong to any real quadric of codimension 1. Note that any 14 points in CP 2 do belong to a real quadric. Theorem 6.2. Let Φα be a family of PH-foliations of dimension 4 in (M 6 , J), α = 1, . . . , 15. If it is quadratically non-degenerate, then J is integrable. Proof. If NJ 6= 0, then the Grassmanian of 4-planes in Tx M , which are invariants with respect to both J and NJ is a real quadric of codimension 2 in NDG case or codimension 1 in DG cases of Ref. 6 (it can be also

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empty, then its codimension is 4). But no 15 generic points in GrC 2 (Tx M ) can belong to a quadric. 3. We can have some intermediate criteria between approach 1, using fewer number of PH-submanifolds though with some integrability assumptions, and approach 2, using larger number of PH-submanifolds but only genericity conditions. For example, assume we have 14 families of 4D PHfoliations of (M 6 , J), which have generic arrangements of tangents at almost every point. Then we have a field of quadrics Qx ⊂ Tx M , x ∈ M . If the structure is non-integrable, this field satisfies certain integrability criteria (Ξ(π∗ ΘH (Π)) = 0 from Ref. 6). This is a binding requirement. Theorem 6.3. Let Φα be a family of PH-foliations of dimension 4 in (M 6 , J), α = 1, . . . , 14 with generic arrangements of tangents a.e. If the corresponding family of quadrics Q is non-integrable, then J is integrable. Acknowledgments The author thanks IHES for hospitality during the visit of January 2007, when this work was initiated. References 1. D. Alekseevskij, V. Lychagin and A. Vinogradov, Basic ideas and concepts of differential geometry (Encyclopedia of mathematical sciences 28, Geometry 1, Springer-Verlag, 1991). 2. R. Bryant, On the geometry of almost complex 6-manifolds, Asian J. Math. 10 (3) (2006) 561–605. 3. M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347. 4. S. Kobayashi, Transformation groups in Differential geometry (SpringerVerlag, 1972). 5. B.S. Kruglikov, Nijenhuis tensors and obstructions for pseudoholomorphic mapping constructions, Math. Notes 63 (4) (1998) 541–561. 6. B.S. Kruglikov, Non-existence of higher-dimensional pseudoholomorphic submanifolds, Manuscripta Mathematica 111 (2003) 51–69. 7. B.S. Kruglikov, Characteristic distributions on 4-dimensional almost complex manifolds, In: Geometry and Topology of Caustics - Caustics ’02 (Banach Center Publications 62, 2004) 173–182. 8. B.S. Kruglikov, Tangent and normal bundles in almost complex geometry, Diff.Geom.Appl. 25 (4) (2007) 399–418.

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9. B.S. Kruglikov and V.V. Lychagin, Invariants of pseudogroup actions: Homological methods and Finiteness theorem, Int. J. Geomet. Meth. Mod. Phys. 3 (5 & 6) (2006) 1131–1165. 10. A. Kumpera, Invariants differentiels d’un pseudogroupe de Lie. I, J. Differential Geometry 10 (2) (1975) 289–345; II, ibid. 10 (3) (1975) 347–416. 11. A. Newlander, L. Nirenberg, Complex analytic coordinates in almost-complex manifolds, Ann. Math. 65, ser. 2, issue 3 (1957) 391–404. 12. M. Verbitsky, Hodge theory on nearly Kahler manifolds; arXiv:math.DG/ 0510618 (2005).

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Hirzebruch signature operator for transitive Lie algebroids Jan Kubarski Institute of Mathematics, Technical University of L´ od´ z, Poland E-mail: [email protected] The aim of the paper is to construct Hirzebruch signature operator for transitive invariantly oriented Lie algebroids. Keywords: Lie algebroid, Hirzebruch signature operator, ∗-Hodge operator, elliptic complex, invariantly oriented Lie algebroid. MS classification: 58A14, 58J10.

1. Signature of Lie algebroids 1.1. Definition of Lie algebroids, Atiyah sequence Lie algebroids appeared as infinitesimal objects of Lie groupoids, principal fibre bundles, vector bundles (Pradines, 1967), TC-foliations and nonclosed Lie subgroups (Molino, 1977), Poisson manifolds (Dazord, Coste, Weinstein, 1987), etc. Their algebraic equivalents are known as Lie pseudoalgebras (Herz, 1953) called also further as Lie-Rinehart algebras (Huebschmann, 1990). A Lie algebroid on a manifold M is a triple A = (A, [[·, ·]], #A ) where A is a vector bundle on M , (Sec A, [[·, ·]]) is an R-Lie algebra, #A : A → T M is a linear homomorphism (called the anchor ) of vector bundles and the following Leibniz condition is satisfied [[ξ, f · η]] = f · [[ξ, η]] + #A (ξ) (f ) · η, f ∈ C ∞ (M ), ξ, η ∈ Sec A. The anchor is bracket-preserving,12 #A ◦[[ξ, η]] = [#A ◦ ξ, #A ◦ η].

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A Lie algebroid is called transitive if #A is an epimorphism. For a transitive Lie algebroid A we have the Atiyah sequence #A

0 −→ g ֒→A −→ T M −→ 0, g := ker #A . The fiber g x of the bundle g in the point x ∈ M is the Lie algebra with the commutator operation being [v, w] = [[ξ, η]](x),

ξ, η ∈ Sec A,

ξ(x) = v, η(x) = w,

v, w ∈ g x .

The Lie algebra g x is called the isotropy Lie algebra of A at x ∈ M . The vector bundle g is a Lie Algebra Bundle (LAB in short), called the adjoint of A, the fibres are isomorphic Lie algebras. T M is a Lie algebroid with id : T M → T M as the anchor, g -finitely dimensional Lie algebra - is a Lie algebroid over a point M = {∗} . 1.2. Cohomology algebra,ellipticity of the complex of exterior derivatives dkA

To a Lie algebroid A we associate the cohomology algebra H (A) defined via the DG-algebra of A-differential forms (with real coefficients) (Ω (A) , dA ) , where ^ ^ Ω (A) = Sec A∗ , - the space of cross-sections of A∗ dA : Ω• (A) → Ω•+1 (A) (dA ω) (ξ0 , ..., ξk ) =

k X j=0

+

X

j

(−1) (#A ◦ ξj ) (ω (ξ0 , ...ˆ ..., ξk )) i+j

(−1)

ω ([[ξi , ξj ]], ξ0 , ...ˆı...ˆ ..., ξk ) ,

i 0}, Aut− (ω) = {A ∈ Aut(ω); K(A) < 0}, we have an exact sequence K

1 → SL(V ; E) → Aut+ (ω0 ) → R+ → 1. Lemma 3.11. The group Aut(ω0 ) is a semidirect product SL(V ; E) ⋉ R∗ . Analogously, the group Aut+ (ω0 ) is a semidirect product SL(V ; E) ⋉ R+ . Proof. In the first case it suffices to find a splitting σ : R∗ → Aut(ω0 ). We use the same bases as in Lemma 3.9. To κ ∈ R∗ we assign an automorphism σ(κ) defined by the formulas 1 e1 , σ(κ)e2 = e2 , κ = en+1 , σ(κ)en+2 = κen+2 ,

σ(κ)e1 = σ(κ)en+1

...,

σ(κ)en = en ,

. . . , σ(κ)e2n = κe2n .

It can be immediately seen that σ is a splitting. It is also obvious that σ(R+ ) ⊂ Aut+ (ω0 ), which means that the same splitting can be used also in the second case. We shall now investigate the group SL(V ; E). Let A ∈ GL(V ; E). Because A is E-linear, it preserves the subspace im A, and consequently induces an automorphism Aˆ of the quotient Vˆ = V / im E. We have the projection π : V → Vˆ , which satisfies π(av) = ρ(a)π(v). This projection induces

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also a projection Λn π : ΛnE V → ΛnR Vˆ ,

v1 ∧E · · · ∧E vn 7→ π(v1 ) ∧R · · · ∧R π(vn ).

We have A∗ (e1 ∧E · · · ∧E en ) = detE A · e1 ∧E · · · ∧E en , Aˆ∗ (π(e1 ) ∧R · · · ∧R π(en )) = detR Aˆ · π(e1 ) ∧R · · · ∧R π(en ),

Aˆ∗ (π(e1 ) ∧R · · · ∧R π(en )) = Aˆ∗ (Λn π)(e1 ∧E · · · ∧E en )

= (Λn π)A∗ (e1 ∧E · · · ∧E en ) = (Λn π)(detE A · e1 ∧E · · · ∧E en )

= ρ(detE A) · (Λn π)(e1 ∧E · · · ∧E en ) = ρ(detE A) · π(e1 ) ∧R · · · ∧R π(en ). We have thus proved the formula ˆ ρ(detE A) = detR A. We denote Q the homomorphism assigning to an automorphism A ∈ GL(V ; E) the induced automorphism Aˆ ∈ GL(Vˆ ). It is easy to see that we get a short exact sequence Q

1 → KerQ → GL(V ; E) → GL(Vˆ ) → 1. Let us denote first B = {B ∈ gl(V ; E); BV ⊂ im E}. If B ∈ B, then B(im E) = 0. Namely, if v ∈ im E, then v = ev ′ for some v ′ ∈ V . Then Bv = B(ev ′ ) = eB(v ′ ) = 0. Consequently, if B, B ′ ∈ B, then BB ′ = 0. Every A ∈ KerQ can be expressed in the form A = I + B, where B ∈ B. On the other hand, every endomorphism of the form I + B with B ∈ B is an automorphism. Namely, (I + B)(I − B) = I − B + B − BB = I. Moreover A = I+B obviously belongs to KerQ. If A = I+B and A′ = I+B ′ we have (I + A)(I + B ′ ) = I + B + B ′ + BB ′ = I + B + B ′ . 2 Hence we can see that KerQ is an abelian group isomorphic with B ∼ = Rn .

Lemma 3.12. The group GL(V ; E) is a semidirect product B ⋉ GL(Vˆ ).

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Proof. We use again the same bases as in the proof of lemma 3.9. Obviously the classes [e1 ], . . . , [en ] constitute a basis of the vector space Vˆ . Any automorphism ϕ ∈ GL(Vˆ ) can be expressed in the form ϕ[ei ] =

n X

aij [ej ],

j=1

where aij are real numbers. We define an automorphism A ∈ GL(V ; E) by the formulas n X aij ej . Aei = j=1

Setting σ(ϕ) = A, we get a splitting σ : GL(Vˆ ) → GL(V ; E).

Let us remind that if v1 , v2 ∈ im E, then θ(v1 , v2 , . . . , vn ) = 0. Namely, we have v1 = ev1′ and v2 = ev2′ , and we get θ(v1 , v2 , v3 , . . . , vn ) = θ(ev1′ , ev2′ , v3 , . . . , vn ) = e2 θ(v1′ , v2′ , v3 , . . . , vn ) = 0. Let A = I + B be again an element of KerQ. We obtain (A∗ θ)(v1 , v2 , . . . , vn ) = θ(v1 + Bv1 , v2 + Bv2 , . . . , vn + Bvn ) n X θ(v1 , . . . , vi−1 , Bvi , vi+1 , . . . , vn ) = θ(v1 , v2 , . . . , vn ) + i=1

= θ(v1 , v2 , . . . , vn ) + tr(B)θ(v1 , v2 , . . . , vn ) = (1+tr(B))θ(v1 , v2 , . . . , vn ).

We have thus proved that if A ∈ KerQ, then detE A = 1 + tr(B). Using the formula ρ(detE A) = detR Aˆ we get another short exact sequence q 1 → Kerq → SL(V ; E) → SL(Vˆ ) → 1,

where q = Q|SL(V ; E). Applying the last determinant formula we find easily that 2 Kerq = B0 = {B ∈ B; tr(B) = 0} ∼ = Rn −1 . Lemma 3.13. The group SL(V ; E) is a semidirect product B0 ⋉ SL(Vˆ ).

Proof. The proof follows the same lines as the proof of Prop. 3.12. Summarizing we have the following proposition. Proposition 3.1. The automorphism group Aut(ω0 ) consists of two connected components Aut+ (ω0 ) and Aut− (ω0 ), where Aut+ (ω0 ) is the connected component of the unit. Moreover dimR Aut(ω0 ) = 2n2 − 1.

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Acknowledgments The author was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503 and by the Grant Agency of the Academy of Sciences of the Czech Republic, Grant No. A100190701. References 1. J. Bureˇs and J. Vanˇzura, Unified Treatment of Multisymplectic 3-Forms in Dimension 6; arXiv:math/0405101. 2. N. Hitchin, The geometry of three-forms in six dimensions, J. Diff. Geom. 55 (3) (2000) 547–576. 3. N. Hitchin, Stable forms and special metrics, In: Global differential geometry: the mathematical legacy of Alfred Gray (Bilbao, 2000, Contemp. Math. 288, Amer. Math. Soc., Providence, RI, 2001) 70–89. 4. D.Z. Djokovi´c, Classification of trivectors of an eight-dimensional real vector space, Linear and Multilinear Algebra 13 (1) (1983) 3–39.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 371–381

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Symmetries of almost Grassmannian geometries Lenka Zalabov´ a ˇ Masaryk University, Eduard Cech Center for Algebra and Geometry, Jan´ aˇ ckovo n. 2a, 662 95 Brno, Czech Republic E-mail: [email protected] We study symmetries of almost Grassmannian and almost quaternionic structures. We generalize the classical definition for locally symmetric spaces and we discuss the existence of symmetries on the homogeneous models. We also conclude some observations on the general curved geometries. Keywords: Cartan geometries, parabolic geometries, almost Grassmannian structures, almost quaternionic structures, symmetric spaces. MS classification: 53C15, 53C05, 53A40, 53C35.

The aim of this paper is to introduce and discuss the symmetries for the almost Grassmannian and almost quaternionic structures. These two geometries belong into the class of so called |1|–graded parabolic geometries and we use the language of Cartan and parabolic geometries, which allows us to generalize the classical concepts. For the class of |1|–graded geometries, the symmetries are defined in the same intuitive way as in the affine geometry. At the same time, much of the classical theory of affine symmetries extends. In particular, the existence of a symmetry at a point kills the torsion of the geometry at this point. In view of the nice general theory of parabolic geometries, this already proves the local flatness of the symmetric geometries for most cases of almost Grassmannian geometries. There are also some more interesting types of almost Grassmannian and almost quaternionic geometries, which can carry some symmetry in the point with nonzero curvature. We show, that there can be at most one symmetry in such point.

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1. Definitions and basic facts Throughout the paper we use the standard notation and concepts.6,12 We recall some of the basic facts and notation related to the parabolic geometries below. 1.1. Cartan and parabolic geometries A Cartan geometry of type (G, P ) on a smooth manifold M is a principal P –bundle p : G → M with a Cartan connection ω ∈ Ω1 (G, g), where the 1–form ω is an absolute parallelism which is P –equivariant and reproduces fundamental vector fields. The simplest examples are so called homogeneous models, which are the P –bundles G → G/P endowed with the (left) Maurer Cartan form ωG . A parabolic geometry is a Cartan geometry (G → M, ω) of type (G, P ), where P is a parabolic subgroup of a semisimple Lie group G. The algebra g of the group G is equipped (up to the choice of Levi factor g0 in p) with a grading of the form g = g−k ⊕ · · · ⊕ g0 ⊕ · · · ⊕ gk such that the algebra p of P is exactly p = g0 ⊕ · · · ⊕ gk . We suppose that the gradation is fixed. We also have the subgroup G0 ⊂ P with Lie algebra g0 of elements such that their adjoint action preserves the gradation. Remark, that P is exactly the group of all elements such that their adjoint action preserves induced filtration given as g = g−k ⊃ g−k+1 ⊃ · · · ⊃ gk = gk , where gi = gi ⊕ · · · ⊕ gk . If the length of the gradation of g is k, then the geometry is called |k|–graded. We are mainly interested in |1|–graded geometries and we formulate most of the facts for them. 1.2. Almost Grassmannian structures We take G = Sl(p + q, R). This group acts on Rp+q ≃ Rp ⊕ Rq and the parabolic subgroup P is exactly the stabilizer of Rp . The reductive subgroup G0 is of the form S(Gl(p, R) × Gl(q, R)). We obtain g = sl(p + q, R) and we get gradation of the form g−1 ≃ L(Rp , Rq ), g0 ≃ s(gl(p, R) ⊕ gl(q, R)) and g1 ≃ L(Rq , Rp ). Thus, this choice of a Lie group corresponds to a |1|–graded geometry and its homogeneous model is the Grassmannian of p-dimensional subspaces of Rp+q . Remind that these geometries are given (as G0 -structures) by the identification of the tangent bundle T M ≃ G0 ×G0 g−1 with tensor product of two bundles T M ≃ E ⊗ F ∗ together with the preferred trivialization of ∧q E⊗∧p F . The rank q bundle E → M and the rank p bundle F → M are of the form of the associated vector bundles E ≃ G0 ×G0 Rq and F ≃ G0 ×G0 Rp for the standard actions of G0 .

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1.3. The curvature The curvature of |1|–graded geometry can be described by the curvature function κ : G → ∧2 g∗−1 ⊗ g, where κ(u)(X, Y ) = [X, Y ] − ω([ω −1 (X), ω −1 (Y )](u)). It is valued in the cochains for the second cohomology H 2 (g−1 , g). This group can be also computed as the homology of the codifferential ∂ ∗ : ∧k+1 g∗−1 ⊗ g → ∧k g∗−1 ⊗ g. The parabolic geometry is called normal if the curvature satisfies ∂ ∗ ◦ κ = 0. If the geometry is normal, we can define the harmonic part of curvature, κH : G → H 2 (g−1 , g), as the composition of the curvature and the projection to the second cohomology group. Thanks to the gradation of g, there are several decompositions of the curvature of the |1|–graded geometry. One of the possibilities is the decomposition into homogeneous components, which is of the form κ = κ(1) + κ(2) + κ(3) , where κ(i) (u)(X, Y ) ∈ gp+q+i for all X ∈ gp , Y ∈ gq and u ∈ G. The parabolic geometry is called regular if the curvature function κ satisfies κ(r) = 0 for all r ≤ 0 and clearly, |1|–graded geometries are always regular. The crucial structural description of the curvature of parabolic geometry is provided by the following theorem:14 Theorem 1.1. The curvature κ of regular normal geometry vanishes if and only if its harmonic part κH vanishes. Moreover, if all homogeneous components of κ of degrees less than j vanish identically and there is no cohomology Hj2 (g− , g), then also the curvature component of degree j vanishes. Another possibility is the decomposition of the curvature according to the values κ = κ−1 + κ0 + κ1 and in an arbitrary frame u we have κj (u) ∈ g−1 ∧ g−1 → gj . The component κ−1 valued in g−1 is the torsion. An important feature of |1|–graded geometries is that κi is equal to κ(i+2) for i = −1, 0, 1. Notice that the Maurer–Cartan equations imply that the curvature of homogeneous model is zero. It can be proved that if the curvature of a Cartan geometry of type (G, P ) vanishes, then the geometry is locally isomorphic with the homogeneous model (G → G/P, ωG ).11 Cartan geometry is called locally flat if the curvature κ vanishes. Homogeneous models are sometimes called flat models.

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1.4. Curvatures of almost quaternionic and almost Grassmannian geometries Next, we come back to the almost Grassmannian and almost quaternionic geometries which are special cases of |1|–graded geometries and we provide the complete list with their non-zero components of the harmonic curvature: • almost Grassmannian structures, g = sl(p + q, R), g0 = s(gl(p, R) × gl(q, R)): p = 1, q = 2; p = 2, q = 1: the projective structures dim = 2, one curvature of homogeneity 3 p = 1, q > 2; p > 2, q = 1: the projective structures dim > 2, one curvature of homogeneity 2 p = 2, q = 2: dim = 4, two curvatures of homogeneity 2 p = 2, q > 2; p > 2, q = 2: dim = pq, one torsion, one curvature of homogeneity 2 p > 2, q > 2: dim = pq, two torsions • almost quaternionic structures, g = sl(p + 1, H):

p = 1: the almost quaternionic geometries, dim = 4, two curvatures of homogeneity 2 p > 1: the almost quaternionic geometries, dim = 4p, one torsion, one curvature of homogeneity 2

• the geometries modeled on quaternionic Grassmannians: two torsions

1.5. Automorphisms and symmetries We shall deal with the automorphisms of Cartan geometries. These are P –bundle morphisms, which preserve Cartan connection on the geometry. They have to preserve the structure given by the existence of the Cartan connection. It is well known that all automorphisms of the homogeneous model (G → G/P, ωG ) of a Cartan geometry are exactly the left multiplications by elements of G.11 We define symmetry on the |1|–graded parabolic geometry in the following way: Definition 1.1. Let (G → M, ω) be a |1|–graded geometry. The symmetry at the point x is a locally defined diffeomorphism sx on M such that:

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• sx (x) = x • Tx sx |Tx M = − idTx M • sx is covered by an automorphism of the Cartan geometry. The geometry is called (locally) symmetric if there is a symmetry at each point x ∈ M . Thus the definition of the symmetries of |1|–graded geometries follows completely the classical intuitive idea. 2. Homogeneous models Homogeneous models are simplest candidates for symmetric geometries. We ask, whether homogeneous models of almost Grassmannian geometries and almost quaternionic geometries are symmetric or not. It can be proved:15,17 Proposition 2.1. All symmetries of homogeneous models of |1|–graded geometries at the origin are exactly left multiplications by elements g ∈ P such that g = g0 exp Z satisfying Adg0 (X) = −X for all X ∈ g−1 and Z ∈ g1 is arbitrary. Thus we are looking for these elements in the models for the latter two geometries. 2.1. Almost Grassmannian structures We have g = sl(p + q, R) and we first take G = Sl(p + q, R), see section Y 1.2. The algebra g consists of block elements ( X Z W ) with block size p and q where tr(X) + tr(W ) = 0 and elements of g−1 are those with X, Y, W vanishing. The adjoint action of some ( S0 T0 ) ∈ G0 on ( V0 00 ) ∈ g−1 is T V S −1 and we look for S and T such that T V = −V S for all V ∈ L(Rp , Rq ). The properties of matrix multiplication give that T and S are diagonal, elements on the diagonal of T are equal, elements on the diagonal of S are equal and elements on the diagonal of T are equal to minus elements
from 0 and S. The
condition on determinant gives that only the elements E0 −E −E 0 satisfy all latter restrictions. 0 E We have to discuss the dependence on p and q to resolve whether some of these two elements belong to Sl(p + q, R) and give a symmetry. We get some symmetry if at least one of p and q is
even. If only p is even, X then all symmetries are given by elements −E for all X ∈ L(Rq , Rp ). 0 E

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X for If only q is even, then all symmetries are given by elements E0 −E all X ∈ L(Rq , Rp ). If both sizes are even, then all latter elements give symmetries. If p and q are odd, then there is no symmetry. The situation where p and q are both even is exactly the situation in non–effective models. In this special case, the
geometry also has nontrivial 0 kernel. This is of the form {( E0 E0 ) , −E 0 −E }. Clearly, the second element belongs to the group Sl(p + q, R) if and only if p and q are both even (or odd, but this case is not interesting). In this case, there are two different elements giving the same symmetry and these two elements differ by the multiplication by −E. We can take G = P Sl(p+q, R) instead of Sl(p+q, R) to get effective geometry. With this choice, each symmetry is given by exactly one class represented by some of the above elements.

2.2. Almost quaternionic structures Now we consider almost quaternionic structures, we have g = sl(m + 1, H). There are again two interesting choices of the groups. We can choose G = Sl(m + 1, H) with the canonical action on Hm+1 . The parabolic subgroup P is the stabilizer of the line spanned by the first basis  aquaternionic 4 m+1 0 vector in H . Then G0 = ( 0 A ) | |a| detR A = 1 . Next, we can take G = P Gl(m + 1, H), the quotient of all invertible quaternionic linear endomorphisms by the subgroup of real multiples of identity. Let P be the (factor of the) stabilizer of the quaternionic line spanned by the first basis vector. The subgroup G0 consists of classes in P of block diagonal matrices which are represented by matrices of the form ( a0 A0 ) such that 0 6= a ∈ H and A ∈ Gl(m, H).
We have g−1 = {( X0 00 ) | X ∈ Hm } and we look for elements q0 B0 such that BX = −Xq for each X. Again, such an element must be diagonal and the elements on the diagonal of B are equal to −q. Suppose that q = a+bi+ cj+dk. If we choose X = ( 0i ) we get (−a−bi−cj−dk)i = −i(a+bi+cj+dk), thus −ai
+ b + ck − dj = −ai + b − ck + dj and so c = d = 0. Then the choice
0 . X = 0j gives that q has to be real. We again get the element 10 −E In the case of P Gl(m + 1, H), this element clearly represents the class giving a symmetry. In the case of Sl(m+1, H) it should again depend on the dimension of the manifold. But the real dimension equals to 4m and also in this case, the
symmetry is well defined. All elements giving symmetries W for all W ∈ Hm∗ . look like 10 −E

Thus we see that homogeneous models are mostly symmetric. If not, we can change the group G to get symmetric geometry.

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3. Torsion restrictions We are mainly interested in almost Grassmannian and almost quaternionic structures and we will formulate most facts only for them. We shall follow the general well known results on all |1|–graded geometries.15,17 We know that curved geometries could be symmetric because the homogeneous models are. We look for restrictions on the curvature of these geometries. The following theorem plays a crucial role for us. Theorem 3.1. Symmetric |1|–graded parabolic geometries are torsion free. The Theorem 1.1 and the whole theory on harmonic curvature of parabolic geometries give as a simple consequence the following corollary. Theorem 3.2. The following symmetric normal geometries have to be locally flat: • almost Grassmannian geometries such that p > 2 and q > 2 • geometries modeled on quaternionic Grassmannians (but not the almost quaternionic ones). The crucial point is that the components in harmonic curvatures are only of homogeneity one, see 1.4. Similar argument applies for geometries where the only available homogeneity is three and we get the following Theorem. Theorem 3.3. Symmetric normal projective geometries of dimension 2 are locally flat. There are also some more interesting cases, which can carry some symmetry in a point with nonzero curvature. We are interested in curved versions of: • projective geometries of dim > 2 • almost quaternionic geometries • almost Grassmannian structures such that p = 2 or q = 2. These geometries have homogeneous component of curvature of degree 2 and we concentrate on it. 4. Further curvature restrictions We use the theory of Weyl structures to study more interesting cases of almost Grassmannian and almost quaternionic geometries. We introduce Weyl structures6 only in the |1|–graded case.

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4.1. Weyl structures Remind that there is the underlying bundle G0 := G/ exp g1 for each parabolic geometry, which is the principal bundle p0 : G0 → M with structure group G0 . At the same time we get the principal bundle π : G → G0 with structure group P+ = exp g1 . The Weyl structure σ for parabolic geometry is a global smooth G0 –equivariant section of the projection π : G → G0 . There exists some Weyl structure σ : G0 → G on an arbitrary parabolic geometry, and for arbitrary two Weyl structures σ and σ ˆ, there is exactly one G0 –equivariant mapping Υ : G0 → g1 such that σ ˆ (u) = σ(u) · exp Υ(u) for all u ∈ G0 . The equivariancy allows to extend Υ to P –equivariant mapping G → g1 and in fact, it is a 1–form on M . Weyl structures form an affine space modeled over the vector space of all 1–forms and in this sense we can write σ ˆ = σ + Υ. The choice of the Weyl structure σ defines the decomposition of G0 – equivariant 1–form σ ∗ ω ∈ Ω1 (G0 , g) such that σ ∗ ω = σ ∗ ω−1 + σ ∗ ω0 + σ ∗ ω1 . The part σ ∗ ω0 ∈ Ω1 (G0 , g0 ) defines the principal connection on p0 : G0 → M , the Weyl connection. For each representation λ : G0 → Gl(V ) we get the induced Weyl connection ∇σ on G0 ×G0 V and for arbitrary two Weyl structures σ and σ ˆ = σ · exp Υ we get explicit formula for the change of corresponding connection ∇σ and ∇σˆ . It can be nicely written in the language of tractor calculi in the following way.4,6 We have ∇σξˆ (s) = ∇σξ (s) + {ξ, Υ} • s for ξ ∈ X(M ) and s ∈ Γ(G0 ×G0 V ). Here { , } is the algebraic bracket of a vector field with a 1–form, which becomes an endomorphism on T M and • is the algebraic action derived from λ. For each automorphism ϕ of the geometry, there is the pullback ϕ∗ σ of the Weyl structure σ. It is again Weyl structure and there is exactly one Υ such that ϕ∗ σ = σ + Υ. In addition, it respects the affine structure, i.e. ϕ∗ (σ + Υ) = ϕ∗ σ + ϕ∗ Υ. Finally we remind the existence of the normal Weyl structure at u. It is the only G0 –equivariant section σu : G0 → G satisfying ω −1 (X)

σu ◦ π ◦ Fl1

ω −1 (X)

(u) = Fl1

(u).

We remark that the pullback of normal Weyl structure is again normal Weyl structure.

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4.2. The Weyl curvature The curvature of each symmetric geometry is of the form κ = κ0 : G → ⊗ g0 , because these geometries are torsion free. If we choose some Weyl structure σ, we can take the decomposition σ ∗ κ0 = σ ∗ κ0 + σ ∗ κ1 . The part ∧2 g∗−1

σ ∗ κ0 : G0 → ∧2 g∗−1 ⊗ g0 does not change, if we change the Weyl structure, because it is the lowest part of decomposition. This part is called Weyl curvature and is usually denoted by W . It is the most interesting part of the curvature an we will focus on it. Next, we summarize here some facts on Weyl structures and Weyl curvature of symmetric |1|–graded geometries.16,17 Proposition 4.1. Let (G → M, ω) be a |1|–graded geometry. Suppose, that there is a symmetry sx in x ∈ M and let ϕ be some covering of sx . Remark, the following properties do not depend on the choice of covering ϕ of sx . (1) There is a Weyl structure σ such that ϕ∗ σ = σ in the point x. (2) There is exactly one normal Weyl structure σ such that ϕ∗ σ = σ over some neighborhood of x. (3) There exists a Weyl connection ∇σ such that ∇σ W = 0 in x. The connection corresponds to the fixed Weyl structure in x. (4) Assume there are two different symmetries in x on a |1|–graded geometry. Then {ξ, Υ} • W = 0 holds in x for any ξ ∈ X(M ) and one fixed 1–form Υ. Here Υ is a suitable multiple of the difference between ’fixed’ Weyl structures of the symmetries. 4.3. The main results We use the fourth property of Proposition 4.1 to prove the following restriction on the Weyl curvature and thus the whole curvature of geometries. Theorem 4.1. Suppose that there exist two different symmetries in x on an almost Grassmannian structure of the type (2, q) and suppose that the latter Υ has maximal rank. Then W vanishes in x and thus the whole curvature vanishes in x.

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If there are two such different symmetries in each point, then the geometry is locally flat. Proof. We write up to a choice of the frame the expression {ξ, Υ} • W = 0 in our concrete geometry. The idea is to choose ξ such that {ξ, Υ} corresponds to a reasonable element of g. 0 1 0 ... 0 T If we write Υ = ( 10 01 ... ... 0 ) and take ξ = ( 0 1 ... 0 ) , we get the decomposition of T M = U ⊕ V such that U and V are eigenspaces with eigenvalues 1 and 2. Clearly, the bracket corresponds to the element  0 0 0  1  0   . . . 0

0 0 1 . . . 0

0 0 0 0 . . . 0

0 0 0 0 . . . 0

... ... ... ... .. . ...

0 0 0 0 0 0   0,0 . . . . . . 0 0

0 0 0 0 . . . 0

1 0 0 0 . . . 0

0 1 0 0 . . . 0

... ... ... ... .. . ...

0  0 0   0  .  . . 0

 −1

  = 

0 0 0 ... 0 −1 0 0 . . . 0 0 1 0 ... 0 0 0 1 ... . . . . . . . . . .. . . . . 0 0 0 0 ...

0 0 0  0 . . . 0

and the action of the bracket on the vector is  −1    

0 0 0 ... 0 0 0 −1 0 0 . . . 0 0 0 0 1 0 ... 0 a   0 0 0 1 ... 0, c . . . . . . . . . . .. . . . . . . . . . 0 0 0 0 ... 0 e

0 0 b d . . . f

0 0 0 0 . . . 0

0 0 0 0 . . . 0

... ... ... ... .. . ...

0  0 0   0  .  . . 0

=



0 0  2a  2c   .. . e

0 0 2b 2d . . . f

0 0 0 0 . . . 0

0 0 0 0 . . . 0

... ... ... ... .. . ...

0 0 0  0 . . . 0

Similarly, T ∗ M decomposes into eigenspaces with eigenvalues −2 and −1. Weyl curvature W lives in a component of T ∗ M ∧ T ∗ M ⊗ T M ⊗ T ∗ M , which splits according to the decomposition of T M . The action of the bracket on each component is an integer from {−1, −2, −3, −4, −5}. Then W has to vanish. In this case, the whole curvature vanishes. Remark, that this argument does not work in the case of singular Υ. There is no ξ such that all the eigenvalues of the bracket are non-zero. Similar arguments and computations work in the other cases. Theorem 4.2. (1) Suppose that there exist two different symmetries in x on an almost Grassmannian structure of the type (p, 2) and suppose that the latter Υ has maximal rank. Then W vanishes in x. (2) Suppose that there exist two different symmetries in x on an almost quaternionic structure of the type (2, q). Then W vanishes in x.

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Acknowledgments Discussions with Jan Slov´ak and Boris Doubrov were very useful during the work on this paper. This research has been supported by the grant GACR 201/05/H005. References 1. T.N. Bailey and M.G. Eastwood, Complex Paraconformal Manifolds - their Differential Geometry and Twistor Theory, Forum Mathematikum 3 (1991) 61–103. ˇ 2. A. Cap, Two constructions with parabolic geometries, In: Proceedings of the 25th Winter School on Geometry and Physics (Srni, 2005, Rend. Circ. Mat. Palermo Suppl. ser. II.). ˇ 3. A. Cap and R. Gover, Tractor Bundles for Irreducible Parabolic Geometries, In: Global analysis and harmonic analysis (SMF, S´eminaires et Congr`es, n.4, 2000) 129–154. ˇ and R. Gover, Tractor Calculi for Parabolic Geometries, Trans. Amer. 4. A. Cap Math. Soc. 354 (2002) 1511–1548. ˇ 5. A. Cap and H. Schichl, Parabolic geometries and canonical Cartan connection, Hokkaido Math. J. 29 (2000) 453–505. ˇ 6. A. Cap and J. Slov´ ak, Weyl Structures for Parabolic Geometries, Math. Scand. 93 (2003) 53–90. ˇ 7. A. Cap, J. Slov´ ak, Parabolic Geometries, Mathematical Surveys and Monographs, AMS Publishing House, to appear. 8. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry. Vol II (John Wiley & Sons, New York-London-Sydney, 1969, pp. 470). 9. I. Kol´ aˇr, P.W. Michor and J. Slov´ ak, Natural Operations in Differential Geometry (Springer-Verlag, 1993, pp. 434). 10. F. Podesta, A Class of Symmetric Spaces, Bulletin de la S.M.F. 117 (3) (1989) 343–360. 11. R.W. Sharpe, Differential geometry: Cartan’s generalization of Klein’s Erlangen program (Graduate Texts in Mathematics 166, Springer-Verlag, 1997). 12. J. Slov´ ak, Parabolic geometries, Research Lecture Notes, Part of DrScdissertation, Masaryk University, 1997, pp. 70, IGA Preprint 97/11, University of Adelaide. ˇ 13. J. Silhan, Cohomology of Lie algebras, algorithm for computation available at http://bart.math.muni.cz/ silhan/lie/. 14. K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Advanced Studies in Pure Mathematics 22 (1993) 413–494. 15. L. Zalabov´ a, Remarks on Symmetries of Parabolic Geomeries, Arch. Math. 42 (Supplement) 357–368. 16. L. Zalabov´ a, Symmetries of Parabolic Geometries, submitted to Differential Geometry and Its Aplications. 17. L. Zalabov´ a, Symmetries of Parabolic Geometries, Ph.D. thesis, 2007 ˇ adn´ık, Generalised Geodesics, Ph.D. thesis, 2003, pp. 65. 18. V. Z´

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 385–396

385

Zeta regularized integral and Fourier expansion of functions on an infinite dimensional torus Akira Asada Sinsyu University, 3-6-21 Nogami, Takarazuka, 665-0022 Japan E-mail: [email protected] Let {H, G} be a pair of a Hilbert space and a Schatten class operator G such that ζ(G, s) = trGs is holomorphic at s = 0. Then regularization of integrals on suitable subsets of H ♯ , an extension of H obtained to add a 1-dimensional space, is defined by using ζ(G, s). In this paper, we apply this regularized integral to periodic functions of H ♯ , and show practical computations of Fourier expansions of periodic functions. Results also show there exists de Rham type cohomology having the Poincar´ e duality exists on suitable infinite dimensional torus. Keywords: Zeta regularization, regularized infinite dimensional integral, infinite dimensional torus. MS classification: 58B25, 42B99, 58J52.

1. Introduction To overcome the difficulty of divergence in the calculus on a Hilbert space H, we have proposed to consider the pair {H, G}, where G is a (positive) Schatten class operator on H, such that its ζ-function ζ(G, s) = trGs is holomorphic at s = 0.1,2 By using G, we introduce the Sobolev k-norm kxkk , x ∈ D(G−k ), by kxkk = kG−k/2 xk = (G−k x, x).

Sobolev space constructed by this norm and D(G−k ) is denoted by W k . The complete ortho-normal basis of H is fixed to be {e1 , e2 , . . .}; Gen = µn en , µ1 ≥ µ2 ≥ · · · > 0. Selection of e1 , e2 , . . . gives an additional structure to {H, G}. But we do not discuss on this point. We set ν = ζ(G, 0) and d the P∞ d/2 location of the first pole of ζ(G, s). Then e∞ = n=1 µn en ∈ W l , l < 0 but not belongs to H. We set \ H ♯ = H ⊕ Ke∞ ⊂ W l. l 0 be eigenvalues of G with normalized eigenfunctions e1 , e2 , . . .; Gen = µn en . We fix the complete orthonormal basis of H to be e1 , e2 , . . .. Note that selecting e1 , e2 , . . . gives an additional structure to {H, G}. But at this stage, we can not obtain any additional informations from this selection. The k-th power Gk of G is defined by Gk en = µkn en . It does not depend on the choice of eigenvectors, but if k < 0, it does not defined on H. The domain of Gk is denoted by D(Gk ). Let (x, y) and kxk be inner product and norm of H. Then we define the Sobolev k-inner product and norm by p (1) (x, y)k = (G−k/2 x, G−k/2 y), kxkk = (x, x)k = kG−k/2 xk.

The Sobolev space constructed by this Sobolev norm and elements of D(Gk/2 ) is denoted by W k (W 0 = H). The complete orthonormal basis k/2 of W k is given by e1,k , e2,k , . . .; en,k = µn en = Gk/2 en . k l As sets, W ⊂ W , l < k. We set ∩l0 W l . We can regard T {Gt |t ∈ R} as the 1-parameter group of operators on W ∞ = l W l . k m We define the operator et(log G) G by log G = lim

et(log G)

k

Gm

en = et(log µn )

If G is the Green operator of D, then etG By definition, if k ≥ 1, we have et(log G)

k

Gm

−m

k

µm n

en .

is the heat kernel of Dm .

= Gt(log G)

k−1

Gm

.

Theorem 4.1. If ζ(G, s) is holomorphic at s = m, then we have detG et(log G) d Proof. Since trGm+s = tr ds eζ

(k)

(G,m+s)



k

Gm

= etζ

(k)

(G,m)

.

(6)

 d m+s , we have G ds

dk

|s=0 = et dsk trG

m+s

= ettr((log G)

k

dk

|s=0 = ettr( dsk G

G

m+s

m+s

)

|s=0

|s=0 = detG et(log G)

k

Gm

.

Hence we have Theorem. For example, we have m

detG etG = etζ(G,m) . Hence if G is the Green operator of D and ζ(G, −m) = 0, regularized determinant of the heat kernel of Dm does not depend on t. If k ≥ 1, we can rewrite (6) as follows; detG Gt(log G)

k−1

Gm

= etζ

(k)

(G,m)

.

(7)

Zeta regularized integral and Fourier expansion of functions

391

By (6), we have :

∞ Y

t(log µn )k µm n

(1 + xn ) := etζ

µn

(k)

(G,m)

n=1

∞ Y

(1 + xn ),

n=1

if ζ(G, s) is holomorphic at s = m and

P∞

n=1

|xn | < ∞.

5. Regularized infinite dimensional integral Let a = (a1 , a2 , . . .) and b = (b1 , b2 , . . .), an < bn , n = 1, 2, . . . be P P a = n an en , b = n bn en ∈ H ♯ . Then we set ∞ X

xn en ∈ H ♯ |an ≤ xn ≤ bn }.

N X

xn en ∈ RN |an ≤ xn ≤ bn }.

Da,b = {

n=1

We also set N Da,b ={

P

n=1

Let ∗ = (∗1 , ∗2 , . . .), n ∗n ∈ Da,b and ∗N = (∗N +1 , ∗N +2 , . . .), and let f be a Frech´ Z et differentiable function on Da,b . Then we define regularized

integral Z

Da,b

f (x) : d∞ x : of f on Da,b by ∞

f (x) : d x := lim

Da,b

N →∞

Z

n Da,b

µs

µs

f (x1 , . . . , xN , ∗N )d(x1 1 ) · · · d(xNN )|s=0 .(8)

It is shown this integral does not depend on the choice of ∗, if it exists. Da,b is also defined for an = −∞ or bn = ∞. Regularized integrals on such domains are similarly defined. Regularized integral is also defined if a, b ∈ W k,♯ . Such integral may be regarded as the integral on Gk/2 Da,b . In general, if Ix is a scaling operator, and Ix♯ f (y) = f (Ix♯ y), where f is a function on Ix D, we have Z Z |detG Ix |−1 f (ξ) : d∞ ξ :, ξ = Ix y. (9) Ix♯ f (y) : d∞ y := D

Ix D

If G is the Green operator of D, taking Ix = G1/2 , we obtain Z 1 e−π(x,Dx) : d∞ x := √ . detD W 1/2,♯

392

A. Asada

If f (x) =

Q∞

n=1

fn (xn ) and

Z

bn

an

fn (xn )dxn = cn , cn 6= 0, ±∞, then by

using scaling transformation Ic , c = (c1 , c2 , . . .), we have Z ∞ ∞ Z bn Y Y fn (xn )dxn : . fn (xn ) : d∞ x :=: Da,b n=1

n=1

(10)

an

6. Periodic functions on H ♯ d/2

We denote Z∞ the abelian group generated by µn en , n = 1, 2, . . .. It is a closed subgroup of H, but not closed as a subgroup of H ♯ . The closure ˆ ∞ of Z∞ in H ♯ is Z ˆ ∞ = Z∞ ⊕ Ze∞ . Z

(11)

ˆ ∞ are discrete subgroups of H and H ♯ , respectively. We can Z∞ and Z d/2 d/2 take D0,µd/2 , 0 = (0, 0, . . .) and µd/2 = (µ1 , µ2 , . . .) as a fundamental ˆ ∞ . Then we can identify a periodic function f ; f (x) = f (x+n), domain of Z ˆ ∞ and a function on f such that n∈Z f (x)|xn =0 = f (x)|xn =µd/2 , n

n = 1, 2, . . . .

(12)

We also set T∞ = H/Z∞ ,

ˆ ∞ = H ♯ /Z ˆ ∞. T

(13)

S 1 = Re∞ /Ze∞ .

(14)

By definitions, we have ˆ ∞ = T∞ × S 1 , T

ˆ ∞ to T∞ and S 1 are denoted by p0 and p∞ , respecThe projections from T tively. ˆ ∞ ) and C 1 (T∞ ) be the space of Frech´et differentiable C 1 -class Let Cb1 (T b ˆ ∞ ) and T∞ ). Then p∗ (C 1 (T∞ )) ⊗ p∗ (C 1 (S 1 )) is dense in functions on T ∞ 0 b ˆ ∞ ). Cb1 (T Taking D0,µd/2 as the fundamental domain, functions of p∗0 (Cb1 (T∞ )) are expansed by finite products of trigonometric functions xn ). (15) cmn = cos(2mn πµ−d/2 n P∞ −d/2 On the other hand, since the variable x∞ of Re∞ is n=1 µn xn , (comP∞ −d/2 plex valued) functions of C 1 (S 1 ) is expansed by exp(2mπ n=1 µn xn ). Hence functions of p∗∞ (C 1 (S 1 )) are expansed by infinite products xn ), smn = sin(2mn πµ−d/2 n

fI (x) =

∞ Y

n=1

∗ mn ,

I = (m1,∗ , m2,∗ , . . .),

Zeta regularized integral and Fourier expansion of functions

393

where ∗ is either of s or c. On H ♯ , fI = 0 unless limn→∞ mn = m∞ exists and except finite factors, ∗ = s, or ∗ = c. We say such functions to be elementary trigonometric functions on H ♯ . ˆ ∞ ) is expansed by elementary trigonoTheorem 6.1. Functions in Cb1 (T ♯ metric functions on H . ˆ∞ 7. Fourier expansions of functions on T ˆ − ∞), and the fundamental domain of T ˆ∞ By theorem 6.1, if f ∈ Cb1 (T is taken to be D0,µd/2 , then we have X f (x) = cI fI (x). (16) I

ˆ this gives Regarding f to be the periodic function with the period Z, ♯ the Fourier expansion of periodic functions on H . Regarding fI to be a function on D0,µd/2 , we have by (10) Z fI fJ : d∞ x : = 0, I 6= J.

(17)

D0,µd/2

Z

D0,µd/2

ǫI =

(

1 (detG)d/2 , 2k 1 (detG)d/2 , 2ν−k

Hence we have cI =

1 ǫI

fI2 : d∞ x : = ǫI , mn = 0 except n ∈ {n1 , . . . , nk },

mn 6= 0 except n ∈ {n1 , . . . , nk }.

Z

f (x)fI (x) : d∞ x : .

(18)

(19)

(20)

D0,µd/2

Therefore we obtain Theorem 7.1. If f is a C 1 -class Frech´et differentiable function on H ♯ having the period (12), then we have X 1 Z f (x)fI (x) : d∞ x : fI (x). (21) f (x) = ǫI D d/2 I

0,µ

As an application, we have Z X |f (x)|2 : d∞ x := ǫI |cI |2 > 0, D0,µd/2

I

(22)

394

A. Asada

if f is a C 1 -class Frech´et differentiable function and f 6= 0. Therefore we ˆ ∞ ) by can define regularized inner product : (f, g) : of f, g ∈ Cb1 (T Z : (f, g) := f g¯ : d∞ x : . (23) D0,µd/2

ˆ ∞ ) and this inner product is denoted The Hilbert space obtained from Cb1 (T ˆ ∞ ). Then we have by L2 (T X X ˆ ∞) ∼ L2 (T cI f I | ǫI |cI |2 < ∞}. (24) ={ I

I

8. Some applications of Fourier expansions ∞ X ∂2 be the Laplacian of H. ∆ can not act on r(x) = kxk. ∂x2n n=1 To avoid this difficulty, we have introduced regularized Laplacian : ∆ : by

Let ∆ =

:∆:f =

∞ X

µ2s n

n=1

For example, we have

: ∆ : r(x) =

∂2f |s=0 . ∂x2n

(25)

ν−1 . r(x)2

We consider the following boundary value problem of : ∆ :. f (x)|xn =0 = f (x)|xn =µd/2 , n

∂f (x) ∂f (x) |xn =0 = | n = 1, 2, . . . d/2 . ∂xn ∂xn xn =µn

Q∞ If f (x) = n=1 fn (xn ) is an eigenfunction of this boundary value problem, it must be fn (xn ) = A sin(2mn πµ−d/2 xn ) + B cos(2mn πµ−d/2 xn ). n n Hence an elementary trigonometric function fI may be an eigenfunction of the above boundary value problem of : ∆ :. Let I = (m1,∗ , m2,∗ , . . .). Then fI is an eigenfunction of : ∆ belongP 2 −d ing to − N k=1 mnk µn , if mn,∗ = 0n,c , except n ∈ {n1 , . . . , nk }. While if limn→∞ mn = m∞ ≥ 1, fI (x) becomes an eigenfunction of : ∆ : provided ζ(G, s) is holomorphic at s = −d. If ζ(G, −d) exists, we have : ∆ : fI (x) = −(m2∞ ζ(G, −d) +

N X

k=1

(m2nk − m2∞ µ−d nk )fI (x).

Zeta regularized integral and Fourier expansion of functions

395

ˆ ∞ ), these eigenfuncSince elementary trigonometric functions spans L2 (T tions and eigenvalues exhaust the eigenvalues and eigenfunctions of the above boundary condition of : ∆ :. We may consider Z

∞

1 : d x :=:

D0,µd/2

∞ Y

µd/2 := (detG)d/2 , n

n=1

ˆ ∞ . This shows existence of regularized to be the regularized volume of T ∞ ˆ . By using regularized volume form, we can define (∞ − volume form of T ˆ ∞ and de Rham type cohomology of T ˆ ∞ having the Poincar´e p)-forms on T duality. Since the commutation rule of a p-form φp and an (∞ − q)-form ψ ∞−q should be φp ∧ ψ ∞−q = (−1)p(ν−q) ψ ∞−q ∧ φp , such cohomology can not defined unless ν is not an integer, as a real cohomology group. Note that to define φp ∧ ψ ∞−q = ep(ν−q)πi ψ ∞−q ∧ φp ,

ψ ∞−q ∧ φp = e−p(ν−q)πi φp ∧ ψ ∞−q ,

ˆ ∞ is well defined (cf. complex coefficients de Rham type cohomology of T Refs. 7,8). But we omit details. References 1. A. Asada, Regularized calculus: An application of zeta-regularization to infinite dimensional geometry and analysis, Int. J. Geom. Meth. Mod. Phys. 1 (2004) 107–157. 2. A. Asada, Regularized volume form of the sphere of a Hilbert space with the determinant bundle, In: Differential Geometry and Its Applications (Proc. DGA2004, (J. Bureˇs, O. Kowalski, D. Krupka and J. Slov´ ak, Eds.) matfyzpress, Prague, 2005) 397–409. 3. A. Asada, Fractional calculus and regularized residue of infinite dimensional space, In: Mathematical Methods in Engineering ((K. Tas, J.A. Tenreiro Machado and D. Baleanu, Eds.) Springer, 2007) 3–11. 4. A. Asada, Regularized integral and Fourier expansion of functions on infinite dimensional tori, preprint. 5. A. Asada and N. Tanabe, Regularization of differential operators of a Hilbert space and meanings of zeta-regularization, Rev. Bull. Cal. Math. Soc. 11 (2003) 45–52. 6. A. Cardona, C. Ducourtioux and S. Paycha, From tracial anomalies to anomalies in quantum filed theory, Commun. Math. Phys. 242 (2002) 31–65. 7. A. Connes, Entire cyclic cohomology of Banach algebra and characters of θ-summable Fredholm module, K-theory 1 (1988) 519–548.

396

A. Asada

8. J. Cuntz, Cyclic Theory, Bivariant K-theory and the Chern-Connes Character, In: Cyclic Cohomology in Non-Commutative Geometry (EMS 121, Springer, 2001) 1–71. 9. P. Gilkey, The residue of global η function at the origin, Adv. Math. 40 (1981) 290–307. 10. S. Paycha, Renormalized trace as a looking glass into infinite dimensional geometry, Infin. Dim. Anal. Quantum Prob. Relat. Top. 4 (2001) 221–266. 11. B. Simon, Trace Ideals and Their Applications (Cambridge, 1979).

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 397–406

397

On the direction independence of two remarkable Finsler tensors S. B´ acs´ o and Z. Szilasi Institute of Informatics, University of Debrecen, Debrecen, Hungary E-mail: [email protected], [email protected] www.inf.unideb.hu Finsler manifolds some of whose characteristic tensors are direction independent provide stimulation for current research. In this paper we show that the direction independence of the Landsberg and the stretch tensor implies the vanishing of these tensors. Keywords: Finsler manifolds, Landsberg tensor, stretch tensor, direction independence. MS classification: 53B40.

1. Basic constructions Throughout this paper, M will be an n-dimensional smooth manifold. C ∞ (M ) denotes the ring of real-valued smooth functions on M . Tp M is the S Tp M is the tangent bundle of M, tangent space to M at p ∈ M , T M := p∈M ◦

τ : T M → M is the natural projection. T M denotes the open subset of the ◦

◦

nonzero tangent vectors to M , τ := τ ↾ T M . X(M ) is the C ∞ (M )-module of (smooth) vector fields on M . Capitals X, Y, . . . will denote vector fields on M, while, usually, Greek letters ξ, η, ζ, . . . will stand for vector fields on T M . iξ is the substitution operator induced by ξ ∈ X(T M ), d denotes the operator of the exterior derivative. All of our considerations will be purely of local character, so we may assume without loss of
generality that our base manifold M admits a global n coordinate system ui i=1 ; this assumption simplifies a little the notation. Then xi , y i


n

i=1

;

xi := ui ◦ τ, y i := dui

398

´ o´ and Z. Szilasi S. Bacs

is a coordinate system for T M . These coordinate systems yield the bases n n   n  ∂ ∂ ∂ ˙i , and =: ∂ , ∂ i ∂ui i=1 ∂xi ∂y i i=1 i=1 of X(M ) and X(T M ), respectively. S r Ts (Tp M ) be the bundle of type Let Tsr M := p∈M

r s




tensors over M ,

and let τsr : T
sr M −→ M be the natural projection. Following Z. I. Szab´o,9 by a type rs Finsler tensor field over M we mean a smooth map ◦

e : T M −→ T r M such that τ r ◦ A e = τ◦ . A s s ◦

These C ∞ (T M )-module, which will be denoted by ◦  ◦ tensor fields form a◦  Tsr τ . In particular, X τ := T01 τ is the module of Finsler vector ◦ fields, and X∗ τ is its dual. In what follows, Finsler tensor fields will simply be mentioned as tensors, or, for obvious reasons, tensors along the ◦ projection τ . Evidently, the construction also works on the whole T M , and leads to the C ∞ (T M )-modules Tsr (τ ). Via restrictions, Tsr (τ ) may be ◦ interpreted as a submodule of Tsr τ ; we shall use this harmless inclusion in what follows. b := X ◦ τ is a Finsler vector field, If X is a vector field on M , then X  

called a basic vector field along τ . In particular,

b ∂ ∂ui

n

i=1

is a base for the

module X(τ ). Besides the class of basic vector fields, a distinguished role is played by the canonical Finsler vector field  δ := 1T M , called classically b ∂ (with sum convention in support element. In coordinates, δ = y i ∂u i force). We have a canonical C ∞ (T M )-linear injection i : X(τ ) −→ X(T M ) and a surjection j : X(T M ) −→ X(τ ) such that !     ∂b ∂b ∂ ∂ ∂ i = = = 0; i ∈ {1, . . . , n} . ; j , j ∂ui ∂y i ∂xi ∂ui ∂y i v (For an intrinsic construction of i and j, see e.g. Ref. 10.  X(T M ) := i (X(τ )) b is the vertical lift is the module of vertical vector fields on T M , X v := i X

of X ∈ X(M ). C := i(δ) is called the Liouville vector field. In coordinates, ∂ C = y i ∂y i . J := i ◦ j is said to be the vertical endomorphism of X(T M ). It follows at once that Im(J) = Ker(J) = Xv (T M ),

J2 = 0.

On the direction independence of two remarkable Finsler tensors

399

We define the dJ -differential of a smooth function f on T M as the one-form ∂f i dJ f := df ◦ J on TM. In coordinates, dJ f = ∂y i dx . e be a Finsler vector field over M. We The formalism can go on. Let X

define a tensor derivation ∇vXe on the algebra of Finsler tensor fields by the following requirements:   e f ; f ∈ C ∞ (T M ). (i) On functions, ∇vXe f := iX h i e η , where η ∈ X(T M ) is such (ii) On Finsler vector fields, ∇vXe Ye := j iX, that j(η) = Ye . e ∈ T r (τ ), then ∇v A e is given by the product rule. (iii) If A s

e X

e ∇vXe is called the (canonical) v-covariant derivative with respect to X. e = ξ i ∂b i , Ye = η i ∂b i , then In coordinates: if X ∂u ∂u ∇vXe f = ξ i

j ∂f ∂ v e i ∂η , ∇ Y = ξ . e i i X ∂y ∂y ∂y j

We see that ∇vXe Ye is well-defined: it does not depend on the choice of the vector field η. We have, in particular, ∇v Yb = 0 for any vector fields X,Y b X

on M .
As a final step, we define the vertical differential of a type rs Finsler
e as the r tensor ∇v A e which ‘collects all the v-covariant tensor field A s+1 e For simplicity, if r = s = 1, then derivatives’ of A.      e α e := ∇v A e α e (iii) e, Ye , X e , Y = ∇v A e X         v e e α e ∇v α e −A e α e A e, Ye − A e , Y e , ∇ Y iX e e X X

e Ye ∈ X(τ ) and α for all
X, e ∈ X∗ (τ ). More generally, if the components of r e an s tensor A are !   d d ∂ ∂ i ...i 1 r d d ci := dui ◦ τ 0 , ir i1 e e du , . . . , js A 1 i1 ...js := A du , . . . , du , j 1 ∂u ∂u ei1 ...ir ; these functions will be denoted e are ∂˙j A then the components of ∇v A i1 ...js ei1 ...ir . We recognize that in components ’vertical differentiation reby A i1 ...js ·j duces to partial differentiation with respect to the fibre coordinates’. Notice that ∇v f and dJ f are related by dJ f = ∇v f ◦ j , f ∈ C ∞ (T M ) .

400

´ o´ and Z. Szilasi S. Bacs

2. Finsler functions. The h-Berwald derivative By a Finsler function we mean a continuous function F : T M −→ [0, ∞[, satisfying the three conditions: ◦

(i) F is smooth on T M . (ii) F is positive-homogeneous of degree 1, i.e., F (λv) = λF (v) for all λ ∈ R∗+ and v ∈ T M . ◦

(iii) The metric tensor g := 12 ∇v ∇v F 2 is pointwise non-degenerate on T M . A manifold endowed with a Finsler function is said to be a Finsler manifold. Quite surprisingly, under these conditions the metric tensor g is positive  b b ∂ ∂ definite, see Ref. 6. The components gij := g ∂u , of g are just the i ∂uj 1 ˙ ˙ functions ∂i ∂j F 2 . 2

In the remainder of the paper, (M,F) will be a Finsler manifold. We show that the Finsler function F and the metric tensor g are related by g (δ, δ) = F 2 .

(1)

Indeed, by the homogeneity of F , we have CF 2 = 2F 2 , and it can easily be checked that ∇v δ = 1X(τ ). Thus


1 1 1 v v 2 ∇ ∇ F (δ, δ) = ∇vδ ∇v F 2 (δ) = C CF 2 − ∇v F 2 (δ) 2 2 2
1 4F 2 − 2F 2 = F 2 . = 2

g (δ, δ) =

In the Finslerian case the canonical vector field δ has a dual 1-form δ ∗ ◦ along τ given by       e := g X, e Ye ; X e ∈ X∗ τ◦ . δ∗ X (2)

Then δ ∗ (δ) = F 2 by (1). The components of δ ∗ can be obtained from the components of δ by index lowering: ! ! b b b ∂ ∂ ∂ =g , y j j = gij y j , i ∈ {1, . . . , n} . yi := δ ∗ ∂ui ∂ui ∂u It follows immediately that yi y i = F 2 .

(3)

On the direction independence of two remarkable Finsler tensors

401


By the Cartan tensor of (M, F ) we mean the type 03 Finsler tensor C♭ := 12 ∇v g. Its components are ! ∂b 1 ∂b ∂b 1 Cijk := C♭ = ∂˙k gij = ∂˙k ∂˙j ∂˙i F 2 , , , ∂ui ∂uj ∂uk 2 4

thus C♭ is totally symmetric. Raising an index, we get the vector-valued Cartan tensor C , metrically equivalent to C♭ . More pedantically, C is defined by         e Ye , Z e = C♭ X, e Ye , Z e ; X, e Ye , Z e ∈ X τ◦ , g C X,

so its components are

i g ir Cjkr =: Cjk ;


g ij := (gij )−1 .

It is a fundamental fact, that F determines a unique spray S : T M −→ T T M via the Euler-Lagrange equation iS ddJ F 2 = −dF 2 . S is called the canonical spray of the Finsler manifold. In coordinates, S = y i ∂i − 2Gi ∂˙i , where  2 2  ∂ F ∂F 2 1 r , i ∈ {1, . . . , n} . y − Gi = g ij 4 ∂xr ∂y j ∂xj ◦

The spray coefficients Gi are of class C 1 on T M , smooth on T M and are positively homogeneous of degree 2. The canonical spray determines the  ◦ canonical Ehresmann connection H : X τ −→ X (T M ) of (M, F ) by

Crampin’s construction 4     b ∈ X τ◦ 7−→ X h := H X b := X

1 2

(X c + [X v , S]) , X ∈ X(M )

(X c denotes the complete lift of X). X h is called the horizontal lift of X. ∂ The horizontal lifts of the coordinate vector fields ∂u j take the form h  ∂ ∂Gi ∂ ∂ = − , j ∈ {1, . . . , n} ; j j ∂u ∂x ∂y j ∂y i the functions Gij := ∂˙j Gi are said to be the Christoffel symbols of H . The horizontal and the vertical projector associated to H are h := H ◦j and v = 1  ◦  − h, respectively. Following Berwald’s terminology,3 we X TM

call the type




1 2

Finsler tensor R defined by

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´ o´ and Z. Szilasi S. Bacs

    b Yb := −v X h , X v ; X, Y ∈ X(M ) iR X,

the fundamental affine curvature of the Finsler manifold. To be in harmony i with Matsumoto’s conventions,8 we define the components Rjk of R by   b b b ∂ ∂ ∂ i i ˙ i Rjk ∂ui = R ∂uk , ∂uj . If Gjk := ∂k Gj , then i Rjk



∂ = ∂uk

h

Gij

−



∂ ∂uj

h

Gik = ∂k Gij − ∂j Gik + Grj Girk − Grk Girj . (4)

In the spirit of Berwald’s above mentioned paper, by the affine curvature
tensor of (M, F ) we mean the type 13 tensor H:= ∇v R. The components b

∂ i of H are determined by Hjkl ∂ui := H

b b ∂ ∂ ∂ul , ∂uk

∂b ∂uj

i i i Hjkl = ∂˙j Rkl = Rkl·j ,

. Obviously,

(5)

We define a further important tensor, the Berwald curvature B, by      b Yb Zb := X v , Y h , Z v ; X, Y, Z ∈ X(M ). iB X,

Its components are Gijkl := ∂˙l Gijk . Following the above scheme, we construct a further tensor derivation on the algebra of Finsler  ◦ tensors, depending on the canonical Ehresmann e connection. Let X ∈ X τ . Define the operator ∇hXe ◦

e , f ∈ C ∞ (T M ); (i) on functions by ∇hXe f := (H X)f h i e iYe ; (ii) on Finsler vector fields by i∇hXe Ye := v H X,
(iii) on type rs tensors by the product rule.

e Its Christoffel ∇hXe is said to be the h-Berwald derivative with respect to X. i symbols are just the functions Gjk , i.e., we have ∇hd ∂

∂uk

∂b ∂b i =: G . jk ∂uj ∂ui

After this the h-Berwald differential ∇h can be defined in the same way as the vertical differential ∇v (formally: replace the canonical injection i by e are the surjection H). As for the index gymnastics, if the components of A i ...i i ...i h 1 r 1 r e e e A j1 ...js , then the components of ∇ A will be denoted by Aj1 ...js ;j . These e As an functions are much more complicated than the components of ∇v A. illustration, we calculate the components of the h-Berwald differential of

On the direction independence of two remarkable Finsler tensors

403

the metric tensor g: gij;k :=∇h g

= =



∂b ∂uk

∂ ∂uk

∂b ∂b ∂b , , ∂ui ∂uj ∂uk !h

h

!

gij − g ∇h ∂b

:=

∂uk



∇h ∂b g ∂uk

∂b ∂b ∂ui ∂uj

!



−g

∂b ∂b , ∂ui ∂uj

!

∂b ∂b h , ∇ b ∂ j ∂ui ∂uk ∂u

!

gij − Grik grj − Grjk gir .

3. Landsberg tensor depending only on the position
By the Landsberg tensor of a Finsler manifold (M, F ) we mean the type tensor

0 3

P := − 12 ∇h g

◦

along τ . Its components are Pijk = − 12 gij;k , where the functions gij;k have just been calculated. The Landsberg tensor and the Cartan tensor C♭ are related by P = ∇hδ C♭ .

(6)

Pijk = Cijk;l y l ,

(7)

In components,

which may easily be shown. A coordinate-free proof of (6) needs a little more effort, see Ref. 10, section 3.11. Now we are in a position to show that the property ∇v P = 0 implies a drastic consequence. Proposition 3.1. If the Landsberg tensor of a Finsler manifold depends only on the position, then it vanishes identically. Proof. Keeping the notation introduced above, suppose that ∂˙l Pijk = Pijk·l = 0. Then differentiation of relation (7) with respect to ∂˙l leads to Cijk;l + Cijk;r·l y r = 0.

(8)

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´ o´ and Z. Szilasi S. Bacs

Now we use the Ricci identity (Ref. 8, 2.5.5) for the h-Berwald derivative and the vertical derivative. Then we obtain Cijk;r·l − Cijk·l;r = −Csjk Gsilr − Cisk Gsjlr − Cijs Gsklr

(9)

(recall that Gijkl = ∂˙l Gijk are the components of the Berwald tensor). Since the functions Gijk are positively homogeneous of degree 0, we have Gijkl y l = 0. Thus, transvection of (9) with y r leads to Cijk;r·l y r = Cijk·l;r y r . Hence (8) takes the form Cijk;l + Cijk·l;r y r = 0.

(10)

Interchanging indices k and l, we obtain Cijl;k + Cijl·k;r y r = 0.

(11)

Since Cijk·l = Cijl·k , if we subtract (11) from (10) we find that Cijk;l − Cijl;k = 0.

(12)

But transvection of (12) with y l yields Cijk;l y l = 0,

(13)

∂g

since Cijl y l = 12 ∂yijl y l = 0 by the 0+ -homogenity of the functions gij , and by the commutation of contractions and covariant derivatives. Relations (13) and (7) imply our assertion P = 0. 4. Stretch tensor depending only on the position Inspired by a manuscript of L. Kozma,5 we define the stretch tensor Σ of a Finsler manifold (M, F ) by     1  e e e e e Ye , Z, e U e − ∇h P X, e Ye , U e, Z e , (14) Σ X, Y , Z, U := ∇h P X, 2 e Ye , Z, e U e are arbiwhere P is the Landsberg tensor discussed above, and X, trary Finsler vector fields. Since the components of ∇h P are !h ∂b Pijk − Gril Prjk − Grjl Pirk − Grkl Pijr , Pijk;l = ∂ul

it follows that the components of Σ are

Σijkl = 2 (Pijk;l − Pijl;k ) .

(15)

On the direction independence of two remarkable Finsler tensors

405

This is just the formula obtained by M. Matsumoto for the stretch tensor in Ref. 7. Notice that the stretch tensor was discovered by L. Berwald.1 He also found an important relation between the affine curvature and the stretch tensor, which may be formulated as follows:        e U e , X, e Ye , X, e Ye , Z, e U e ∈ X τ◦ . (16) e Ye , Z, e U e = −δ ∗ ∇v H Z, Σ X,

In terms of tensor components, (16) leads to

r , Σijkl = −yr ∂˙j Hikl

(17)

2

this is just formula (14) of Berwald’s paper. From (5) and (17) it follows that we also have r . Σijkl = −yr ∂˙j ∂˙i Rkl

(18)

Relations (16) and (18) imply that the stretch tensor vanishes, if the fundamental affine curvature, or, equivalently, the affine curvature tensor of the Finsler manifold depends only on the position. Now we shall show that this conclusion is also true, if Σ itself has this property. Proposition 4.1. If the stretch tensor of a Finsler manifold depends only on the position, then it vanishes identically. Proof. We use the same tactics as in the previous proof. By our condition, Σijkl·m = ∂˙m Σijkl = 0 , so from (15) we get for the Landsberg tensor Pijk;l·m − Pijl;k·m = 0.

(19)

Using the Ricci identity for Pijk;l·m we get Pijk;l·m = Pijk·m;l − Prjk Griml − Pirk Grjml − Pijr Grkml .

(20)

Pijk;l·m y i = Pijk·m;l y i

(21)

Transvection of (20) with y i leads to

(7)

because of Pijk y i = Cijk;l y i y l = 0. In the same way we obtain Pijl;k·m y i = Pijl·m;k y i .

(22)

Relations (19), (21) and (22) imply that Pijk·m;l y i = Pijl·m;k y i .

(23)

On the other hand, from the identity Prjk y r = 0 we obtain by repeated covariant differentiation

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0 = (Prjk y r )·i;l = Pijk;l + Prjk·i;l y r . Interchanging indices k and l, we get Pijl;k + Prjl·i;k y r = 0 . (23) and the last two relations imply that Pijk;l − Pijl;k = 0. Hence, by (15), Σijkl = 0. Acknowledgments The first author was supported by National Science Research Foundation OTKA No. T48878. References ¨ 1. L. Berwald, Uber Parallel¨ ubertragung in R¨ aumen mit allgemeiner Massbestimmung, Jber. Deutsch Math.-Verein 34 (1926) 213–220. 2. L. Berwald, Parallel¨ ubertragung in allgemeinen R¨ aumen, Atti. Congr. Intern. Mat. Bologna 4 (1928) 263–270. 3. L. Berwald, Ueber Finslersche und Cartansche Geometrie IV, Ann. Math. 48 (1947) 755–781. 4. M. Crampin, On horizontal distribution on the tangent bundle of a differentiable manifold, J. London Math. Soc. (2) 3 (1971) 178–182. 5. L. Kozma, Unpublished manuscript. 6. R.L. Lovas, A note on Finsler-Minkowski norms, Houston J. Math. 33 (2007) 701–707. 7. M. Matsumoto, On the stretch curvature of a Finsler space and certain open problems, J. Nat. Acad. Math. India 11 (1997) 22–32. 8. M. Matsumoto, Finsler Geometry in the 20th-Century, In: Handbook of Finsler Geometry ((P. Antonelli, Ed.) Kluwer Academic Publishers, Dordrecht, 2003). ¨ 9. Z.I. Szab´ o, Uber Zusammenh¨ ange vom Finsler-Typ, Publ. Math. Debrecen 27 (1980) 77–88. 10. J. Szilasi, A Setting for Spray and Finsler Geometry, In: Handbook of Finsler Geometry ((P. Antonelli, Ed.) Kluwer Academic Publishers, Dordrecht, 2003).

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 407–417

407

Lie systems and integrability conditions of differential equations and some of its applications J.F. Cari˜ nena and J. de Lucas Departamento de F´ısica Te´ orica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain E-mail: [email protected], [email protected] The geometric theory of Lie systems is used to establish integrability conditions for several systems of differential equations, in particular some Riccati equations and Ermakov systems. Many different integrability criteria in the literature will be analysed from this new perspective, and some applications in physics will be given. Keywords: Integrability criteria, Riccati, Ermakov systems, Milne–Pinney equation, superposition rules, Lie systems. MS classification: 34A26.

1. Introduction Non-autonomous systems of first-order and second-order differential equations appear in many places in physics. For instance, Hamilton equations are systems of first-order differential equations, while Euler–Lagrange equations for regular Lagrangians are systems of second-order differential equations. A system of second-order differential equations in n variables of the form x¨i = F i (x, x, ˙ t), with i = 1, . . . , n, is related with a system of first-order equations in 2n variables:  i x˙ = v i , i = 1, . . . n . (1) v˙ i = F i (x, v, t) Therefore, it is enough to restrict ourselves to study systems of first-order differential equations. From the geometric viewpoint a system x˙ i = X i (x, t) ,

i = 1, . . . , n ,

(2)

is associated with the t-dependent vector field X = X i (x, t) ∂/∂xi whose integral curves are determined by the solutions of the system.

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Unfortunately, there is no general method for solving such equations. Relevant questions about integrability are how to find a particular solution (determined by x(0) = x0 ), or a r-parameter family of solutions, or even the general solution (a n-parameter family of solutions). Finally, when is it possible to find and how to determine a superposition rule for solutions? We shall understand that to find a solution means to reduce the problem to carry out some quadratures. For instance, the general solution of the inhomogeneous linear differential equation dx/dt = b0 (t) + b1 (t)x can be found with two quadratures and it is given by ! ! Z t  Z t′ Z t x(t) = exp b1 (s) ds dt′ . b0 (t′ ) exp − b1 (s) ds × x0 + 0

0

0

Actually, when the systems we are dealing with are linear, there is a linear superposition principle allowing us to find the general solution as a linear combination of n particular solutions. For instance, for the harmonic oscillator with a t-dependent angular frequency ω(t):  x˙ = v x ¨ = −ω 2 (t) x ⇐⇒ , v˙ = −ω 2 (t) x whose solutions are the integral curves of the t-dependent vector field X = v∂/∂x − ω 2 (t)x ∂/∂v, if we know a particular solution, the general solution can be found by means of one quadrature and if we know two particular solutions, x1 and x2 , the general solution is a linear combination (no quadrature is needed) x(t) = k1 x1 (t) + k2 x2 (t). There are systems whose general solution can be written as a nonlinear function of some particular solutions. For instance, for Riccati equation: if a particular solution is known, the general solution is obtained by two quadratures, if two particular solutions are known the problem reduces to one quadrature and, finally, when three particular solutions are known, x1 , x2 and x3 , the general solution can be found from the cross-ratio relation x − x1 x3 − x1 : =k , x − x2 x3 − x2 which provides us a nonlinear superposition rule.1 There also exist cases in which we can superpose solutions of one system for finding solutions of another one. We have seen one example: the general solution of the inhomogeneous linear equation can be written as x(t) = x1 (t) + C x0 (t), where x0 (t) is a solution of the associated homogeneous equation and x1 (t) is a particular solution of the inhomogeneous linear one.

Lie systems and integrability conditions of differential equations

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Milne-Pinney equation2 x ¨ = −ω 2 (t)x + k/x3 is usually studied together with the time-dependent harmonic oscillator y¨ + ω 2 (t)y = 0 and the system is called Ermakov system. Pinney showed in a short paper2 that the general solution of the first equation can be written as a nonlinear superposition of two solutions of the associated harmonic oscillator. All these properties can be better understood in the framework of Lie systems, conveniently extended in some cases to include systems of second-order differential equations. These systems have a lot of applications not only in mathematics but also in many different branches of classical and quantum physics. Let us look for systems admitting a (maybe nonlinear) superposition rule. The main result was given by Lie:3,4 Theorem 1.1. Given (2) a necessary and sufficient condition for the existence of a function Φ : Rn(m+1) → Rn such that the general solution is x = Φ(x(1) , . . . , x(m) ; k1 , . . . , kn ), with {x(a) | a = 1, . . . , m} being a set of particular solutions of the system and k1 , . . . , kn , are n arbitrary constants, is that the system can be written as dxi = Z 1 (t)ξ1i (x) + · · · + Z r (t)ξri (x), dt where Z 1 , . . . , Z r , are r functions depending only on t and ξαi , α = 1, . . . , r, are functions of x = (x1 , . . . , xn ), such that the r vector fields in Rn given n X ξαi (x1 , . . . , xn )∂/∂xi , α = 1, . . . , r, close on a real finiteby Xα ≡ i=1

dimensional Lie algebra, i.e. the Xα are l.i. and there are r3 real numbers, r X cαβ γ , such that [Xα , Xβ ] = cαβ γ Xγ . The number r satisfies r ≤ mn. γ=1

The condition in the Theorem 1.1 is that X(x, t) can be written as r X X(x, t) = Z α (t)Xα (x), with Xα as mentioned above. α=1

Non-autonomous systems corresponding to such t-dependent vector fields will be called Lie systems. One instance of Lie system is the Riccati equation1 dx(t) = b2 (t) x2 (t) + b1 (t) x(t) + b0 (t) , dt

(3)

for which m = 3 and the superposition principle comes from the relation x − x1 x3 − x1 k x1 (x3 − x2 ) + x2 (x1 − x3 ) : = k =⇒ x = . x − x2 x3 − x2 k (x3 − x2 ) + (x1 − x3 )

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The associated Lie algebra is generated by X0 , X1 and X2 given by X0 = which close on a

∂ , ∂x

X1 = x

∂ , ∂x

X2 = x2

∂ , ∂x

sl(2, R) 3-dimensional real Lie algebra, because

[X0 , X1 ] = X0 ,

[X0 , X2 ] = 2X1 ,

[X1 , X2 ] = X2 .

(4)

The time-dependent harmonic oscillator is also an example of physical relevance. It is described by a Hamiltonian H=

1 1 p2 + m(t)ω 2 (t)x2 , 2 m(t) 2

which gives rise to the dynamics defined by the t-dependent vector field X(x, p, t) =

1 ∂ ∂ p − m(t)ω 2 (t) x . m(t) ∂x ∂p

If we consider the set of vector fields   ∂ 1 ∂ ∂ x , −p X0 = p , X1 = ∂x 2 ∂x ∂p

X2 = −x

∂ , ∂p

which close on a sl(2, R) Lie algebra with the same commutation relations as (4), the corresponding t-dependent vector field X can be written as a linear combination X(·, t) = m(t)ω 2 (t) X2 (·) + (1/m(t)) X0 (·), i.e. it is a 2 X linear combination with t-dependent coefficients X(·, t) = bα (t)Xα (·) α=0

with b0 (t) = 1/m(t), b1 (t) = 0 and b2 (t) = m(t)ω 2 (t). The prototype of Lie system is the time-dependent right-invariant vector fields in a Lie group G. Let {a1 , . . . , ar } denote a basis of Te G. A rightinvariant vector field X R is one such that X R (g) = Rg∗e Xe . Define XαR by XαR (g) = Rg∗e aα . The t–dependent right-invariant vector field ¯ t) = − X(g,

r X

bα (t)XαR (g) .

α=1

defines a Lie system in G whose integral curves are solutions of the system Pr g˙ = − α=1 bα (t) XαR (g), and when applying Rg−1 to both sides we see that g(t) satisfies Rg−1 (t)∗g(t) g(t) ˙ =−

r X

α=1

bα (t)aα ∈ Te G .

(5)

Let H be a closed subgroup of G and consider the homogeneous space M = G/H. Then, G can be seen as a principal bundle over G/H: (G, τ, G/H).

Lie systems and integrability conditions of differential equations

411

The XαR are τ -projectable on the corresponding fundamental vector fields of the left-action λ : (g, g ′ H) ∈ G × M → (gg ′ H) ∈ M given by −Xα = −Xaα with τ∗g XαR (g) = −Xα (gH), the projected vector field in M will Pr be X(x, t) = α=1 bα (t)Xα (x), and its integral curves are the solutions of Pr the system of differential equations: x˙ = α=1 bα (t)Xα (x). The solution of this last system starting from x0 is x(t) = λ(g(t), x0 ), with g(t) being the solution of (5) such that g(0) = e. This means that solving such Lie system in G we are simultaneously solving the corresponding problems in all its homogeneous spaces. 2. SODE Lie systems A system of second order differential equations can be studied through the corresponding system of first-order differential equations as indicated in (1). We call SODE Lie systems those for which the associated first-order one is a Lie system, i.e. it can be written as a linear combination with t-dependent coefficients of vector fields closing on a finite-dimensional real Lie algebra. An example is the 1-dimensional harmonic oscillator with timedependent frequency, but the same is true for the 2-dimensional isotropic harmonic oscillator with time-dependent frequency, with an associated vector field ∂ ∂ ∂ ∂ − ω 2 (t)x1 + v2 − ω 2 (t)x2 , X = v1 ∂x1 ∂v1 ∂x2 ∂v2 which is a linear combination X = X2 − ω 2 (t)X1 with X1 = x1

∂ ∂ + x2 , ∂v1 ∂v2

X2 = v1

∂ ∂ + v2 , ∂x1 ∂x2

and then they close once again on a Lie algebra isomorphic to [X1 , X2 ] = 2 X3 , with X3 =

1 2

[X1 , X3 ] = −X1 ,

sl(2, R):

[X2 , X3 ] = X2 ,

(6)

  ∂ ∂ ∂ ∂ x1 . − v1 + x2 − v2 ∂x1 ∂v1 ∂x2 ∂v2

The search for the superposition rule for a Lie system consists on looking for enough number of first integrals, independent of the time-dependent coefficients, in an extended space in which we consider several replicas of the given vector field. The 2-dimensional case admits an invariant F given by the first integral F (x1 , x2 , v1 , v2 ) = x1 v2 − x2 v1 , which can be seen as a partial superposition

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rule.5 Actually, if x1 (t) is a solution of the first equation, then we obtain for each real number k the first-order differential equation for the variable x2 , x1 (t) dx2 /dt = k + x˙ 1 (t)x2 , from where x2 can be found to be given Rt by x2 (t) = k ′ x1 (t) + k x1 (t) x−2 1 (ζ) dζ. With three copies of the same harmonic oscillator, i.e. X1 and X2 given by X1 = v1

∂ ∂ ∂ + v2 +v , ∂x1 ∂x2 ∂x

X2 = x1

∂ ∂ ∂ + x2 +x , ∂v1 ∂v2 ∂v

there exist two independent first integrals F1 (x1 , x2 , x, v1 , v2 , v) = xv1 −x1 v and F2 (x1 , x2 , x, v1 , v2 , v) = xv2 − x2 v, from where we obtain the expected superposition rule: x = k1 x1 + k2 x2 ,

v = k1 v1 + k2 v2 .

Another interesting non-linear example is the Pinney equation, the second order non-linear differential equation: k , x3 where k is a constant, with associated t-dependent vector field   ∂ ∂ k X=v + −ω 2 (t)x + 3 , ∂x x ∂v x¨ = −ω 2 (t)x +

which is a Lie system because it can be written as X = L2 − ω 2 (t)L1 , where L1 := x ∂/∂v and L2 = (k/x3 ) ∂/∂v + v ∂/∂x generate a threedimensional real Lie algebra isomorphic to sl(2, R) with nonzero defining relations similar to (6) with L3 = (1/2) (x ∂/∂x − v ∂/∂v). Note that this isotonic oscillator shares with the harmonic one the property of having a period independent of the energy, i.e. they are isochronous, and in the quantum case they have a equispaced spectrum. The fact that they have the same associated Lie algebra means that they can be solved simultaneously in the group SL(2, R) by the same equation Rg−1 ∗g g˙ = ω 2 (t) a1 − a2 ,

g(0) = e .

3. Ermakov systems We can consider the generalised Ermakov system given by:  1  x ¨ = 3 f (y/x) − ω 2 (t)x x  y¨ = 1 g(y/x) − ω 2 (t)y  y3

which when f (u) = k and g(u) = 0 reduces to the Ermakov system.

Lie systems and integrability conditions of differential equations

413

This system is described by the t-dependent vector field     ∂ 1 1 ∂ ∂ ∂ + −ω 2 (t)x+ 3 f (y/x) + −ω 2 (t)y+ 3 g(y/x) , +vy X = vx ∂x ∂vy x ∂vx y ∂vy which can be written as a linear combination X = N2 − ω 2 (t) N1 , where N1 and N2 are the vector fields N1 = x

∂ ∂ +y , ∂vx ∂vy

N2 = vx

∂ ∂ ∂ ∂ 1 1 + vy , + f (y/x) + g(y/x) ∂x x3 ∂vx ∂y y 3 ∂vy

that generate a three-dimensional real Lie algebra isomorphic to with a third generator   ∂ 1 ∂ ∂ ∂ x . N3 = +y − vx − vy 2 ∂x ∂vx ∂y ∂vy

sl(2, R)

There exists a first integral for the motion, F : R4 → R, for any ω 2 (t), which satisfies Ni F = 0 for i = 1, . . . , 3, but as [N1 , N2 ] = 2N3 it is enough to impose N1 F = N2 F = 0. The condition N1 F = 0, implies that there exists a function F¯ : R3 → R such that F (x, y, vx , vy ) = F¯ (x, y, ξ = xvy − yvx ). Then using the method of the the characteristics in condition N2 F = 0, we can obtain the first integral:     Z x/y  1 1 1 1 − 3f +ug du . F (x, y, vx , vy ) = (xvy − yvx )2 + 2 u u u For the Ermakov system with f (1/u) = k and g(1/u) = 0 we obtain the known Ermakov invariant k  y 2 1 F (x, y, vx , vy ) = + (xvy − yvx )2 2 x 2 We can now consider a system made up by a Pinney equation with two associated harmonic oscillator equations, with associated t-dependent vector field   ∂ ∂ ∂ ∂ ∂ ∂ k ∂ 2 X = vx − ω (t) x +y +z + vy + vz + 3 ∂x ∂y ∂z y ∂vy ∂vx ∂vy ∂vz which can be expressed as X = N2 − ω 2 (t)N1 where N1 and N2 are: N1 = y

∂ ∂ ∂ +x +z , ∂vy ∂vx ∂vz

N2 = vy

∂ ∂ ∂ 1 ∂ + vx + + vz , ∂y y 3 ∂vy ∂x ∂z

These vector fields generate a three-dimensional real Lie algebra isomorphic to sl(2, R) with the vector field N3 given by   ∂ ∂ ∂ ∂ ∂ ∂ 1 x . +y +z − vx − vy − vz N3 = 2 ∂x ∂vx ∂y ∂vy ∂z ∂vz

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In this case there exist three first integrals for the distribution generated by these fundamental vector fields: The Ermakov invariant I1 of the subsystem involving variables x and y, the Ermakov invariant I2 of the subsystem involving variables y and z, and finally, the Wronskian W of the subsystem involving variables x and z. They are given by W = xvz − zvx ,  2 !  2 ! 1 x z 1 2 2 (yvx − xvy ) + k , I2 = (yvz − zvy ) + k . I1 = 2 y 2 y In terms of these three integrals we can obtain an explicit expression of y in terms of x, z and the integrals I1 , I2 , W : √  1/2 p 2 I2 x2 + I1 z 2 ± 4I1 I2 − cW 2 xz . y= W This can be interpreted as saying that there is a superposition rule allowing us to express the general solution of the Pinney equation in terms of two independent solutions of the corresponding harmonic oscillator with time-dependent frequency. 4. The reduction method and integrability criteria Given an equation (5) on a Lie group, it may happen that the only nonvanishing coefficients are those corresponding to a subalgebra h of g and then the equation reduces to a simpler equation on a subgroup, involving less coordinates. An important result is that if we know a particular solution of the problem associated in a homogeneous space, the original solution reduces to one on the isotopy subgroup. One can show that there is an action of the group G of curves in G on the set of right-invariant Lie systems in G (see e.g. Ref. 6 for a geometric justification), and we can take advantage of such an action for transforming a given Lie system into another simpler one. So, if g(t) is a solution of the given Lie system and we choose a curve g ′ (t) in the group G, and define a curve g(t) by g(t) = g ′ (t)g(t), then the new curve in G, g(t), determines a new Lie system. Indeed, ˙ Rg(t)−1 ∗g(t) (g(t)) = Rg′ −1 (t)∗g′ (t) (g˙ ′ (t)) −

r X

bα (t)Ad (g ′ (t))aα ,

α=1

which is similar to the original one, with a different right-hand side. Therefore, the aim is to choose the curve g ′ (t) in such a way that the new equation be simpler. For instance, we can choose a subgroup H and look for a choice of g ′ (t) such that the right hand side lies in Te H, and hence g(t) ∈ H for all

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t. This can be done when we know a solution of the associated Lie system in G/H allows us to reduce the problem to one in the subgroup H.7 Theorem 4.1. Each solution of (5) on the group G can be written in the form g(t) = g1 (t) h(t), where g1 (t) is a curve on G projecting onto a solution g˜1 (t) for the left action λ of G on the homogeneous space G/H and h(t) is a solution of an equation but for the subgroup H, given explicitly by (Rh−1 ∗h h˙ )(t) = −Ad (g1−1 (t))

r X

α=1

!

bα (t)aα + (Rg−1 ∗g1 g˙ 1 )(t) 1

∈ Te H .

This fact is very important because one can show that Lie systems associated with solvable Lie algebras are solvable by quadratures and therefore, given a Lie system with an arbitrary G having a solvable subgroup, we should look for a possible transformation from the original system to one which reduces to the subalgebra and therefore integrable by quadratures. By the last Theorem there always exists a curve in G that transforms the initial Lie system into a new one related with solvable a Lie subgroup of G. Nevertheless, it can be difficult to find out a solution of the equation in M that determines this transformation. Then, to be able to obtain one is more interesting to suppose also that this transformation is a curve in a certain subset of G, i.e. a one-dimensional subgroup. It would be easier to obtain a transformation but it may be that such a transformation does not exist. In summary: The conditions for the existence of such a transformation of a certain form are integrability conditions for the system. We could choose for showing this assertion a particular example: Riccati equation. One can find in the literature a lot of integrability criteria for Riccati equation,8–10 all of them particular examples of the above method.11 We can also consider other equivalent examples as the Pinney equation, the (generalized) Ermakov system or more relevant examples in Physics, for instance, time-dependent harmonic oscillators. The results obtained for one system are valid for the other; they are essentially conditions for the equation in the group, and all are examples of Lie systems associated with the same Lie group: SL(2, R). Consider, for instance, the Riccati equation (3). The group SL(2, R) contains the affine group (either the one generated by X0 and X1 or the one generated by X1 and X2 ), which is SOLVABLE. Therefore, a transformation from the given equation to one of this subgroup allows us to express the general solution in terms of quadratures. This happens when we know a particular solution x1 of the given equation:

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x = x1 + z, what corresponds to choose   1 −x1 g¯(t) = 0 1

reduces the equation to dz/dt = (2 b2 x1 + b1 )z + b2 z 2 . The reduction by the knowledge of two or three quadratures has also been studied from this perspective and similarly for the Strelchenya criterion.10 Each Riccati equation can be considered as a curve in R3 and we can transform every function in R, x(t), under an element of the group G of smooth SL(2, R)-valued curves Map(R, SL(2, R)), as follows: δ(t) if x(t) 6= − γ(t) , Θ(A, x(t)) = α(t)x(t)+β(t) γ(t)x(t)+δ(t) , Θ(A, ∞) = α(t)/γ(t) , Θ(A, −δ(t)/γ(t)) = ∞ ,   α(t) β(t) when A = ∈G . γ(t) δ(t) ¯ The image x′ (t) = Θ(A(t), x(t)) of a curve x(t) solution of the given Riccati

equation satisfies a new Riccati equation with the coefficients b′2 , b′1 , b′0 : ¯γ b1 + γ¯ 2 b0 + γ¯δ¯˙ − δ¯γ¯˙ , b′2 = δ¯2 b2 − δ¯ ′ ¯γ ) b1 − 2 α b1 = −2 β¯δ¯ b2 + (¯ αδ¯ + β¯ ¯ γ¯ b0 + δ¯α ¯˙ − α ¯ δ¯˙ + β¯γ¯˙ − γ¯ β¯˙ , b′ = β¯2 b − α ¯˙ . ¯ β¯ b + α ¯2 b + α ¯ β¯˙ − β¯α 0

2

1

0

This expression defines an affine action of the group G on the set of Riccati equations or analogous Lie systems. P2 Lie systems in SL(2, R) defined by a constant curve, a(t) = α=0 cα aα , are integrable and  the same happens for curves of the form a(t) = P 2 c a D(t) α=0 α α , where D is an arbitrary function, because a time reparametrisation reduces the problem to the previous one, i.e. the system is essentially a Lie system on a one-dimensional Lie group. We can prove the following theorem which is valid for both Riccati equation and any other Lie system with Lie group SL(2, R): Theorem 4.2. The necessary and sufficient conditions for the existence α(t) 0 ¯ of a transformation: y ′ = G(t)y, i.e. A(t) = , relating the 0 α−1 (t) Riccati equation (3) with (for b0 b2 6= 0, with an integrable one given by

dy ′ = D(t)(c0 + c1 y ′ + c2 y ′2 ) , (7) dt where ci are real numbers, ci ∈ R, is that c0 c2 6= 0, and: !! r 1 b˙ 2 (t) b˙ 0 (t) c0 c2 2 D (t)c0 c2 = b0 (t)b2 (t), b1 (t) + = c1 . − b0 (t)b2 (t) 2 b2 (t) b0 (t)

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The unique transformation is then y ′ = (b2 (t)c0 )1/2 (b0 (t)c2 )−1/2 y. As a consequence, given (7) if there are constants K, L such that s !! 1 b˙ 2 (t) b˙ 0 (t) L b1 (t) + =K − b0 (t)b2 (t) 2 b2 (t) b0 (t) then there exists a time-dependent linear change of variables transforming the given equation into the solvable Riccati equation (7) with c1 = K, c0 c2 = L and D(t) is given as above. The existence of such constant K can be considered a sufficient condition for integrability of the given Riccati equation or the corresponding Milne– Pinney equation. References 1. J.F. Cari˜ nena and A. Ramos, Integrability of the Riccati equation from a group theoretical viewpoint, Int. J. Mod. Phys. A 14 (1999) 1935–1951. 2. E. Pinney, The nonlinear differential equation y ′′ + p(xy + cy −3 = 0, Proc. Am. Math. Soc. 1 (1950) 681. 3. S. Lie and G. Scheffers, Vorlesungen u ¨ber continuierliche Gruppen mit geometrischen und anderen Anwendungen (Edited and revised by G. Scheffers, Teubner, Leipzig, 1893). 4. J.F. Cari˜ nena, J. Grabowski and G. Marmo, Lie–Scheffers systems: a geometric approach (Bibliopolis, Napoli, 2000). 5. J.F. Cari˜ nena, J. Grabowski and G. Marmo, Superposition rules, Lie theorem and partial differential equations, Rep. Math. Phys. 60 (2007) 237–58; arXiv:math-ph/0610013. 6. J.F. Cari˜ nena and A. Ramos, Lie systems and Connections in fibre bundles: Applications in Quantum Mechanics, In: 9th Int. Conf. Diff. Geom and Appl. (2004, (J. Bureˇs et al., Eds.) Matfyzpress, Prague, 2005) 437–452. 7. J.F. Cari˜ nena, J. Grabowski and A. Ramos, Reduction of time-dependent systems admitting a superposition principle, Acta Appl. Math. 66 (2001) 67–87. 8. E. Kamke, Differentialgleichungen: L¨ osungsmethoden und L¨ osungen (Akademische Verlagsgeselischaft, Leipzig, 1959). 9. G.M. Murphy, Ordinary differential equations and their solutions (Van Nostrand, New York, 1960). 10. V.M. Strelchenya, A new case of integrability of the general Riccati equation and its application to relaxation problems, J. Phys. A: Math. Gen. 24 (1991) 44965–4967. 11. J.F. Cari˜ nena, A. Ramos and J. de Lucas, A geometric approach to integrability conditions for Riccati equations, Electron. J. Diff. Eqns. 122 (2007) 1–14.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 419–429

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Transverse Poisson structures: The subregular and minimal orbits P.A. Damianou Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus E-mail: [email protected] H. Sabourin and P. Vanhaecke Laboratoire de Mathematiques, UMR 6086 du CNRS, Universit´ e de Poitiers, 86962 Futuroscope Chasseneuil Cedex France E-mail: [email protected], [email protected] We study the transverse Poisson structures to adjoint orbits in complex simple Lie algebras with special emphasis on the subregular and minimal orbits. Keywords: Poisson manifolds, coadjoint orbits, simple Lie algebras, singularities. MS classification: 53D17, 17B10, 14J17.

1. Some general results In this first section we prove some general results on the transverse Poisson structures to coadjoint orbits. In the final two sections we specialize to the two extreme and most interesting cases, i.e. the subregular and minimal orbits. With the exception of the results on the minimal orbit which are new we only give the statements of the theorems and we refer to Ref. 5 for detailed proofs. 1.1. Transverse Poisson Structures to adjoint orbits The definition of the transverse Poisson structure to a symplectic leaf in a Poisson manifold goes back to A.Weinstein.8 It is given in the following splitting theorem: Theorem 1.1 (A. Weinstein, 1983). Let x0 be a point in a Poisson manifold M . Then near x0 , M is isomorphic to a product S × N where

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S is the symplectic leaf of M , passing through x0 and N is a submanifold of M transverse to S at x0 , which inherits a Poisson structure from M vanishing at x0 . This Poisson structure on N is called the transverse Poisson structure at x0 . When M is the dual g∗ of a complex Lie algebra g, equipped with its standard Lie-Poisson structure, we know that the symplectic leaf through µ ∈ g∗ is the co-adjoint orbit G · µ of the adjoint Lie group G of g. In this case, a natural transverse slice to G · µ is obtained in the following way: we choose any complement n to the centralizer g(µ) of µ in g and we take N to be the affine subspace µ + n⊥ of g∗ . Since g(µ)⊥ = ad∗g µ we have Tµ (g∗ ) = Tµ (G · µ) ⊕ Tµ (N ), so that N is indeed a transverse slice to G · µ at µ. Furthermore, defining on n⊥ any system of linear coordinates (q1 , . . . , qk ), and using the explicit formula for Dirac reduction, one can write down explicit formulas for the Poisson matrix ΛN := ({qi , qj }N )1≤i,j≤k of the transverse Poisson structure, from which it follows easily that the coefficients of ΛN are actually rational functions in (q1 , . . . , qk ). As a corollary, in the Lie-Poisson case, the transverse Poisson structure is always rational. One immediately wonders in which cases the Poisson structure on N is polynomial; more precisely, for which Lie algebras g, for which co-adjoint orbits, and for which complements n. In 1989 P. A. Damianou4 made the conjecture that in gln , for a specific choice of slice (orthogonal to the orbit with respect to the Killing form) the transverse Poisson structure is always polynomial. The conjecture was verified for all nilpotent orbits of gln , for n ≤ 7 and was proved for some special cases i.e. subregular and minimal orbits. In 2002 R. Cushman and M. Roberts3 proved that there exists for any nilpotent adjoint orbit of a semi-simple Lie algebra a special choice of a complement n such that the corresponding transverse Poisson structure is polynomial. In 2005 H. Sabourin in Ref. 6 gave a more general class of complements where the transverse structure is polynomial, using in an essential way the machinery of semisimple Lie algebras. In this paper the transverse slice is always chosen to lie in the class of complements prescribed by Sabourin. 1.2. Reduction to nilpotent orbits It turns out that the transverse Poisson structure to any adjoint orbit G · x of a semi-simple (or reductive) algebra g is essentially determined by

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the transverse Poisson structure to the underlying nilpotent orbit G(s) · e defined by its Jordan decomposition x = s+ e. In fact we have the following result:5 Theorem 1.2. Let x ∈ g be any element, G · x its adjoint orbit and x = s + e its Jordan-Chevalley decomposition. Given any complement ne of g(x) in g(s) and putting n := ns ⊕ ne , where ns = g(s)⊥ , the parallel affine spaces Nx := x + n⊥ and N := e + n⊥ are respectively transverse slices to the adjoint orbit G · x in g and to the nilpotent orbit G(s) · e in g(s). The translation which sends Nx to N realises an isomorphism between the transverse Poisson structure on Nx and N . The Poisson structure on both transverse slices is given by the same Poisson matrix in terms of the same affine coordinates restricted to the corresponding transverse slice. 1.3. The polynomial and quasi-homogeneous character of the transverse Poisson structure The next general result is that the transverse Poisson structure is a quasi-homogeneous Poisson structure of degree −2 with respect to a set of weights that arise from the representation theory of the corresponding simple Lie algebra. We begin with the definition of quasi-homogeneous Poisson structure; see e.g. Ref. 1. Let ν = (ν1 , . . . , νd ) be non-negative integers. A polynomial P ∈ C[x1 , . . . , xd ] is said to be quasi-homogeneous (relative to ν) if for some integer κ, ∀t ∈ C, P (tν1 x1 , . . . , tνd xd ) = tκ P (x1 , . . . , xd ). The integer κ is then called the quasi-degree of P , denoted by ̟(P ). Similarly, a polynomial Poisson structure {· , ·} on C[x1 , . . . , xd ] is said to be quasi-homogeneous (relative to ν) if there exists κ ∈ Z such that, for every quasi-homogeneous polynomials F and G, their Poisson bracket {F, G} is quasi-homogeneous of degree ̟({F, G}) = ̟(F ) + ̟(G) + κ; Before stating the result, we need some notions from the representation theory of simple Lie algebras. First, we choose a Cartan subalgebra h of the semi-simple Lie algebra g, with corresponding root system ∆(h), from which a basis Π(h) of simple roots is selected. Let O be a nilpotent orbit. According to the Jacobson-Morosov-Kostant correspondence, there exists a

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canonical triple (h, e, f ) of elements of g, associated with O and completely determined by the following properties: (1) (h, e, f ) is a sl2 -triple, i.e., [h, e] = 2e, [h, f ] = −2f and [e, f ] = h; (2) h is the characteristic of O, i.e., h ∈ h and α(h) ∈ {0, 1, 2} for every simple root α ∈ Π(h). (3) O = G · e. The triple (h, e, f ) leads to two decompositions of g: (1) A decomposition of g into eigenspaces relative to adh . Each eigenvalue being an integer we have M g= g(i), i∈Z

where g(i) is the eigenspace of adh that corresponds to the eigenvalue i. For example, e ∈ g(2) and f ∈ g(−2).

(2) Let s be the Lie subalgebra of g isomorphic to sl2 , which is generated by h, e and f . The Lie algebra g is an s-module, hence it decomposes as g=

k M

Vnj ,

j=1

where each Vnj is a simple s-module, with nj + 1 = dim Vnj and with adh -weights nj , nj − 2, nj − 4, . . . , −nj . Let Nh be the set of adh -invariant complements to g(e). Theorem 1.3. Let g be a semi-simple Lie algebra, let O be a nilpotent orbit of g with canonical triple (h, e, f ), and let n be in Nh . The transverse Poisson structure on N := e + n⊥ is a polynomial Poisson structure that is quasi-homogeneous of degree −2, relative to the quasi-degrees n1 + 2, . . . , nk + 2, where n1 , . . . , nk denote the highest weights of g as an s-module. A transverse Poisson structure given by theorem 1.3 will be called an adjoint transverse Poisson structure, or ATP-structure. 2. The subregular case We will give an explicit description of the Transverse Poisson structure in the case of the subregular orbit Osr ⊂ g, where g is a semi-simple Lie algebra. Recall that an element Z in g is subregular if dim g(Z) = Rk(g)+2. In this case, the generic rank of the ATP-structure on N is 2 and we know

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dim N − 2 independent Casimirs, namely the basic Ad-invariant functions on g, restricted to N . It follows that the ATP-structure is the determinantal structure (also called Nambu structure), determined by these Casimirs, up to multiplication by a function. What is much less trivial to show is that this function is actually just a non-zero constant. For this we will use Brieskorn’s theory of simple singularities. 2.1. Invariant functions and Casimirs Let Osr = G · e, be a subregular orbit in the semi-simple Lie algebra g of rank ℓ, let (h, e, f ) be the corresponding canonical sl2 -triple and consider the transverse slice N := e + n⊥ to G · e, where n is an adh invariant complement to g(e). We know that the transverse structure on N , equipped with the linear coordinates q1 , . . . , qk , is a quasi-homogeneous polynomial Poisson structure of generic rank 2. Let S(g∗ )G be the algebra of Ad-invariant polynomial functions on g. By a classical theorem due to Chevalley, S(g∗ )G is a polynomial algebra, generated by ℓ homogeneous polynomials (G1 , . . . , Gℓ ), whose degree di := deg(Gi ) = mi + 1, where m1 , . . . , mℓ are the exponents of g. These functions are Casimirs of the Lie-Poisson structure on g. If we denote by χi the restriction of Gi to the transverse slice N , then it follows that these functions are independent Casimirs of the transverse Poisson structure. 2.2. Simple singularities Let h be a Cartan subalgebra of g. The Weyl group W acts on h and the algebra S(g∗ )G of Ad-invariant polynomial functions on g is isomorphic to S(h∗ )W , the algebra of W-invariant polynomial functions on h∗ . The inclusion homomorphism S(g∗ )G ֒→ S(g∗ ), is dual to a morphism g → h/W, called the adjoint quotient. Concretely, the adjoint quotient is given by G : g → Cℓ x 7→ (G1 (x), G2 (x), . . . , Gℓ (x)). The zero-fiber G−1 (0) of G is exactly the nilpotent variety N of g. We are interested in N ∩ N = N ∩ G−1 (0) = χ−1 (0), which is an affine surface with an isolated, simple singularity. Up to conjugacy, there are five types of finite subgroups of SL2 = SL2 (C), the cyclic, dihedral and three exceptional types, denoted by

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Cp , Dp , T , O and I. Given such a subgroup F, one looks at the corresponding ring of invariant polynomials C[u, v]F . In each of the five cases, C[u, v]F is generated by three fundamental polynomials X, Y, Z, subject to only one relation R(X, Y, Z) = 0, hence the quotient space C2 /F can be identified, as an affine surface, with the singular surface in C3 , defined by R = 0. The origin is its only singular point; it is called a (homogeneous) simple singularity. The exceptional divisor of the minimal resolution of C2 /F is a finite set of projective lines. If two of these lines meet, then they meet in a single point, and transversally. Moreover, the intersection pattern of these lines forms a graph that coincides with one of the simply laced Dynkin diagrams of type Aℓ , Dℓ , E6 , E7 or E8 . This type is called the type of the singularity. For the other simple Lie algebras (of type Bℓ , Cℓ , F4 or G2 ), there exists a similar correspondence. By definition, an (inhomogeneous) simple singularity of type ∆ is a couple (V, Γ) consisting of a homogeneous simple singularity V = C2 /F and a group Γ = F′ /F of automorphisms of V . We can now state the following extension of a theorem of Brieskorn, which is due to Slodowy7 Theorem 2.1. Let g be a simple complex Lie algebra, with Dynkin diagram of type ∆. Let Osr = G · e be the subregular orbit and N = e + n⊥ a transverse slice to G · e. The surface N ∩ N = χ−1 (0) has a (homogeneous or inhomogeneous) simple singularity of type ∆. 2.3. The determinantal Poisson structure In terms of linear coordinates q1 , q2 , . . . , qℓ+2 on Cℓ+2 , the formula {f, g}det :=

df ∧ dg ∧ dχ1 ∧ · · · ∧ dχℓ dq1 ∧ dq2 ∧ · · · ∧ dqℓ+2

defines a Poisson bracket on Cℓ+2 with Casimirs χ1 , . . . , χℓ . In our case it means that we have two polynomial Poisson structures on the transverse slice N which have χ1 , . . . , χℓ as Casimirs on N ∼ = Cℓ+2 , namely the transverse Poisson structure and the determinantal structure, constructed by using these Casimirs. The fact that both structures have the same quasi-degree -2, combined with the fact that the singularity of χ−1 (0) is isolated yields the following result : Theorem 2.2. Let Osr be the subregular nilpotent adjoint orbit of a complex semi-simple Lie algebra g and let (h, e, f ) be the canonical triple, associated to Osr . Let N = e + n⊥ be a transverse slice to Osr , where n is

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an adh -invariant complementary subspace to g(e). Let {· , ·}N and {· , ·}det denote respectively the transverse Poisson structure and the determinantal structure on N . Then {· , ·}N = c {· , ·}det for some c ∈ C∗ . 2.4. Reduction to a 3 × 3 Poisson matrix Let Osr be the subregular nilpotent adjoint orbit of a complex semisimple Lie algebra g of rank ℓ. Our goal now is to show that, in well-chosen coordinates, the transverse Poisson structure {· , ·}N on N is essentially given by a 3×3 skew-symmetric matrix, closely related to the polynomial that defines the singularity. The non-Poisson part of the following theorem is due to Brieskorn in the case of ADE singularities and was extended later to the other types of simple Lie algebras by Slodowy.7 Brieskorn’s theorem says that the map χ : N → Cℓ , which is the restriction of the adjoint quotient to the slice N , is a semiuniversal deformation of the singular surface N ∩ N . Using these results and the determinantal formula we obtain the following result: Theorem 2.3. After possibly relabeling the coordinates qi and the Casimirs χi , the ℓ + 2 functions χi , 1 ≤ i ≤ ℓ − 1,

and

qℓ , qℓ+1 , qℓ+2

form a system of coordinates on the affine space N . The Poisson matrix of the transverse Poisson structure on N takes, in terms of these coordinates, the form

ΛN

  0 0 = , 0Ω

where 

 0    Ω = − ∂χℓ  ∂qℓ+2   ∂χ ℓ ∂qℓ+1

 ∂χℓ ∂χℓ − ∂qℓ+2 ∂qℓ+1    ∂χℓ  .  0 ∂qℓ    ∂χℓ − 0 ∂qℓ

It has the polynomial χℓ as Casimir, which reduces to the polynomial which defines the singularity, when setting χj = 0 for j = 1, 2, . . . , ℓ − 1.

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3. The minimal orbit In this section we consider the transverse Poisson structure to the minimal orbit Om in an arbitrary semi-simple Lie algebra g, whose Killing form will be denoted by h· | ·i. This orbit is the nilpotent orbit of minimal dimension (besides the trivial orbit {0}). It is unique and is generated by a root vector Em , associated to a highest root, with respect to a fixed Cartan subalgebra h and a choice of simple roots. Let (Hm , Em , Fm ) denote the canonical triplet, associated to Om and let gEm denote the centralizer of Em in g. 3.1. Properties of the minimal orbit We first list the properties of the minimal orbit that we will use (see Ref. 2 for proofs). The Lie algebra g decomposes in eigenspaces, relatively to adHm , as follows: g = g(−2) ⊕ g(−1) ⊕ g(0) ⊕ g(1) ⊕ g(2). This decomposition has the following properties. (F1) g(2) = C.Em ; (F2) g(−2) = C.Fm ; (F3) g(0) is a reductive subalgebra of g and g(0) = gEm (0) ⊕ C.Hm , where gEm (0) := gEm ∩ g(0); (F4) gEm (0) ⊥ C.Hm ; (F5) gEm (0) is reductive, gEm (0) = Ze ⊕ me where Ze ⊂ h is the center of gEm (0) and me its semi-simple part; + − (F6) Let n+ e := me ∩ n = hXα , α(Hm ) = 0i, let ne denote its opposite and + − let he := me ∩ h. Then me = ne ⊕ he ⊕ ne and h = he ⊕ Ze ⊕ C.Hm . (F7) Let Kf be the skew-symmetric bilinear form defined by Kf (X, Y ) := hFm |[X, Y ]i. The space g(1), equipped with Kf , is a symplectic space of dimension 2s. Moreover, we can choose a basis (Z1 , . . . , Zs , Zs+1 , . . . , Z2s ) such that each vector Zi is a root vector Xαi , associated to a positive root, and [Zi , Zj ] = 0 = [Zi+s , Zj+s ],

[Zi , Z2s+1−j ] = δij Em ,

for all i, j with 1 ≤ i, j ≤ s; (F8) The same result as in (F7) holds for the space g(−1) equipped with the bilinear form Ke (X, Y ) = hEm |[X, Y ]i. The corresponding basis, defined by the same properties as in (F7), will be denoted by X1 , . . . , X2s .

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3.2. The ATP-structure associated to Om Since the centralizer gEm is given by gEm = gEm (0) ⊕ g(1) ⊕ g(2),

we have a decomposition g = gEm ⊕ n, where n = g(−2) ⊕ g(−1) ⊕ C.Hm . It is clear that the Lie subalgebra n of g is adHm -invariant, has dimension 2s+2 and that its orthogonal is given by n⊥ = g(−2) ⊕ g(−1) ⊕ gEm (0). Thus, we choose Nm := Em + n⊥ as transverse slice to Om . The Poisson matrix of the corresponding ATP-Poisson structure on Nm is given, at n ∈ Nm , by the Dirac formula Λm (n) = A(n) + B(n)C −1 (n)B(n)T , where the matrices A(n), B(n), C(n) are constructed as follows. Let Z2s+2 , . . . , Z2s+p+1 be a basis of gEm (0) and let Z2s+1 := Em , so that Z1 , . . . , Z2s+p+1 is a basis of gEm . Also, set X2s+1 := 12 Hm and X2s+2 := Fm . Then, X1 , . . . , X2s+2 is a basis of n. In terms of these bases, the matrices A(n), B(n) and C(n) are given by A(n)ij = hn|[Zi , Zj ]i,

B(n)ik = hn|[Zi , Xk ]i,

C(n)kl = hn|[Xk , Xl ]i,

where 1 ≤ i, j ≤ 2s + p + 1 and 1 ≤ k, l ≤ 2s + 2. In terms of the 2q × 2qmatrices Jq , defined by   0 ... 0 1 .  . . 0     . .   . . . 1   . Jq =  , . .  ..    −1 . .  .    0 .. −1 0 ... 0 we have:

  Js 0 C(n) = 0 −J2

and

C −1 (n) = −C(n).

Proposition 3.1. The ATP-structure of the minimal orbit Om is the sum of two Poisson structures Λm = A + Q, where 1 A is a linear Poisson structure, isomorphic to the Lie-Poisson structure on the dual of the Lie algebra gEm ; 2 Q is a quadratic Poisson bracket, whose Poisson matrix at n ∈ Nm is given by Q(n) := B(n)C −1 (n)B(n)T . Moreover, its generic rank is dim Om − 2.

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P.A. Damianou, H. Sabourin and P. Vanhaecke

Proof. The first statement is clear because the matrix A(n) in the Dirac formula is the matrix of the Lie-Poisson structure on (gEm )∗ . Both matrices A and Λm = A + Q are Poisson. Moreover, since C(n) is constant (independent of n ∈ Nm ), all entries of Q are quadratic polynomials. Since Q is the highest degree term of the Poisson matrix Λ, it is also a Poisson matrix. Therefore A and Q are compatible Poisson structures. We show that the rank of Q is dim Om − 2. To do this let n be any element in Nm . We will restrict our attention now to the matrix B(n). From the definitions, we get [Zi , Fm ] = [Zi , Hm ] = 0,

[Zi , Xk ] ∈ g(−1),

for i, k such that 2s + 2 ≤ i ≤ 2s + p + 1 and 1 ≤ k ≤ 2s. It implies that B(n)ik = 0 for the latter values of i and k. Thus, the last p rows of B(n) are zero,   D(n) B(n) = , 0 where D(n) is the (2s + 1) × (2s + 2)-matrix, whose entries are given by D(n)ik = hn | [Zi , Xk ]i, where 1 ≤ i ≤ 2s + 1 and 1 ≤ k ≤ 2s + 2. If 1 ≤ i ≤ 2s then [Zi , X2s+2 ] ∈ g(−1) and [Em , X2s+2 ] = Hm . Using (F4), it follows that the last column of the matrix D(n) is zero,
D(n) = D′ (n) 0 .

Thus, for n ∈ Em + n⊥ , we have    ′  D(n)C −1 (n)D(n)T 0 D (n)C ′ (n)D′ (n)T 0 Q(n) = = , 0 0 0 0

where C ′ (n) is the submatrix of C −1 (n), obtained by removing its last column and its last row. This implies that Rk(Q(n)) = Rk(D′ (n)C ′ (n)D′ (n)T ) ≤ 2s + 1, for all n ∈ Em + n⊥ . Since Q is skew-symmetric, its rank is at most 2s = dim Om − 2. We show that there exists a point n ∈ Nm where the rank of Q is 2s. Recall that the symplectic vector space g(1) is generated by root vectors, Zi = Xαi , for 1 ≤ i ≤ 2s. So, we can define Pi := n⊥ ∩ Hα⊥i , for 1 ≤ i ≤ 2s, which are hyperplanes of n⊥ . Let P denote their union. Let n ∈ n⊥ \P + Em . Then + ′ 1. If 1 ≤ i 6= k ≤ 2s then [Zi , Xk ] ⊂ n− e ⊕ ne ⊂ V , so D (n)ik = hn | [Zi , Xk ]i = 0;

Transverse Poisson structures: The subregular and minimal orbits

429

6 0; 2. For all i with 1 ≤ i ≤ 2s, D′ (n)ii = hn | Hαi i = 3. For all k with 1 ≤ k ≤ 2s + 1, [Z2s+1 , Xk ] ∈ g(1) ⊕ g(2), so that D′ (n)2s+1,k = 0. Consider the submatrix C ′′ of C ′ , and similarly D′′ of D′ , obtained by removing its last row and its last column. Then the
upper left 2s× 2s-minor of D′ (n)C ′ (n)D′ (n)T is det D′′ (n)C ′′ (n)D′′ (n)T , which is non-zero, This proves that Rk(Q(n)) = 2s = dim Om − 2, so that Rk(Q) = dim Om − 2. References 1. M. Adler, P. Van Moerbeke and P. Vanhaecke, Algebraic integrability, Painlev´e geometry and Lie algebra ( Ergebnisse der Mathematik und ihrer grenzgebiete 47, Springer-Verlag, Berlin Heidelberg, 2004). 2. D.H. Collingwood and W. Mc Govern, Nilpotent orbits in semisimple Lie algebras (Van Nostrand Reinhold, Mathematics series, 1993). 3. R. Cushman and M. Roberts, Poisson structures transverse to coadjoint orbits, Bull.Sci.Math. 126 (2002) 525–534. 4. P.A. Damianou, Nonlinear Poisson brackets, Ph.D. Dissertation, University of Arizona, 1989. 5. P.A. Damianou, H. Sabourin and P. Vanhaecke, Transverse Poisson structures to adjoint orbits in semi-simple Lie algebras, Pac. J. of Math. 232 (2007) 111–139. 6. H. Sabourin, Sur la structure transverse ` a une orbite nilpotente adjointe, Canad.J.Math 57 (2005) 750–770. 7. P. Slodowy, Simple singularities and simple algebraic groups (Lect.Notes in Math. 815, Springer Verlag, Berlin 1980). 8. A. Weinstein, Local structure of Poisson manifolds, J. Diffential Geom. 18 (1983) 523–557.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 431–444

431

On first-order differential invariants of the non-conjugate subgroups of the Poincar´ e group P (1, 4) V.M. Fedorchuk Institute of Mathematics, Pedagogical Academy, Podchor¸ az˙ ych 2, 30-084, Krak´ ow, Poland; Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of the National Ukrainian Academy of Sciences, Naukova Street 3b, Lviv, 79601, Ukraine E-mail: [email protected] V.I. Fedorchuk Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of the National Ukrainian Academy of Sciences, Naukova Street 3b, Lviv, 79601, Ukraine E-mail: [email protected] The number of non-equivalent functional bases of the first-order differential invariants for all non-conjugate subgroups of the group P (1, 4) is established. All these bases have been classified according to their dimensions. Some application of the results obtained are considered. Keywords: Poincar´ e group P (1, 4), Lie algebra of the group P (1, 4), nonconjugate subgroups of the group, first-order differential invariants, nonequivalent functional bases. MS classification: 17B05, 17B81.

1. Introduction It is well known that the differential invariants of Lie groups of point transformations play an important role in geometry, group analysis of differential equations, theoretical and mathematical physics, gas dynamics, etc. (see, for example, Refs. 1–23). The development of theoretical and mathematical physics required various extensions of the four-dimensional Minkowski space M (1, 3) and, correspondingly, various extensions of a Poincar´e group P (1, 3). The natural extension of this group is a generalized Poincar´e group P (1, 4). The group P (1, 4) is a group of rotations and translations of the five-dimensional

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Minkowski space M (1, 4). This group has many applications in theoretical and mathematical physics (see, for example, Refs. 24–26). The group P (1, 4) has many subgroups used in theoretical physics.27–31 Among these subgroups, there are the Poincar´e group P (1, 3) (the symmetry group e 3) (see also of relativistic physics) and the extended Galilei group G(1, Ref. 32) (the symmetry group of non-relativistic physics). Therefore, the results obtained with the help of the subgroup of the group P (1, 4) will be useful in relativistic and non-relativistic physics. The articles Refs. 33–39 are devoted to the construction and the classification of functional bases of the first-order differential invariants of the splitting and non-splitting subgroups of the group P (1, 4). It should be noted that these results were obtained for the splitting and non-splitting subgroups separately. The purpose of this paper is to present some results obtained, which would be valid to all non-conjugate subgroups of the group P (1, 4). We plan to find the number of the non-equivalent functional bases of the firstorder differential invariants for all non-conjugate subgroups of the group P (1, 4). All these bases will be classified according to their dimensions. In order to present some of the results obtained, we have to consider the Lie algebra of the group P (1, 4). 2. The Lie algebra of the group P (1, 4) and its non-conjugate subalgebras The Lie algebra of the group P (1, 4) is given by the 15 basis elements Mµν = −Mνµ (µ, ν = 0, 1, 2, 3, 4) and Pµ′ (µ = 0, 1, 2, 3, 4), satisfying the commutation relations  ′ ′  ′  Pµ , Pν = 0, Mµν , Pσ′ = gµσ Pν′ − gνσ Pµ′ , 

 ′ ′ ′ ′ ′ ′ = gµρ Mνσ + gνσ Mµρ − gνρ Mµσ − gµσ Mνρ , Mµν , Mρσ

where g00 = −g11 = −g22 = −g33 = −g44 = 1, gµν = 0, if µ 6= ν. ′ Here and in what follows, Mµν = iMµν . We consider the following representation of the Lie algebra of the group P (1, 4): P0′ =

∂ , ∂x0

P4′ = −

∂ , ∂x4

P1′ = −

∂ , ∂x1

P2′ = −

∂ , ∂x2


′ Mµν = − xµ Pν′ − xν Pµ′ .

P3′ = −

∂ , ∂x3

On first-order differential invariants of the non-conjugate subgroups of P (1, 4)

433

In order to study the non-conjugate subalgebras of the Lie algebra of the group P (1, 4), we used the method proposed in Ref. 40. These subalgebras have been described in Refs. 27–31. They have been divided by splitting and non-splitting ones. Splitting subalgebras Pi,a of the Lie algebra of the group P (1, 4) can be written in the following form: ◦

Pi,a = Fi + Nia , where Fi are subalgebras of the Lie algebra of the group O(1, 4), Nia are subalgebras of the Lie algebra of the translations group T (5) ⊂ P (1, 4). Non-splitting subalgebras Pej,k are subalgebras, for which the basis can be choosen in the form: X X ek = Bk + drj Xj , cki Xi , B i

j

where cki and drj are fixed real constants (not equal zero simultaneously). Bk are bases of subalgebras of the Lie algebra of the group O(1, 4), Xi are bases of subalgebras of the Lie algebra of the group T (5). One of the important consequences of the study of the non-conjugate subalgebras of the Lie algebra of the group P (1, 4) is that the Lie algebra of the group P (1, 4) contains, as subalgebras, the Lie algebras of the following e 3) important for theoretical and mathematical physics groups: P (1, 3), G(1, (see also Ref. 32), O(1, 4), O(4), E(4), etc. ′ ′ Further, instead of Mµν = −Mνµ (µ, ν = 0, 1, 2, 3, 4) and Pµ′ (µ = 0, 1, 2, 3, 4), we will use the following basis elements: ′ G = M40 ,

′ L1 = M32 ,

′ ′ Pa = M4a − Ma0 ,

X0 =

1 ′ (P − P4′ ) , 2 0

′ L2 = −M31 ,

′ ′ Ca = M4a + Ma0 ,

Xk = Pk′

(k = 1, 2, 3),

′ L3 = M21 ,

(a = 1, 2, 3),

X4 =

1 ′ (P + P4′ ) . 2 0

e 3) is generated by the following basis The Lie algebra of the group G(1, elements: L1 , L2 , L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 .

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3. The non-equivalent functional bases of first-order differential invariants of non-conjugate subgroups of the group P (1, 4) For all non-conjugate subgroups of the group P (1, 4), the functional bases of the first-order differential invariants have been constructed. In the construction of these bases, it has turned out that different non-conjugate subalgebras of the Lie algebra of the group P (1, 4) may have the same ones. Consequently, there is no one-to-one correspondence between the nonconjugate subalgebras of the Lie algebra of the group P (1, 4) and their respective functional bases of the first-order differential invariants. Moreover, some of the functional bases (which are of the same dimension) may be equivalent. Our aim is to obtain non-equivalent functional bases only. (1) (1) (1) (2) (2) (2) Let {J1 , J2 , ..., Jt } and {J1 , J2 , ..., Jt } be the functional bases of the first-order differential invariants, which correspond to the subalgebras L1 and L2 of the Lie algebra of the group P (1, 4). (1)

(1)

(1)

Definition 3.1. We say that the functional bases {J1 , J2 , ..., Jt } and (2) (2) (2) {J1 , J2 , ..., Jt } be equivalent if there exist smooth functions f1 , f2 , ..., ft and g1 , g2 , ..., gt such that (2)

(2)

(2)

= g2 (J1 , J2 , ..., Jt )

J1

(1)

(1)

(1)

J2

= f2 (J1 , J2 , ..., Jt )

(2)

(1)

(1)

(2)

J2

(2)

= g1 (J1 , J2 , ..., Jt )

(1)

= f1 (J1 , J2 , ..., Jt )

(2)

(1)

(1)

(2)

J1

........................................ and ............................................ (2) (2) (2) (1) (1) (1) (1) Jt = gt (J1 , J2 , ..., Jt ). = ft (J1 , J2 , ..., Jt )

(2)

Jt

(1)

(1)

(1)

(2)

(2)

Lemma 3.1. Two functional bases {J1 , J2 , ..., Jt } and {J1 , J2 , ..., (2) Jt } are equivalent if and only if they satisfy the following conditions: (2) er(1) e (1) J (2) = 0, X e (1) J (2) = 0, ..., X =0 X 1 Jt 1 1 1 2

(1) e (2) J (1) = 0, X e (2) J (1) = 0, ..., X er(2) =0 X 2 Jt 1 1 1 2

(∗)

er(2) e (2) e (2) , ..., X e (1) , X e (1) , ..., X er(1) where {X 2 } are the first-prolonged 1 }, {X1 , X2 1 2 bases operators of the Lie subalgebra L1 and L2 , respectively; r1 , r2 are the dimensions of the subalgebras L1 and L2 . Proof. The Proof of this Lemma can be found in Ref. 39. Proposition 3.1. There exist 498 non-equivalent functional bases of the first-order differential invariants for the non-conjugate subgroups of the group P (1, 4).

On first-order differential invariants of the non-conjugate subgroups of P (1, 4)

435

Proof. The list of the non-conjugate (the conjugation was considered under the group P (1, 4)) subalgebras of the Lie algebra of the group P (1, 4) contains 555 ones.41 As resulting from the calculation of the general ranks of matrices, which contain coordinates of the one-prolonged basis elements of the subalgebras of the Lie algebra considered, and using the theorem on number of invariants of the Lie group of the point transformations (see, for example, Refs. 6,8) we make sure that all of the non-conjugate subalgebras of the Lie algebra of the group P (1, 4) have the functional bases of the first-order differential invariants. Thus, there are 555 functional bases of the first-order differential invariants. Among them, there are equivalent ones. Equivalent functional bases can only be among those, which have the same dimensions. Let L1 be a non-conjugate subalgebra of the Lie algebra of the group P (1, 4), which has a t-dimensional functional basis of the first-order differ(1) (1) (1) ential invariants {J1 , J2 , ..., Jt }. To find the bases, which are equivalent (2) (2) (1) (2) (1) (1) to {J1 , J2 , ..., Jt }, we use the Lemma. Let {J1 , J2 , ..., Jt } be the t-dimensional functional basis of the first-order differential invariants of the other non-conjugate subalgebra L2 . Based on the Lemma, if these functional bases satisfy the conditions (*), then, the considered bases are equivalent. Otherwise the considered bases are not equivalent. In the analogous manner, we check whether or not other t-dimensional functional bases of the (1) (1) (1) first-order differential invariants are equivalent to the {J1 , J2 , ..., Jt }. In this way, we obtain all t-dimensional functional bases, which are equiv(1) (1) (1) alent to {J1 , J2 , ..., Jt }. In the analogous manner, we construct classes of the equivalent functional bases of other dimensions. The direct check provides 498 non-equivalent functional bases of the first-order differential invariants for the non-conjugate subgroups of the group P (1, 4). The Proposition is proved.

It is impossible to present here all non-equivalent functional bases. Therefore, below we only give a short review of the results obtained. Let N be a number of non-equivalent functional bases invariant under non-conjugate subgroups of the group P (1, 4), Ns be a number of non-equivalent functional bases invariant under splitting non-conjugate subgroups of the group P (1, 4), Nns be a number of non-equivalent functional bases invariant under non-splitting non-conjugate subgroups of the group P (1, 4).

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3.1. One-dimensional non-equivalent functional bases There are no one-dimensional non-equivalent functional bases. 3.2. Two-dimensional non-equivalent functional bases There exists one two-dimensional non-equivalent functional basis. In this case Nns = 0. Thus, N = Ns = 1. The functional basis is invariant under the non-galilean subalgebras of the Lie algebra of the group P (1, 4). Below, for some of the non-conjugate subalgebras of the Lie algebra of the group P (1, 4), we write their basis elements and the corresponding functional bases of differential invariants. hG, P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i, hL3 + eG, P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 , e > 0i, hG, L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i, hG, L1 , L2 , L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i, hG, C1 , C2 , C3 , L1 , L2 , L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i, J1 = u,

J2 = u20 − u21 − u22 − u23 − u24 ; uµ ≡

∂u , µ = 0, 1, 2, 3, 4. ∂xµ

3.3. Three-dimensional non-equivalent functional bases There exist N = 7 non-equivalent three-dimensional functional bases. In this case Ns = 7, Nns = 4. Thus, Ns + Nns − N = 4 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example: hG, P1 , P2 , X0 , X1 , X2 , X3 , X4 i, hL3 + eG, P1 , P2 , X0 , X1 , X2 , X3 , X4 , e > 0i, hG, L3 , P1 , P2 , X0 , X1 , X2 , X3 , X4 i, hG + a3 X3 , L3 + d3 X3 , P1 , P2 , X0 , X1 , X2 , X4 , a3 < 0, d3 < 0i, hG, L3 + d3 X3 , P1 , P2 , X0 , X1 , X2 , X4 , d3 < 0i, J1 = u, J2 = u3 , J3 = u20 − u21 − u22 − u24 . It should be noted that among the three-dimensional functional bases there is one basis invariant under the subalgebras of the Lie algebra of the

On first-order differential invariants of the non-conjugate subgroups of P (1, 4)

437

e 3) and 6 bases invariant under the non-galilean subalgebras. group G(1, 3.4. Four-dimensional non-equivalent functional bases There exist N = 25 non-equivalent four-dimensional functional bases. In this case Ns = 18, Nns = 10. Thus, Ns + Nns − N = 3 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example: hP1 , P2 , P3 , X1 , X2 , X3 , X4 i, hL3 − P3 , P1 , P2 , X1 , X2 , X3 , X4 i, hL3 , P1 , P2 , P3 , X1 , X2 , X3 , X4 i, hL1 , L2 , L3 , P1 , P2 , P3 , X1 , X2 , X3 , X4 i, hL3 + d3 X3 , P1 , P2 , P3 , X1 , X2 , X4 , d3 < 0i, J1 = x0 + x4 , J2 = u, J3 = u0 − u4 , J4 = u20 − u21 − u22 − u23 − u24 . It should be noted that among the four-dimensional functional bases there are 7 bases invariant under the subalgebras of the Lie algebra of the e 3) and 18 bases invariant under the non-galilean subalgebras. group G(1, 3.5. Five-dimensional non-equivalent functional bases There exist N = 63 non-equivalent five-dimensional functional bases. In this case Ns = 37, Nns = 28. Thus, Ns + Nns − N = 2 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example: hP1 , P2 , X1 , X2 , X3 , X4 i, hL3 , P1 , P2 , X1 , X2 , X3 , X4 i, hL3 + d3 X3 , P1 , P2 , X1 , X2 , X4 , d3 < 0i, J1 = x0 + x4 , J5 =

u20

−

u21

−

J2 = u, u22

−

u24

J3 = u3 ,

J4 = u0 − u4 ,

.

It should be noted that among five-dimensional functional bases there are 22 bases invariant under the subalgebras of the Lie algebra of the group e 3) and 41 bases invariant under the non-galilean subalgebras. G(1,

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V.M. Fedorchuk and V.I. Fedorchuk

3.6. Six-dimensional non-equivalent functional bases There exist N = 108 non-equivalent six-dimensional functional bases. In this case Ns = 51, Nns = 57. Thus, Ns + Nns − N = 0 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example of the functional basis, which is invariant under the non-splitting non-galilean subalgebras: hL3 + dG + α3 X3 , P3 , X1 , X2 , X4 , d > 0, α3 < 0i, J1 = u,

J2 =

x0 + x4 , u0 − u4

J3 = d arctan

J4 = dx3 −α3 ln(x0 +x4 )+du3

x0 + x4 , u0 − u4

u2 − ln(x0 + x4 ), u1 J5 = u21 +u22 ,

J6 = u20 −u23 −u24 .

It should be noted that among the six-dimensional functional bases there are 53 bases invariant under the subalgebras of the Lie algebra of the group e 3) and 55 bases invariant under the non-galilean subalgebras. G(1, 3.7. Seven-dimensional non-equivalent functional bases

There exist N = 136 non-equivalent seven-dimensional functional bases. In this case Ns = 58, Nns = 78. Thus, Ns + Nns − N = 0 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example of the functional basis, which is invariant under the splitting non-galilean subalgebras: hG, X1 , X2 , X3 i, J1 = (x20 − x24 )1/2 , J5 = u2 ,

J6 = u3 ,

J2 = u,

J3 = (x0 + x4 )(u0 + u4 ),

J4 = u1 ,

J7 = u20 − u24 .

It should be noted that among the seven-dimensional functional bases there are 75 bases invariant under the subalgebras of the Lie algebra of the e 3) and 61 bases invariant under the non-galilean subalgebras. group G(1, 3.8. Eight-dimensional non-equivalent functional bases

There exist N = 89 non-equivalent eight-dimensional functional bases. In this case Ns = 40, Nns = 49. Thus, Ns + Nns − N = 0 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example of the functional basis, which is invariant under

On first-order differential invariants of the non-conjugate subgroups of P (1, 4)

439

the splitting galilean subalgebras: hL3 , X3 , X4 i, J1 = x0 + x4 , J5 = u0 ,

J2 = (x21 + x22 )1/2 ,

J6 = u3 ,

J7 = u4 ,

J4 = x1 u2 − x2 u1 ,

J3 = u,

J8 =

u21

+

u22

.

It should be noted that among the eight-dimensional functional bases there are 52 bases invariant under the subalgebras of the Lie algebra of the e 3) and 37 bases invariant under the non-galilean subalgebras. group G(1,

3.9. Nine-dimensional non-equivalent functional bases

There exist N = 49 non-equivalent nine-dimensional functional bases. In this case Ns = 21, Nns = 28. Thus, Ns + Nns − N = 0 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example of the functional basis, which is invariant under the non-splitting galilean subalgebras: hP3 + X2 , X1 i, J1 = x0 + x4 , J4 = u,

J2 =

x3 + x2 , x0 + x4

J3 = (x20 − x23 − x24 )1/2 ,

J5 = (x0 + x4 )u3 + (u0 − u4 )x3 ,

J8 = u0 − u4 ,

J9 =

u20

−

u23

−

u24

J6 = u1 ,

J7 = u2 ,

.

It should be noted that among the nine-dimensional functional bases there are 30 bases invariant under the subalgebras of the Lie algebra of the e 3) and 19 bases invariant under the non-galilean subalgebras. group G(1,

3.10. Ten-dimensional non-equivalent functional bases

There exist N = 20 non-equivalent ten-dimensional functional bases. In this case Ns = 10, Nns = 10. Thus, Ns + Nns − N = 0 functional bases are invariant under splitting and non-splitting subalgebras. Let us give an example of the functional basis, which is invariant under the non-splitting non-galilean subalgebras: hP3 + C3 + eL3 + α(X0 + X4 ), e > 2, α < 0i, J1 = (x21 + x22 )1/2 , J4 = u,

J2 = (x23 + x24 )1/2 ,

J5 = x1 u2 − x2 u1 ,

J3 = ex0 − α arctan

J6 = x3 u4 − x4 u3 ,

x1 , x2

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V.M. Fedorchuk and V.I. Fedorchuk

J7 = 2x0 + α arctan

u4 , u3

J8 = u0 ,

J9 = u21 + u22 ,

J10 = u23 + u24 .

It should be noted that among the ten-dimensional functional bases there are 12 bases invariant under the subalgebras of the Lie algebra of the e 3) and 8 bases invariant under the non-galilean subalgebras. group G(1, 4. On some applications of the results obtained

It is well known (see, for example, Refs. 4,6,8,24–26,41) that differential equations with non-trivial symmetry groups play an important role in theoretical and mathematical physics, mechanics, gas dynamics etc. Therefore, the construction and investigation of equations of this type are important from the physical and mathematical points of view. In particular, the results obtained can be used in order to construct the first-order differential equations in the space M (1, 4) × R(u), which are invariant under the nonconjugate subgroups of the group P (1, 4). Here, and in what follows, R(u) is the number axis of the dependent variable u. Indeed, (see, for example, Refs. 1,6,8), in many cases these equations can be written in the following form: F (J1 , J2 , ..., Jt ) = 0, where F is an arbitrary smooth function of its arguments, {J1 , J2 , ..., Jt } are functional bases of the first-order differential invariants of the corresponding non-conjugate subalgebras of the Lie algebra of the group P (1, 4). In this way, we have constructed 498 classes of the first-order differential equations in the space M (1, 4) × R(u) with non-trivial symmetry. Now, let us present some of the obtained classes of the first-order differential equations, which are invariant under some important for theoretical and mathematical physics subalgebras of the Lie algebra of the group P (1, 4). Below, for each of the such subalgebras we write their basis elements and corresponding classes of the first-order differential equations in the space M (1, 4) × R(u). (1) hL1 , L2 , L3 , P1 + C1 , P2 + C2 , P3 + C3 i(∼ = O(4)), F x0 , (x21 + x22 + x23 + x24 )1/2 , u, x1 u1 + x2 u2 + x3 u3 + x4 u4 ,
u0 , u21 + u22 + u23 + u24 = 0 ;

(2) hL1 , L2 , L3 , P1 +C1 , P2 +C2 , P3 +C3 , X1 , X2 , X3 , X0 −X4 i(∼ = E(4)), hL1 +

1 1 1 (P1 + C1 ) , L2 + (P2 + C2 ) , L3 + (P3 + C3 ) , X1 , X2 , 2 2 2

On first-order differential invariants of the non-conjugate subgroups of P (1, 4)

441

X3 , X0 − X4 i,

1 1 1 (P1 + C1 ) , L2 + (P2 + C2 ) , L3 + (P3 + C3 ) , 2 2 2 1 L3 − (P3 + C3 ) , X1 , X2 , X3 , X0 − X4 i, 2
F x0 , u, u0 , u21 + u22 + u23 + u24 = 0 ; hL1 +

(3) hL1 , L2 , L3 , P1 − C1 , P2 − C2 , P3 − C3 , X1 , X2 , X3 , X0 + X4 i(∼ = P (1, 3)),


F x4 , u, u4 , u20 − u21 − u22 − u23 = 0 ;

(4) hG, C1 , C2 , C3 , L1 , L2 , L3 , P1 , P2 , P3 i(∼ = O(1, 4)), F (x20 − x21 − x22 − x23 − x24 )1/2 , u, x0 u0 + x1 u1 + x2 u2 + x3 u3 + x4 u4 ,
u20 − u21 − u22 − u23 − u24 = 0 ;

e 3)), (5) hL1 , L2 , L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i(∼ = G(1, hP1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i,

hL3 − P3 , P1 , P2 , X0 , X1 , X2 , X3 , X4 i, hL3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i, hL3 − X0 , P1 , P2 , P3 , X1 , X2 , X3 , X4 i, hP1 , P2 , P3 + X0 , L3 + βX0 , X1 , X2 , X3 , X4 , β < 0i,
F u, u0 − u4 , u20 − u21 − u22 − u23 − u24 = 0 ;

(6) hG, C1 , C2 , C3 , L1 , L2 , L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i(∼ = P (1, 4)),

hG, P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i, hL3 + eG, P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 , e > 0i, hG, L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i, hG, L1 , L2 , L3 , P1 , P2 , P3 , X0 , X1 , X2 , X3 , X4 i,
F u, u20 − u21 − u22 − u23 − u24 = 0 .

Since the Lie algebra of the group P (1, 4) contains, as subalgebras, the Lie algebra of the Poincar´e group P (1, 3) and the Lie algebra of the e 3) (see also Ref. 32), the obtained differential extended Galilei group G(1,

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V.M. Fedorchuk and V.I. Fedorchuk

equations can be used in relativistic and non-relativistic physics. References ¨ 1. S. Lie, Uber Differentialinvarianten, Math. Ann. 24 (1) (1884) 537–578. 2. S. Lie, Vorlesungen u ¨ber continuierliche gruppen (Teubner, Leipzig, 1893). 3. A. Tresse, Sur les invariants differentiels des groupes continus de transformations, Acta math. 18 (1894) 1–88. 4. E. Vessiot, Sur l’integration des sistem differentiels qui admittent des groupes continus de transformations, Acta math. 28 (1904) 307–349. 5. A. Kumpera, Invariants diff´erentiels d’un pseudogroupe de Lie, G´eom´etrie differentielle, In: Lecture Notes in Math. (V.392, Springer, Berlin, 1974) 121– 162. 6. L.V. Ovsiannikov, Group Analysis of Differential Equations (Academic Press, New York, 1982). 7. E.I. Chakyrov, Differential invariants of certain extentions of the Galilei group, Dinamika Sploshn. Sredy (69) (1985) 123–149. 8. P.J. Olver, Applications of Lie Groups to Differential Equations (SpringerVerlag, New York, 1986). 9. W.I. Fushchych and I.A. Yehorchenko, Differential Invariants for Galilei Algebra, Dokl. Acad. Nauk. Ukr. SSR, Ser.A (4) (1989) 19–34. 10. W.I. Fushchych and I.A. Yehorchenko, Differential Invariants for Poincar´e Algebra and Conformal Algebra, Dokl. Acad. Nauk. Ukr. SSR, Ser.A (5) (1989) 46–53. 11. D. Krupka and J. Janyˇska, Lectures on Differential Invariants (Brno University, Czech Republic, 1990). 12. P.J. Olver, Differential invariants and invariant differential equations, Lie Groups and their Appl. 1 (1994) 177–192. 13. P.J. Olver, Differential invariants. Geometric and algebraic structures in differential equations, Acta Applicandae Math. 41 (1-3) (1995) 271–284. 14. D.Q. Chao and D. Krupka, 3rd order differential invariants of coframes, Preprint 10/1997, Slezsk´ a univerzita v Opavˇe, Matematick´ yu ´stav v Opavˇe, 1997. 15. X. Xiaoping, Differential invariants of classical groups, Duke Math. J. 94 (3) (1998) 543–572. 16. G.M. Beffa and P.J. Olver, Differential Invariants for parametrized projective surfaces, Comm. Anal. Geom. 7 (4) (1999) 807–839. ˇ enkov´ 17. J. Sedˇ a, Differential invariants of the meric tensor, In: Proceedings of the Seminar on Differential Geometry, Mathematical Publications (V. 2, Silesian University in Opava, Opava, 2000) 145–158. 18. Y. Nutku and M.B. Sheftel, Differential Invariants and group foliation for the complex Monge-Amp`ere equation, J. Phys. A: Math. Gen. 34 (2001) 137–156. 19. R.O. Popovych and V.M. Boyko, Differential Invariants and Application to Riccati-Type Systems, In: Proc. of the Fourth Internat. Conf. Symmetry in Nonlinear Mathematical Physics (Ukr. Math. Congr. Dedicated to 200th An-

On first-order differential invariants of the non-conjugate subgroups of P (1, 4)

20.

21.

22.

23.

24.

25. 26. 27. 28. 29.

30. 31.

32.

33.

34.

35.

443

niversary of Mykhailo Ostrograds’kyi, 9–15 July 2001, Kyiv, Ukraine, Proc. of Inst. of Math. of NAS of Ukraine, Kyiv, 43, Part 1, 2002) 184–193. A.P. Chupakhin, Differential invariants: theorem of commutativity, Communications in Nonlinear Science and Numerical Simulation 9 Issue 1, (2004) 25–33. S.V. Golovin, Applications of the differential invariants of infinite dimensional groups in hydrodynamics Communications in Nonlinear Science and Numerical Simulation 9 Issue 1, (2004) 35–51. N.H. Ibragimov, M. Torrisi and A. Valenti, Differential invariants of nonlinear equations νtt = f (x, νx )νxx +g(x, νx ), Communications in Nonlinear Science and Numerical Simulation 9 Issue 1, (2004) 69–80. R. Tracina, Invariants of a family of nonlinear wave equations, Communications in Nonlinear Science and Numerical Simulation 9 Issue 1, (2004) 127–133. W.I. Fushchych, Representations of full inhomogeneous de Sitter group and equations in five-dimensional approach. I, Teoret. i mat. fizika 4 (3) (1970) 360–367. V.G. Kadyshevsky, New approach to theory electromagnetic interactions, Fizika elementar. chastitz. i atomn. yadra 11 (1) (1980) 5–39. W.I. Fushchych and A.G. Nikitin, Symmetries of Equations of Quantum Mechanics (Allerton Press Inc., New York, 1994). V.M. Fedorchuk, Continuous subgroups of the inhomogeneous de Sitter group P (1, 4), Preprint, N 78.18, Inst. Matemat. Acad. Nauk Ukr. SSR, Kyiv, 1978. V.M. Fedorchuk, Splitting subalgebras of the Lie algebra of the generalized Poincar´e group P (1, 4), Ukr. Mat. Zh. 31 (6) (1979) 717–722. V.M. Fedorchuk and W.I. Fushchych, On subgroups of the generalized Poincar´e group, In: Proceedings of the International Seminar on Group Theoretical Methods in Physics (V.1, Nauka, Moscow, 1980) 61–66. V.M. Fedorchuk, Nonsplitting subalgebras of the Lie algebra of the generalized Poincar´e group P (1, 4), Ukr. Mat. Zh. 33 (5) (1981) 696–700. W.I. Fushchich, A.F. Barannik, L.F. Barannik and V.M. Fedorchuk, Continuous subgroups of the Poincar´e group P (1, 4), J. Phys. A: Math. Gen. 18 (14) (1985) 2893–2899. W.I. Fushchich and A.G. Nikitin, Reduction of the representations of the generalized Poincar´e algebra by the Galilei algebra, J. Phys.A: Math. and Gen. 13 (7) (1980) 2319–2330. V.M. Fedorchuk and V.I. Fedorchuk, Differential invariants of first-order of splitting subgroups of the generalized Poincar´e group P (1, 4), Mat. Metody i Fiz.- Mekh. Polya 44 (1) (2001) 16–21. V.M. Fedorchuk and V.I. Fedorchuk, On first-order differential invariants for splitting subgroups of the generalized Poincar´e group P (1, 4), Dopov. Nats. Akad. Nauk Ukrainy (5) (2002) 36–42. V.M. Fedorchuk and V.I. Fedorchuk, On Differential Invariants of First - and Second-Order of the Splitting Subgroups of the Generalized Poincar´e Group P (1, 4) In: Proc. 4th Internat. Conf. Symmetry in Nonlinear Mathematical Physics (Ukr. Math. Congr, Dedicated to 200th Anniversary of Mykhailo

444

36.

37.

38.

39.

40.

41.

V.M. Fedorchuk and V.I. Fedorchuk

Ostrohrads’kyi, 9–15 July, 2001, Kyiv, Ukraine, Proc. of Inst. of Math. of NAS of Ukraine, Kyiv, 43, Part 1, 2002) 140–144. V.M. Fedorchuk and V.I. Fedorchuk, On the Differential First- Order Invariants of the Non-Splitting Subgroups of the Poincare group P (1, 4), In: Proc. of Inst. of Math. of NAS of Ukraine (Kyiv, 50, Part 1, 2004) 85–91. V.M. Fedorchuk and V.I. Fedorchuk, On the differential first-order invariants for the non-splitting subgroups of the generalized Poincare group P (1, 4), Annales Academiae Paedagogicae Cracoviensis Studia Mathematica 4 (2004) 65–74. V.M. Fedorchuk and V.I. Fedorchuk, On functional bases of first-order differential invariants of continuous subgroups of the Poincar´e group P (1, 4), Mat. Metody i Fiz. - Mekh. Polya 48 (4) (2005) 51–58. V.M. Fedorchuk and V.I. Fedorchuk, First-Order Differential Invariants of the splitting subgroups of the Poincar´e group P (1, 4), Universitatis Iagellonicae Acta Mathematica 44 (2006) 21–30. J. Patera, P. Winternitz and H. Zassenhaus, Continuous subgroups of the fundamental groups of physics. I. General method and the Poincar´e group, J. Math. Phys. 16 (8) (1975) 1597–1614. W. Fushchych, L. Barannyk and A. Barannyk, Subgroup analysis of the Galilei and Poincar´e groups and reductions of nonlinear equations (Naukova Dumka, Kiev, 1991).

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 445–453

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On the flag curvature of the dual of Finsler metric Dragos Hrimiuc Department of Mathematics, University of Alberta, Edmonton,Canada, T6G 2G1 E-mail: [email protected] We define the flag curvature of the dual of a Finsler metric and study its relationship with the flag curvature of the initial Finsler metric via Legendre transformation. Keywords: Finsler metrics, nonlinear connection, Legendre transformation, flag curvature. MS classification: 53B40, 53C60, 53Z05.

1. Introduction The flag curvature of a Finsler metric, an extension of the sectional curvature in Riemannian geometry, is a significant invariant in Finsler geometry that captured the attention of an important number of researchers. I would like merely to mention the recent work of Bao and Robles,1 Shen,2 Bao, Robles and Shen,3 Bryant.4 It is very known that some problems written on the tangent bundle can be simplified if they are re-written by using the machinery of the dual. For example the geodesics of a Finsler metric (the integral curves of a certain second order vector field) can be viewed as the projections of the geodesics of the dual metric (integral curves of a Hamiltonian vector field) on the base manifold. There are other important cases when the metric could have a complicated format or is implicitly given, while its dual could be simpler. Some of such metrics will be presented in the next section. The flag curvature for the dual metric is introduced in the last section and its connection with the curvature of initial metric is given by using the Legendre transformation.

Dedicated to Professor Radu Miron on the occasion of his 80th birthday

446

D. Hrimiuc

2. The dual of a Finsler metric Let M be a smooth manifold and (T M ,τ ), (T ∗ M ,τ ∗ ) its tangent and cotangent bundles. Let A ⊂ T M be an open set and Ax := A ∩ Tx M , x ∈ M. The fiber derivative of a differentiable function f : A → R at y ∈ Ax in the direction of u ∈ Tx M is the mapping Df : A → T ∗ M , Df (y)u = d f (y + tu)|t=0 . hDf (y), ui := dt The second fibre derivative of a twice differentiable function f : A → R at y ∈ Ax is the mapping D2 f : A → L2sym (T M, R) given by ∂2f (y + su + tv)) |s=t=0 , u, v ∈ Tx M , x ∈ M ∂s∂t 2 Notice that for each y∈

A2x we have  the 2linear mapping D f (y) : Tx M → ∗ 2 Tx M , D f (y)u, v = D f (y)v, u = D f (y) (u, v), u, v ∈ Tx M , x ∈ M . D2 f (y) (u, v) =

Definition 2.1. A Finsler metric on T M is a continuous function F : ◦ T M → [0, ∞) , C ∞ on the slit tangent bundle T M := T M \ {0} that is a also a Minkowski norm on each fibre i.e. (a) F (λy) = λF (y), λ > 0, y ∈
Tx M (b) For each y ∈ Tx M , D2 12 F 2 (y) is an inner product on Tx M , x ∈ M .

If set L := 12 F 2 , then the following formulae are direct consequences of the above definition:5

 F 2 (y) = D2 (L) (y)y, y , y ∈ Tx M , x ∈ M (1) hDF (y), ui ≤ F (u), y, u ∈ Tx M, x ∈ M

(2)

F (u + v) ≤ F (u) + F (v), u, v ∈ Tx M , x ∈ M

(3)

Let F be a Finsler metric on T M and ∂ΩF,x := {y ∈ Tx M ; F (y) = 1} the sphere (indicatrix) of the metric at each x ∈ M. Definition 2.2. The dual of the Finsler metric F is is the function F ∗ : T ∗ M → R defined by F ∗ (p) =

sup hp, ui = sup

u∈∂ΩF,x

u6=0

hp, ui , p ∈ Tx∗ M , x ∈ M u

Definition 2.3. The Legendre transformation generated by F , denoted LF , is the following bundle map ( not necessarily linear) over the identity:   ◦ ◦ 1 2 ∗ LF : T M → T M , LF = D (L) = D F 2

On the flag curvature of the dual of Finsler metric

447

Notice that LF (y) = D (L) (y) = D2 (L) (y) y, F 2 (y) = hLF (y) , yi

(4)

and from (2) we get: hLF (y) , ui ≤ F (y)F (u)

(5)

By using (1)-(5) we can prove the following result: ◦

◦

Proposition 2.1. (a) LF : T M → T ∗ M is a smooth 1-positive homogeneous diffeomorphism (b) F ∗ = F ◦ L−1 F (c) F ∗ is a Finsler metric on T ∗ M . Notice that LF can be extended to a continuous mapping on T M by taking LF (0) = 0. Remark 2.1. The following equations hold: F ∗ (p) = F (y), p = LF (y)

(6)

1 ∗2 F (7) 2 We also mention that LF is a 1-positive homogeneous diffeomorphism that preserves the norms F and F ∗ and LF is linear if and only if F (or F ∗ ) is Riemannian. L−1 F = D (H) , H :=

Theorem 2.1. Let F be a Finsler metric on T M , w a smooth vector field on M , such that F (w) < 1 and let β = LF (w) . The following properties hold: (a) Fe (y) = F (y) − hβ, yi is a Finsler metric on T M (b) F ∗ (p) = Fe ∗  (p − F ∗ (p) β) (c) Fe ∗ (p) = F ∗ p + Fe∗ (p) β .

Proof. (a) The proof is quite similar to that given for Randers metrics5 (b) We have:   LF (y) e e −β LFe (y) = F (y) DF (y) = (F (y) − hβ, yi) F (y)   p = (F ∗ (p) − hβ ◦ LF ∗ , pi) −β ∗ F (p)   hβ ◦ LF ∗ , pi p + (−F ∗ (p) + hβ ◦ LF ∗ , pi) β = 1− F ∗ (p)

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If pe := LFe (y) then

Fe ∗ (e p) = Fe (y) = F (y) − hβ, yi = F ∗ (p) − hβ ◦ LF ∗ , pi

Hence

and therefore

pe =

  Fe ∗ (e p) p e∗ (e e ∗ (e p − F p ) β = F p ) − β F ∗ (p) F ∗ (p) Fe ∗ (e p) = Fe ∗ (e p) Fe ∗



p −β F ∗ (p)



and (b) follows. Now (c) can be obtained from (b). We remark that generally Fe ∗ can not be found in terms of F ∗ and β. However, when F is Riemannian we have:6

Corollary 2.1. Let F (y) = |y| be a Riemannian metric on M and Fe(y) =

|y| − L|·| (w) , y , |w| < 1 ( a Randers metric). Then q 2 2 ∗2 hw, pi + (1 − |w| ) |p| hw, pi Fe∗ (p) = + (8) 2 2 1 − |w| 1 − |w| ∗

where |·| is the dual of the Riemannian norm |·|.

Proof. Use equation (c) from the above proposition. Remark 2.2. If F is a Finsler metric on T M and w is a smooth vector field on M , such that F (w) < 1 we can also define Fb (y) = F (y) − hLF (y) , wi . Fe ≡ Fb if and only if F is Riemannian. It is still an open question if Fb is or is not a Finsler metric.

Let F is a Finsler metric on T M and w is a smooth vector field on M , such that F (w) < 1. Then, every one of the following equations F (y) = F (y − F (y)w)

(9)

F (y) = F (y + F (y)w)

(10)

2 7

defines a new Finsler metric F on T M . ’ This Finsler metric is related to so called Zermello navigation problem used by Shen,2 Bao, Robles and Shen.3 Notice the similarity between these equations and those of Theorem 2.1. We remark that F can not be generally obtained in terms of F . However, we have:

449

On the flag curvature of the dual of Finsler metric

Theorem 2.2. The dual of F is ∗

F (p) = F ∗ (p) − hp, wi , p ∈ Tx∗ M, w ∈ Tx M, x ∈ M.

(11)

Proof. Let ϕ : T M → T M, ϕ(y) = y − F (y)w, y, w ∈ Tx M ; then F = F ◦ ϕ−1 .Thus ∗

= F ◦ (LF ◦ ϕ) F = F ◦ L−1 F

−1

and also LF (ϕ(y)) = LF (y) ◦ (Dϕ)−1 On the other hand −1

Dϕ(y) = id − w ⊗ DF (y) and (Dϕ(y))

=id +

1 w ⊗ DF (y) 1 − B(y)

1 where B(y) := DF (y) w = F (y) LF (y)w. Hence   1 1 LF (ϕ(y)) = LF (y) ◦ id + w ⊗ DF (y) = LF (y) 1 − B(y) 1 − B (y)

and −1

(1 − B(y))y = (LF ◦ ϕ)

◦ LF (y)

Therefore ∗

−1

F (LF (y)) = F ◦ (LF ◦ ϕ)

◦ LF (y) = F ((1 − B(y))y) = F (y) − LF (y)w

and (11) follows. 3. Finsler metrics and flag curvature Let (F, M ) be a Finsler space. The canonical connection(nonlinear) associated to (F, M ) is the almost product structure N = −Lξ J

(12)

where J is the canonical tangent structure of T M and ξ is the second order vector field associated to F , i.e. the solution of the equation iξ ω = dL

(13)

with ω = −ddJ (L) and L = 12 F 2 . This connection produces the standard splitting T T M = V T M ⊕ HT M

(14)

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D. Hrimiuc

where V T M = Ker (id + N ) is the vertical bundle and HT M = Ker (id − N ) is the horizontal bundle. The horizontal and vertical projectors induced by N are h = 12 (id + N ) and v = 12 (id − N ). If [h, h] denotes the Nijenhuis bracket, the curvature of N defined as 1 Ω = − [h, h] 2 gives the obstruction against the integrability of HT M . For each y ∈ Tx M , x ∈ M let us consider v

h

(·)y : Tx M → Vy T M , (·)y : Tx M → Hy T M

(15)

(16) v

the vertical and respectively the horizontal lift mappings and let C = (y)y be the Liouville vector field. For a more detailed description of the above concepts please see Miron,8 Szilasi.9 Now we are ready to define an essential concept: Definition 3.1. The Riemannian curvature tensor at y ∈ Tx M is given by h

Ry = −iξ Ω ◦ (·)y

(17)

We mention that the Riemannian curvature tensor is defined here globally, in terms of the curvature tensor of the nonlinear connection and not in terms of Chern’s linear connection as in Bao.5 ∨ The Finsler metric F induces a metric tensor g on the vertical bundle V T M defined by the following equation: ∨

g ◦ (J × J) = −ωF ◦ (J × id)

(18)

Definition 3.2. Let y ∈ Tx M , x ∈ M . The flag curvature at u ∈ Tx M is defined by   ∨ ∨ g Ry (u) , u (19) Ky (u) =  h∨  ∨i2  ∨ ∨ ∨ ∨ g (C, C) g u, u − g C, u ∨

v

where u := (u)y .

Notice that the way we define the flag curvature is slightly different from the classical one. This definition seems to be more convenient for our approach that intensively makes use of Legendre transformation. Let XH := (LF )∗ ξ and J ∗ := (LF )∗ J be the push-forward of ξ an respectively J by LF . XH is the Hamiltonian vector field associated to the

On the flag curvature of the dual of Finsler metric

451

Hamiltonian H = 12 F ∗2 and J ∗ is a tangent structure on T ∗ M that will be called the dual of J. Proposition 3.1. N ∗ = −LXH J ∗ is a connection (nonlinear) on T ∗ M . Proof. It is enough to verify the that J ∗ N ∗ = J ∗ and N ∗ J ∗ = −J ∗ . These equalities are obtained from those verified by N by using Legendre transformation. The connection defined above is called the dual of N . See also Ref. 6. Let T T ∗M = V T ∗ M ⊕ HT ∗ M

(20)

the splitting generated by N ∗ . Theorem 3.1. (a) (LF )∗ (HT M ) = HT ∗ M and (LF )∗ (V T M ) = V T ∗ M (b) The horizontal and vertical projectors h∗ and respectively v ∗ induced by the splitting(20)are the duals of h and v i.e. h∗ = (LF )∗ h =(LF )∗ ◦ h ◦ −1 ∗ (LF )−1 ∗ and v = (LF )∗ v = (LF )∗ ◦ v ◦ (LF )∗ . Notice that based on part (a) of the above theorem we obtain that the duals (the push forward by Legendre transformation) of the vertical and horizontal lift mappings (16) at p = LF (y) , y ∈ Tx M are: ∗v

∗h

(·)p : Tx∗ M → Vp T ∗ M , (·)p : Tx∗ M → Hp T ∗ M

(21)



∗v −1 ∗ ∗ h ∗ ∗h ∗ v i.e., (u∗ )p = (LF )∗ L−1 F (u ) y and (u )p = (LF )∗ LF (u ) y , u ∈ ∗ Tx M . Theorem 3.2. (a) The dual of Ω is just the curvature of N ∗ 1 Ω∗ = (LF )∗ Ω = − [h∗ , h∗ ] 2

(22)

(b) The dual of the Riemannian curvature tensor (17) of N at y ∈ Tx M is the Riemannian curvature tensor of N ∗ at p = LF (y) ∈ Tx∗ M i.e. ∗h

∗ Rp∗ = (LF )∗ Ry = (LF )∗ ◦ Ry ◦ L−1 F = −iXH Ω ◦ (·)p

(23)


Proof. (a) − 12 [h∗ , h∗ ] = − 21 [(LF )∗ (h) , (LF )∗ (h)] = (LF )∗ − 12 [h, h] = (LF )∗ Ω = Ω∗ .

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(b)Rp∗ (u∗ )

=

∗h −iXH Ω∗ ((u∗ )p ).

∗ (LF )∗ (Ry L−1 F (u ))


∗ h (LF )∗ (−iξ Ω L−1 F (u ) y

=

=

∨

We mention that the dual of the metric tensor g is ∨∗

∨

g = (LF )∗ g

(24)

g ◦ (J ∗ × J ∗ ) = ω ◦ (id × J ∗ )

(25)

and we have ∨∗

with ω the canonical two form on T T ∗ M . Definition 3.3. Let p ∈ Tx∗ M , x ∈ M . The co-flag curvature tensor at u∗ ∈ Tx∗ M is defined by   ∨ ∨∗ g Rp∗ (u∗ ) , u∗ Kp∗ (u∗ ) = 2     ∨ ∨ ∨ ∨∗ ∨∗ ∨∗ g (C ∗ , C ∗ ) g u∗ , u∗ − g C ∗ , u∗ ∨

∗v

where u∗ := (u∗ )p and C ∗ := (LF )∗ (C). By using Theorem 3.2 and (19), (24) we cam prove the following Theorem 3.3. If y, u ∈ Tx M then KL∗ F (y) (LF (u)) = Ky (u) i.e. the flag curvature is LF invariant. Corollary 3.1. (F, M ) is of scalar curvature(constant) if and only if (F ∗ , M ) is of scalar(constant) curvature. References 1. D. Bao and C. Robles, Ricci and Flag Curvatures in Finsler Geometry, A sampler of Riemann-Finsler Geometry, MSRIP 50 (2004) 198–256. 2. Z. Shen, Finsler metric with K = 0 and S = 0, Canadian J. Math. 55 (2003) 112–132. 3. D. Bao , C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geometry 66 (2004) 391–449. 4. R.L. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math. 28 (2002) 221–262. 5. D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry (Springer, 2000). 6. D. Hrimiuc and H. Shimada, On the L-duality between Lagrange and Hamilton manifolds, Nonlinear World 3 (1996) 613–641.

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7. D. Hrimiuc, On the affine deformation of a Minkowski norm, Periodica Mathematica Hungarica 48 (2004) 49–60. 8. R. Miron, D. Hrimiuc, H. Shimada and S.V. Sabau, The Geometry of Hamilton and Lagrange Spaces (Kluwer, 2001). 9. J. Szilasi, A setting for a spray and Finsler geometry, In: Handbook of Finsler Geometry (Vol 2, (P.L. Antonelli, Ed.) Kluwer Academic Publishers, 2003) 1185–1426.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 455–462

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A note on the height of the canonical Stiefel–Whitney classes of the oriented Grassmann manifolds J´ ulius Korbaˇs Department of Algebra, Geometry, and Mathematical Education, Faculty of Mathematics, Physics, and Informatics, Comenius University, Bratislava 4, SK-842 48 Bratislava 4, Slovakia or Mathematical Institute, Slovak Academy of Sciences, ˇ anikova 49, SK-814 73 Bratislava 1, Slovakia Stef´ E-mail: [email protected] We derive a functional upper bound κ ˜ i (n, k) for the Z2 -height of the ith Stiefel– Whitney class of the canonical oriented k-plane bundle over the oriented Grass˜ n,k of oriented k-dimensional vector subspaces in Euclidean mann manifold G n-space. It turns out that κ ˜ 2 (n, k) is for many pairs (n, k) better than the upper bound implied by Dutta and Khare’s (2002) results on the height of the second Stiefel–Whitney class of the canonical k-plane bundle over the Grassmann manifold Gn,k of k-dimensional vector subspaces in Euclidean n-space. In addition to this, κ ˜ 2 (n, 2) always coincides with the exact value of the height (well ˜ n,2 ). We also prove that κ known for G ˜ 2 (22t , 3) (t ≥ 2) implies an upper bound which differs from the exact value of the height of the second Stiefel–Whitney class by at most one. Keywords: Oriented Grassmann manifold, Stiefel–Whitney class, cup product, height. MS classification: 57R19, 53C30, 55M30, 57R20, 57T15.

1. Introduction and statement of the main results ∼ SO(n)/SO(k) × SO(n − k) ˜ n,k = The oriented Grassmann manifold G (2k ≤ n) consists of oriented k-dimensional vector subspaces in Rn ; its dimension is k(n − k). We shall suppose that k ≥ 2, to exclude the spheres ˜ n,1 ∼ G = S n−1 from our considerations. ˜ n,k is a universal double covering space As is well known, the manifold G ∼ of the Grassmann manifold Gn,k = O(n)/O(k) × O(n − k), the latter con˜ n,k → Gn,k sisting of all k-dimensional vector subspaces in Rn . Let p : G be the obvious covering projection. For the tangent bundles we have ˜ n,k ∼ TG = p∗ (T Gn,k ).

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As is usual for smooth manifolds, by the ith Stiefel–Whitney class of ˜ Gn,k we understand the ith Stiefel–Whitney class of its tangent bundle, ˜ n,k ) = wi (T G ˜ n,k ) ∈ H i (G ˜ n,k ; Z2 ); at the same time we hence we have wi (G i have wi (Gn,k ) = wi (T Gn,k ) ∈ H (Gn,k ; Z2 ). ˜ n,k , Let γ˜n,k (briefly γ˜ ) be the canonical oriented k-plane bundle over G and let γn,k (briefly γ) be the canonical k-plane bundle over Gn,k (see, e.g., ˜ n,k (resp. Gn,k ) we Ref. 9). By the ith canonical Stiefel–Whitney class of G mean the ith Stiefel–Whitney class of the canonical bundle, hence the class wi (˜ γn,k ), briefly denoted by w ˜i when n and k are clear from the context (resp. wi (γn,k ), briefly denoted by wi when n and k are clear from the context). Given a topological space X, we recall that the Z2 -height (briefly height) of a cohomology class y ∈ H ∗ (X; Z2 ) is defined to be sup{t; y t 6= 0 ∈ H ∗ (X; Z2 )}, and we denote it by height(y). For instance, height(w1 (γn,k )) was calculated by R. Stong in Ref. 10, J. L¨ orinc in Ref. 8, extending earlier results of S. Ilori and D. Ajayi (Ref. 5), calculated the height of the first Stiefel–Whitney characteristic class of any flag manifold O(n1 +· · ·+nq )/O(n1 )×· · ·×O(nq ), and S. Dutta and Khare in Ref. 4 calculated height(w2 (γn,k )). For some purposes, it is desirable to know the height of wi (˜ γn,k ). Our interest in this invariant stems from the intention to derive results on the ˜ n,k , procup-length and Lyusternik-Shnirel’man category of the manifolds G ceeding in the spirit of Ref. 7. To calculate height(w2 (˜ γn,2 )), it is enough to use the knowledge of the cohomology algebra H ∗ (Gn,2 ; Z2 ) (see, e.g., Ref. 2), the Gysin exact sequence associated with the covering projection mentioned above (see, e.g., S12 in Ref. 9), and elementary considerations. So one readily verifies (but it is also well known) that ( n−3 if n is odd, 2 (1) height(w2 (˜ γn,2 )) = n−2 if n is even. 2 ˜ n,k ; Z2 ) is in For k ≥ 3, as in contrast to k = 2, the algebra H ∗ (G general unknown. In spite of this drawback, again the knowledge of the algebra H ∗ (Gn,k ; Z2 ) and the corresponding Gysin exact sequence make it possible — in theory — to calculate the height of any wi (˜ γn,k ). But in practice the matter is different: the computations are quite complicated, and with growing n and k they become unmanageable, even when one uses a computer.

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In view of this, it would be desirable to have some explicit formulae, hence something like (1). But in general no formulae for height(wi (˜ γn,k )) with k ≥ 3 are available up to now. As a step on the way to desired formulae, it seems reasonable to look for some good general estimates for height(wi (˜ γn,k )). Having this in mind, we mainly derive here some upper bounds. We first define two functions: κ ˜i (n, k) for positive integers i, n, k, and κ2 (n, k) for positive integers n and k such that 6 ≤ 2k ≤ n. We put   ⌊ (k−1)(n−k) ⌋ if n is odd, i κ ˜i (n, k) =  ⌊ (k−1)(n−k+1) ⌋ if n is even, i where ⌊a⌋ denotes the integer part of a ∈ R, and

 s 2 −1 if n = 2s + 1,    s  if n = 2s + 2, 2 κ2 (n, 3) = 2s + 2p+1 − 2 if n = 2s + 2p + 1, s > p ≥ 1,    2s + 2p+1 − 1 if n = 2s + 2p + t + 1, s > p ≥ 1,   1 ≤ t ≤ 2p − 1;  s −1 if n = 2s + 1,   2s+1  2 − 4 if n = 2s + 2, κ2 (n, 4) = s+1  2 − 4 if n = 2s + 3,   s+1 2 − 1 if 2s + 4 ≤ n ≤ 2s+1 ;  s 2 −1 if n = 2s + 1, k ≥ 5, κ2 (n, k) = 2s+1 − 1 if 2s + 2 ≤ n ≤ 2s+1 , k ≥ 5.

We note that height(w2 (γn,k )) = κ2 (n, k) by Ref. 4. We now state the following upper bounds for height(wi (˜ γn,k )). ˜ n,k with 4 ≤ Proposition 1.1. For the oriented Grassmann manifold G 2k ≤ n we have height(wi (˜ γn,k )) ≤ κ ˜ i (n, k) for i = 2, . . . , k. In some situations, this proposition gives interesting upper bounds. Indeed, one can see immediately (from Proposition 1.1 and the formula (1)) that height(w2 (˜ γn,2 )) = κ ˜ 2 (n, 2). In addition to this, the following corollary exhibits an infinite family of manifolds for which κ ˜ leads to an upper bound differing from the exact value by at most one.

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˜ 22t ,3 , t ≥ 2, we Corollary 1.1. For the oriented Grassmann manifolds G have 22t − 4 ≤ height(w2 (˜ γ22t ,3 )) ≤ 22t − 3. For the second canonical Stiefel–Whitney class, the upper bound for its height given by Proposition 1.1 can sometimes be improved by the following. ˜ n,k , 6 ≤ 2k ≤ n, Proposition 1.2. For the oriented Grassmann manifold G we have height(w2 (˜ γn,k )) ≤ min{˜ κ2 (n, k), κ2 (n, k)}. Applications of our results to calculations of the cup-length and Lyusternik-Shnirel’man category of the oriented Grassmann manifolds will be presented in a forthcoming paper. 2. Proofs of the results In this section, we prove our results stated in Sec. 1. Before passing to the proofs, we recall (see, e.g., S12 in Ref. 9) that besides the compact ˜ n,k we also have their limit space G ˜ ∞,k ; this infinite CWCW-complexes G complex can be identified with the classifying space BSO(k). Let γ˜∞,k ˜ ∞,k . Recall that the cohomology denote the canonical k-plane bundle over G ∗ ˜ algebra H (G∞,k ; Z2 ) is a Z2 -polynomial algebra; we can write ˜ ∞,k ; Z2 ) = Z2 [w2 (˜ H ∗ (G γ∞,k ), . . . , wk (˜ γ∞,k )]. As is well known (or a proof can be obtained by mimicking the corresponding part of the proof of Lemma 3.1 in Ref. 1), the restriction ho˜ ∞,k ; Z2 ) → H ∗ (G ˜ n,k ; Z2 ), induced by the obvious momorphism j ∗ : H ∗ (G ˜ ˜ “inclusion” j : Gn,k → G∞,k , is an isomorphism in dimensions ≤ n − k − 1 and a monomorphism in dimension n − k. As a consequence (we have ˜ n,k ; Z2 ) do not obey j ∗ (wi (˜ γ∞,k )) = w ˜i ), the classes w ˜2 , . . . , w ˜k ∈ H ∗ (G any relations in dimensions ≤ n − k. Proof of Proposition 1.1. Let us suppose that n is odd. By an obvious adjustment of 3.6.2 of Ref. 1 or by Theorem 1.1 in Ref. 6, the Stiefel– ˜ n,k ) (i ≤ k ≤ n − k) can now be expressed uniquely Whitney class wi (G as ˜ n,k ) = w wi (G ˜i + Qi (w˜2 , . . . , w ˜i−1 ), where Qi is a Z2 -polynomial.

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An easy induction shows that ˜ n,k ) + Pi (w2 (G ˜ n,k ), . . . , wi−1 (G ˜ n,k )), w ˜i = wi (G ˜ n,k ) where Pi is a polynomial, for i = 2, . . . , k. Indeed, we have w ˜2 = w2 (G (it is enough to use the formula for w2 (Gn,k ) from Ref. 1 together with the ˜ n,k ∼ fact that p∗ (γ) = γ˜ and T G = p∗ (T Gn,k )). By what we have said above, one has ˜ n,k ) + a polynomial in w w ˜j = wj (G ˜2 , . . . , w ˜j−1 for j ≥ 2. The induction hypothesis then implies that

˜ n,k ) + Pj (w2 (G ˜ n,k ), . . . , wj−1 (G ˜ n,k )) w ˜j = wj (G

for some polynomial Pj . ˜ n,k ; Z2 ) is in dimensions ≤ n − k − 1 generSo the algebra H ∗ (G ated by the canonical Stiefel–Whitney classes w ˜2 , . . . , w ˜k , or — thanks to the fact that n is odd — alternatively by the Stiefel–Whitney classes ˜ n,k ), . . . , wk (G ˜ n,k ). w2 (G 1+⌊

(k−1)(n−k)

⌋

i = 0. Let us suppose that the We need to prove that w ˜i contrary is true. Then, by Poincar´e duality, there is a cohomology class

y ∈ H k(n−k)−i−i⌊

(k−1)(n−k) ⌋ i

˜ n,k ; Z2 ) (G

such that the cup product ⌋ 1+⌊ (k−1)(n−k) i

w ˜i

˜ n,k ; Z2 ) ∼ y ∈ H k(n−k) (G = Z2

is nonzero. One readily verifies that the number k(n − k) − i − i⌊ (k−1)(n−k) ⌋ i does not exceed n− k − 1. Hence the cohomology class y can be expressed in ˜ n,k ), . . . , wk (G ˜ n,k ), and one sees terms of the Stiefel–Whitney classes w2 (G that the value of the product ⌋ 1+⌊ (k−1)(n−k) i

w ˜i

˜ n,k ; Z2 ) y ∈ H k(n−k) (G

˜ n,k is nothing but at the fundamental Z2 -homology class of the manifold G ˜ n,k . But each G ˜ n,k is bordant to zero (this a Stiefel–Whitney number of G is well known; if needed, consult, e.g., Theorem 31.1 of Ref. 3 keeping in ˜ n,k reversing the orientation of oriented kmind that the involution on G dimensional vector subspaces in Rn has no fixed point). Therefore, by the Pontrjagin-Thom theorem (see, e.g., Ref. 9) all Stiefel–Whitney numbers ˜ n,k vanish. As a result, also the product of G ⌋ 1+⌊ (k−1)(n−k) i

w ˜i

˜ n,k ; Z2 ) y ∈ H k(n−k) (G

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must

be

zero.

⌋ 1+⌊ (k−1)(n−k) i w ˜i

This

contradiction

proves

that

we

= 0, and therefore the height of w ˜i is at most

really

have

⌋, ⌊ (k−1)(n−k) i

as claimed. ˜ n,k → G ˜ n+1,k , Now let n be even. We have the obvious “inclusion” j : G ∗ and for the pullback bundles we obtain j (˜ γn+1,k ) = γ˜n,k . Hence the cohomology homomorphism induced by j maps any canonical Stiefel–Whitney class to the corresponding canonical Stiefel–Whitney class, j ∗ (wi (˜ γn+1,k )) = wi (˜ γn,k ). ˜ n+1,k ; Z2 ) → H ∗ (G ˜ n+1,k ; Z2 ) is a homomorMore precisely, since j ∗ : H ∗ (G phism of algebras, we have j ∗ (wi (˜ γn+1,k )a ) = wi (˜ γn,k )a for any a. So we see that if wi (˜ γn,k )a 6= 0, then also wi (˜ γn+1,k )a 6= 0. Therefore we always have height(wi (˜ γn,k )) ≤ height(wi (˜ γn+1,k )). Since n + 1 is odd, by what we have proved we obtain that height(wi (˜ γn,k )) ≤ height(wi (˜ γn+1,k )) ≤

(k − 1)(n − k + 1) , i

and the proof of Proposition 1.1 is complete. Proof of Corollary 1.1. It is known (see p. 104 in Ref. 10) that the cup product 2t

w12

−1

2t

w22

−4

2t

∈ H 3·2

−9

(G22t ,3 ; Z2 ) ∼ = Z2

does not vanish. Since (see p. 103 in Ref. 10) height(w1 ) = 22t − 1, this 2t means that the class w22 −4 is not a multiple of w1 . So the standard Gysin 2t sequence argument implies that we have w ˜22 −4 6= 0, and therefore height(w ˜2 ) ≥ 22t − 4, as claimed. In addition to this, we know (see Proposition 1.1) that

2t

But w ˜22

height(w ˜2 ) ≤ 22t − 2. −2

= 0, which means that height(w ˜2 ) ≤ 22t − 3,

as claimed.

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2t

Indeed, let us suppose that w ˜22 −2 6= 0. Then by Poincar´e duality there 2t ˜ 22t ,3 ; Z2 ) such that the cup product exists a cohomology class z ∈ H 2 −5 (G 2t

w ˜22

−2

˜ 22t ,3 ; Z2 ) ∼ z ∈ H top (G = Z2

does not vanish. By what we have said above, since 22t − 5 < 22t − 4, the element z must be decomposable in the sense that z = w ˜2a1 w ˜3b1 + . . . w ˜2ar w ˜3br for some a1 , . . . , ar , b1 , . . . , br . We conclude that there are some numbers a and b such that the element 2t

w ˜22

−2+a

˜ 22t ,3 ; Z2 ) w ˜3b ∈ H top (G

does not vanish. Since height(w ˜2 ) ≤ 22t − 2, it must be a = 0, and b must be a positive integer. In other words, we have, for some b > 0, a nonzero element 2t

w ˜22

−2

w ˜3b ∈ H 3(2

2t

−3)

˜ 22t ,3 ; Z2 ) ∼ (G = Z2 .

By counting dimensions we obtain that 22t+1 − 4 + 3b = 3 · 22t − 9, hence 22t −5 = 3b. But the latter is impossible: indeed, we have 22t −5 = 22(t+1) − 5−3·22t, hence if 3 does not divide 22t −5, then 3 does not divide 22(t+1) −5; 2t of course, 3 does not divide 24 − 5. This shows that w ˜22 −2 = 0, and the proof of the corollary is complete. Proof of Proposition 1.2. By Ref. 4, we know that height(w2 (γn,k )) = κ2 (n, k). a a The fact that p∗ (w2 (γn,k )) = w2 (˜ γn,k ) implies that we always have

height(w2 (˜ γn,k )) ≤ height(w2 (γn,k )) = κ2 (n, k). At the same time, by Proposition 1.1 we have height(w2 (˜ γn,k )) ≤ κ ˜ 2 (n, k). One readily verifies that the bound κ ˜ 2 (n, k) is for many pairs (n, k) better than κ2 (n, k); but it is not so for all pairs (n, k). That is why we always take the minimum of the two upper bounds. Proposition 1.2 is proved. Acknowledgments The author was supported in part by two grants of VEGA (Slovakia). He thanks Tibor Macko for his comments.

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References 1. V. Bart´ık and J. Korbaˇs, Stiefel–Whitney characteristic classes and parallelizability of Grassmann manifolds, Rend. Circ. Mat. Palermo (2) 33 (Suppl. 6) (1984) 19–29. 2. A. Borel, La cohomologie mod 2 de certains espaces homog`enes, Comment. Math. Helvetici 27 (1953) 165–193. 3. P. Conner, Differentiable Periodic Maps (2nd edition, LNM 738, Springer, Berlin, 1979). 4. S. Dutta and S.S. Khare, On second Stiefel–Whitney class of Grassmann manifolds and cuplength, J. Indian Math. Soc. 69 (2002) 237–251. 5. S. Ilori and D. Ajayi, The height of the first Stiefel–Whitney class of the real flag manifolds, Indian J. Pure Appl. Math. 31 (2000) 621–624. 6. J. Korbaˇs, Some partial formulae for Stiefel–Whitney classes of Grassmannians, Czechoslovak Math. J. 36 (111) (1986) 535–540. 7. J. Korbaˇs, Bounds for the cup-length of Poincar´e spaces and their applications, Topology Appl. 153 (2006) 2976–2986. 8. J. L¨ orinc, The height of the first Stiefel–Whitney class of any nonorientable real flag manifold, Math. Slovaca 53 (2003) 91–95. 9. J. Milnor and J. Stasheff, Characteristic Classes (Princeton University Press, Princeton, N. J., 1974). 10. R.E. Stong, Cup products in Grassmannians, Topology Appl. 13 (1982) 103– 113.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 463–473

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Differential invariants of velocities and higher order Grassmann bundles D. Krupka and Z. Urban Department of Algebra and Geometry, Palack´ y University, Olomouc, Czech Republic E-mail: [email protected], [email protected] http://globanal.upol.cz In this paper we present the theory of higher order velocities and their scalar differential invariants. We consider a natural action of a differential group on manifolds of higher order velocities, and study properties of its orbits (contact elements) and orbit spaces (higher order Grassmann bundles). We show that this action defines on a manifold of regular velocities the structure of a principal bundle with structure group the differential group. The bundle projection is then naturally interpreted as the basis of scalar invariants of higher order velocities. We give a recurrence formula for differential invariants. Explicit description of the basis is given for velocities of order ≤ 3. Analogous methods can be applied to the problem of finding bases of differential invariants of different geometric objects. Keywords: Jet, velocity, differential invariant, Grassmann bundle. MS classification: 53A55, 58A20, 58A32.

1. Introduction This paper represents a shortened version of a plenary lecture, delivered by the first author at the conference Differential Geometry and its Applications, Olomouc, Czech Republic, August 2007. Its aim is to describe the structure of scalar differential invariants of (higher order) regular velocities. By the r-th differential group Lrn of Rn we mean the Lie group of invertible r-jets with source and target at the origin 0 ∈ Rn . Let Y be a manifold of dimension n + m. By the (r, n)-velocity with values in Y we mean an r-jet with source at 0 ∈ Rn and target in Y ; we denote the set of velocities by Tnr Y . Lrn acts naturally on Tnr Y , the action being induced by the composition of jets. A real-valued function, constant on orbits of this action, is said to be a scalar differential invariant. Every scalar differential invariant is completely described, in the well-known sense, by means of the quotient

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projection of Tnr Y onto the quotient space Tnr Y /Lrn . Our analysis of the group action of Lrn on Tnr Y shows that the manifold of regular velocities imm Tnr Y ⊂ Tnr Y is a principal Lrn -bundle over imm Tnr Y /Lrn . We call imm Tnr Y /Lrn the Grassmann prolongation of Y , and the bundle projection the basis of scalar differential invariants of (r, n)velocities. We find explicit chart description for the quotient projections of regular velocities of order ≤ 3. The concepts, used in this paper, follow basic ideas of the Ehresmann’s theory of jets and contact elements1 (for generalities see also Ref. 3 and Ref. 5). It should be pointed out that the concept of a higher order Grassmann fibrations has also been considered in a different framework by Olver.8 The exposition, used in this paper, is based on Grigore, D. Krupka, and M. Krupka.2,4,6,7 Explicit analysis of the differential invariants of order r ≤ 3 is taken from Ref. 9; the proofs can be found in Ref. 6 and Ref. 9.

2. Differential invariants Recall that if G is a group acting on two sets P and Q on the left, then a mapping f : P → Q is said to be G-equivariant, if f (g · p) = g · f (p) for all g ∈ G and p ∈ P . G-equivariant mappings are also called invariants of the group G. If Q is the real line R, endowed with the trivial action of G, an equivariant mapping f : P → R is called a scalar invariant. These concepts can be applied to the case when G is a differential group. By the r-th differential group Lrn of Rn we mean the Lie group of invertible r-jets with source and target at the origin 0 ∈ Rn . Recall that Lrn as the set consists of invertible r-jets J0r α of diffeomorphisms α of open sets U ⊂ Rn , containing the origin 0 ∈ Rn , such that α(0) = 0. The multiplication Lrn ∋ (J0r α, J0r β) → J0r α·J0r β = J0r (α◦β) ∈ Lrn is given by the composition of jets, and defines the structure of a Lie group on Lrn . Clearly, L1n is canonically identified with the general linear group GLn (R), and the canonical jet projection of Lrn onto L1n is a Lie group homomorphism. Denoting by Knr its kernel, we can represent Lrn as the semi-direct product of L1n and Knr , Lrn = L1n ×s Knr ; Knr is a nilpotent normal subgroup of Lrn . If P and Q are two left Lrn -manifolds, then a mapping f : P → Q is said to be a differential invariant, if for all J0r α ∈ Lrn and p ∈ P , f (J0r α · p) = J0r α · f (p).

(1)

If π : P → P/Lrn is the quotient projection and we take for Q the real

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line R with trivial action of the group Lrn , we have a commutative diagram f

/ /R y< < y yy π yy   yy f0 P/Lrn P

(2)

defining a mapping f0 : P/Lrn → R. The decomposition f = f0 ◦ π

(3)

then allows us to express the scalar differential invariant f in terms of π and f0 . If, moreover, the orbit space P/Lrn has a manifold structure such that π is a submersion, then f0 is differentiable if and only if f is differentiable. We call π the basis of differential invariants on the left Lrn -manifold P . Note that if Q is a left L1n -manifold, e.g. a vector space of tensors over n R , endowed with the tensor action of the group L1n , and f : P → Q is a differential invariant, then the restriction of the group action of Lrn to Knr yields the condition f (J0r α · p) = f (p). This observation indicates the meaning of the semi-direct product structure of Lrn for computation of differential invariants (the orbit reduction method ): Each differential invariant is expressible as the composite of a scalar invariant of the group Knr and an invariant of the general linear group L1n . 3. Velocities r From now on, m, n ≥ 1 and r ≥ 0 are integers. J(x,y) (X, Y ) denotes the set of r-jets with source x in a manifold X and target y in a manifold Y . The composition of jets is denoted ◦. Let Y be a manifold of dimension m + n. By an n-velocity of order r at r (Rn , Y ), P = J0r ζ. We denote a point y ∈ Y we mean an r-jet P ∈ J(0,y) [ r Tnr Y = (Rn , Y ), (4) J(0,y) y∈Y

and define surjective mappings τnr,s : Tnr Y → Tns Y , where 0 ≤ s ≤ r, by τnr,s (J0r ζ) = J0s ζ. The set Tnr Y is endowed with a right action of the group Lrn , defined by the jet composition Tnr Y × Lrn ∋ (P, A) → P ◦ A ∈ Tnr Y.

(5)

For every chart (V, ψ), ψ = (y K ), on Y we set Vnr = (τnr,0 )−1 (V ), and ψnr = (y K , yiK1 , yiK1 i2 , . . . , yiK1 i2 ...ir ), where 1 ≤ K ≤ n + m, 1 ≤ i1 ≤ i2 ≤ . . . ≤ ir ≤ n, and for every P ∈ Vnr , P = J0r ζ, yiK1 i2 ...il (P ) =

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Di1 Di2 . . . Dil (y K ζ)(0). Note that this formula can also be written in a different way. Denote by trξ the translation of Rn+m , expressed by the equation trξ (x) = x − ξ. Writing trK ξ for the components of the translation, we can write yiK1 i2 ...il (P ) = Di1 Di2 . . . Dil (trK ψζ(0) ψζ)(0). r r In the following lemma we denote Ln,m = J(0,0) (Rn , Rm ). Lemma 3.1. There exists one and only one smooth structure on Tnr Y such that for every chart (V, ψ), the pair (V
nr , ψnr ) is a chart on Tnr Y . The dir mension of Tnr Y is N = (n + m) n+r n . In this smooth structure Tn Y is r r,0 a smooth fibration with fiber Ln,m , base Y , and projection τn . The group action (5) is smooth. The set Tnr Y endowed with the smooth structure and with the group action, defined by Lemma 3.1, is called the manifold of n-velocities of order r over Y . The chart (Vnr , ψnr ) on Tnr Y is said to be associated with the chart (V, ψ). The group action (5) can be easily determined in the canonical coordinates on Lrn and a chart (V, ψ) on Y . Using the associated chart (Vnr , ψnr ), we get the equations yK = yK ,

yK i1 i2 ...il =

l X p=1

yjK1 j2 ...jp

X

j

ajI11 ajI22 . . . aIpp ,

(6)

(I1 ,I2 ,...,Ip )

where the second sum is extended to all partitions (I1 , I2 , . . . , Ip ) of the set {i1 , i2 , . . . , il }. Let γ be a smooth mapping of an open set U ⊂ Rn into Y . Then for any t ∈ U , the mapping x → γ ◦ tr−t (x) is defined on a neighbourhood of the origin 0 ∈ Rn so the mapping U ∋ t → (T0r γ)(t) = J0r (γ ◦ tr−t ) ∈ Tnr Y is defined. This mapping is called the r-prolongation of γ. Its chart expression is given by yiK1 i2 ...il ◦ T0r γ(t) = Di1 Di2 . . . Dil (y K γ)(t).

(7)

T0 T0r−1 ζ · ξ = ξ i di (P ),

(8)

Let P ∈ Tnr Y , P = J0r ζ. A representative ζ of P defines the tangent mapping T0 T0r−1 ζ, which sends a tangent vector ξ ∈ T0 Rn to the tangent vector T0 T0r−1 ζ · ξ of Tnr−1 Y at τnr,r−1 (P ) = J0r−1 ζ. If ξ = ξ i (∂/∂ti )0 , then by (7),

where di (P ) =

r−1 X

X

l=0 i1 ≤i2 ≤...≤il

yiK1 i2 ...il i (P )

∂ ∂yiK1 i2 ...il

!

(9) J0r−1 ζ

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defines the i-th formal derivative morphism Tnr Y ∋ P → di (P ) ∈ T Tnr−1Y over Tnr−1 Y . The tangent vectors (9) are defined independently of the chart. Note that ∂/∂yiK1 i2 ...il are understood as tangent vectors to Tnr−1 Y ; (9) is not a vector field. The i-th formal derivative of a function f : Vnr−1 → R is defined by di f =

r−1 X

X

yiK1 i2 ...il i

l=0 i1 ≤i2 ≤...≤il

∂f ∂yiK1 i2 ...il

.

(10)

Since by (7), di f ◦ T0r γ = Di (f ◦ T0r−1 γ), we have di dj f = dj di f . Note that if we take f = yiL1 i2 ...il in (10), we get di yiL1 i2 ...il = yiL1 i2 ...il i .

(11)

We give explicit transformation formulas between the associated charts on Tnr Y . Suppose we have the transformation equations between two charts (V, ψ) to (V , ψ) Our aim now will be to derive explicit transformation formulas between the induced charts on Tnr Y . Let us write the transformation equations from (V, ψ) and (V , ψ), y K = F K (y L ). We wish to deter, defining induced transformations , . . . , FiK , FiK mine the functions FiK 1 i2 ...ir 1 i2 1 L L K K L L y i1 i2 ...il = Fi1 i2 ...il (y , yj1 , yj1 j2 , . . . , yj1 j2 ...jl ) from (Vnr , ψnr ) to (V rn , ψ rn ). L Note that since di = di , we have from (11), y L i1 i2 ...il il+1 = dil+1 y i1 i2 ...il = L L dil+1 dil yi1 i2 ...il−1 = . . . = dil+1 dil . . . di1 y . In the following formula we sum through all partitions (I1 , I2 , . . . , Ip ) of the set {i1 , i2 , . . . , il }. Lemma 3.2. The functions FiK are given by 1 i2 ...il FiK 1 i2 ...il

=

l X

X

p=1 (I1 ,I2 ,...,Ip )

L

yIL11 yIL22 . . . yIpp

∂ pF K . ∂y L1 ∂y L2 . . . ∂y Lp

(12)

4. Regular velocities We need a convention on splitting of the sequence (1, 2, . . . , n, n + 1, . . . , n+m) into two complementary subsequences. Any subsequence, consisting of n integers, is called an n-subsequence. Given an n-subsequence (i) = (i1 , i2 , . . . , in ), we also have the complementary subsequence (σ) = (σ1 , σ2 , . . . , σm ). By definition, i1 < i2 < . . . < in and σ1 < σ2 < . . . < σm . A velocity P ∈ Tnr Y is said to be regular, if P = J0r ζ, where ζ is an immersion at the origin 0 ∈ Rn . This condition is equivalent with the existence of a chart (V, ψ), ψ = (y K ), at ζ(0) and an n-subsequence

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(i) = (i1 , i2 , . . . , in ) of the sequence (1, 2, . . . , n, n + 1, . . . , n + m) such that, with i ∈ (i1 , i2 , . . . , in ),

(13) det yji (P ) = det Dj (y i ◦ ζ)(0) 6= 0.

The subset of regular velocities in Tnr Y is denoted by imm Tnr Y .

Lemma 4.1. The set imm Tnr Y is an open, dense, Lrn -invariant subset of Tnr Y . imm Tnr Y is called the manifold of regular n-velocities of order r over Y . Our aim now will be to analyze the equivalence ⊂ imm Tnr Y × r imm Tn Y , defined by the group action (5), “there exists A ∈ Lrn such that P = Q◦A”. Fix a chart (V, ψ), ψ = (y K ), on Y , and consider the associated chart (Vnr , ψnr ), ψnr = (y K , yiK1 , yiK1 i2 , . . . , yiK1 i2 ...ir ), on imm Tnr Y . We set for every n-subsequence (i) = (i1 , i2 , . . . , in ) of (1, 2, . . . , n, n + 1, . . . , n + m) 
W (i) = P ∈ Vnr | det yji (P ) 6= 0 . (14)

R

W (i) is an open, Lrn -invariant subset of the set Vnr . Shrinking the coordinates ψnr = (y K , yiK1 , yiK1 i2 , . . . , yiK1 i2 ...ir ) to W (i) , we get a chart, associated with (V, ψ) and (i), and denoted by (W (i) , χ(i) ). Charts of this type clearly cover Vnr , and constitute an atlas on imm Tnr Y . The coordinate transformations between (W (i) , χ(i) ) and (W (j) , χ(j) ) coincides with the restriction of the identity mapping of Vnr to W (i) ∩ W (j) . We introduce a collection of functions zik : W (i) → R by zik yqi = δqk ,

(15)

where i ∈ (i) and 1 ≤ k, q ≤ n. Existence of these functions is guaranteed by the definition of W (i) ; zik is a rational function of yqi . Lemma 4.2. Let (P, Q) ∈ imm Tnr Y × imm Tnr Y . The following conditions are equivalent: (a) (P, Q) ∈ . (b) There exist a chart (V, ψ), ψ = (y K ), and an n-subsequence (i) of the sequence (1, 2, . . . , n, n + 1, . . . , n + m), such that P, Q ∈ W (i) , the j coordinates yiK1 i2 ...il (resp. y K i1 i2 ...il , resp. aI ) of P (resp. Q, resp. A) satisfy

R

yK = yK ,

yK i1 i2 ...il =

l X p=1

yjK1 j2 ...jp

X

(I1 ,I2 ,...,Ip )

j

ajI11 ajI22 . . . aIpp ,

(16)

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469

where 1 ≤ i1 , i2 , . . . , il , j1 , j2 , . . . , jp ≤ n, 1 ≤ l, p ≤ r, and the recurrence formula   l X X j ajI11 ajI22 . . . aIpp  , aqi1 i2 ...il = ziq yii1 i2 ...il − yji1 j2 ...jp (17) p=2

(I1 ,I2 ,...,Ip )

where i ∈ (i), and (σ) is the complementary subsequence of (i).

We now introduce new charts on imm Tnr Y , adapted to the group action (5). Let P ∈ W (i) , P = J0r ζ. Any representative ζ defines the (r − 1)-prolongation Tnr−1ζ(t) = J0r−1 (ζ ◦ tr−t ). The mapping ψ (i) ◦ ζ = (y i1 ζ, y i2 ζ, . . . , y in ζ), where (i) = (i1 , i2 , . . . , in ), is a diffeomorphism at 0 ∈ Rn . We have a well-defined mapping Tnr−1ζ ◦ (ψ (i) ◦ ζ)−1 ◦ ψ (i) ◦ τnr,0 , and its tangent mapping at P , denoted h(i) : TJ0r ζ imm Tnr Y → TJ r−1 ζ imm Tnr−1Y. 0

Let ξ be a tangent vector to imm Tnr Y at P , ξ=

r X

ξiK1 i2 ...il

l=0

After some computation

∂ ∂yiK1 i2 ...il

!

.

(18)

(19)

P

h(i) (ξ) = ξ j ∆j (P ),

(20)

∆j = zjq dq .

(21)

where

Lemma 4.3. (a) For every i, k ∈ (i), ∆i ∆k = ∆k ∆i . (b) Let (i) and (j) be two n-subsequences of the sequence (1, 2, . . . , n + m). For any two charts (V, ψ) and (V , ψ), and the associated charts (W (i) , χ(i) ) and (W (j) , χ(j) ), ∆j = z sj ysi ∆i . Theorem 4.1. Let (V, ψ), ψ = (y K ), be a chart on Y , let (i) be an nsubsequence of the sequence (1, 2, . . . , n + m), (σ) the complementary subsequence, and let (W (i) , χ(i) ), χ(i) = (y K , yiK1 , yiK1 i2 , . . . , yiK1 i2 ...ir ), be the chart, associated with (V, ψ) and (i). (a) There exist unique functions wσ , wiσ1 , wiσ1 i2 , . . . , wiσ1 i2 ...ir , with i1 , i2 , . . . , ir ∈ (i), σ ∈ (σ), defined on W (i) , symmetric in the subscripts, such that y σ = wσ ,

ypσ1 p2 ...pk =

k X

X

q=1 (I1 ,I2 ,...,Iq )

i

yIi11 yIi22 . . . yIqq wiσ1 i2 ...iq

(22)

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(summation through partitions (I1 , I2 , . . . , Iq ) of the set {p1 , p2 , . . . , pk }). These functions are Lrn -invariant, and satisfy the recurrence formulas wiσ1 i2 ...ik ik+1 = ∆ik+1 wiσ1 i2 ...ik .

(23)

(b) The pair (W (i) , Ψ(i) ), where Ψ(i) = (y i , ypi 1 , ypi 1 p2 , . . . , ypi 1 p2 ...pr , wσ , wiσ1 , wiσ1 i2 , . . . , wiσ1 i2 ...ir ),

(24)

is a chart on imm Tnr Y . (c) The group action (5) is described on W (i) by the equations yi = yi , y ip1 p2 ...pk =

w σ = wσ ,

k X

X

wσi1 i2 ...ik = wiσ1 i2 ...ik , j

ajI11 ajI22 . . . aIqq yji1 j2 ...jq ,

(25)

q=1 (I1 ,I2 ,...,Iq )

where i, i1 , i2 , . . . , ik ∈ (i), σ ∈ (σ), and 1 ≤ k ≤ r. Equations of the orbits are wiσ1 i2 ...ik = cσi1 i2 ...ik ,

(26)

where cσi1 i2 ...ik ∈ R. It is now easy to prove the following result. Theorem 4.2. If Y is Hausdorff, then the right action (5) defines the structure of a right principal Lrn -bundle on imm Tnr Y . If m = 0, then dim Y = n, and a regular n-velocity of order r at a point y ∈ Y is called an r-frame at y. In this case we usually write imm Tnr Y = F r Y , and call the principal Lrn -bundle F r Y the bundle of r-frames over Y . If r = 1, we get the principal GLn (R)-bundle F Y = F 1 Y of linear frames. An Lrn -orbit, passing through a point P ∈ imm Tnr Y , is called an ncontact element of order r; if P = J0r ζ, we usually denote this contact element by Gr0 ζ. The orbit manifold Grn Y = imm Tnr Y /Lrn , i.e., the base of the principal Lrn -bundle imm Tnr Y , is the manifold of n-contact elements of order r; Grn Y is also called the (n, r)-Grassmann prolongation of Y . 5. Grassmann prolongations of manifolds Denote by πnr the quotient projection of imm Tnr Y onto Grn Y . From the construction of the Grassmann prolongation, there exists a unique mapping

Differential invariants of velocities and higher order Grassmann bundles

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ρrn of Grn Y onto Y such that the following diagram r πn

/ / Grn Y s s s ss ss ρrn s   ysy s Y

imm Tnr Y

(27)

commutes. We now describe explicitly the structure of Grn Y and ρrn . Consider a chart (V, ψ), ψ = (y K ), on Y , and the associated chart (Vnr , ψnr ), ψnr = (y K , yiK1 , yiK1 i2 , . . . , yiK1 i2 ...ir ), on imm Tnr Y . Fix an n-subsequence (i) of the sequence (1, 2, . . . , n + m), and con(i) sider, as in Section 4, Theorem 4.1, the chart (W , Ψ(i) ), Ψ(i) =
i i i i σ σ σ σ y , yp1 , yp1 p2 , . . . , yp1 p2 ...pr , w , wi1 , wi1 i2 , . . . , wi1 i2 ...ir , on imm Tnr Y . We have wiσ1 i2 ...ik−1 ik = ∆ik ∆ik−1 . . . ∆i2 ∆i1 wσ ,

(28)

where r−1

∆i =

+

X ∂ + i ∂y

X

wiσ1 i2 ...il i

l=0 i1 ≤i2 ≤...≤il

r−1 X

X

zis yik1 i2 ...il s

l=1 i1 ≤i2 ≤...≤il

(i)

∂ ∂wiσ1 i2 ...il

∂ ∂yik1 i2 ...il

(29)

.

(i)

Denoting WG = πnr (W (i) ), ΨG = (y i , wσ , wiσ1 , wiσ1 i2 , . . . , wiσ1 i2 ...ir ), we obtain the associated chart on the Grassmann prolongation Grn Y . Transformation equations between coordinates of these charts are considered in the following lemma. Lemma 5.1. Let (V, ψ), ψ = (y K ), and (V , ψ), ψ = (y K ), be two charts on Y such that V ∩ V 6= ∅, with transformation equations yj = F j (y i , wσ ), Then w νj1

=

z sj1 ysi



w ν = F ν (y i , wσ ).

ν ∂F ν σ ∂F + w i ∂y i ∂wσ



.

(30)

(31)

To obtain transformation formulas for further coordinates, one should apply the recurrence formula (28). r As before, denote Lrn,n+m = J(0,0) (Rn , Rn+m ), and consider the subset r r imm Ln,n+m of Ln,n+m , formed by jets of immersions at 0 ∈ Rn . Formula (7) defines on imm Lrn,n+m a right action of the differential group Lrn ; we

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denote Grn,n+m = imm Lrn,n+m /Lrn . Grn,n+m is a manifold, called the nGrassmannian of order r over Rn+m . Grn,n+m is endowed with a left action of the differential group Lrn+m . If Gr0 ζ is the contact element, containing an r-jet J0r ζ, and A ∈ Lrn+m , A = J0r α, then Gr0 (α◦ζ) is defined, and depends on A only. We set A·Gr0 ζ = Gr0 (α ◦ ζ). Lrn+m also acts on the product F r Y × Grn,n+m ; we have the right action F r Y × Grn,n+m × Lrn+m ∋ (J0r µ, Gr0 ζ, J0r α) →
→ J0r (µ ◦ α), Gr0 (α−1 ◦ ζ) ∈ F r Y × Grn,n+m .

(32)

Let F r Y denote the bundle of r-frames over Y . The principal Lrn+m bundle structure of F r Y and the structure of the Grassmannian Grn,n+m allow us to define the mapping F r Y × Grn,n+m ∋ (J0r µ, Gr0 ζ) → Gr0 (µ ◦ ζ) ∈ Grn Y ; it is immediately seen that this mapping is constant on Lrn+m orbits, and is smooth, i.e., is a frame mapping. Thus, we have the following assertion. Theorem 5.1. Grn Y has the structure of a fiber bundle over Y with fiber Grn,n+m , associated with the principal bundle of frames F r Y . 6. Example: Scalar differential invariants of order ≤ 3 The action (5) is for r = 3 expressed by the equations yK = y K ,

K k1 k1 k2 K yK p1 p2 = ap1 ap2 yk1 k2 + ap1 p2 yk1 ,
+ akp22 p3 akp11 + akp11 p3 akp22 + akp11 p2 akp32 ykK1 k2 (33)

k1 K yK p1 = ap1 yk1 ,

k1 k2 k3 K yK p1 p2 p3 = ap1 ap2 ap3 yk1 k2 k3

+ akp11 p2 p3 ykK1 ,

where K ∈ (i), (σ), 1 ≤ p1 , p2 , p3 ≤ n, 1 ≤ k1 , k2 , k3 ≤ n. Computing the projection πn3 : imm Tnr Y → G3n Y explicitly, we get the following theorem, characterizing all scalar differential invariants of velocities of order ≤ 3 in terms of their basis. Theorem 6.1. Every scalar differential invariant of regular velocities of order ≤ 3 depends only on the following functions: wqσ1 = zqk11 ykσ1 ,


wqσ1 q2 = zqk11 zqk22 ykσ1 k2 − zil ylσ yki 1 k2 ,


wqσ1 q2 q3 = zqk11 zqk22 zqk33 ykσ1 k2 k3 − zil ylσ yki 1 k2 k3 − zqk22 zqk33 ykj 2 k3 zqk11 zjp

i σ − zil ylσ ypk · ykσ1 p − zil ylσ yki 1 p − zqk11 zqk33 ykj 1 k3 zqk22 zjp ypk 2 2
i σ − zil ylσ ypk . − zqk11 zqk22 ykj 1 k2 zqk33 zjp ypk 3 3

(34)

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Acknowledgments The authors acknowledge support of grants 201/06/0922 (GACR) and MSM 6198959214 (Czech Ministry of Education). References 1. C. Ehresmann, Les prolongements d’une variete diff´erentiable I–V, C. R. Acad. Sc. Paris 223 (1951) 598–600, 777–779, 1081–1083; 234 (1952) 1028– 1030, 1424–1425. 2. D.R. Grigore and D. Krupka, Invariants of velocities and higher order Grassmann bundles, J. Geom. Phys. 24 (1998) 244–264. 3. I. Kol´ aˇr, P. W. Michor and J. Slov´ ak, Natural Operations in Differential Geometry (Springer-Verlag, Berlin, 1993). 4. D. Krupka, Natural Lagrangian structures, In: Semester on Diff. Geom. (Banach Center, Warsaw, 1979; Banach Center Publications 12, 1984) 185–210. 5. D. Krupka and J. Janyˇska, Lectures on Differential Invariants (J. E. Purkynˇe University, Faculty of Science, Brno, Czechoslovakia, 1990, pp. 193). 6. D. Krupka and M. Krupka, Jets and contact elements, In: Proc. Sem. on Diff. Geom. (MATH. Publications Vol. 2, Silesian Univ. in Opava, Opava, Czech Republic, 2000) 39–85. 7. M. Krupka, Orientability of higher order grassmannians, Math. Slovaca 44 (1994) 107–115. 8. P.J. Olver, Symmetry groups and group invariant solutions of partial differential equations, J. Diff. Geom. 14 (1979) 497–542. 9. Z. Urban, Grassmann prolongation, thesis, Palack´ y University, Olomouc, Czech Republic, 2005.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 475–487

475

Like jet prolongation functors of affine bundles J. Kurek Institute of Mathematics, Maria Curie-Sklodowska University, Lublin, Poland E-mail: [email protected] W.M. Mikulski Institute of Mathematics, Jagiellonian University, Krak´ ow, Poland E-mail: [email protected] We present a complete description of all fiber product preserving gauge bundle functors F on the category ABm of affine bundles with m-dimensional bases and affine bundle maps with local diffeomorphisms as base maps. Keywords: (fiber product preserving) gauge bundle functors, natural transformations, jets, Weil algebras. MS classification: 58A05, 58A20.

1. Introduction It is well-known that product or fiber product preserving (gauge) bundle functors play a very important role in differential geometry. To such bundle functors one can lift some geometric structures as vector fields, forms, connections, e.t.c. The product preserving bundle functors on the category Mf of manifolds and maps have been classified by means of Weil algebras.3 The fiber product preserving bundle functors on the category F Mm of fibred manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps have been classified in Ref. 4, and studied in Refs. 1,2. The fiber product preserving gauge bundle functors on the category VBm of vector bundles with m-dimensional bases and their vector bundle maps with local diffeomorphisms as base maps have been classified in Ref. 5. The purpose of the present paper is to describe all fiber product pre-

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serving gauge bundle functors on the category ABm of affine bundles with m-dimensional bases and affine bundle maps with local diffeomorphisms as base maps. Let us recall the following definitions (see for ex. Ref. 3). Let F : ABm → F M be a covariant functor into the category F M of fibred manifolds and their fibred maps. Let BABm : ABm → Mf and BF M : F M → Mf be the respective base functors. A gauge bundle functor on ABm is a functor F as above satisfying: (i) (Base preservation) BF M ◦ F = BABm . Hence the induced projections form a functor transformation π : F → BABm . (ii) (Localization) For every inclusion of an open affine subbundle iE|U : E|U → E, F (E|U ) is the restriction π −1 (U ) of π : F E → BABm (E) over U and F iE|U is the inclusion π −1 (U ) → F E. (iii) (Regularity) F transforms smoothly parametrized systems of ABm -morphisms into smoothly parametrized systems of F M-morphisms.

A gauge bundle functor F : ABm → F M is of finite order r if from jxr f = jxr g it follows that Fx f = Fx g for any ABm -objects E1 → M1 , E2 → M2 , any ABm -maps f, g : E1 → E2 and any x ∈ M1 . Given two gauge bundle functors F1 , F2 on ABm , by a natural transformation µ : F1 → F2 we shall mean a system of base preserving fibred maps µ : F1 E → F2 E for every affine bundle E from ABm satisfying F2 f ◦ µ = µ ◦ F1 f for every ABm -morphism f : E → G. A gauge bundle functor F on ABm is fiber product preserving if for pr2 pr1 every fiber product projections E1 ←−− E1 ×M E2 −−→ E2 in the category F pr1 F pr2 ABm the mappings F E1 ←−−− F (E1 ×M E2 ) −−−→ F E2 are fiber product projections in the category F M. In other words F (E1 ×M E2 ) = F (E1 )×M F (E2 ) modulo the restriction of (F pr1 , F pr2 ). A simple example of fiber product preserving gauge bundle functor on ABm is the functor ( )→ : ABm → F M sending any ABm -object E → M into the corresponding vector bundle E → → M and any ABm -map f : E1 → E2 into the corresponding vector bundle map f → : E1→ → E2→ . In fact, ( )→ : ABm → VBm . If we compose ( )→ : ABm → VBm with a fiber product preserving gauge bundle functor T (V,H,t) for so called admissible triple (V, H, t) of some finite order r, see Ref. 5, we obtain fiber product preserving gauge bundle functor T (V,H,t) ◦ ( )→ on ABm . The most important example of fiber product preserving gauge bundle functor on ABm is the r-jet prolongation functor J r : ABm →

Like jet prolongation functors of affine bundles

477

F M, where for an ABm -object p : E → M we have J r E = {jxr σ | σ is a local section of E, x ∈ M } and for a ABm -map f : E1 → E2 covering f : M1 → M2 we have J r f : J r E1 → J r E2 , J r f (jxr σ) = jfr (x) (f ◦ σ ◦ f −1 ), jxr σ ∈ J r E1 . This functor plays an important role in the theory of higher order connections, Lagrangians, differential equations, e.t.c. Another example is the so called vertical r-jet prolongation functor Jvr : ABm → F M, where for a ABm -object p : E → M we have Jvr E = {jxr γ | γ is a local map M → Ex , x ∈ M } and for a ABm -map f : E1 → E2 covering f : M1 → M2 we have Jvr f : Jvr E1 → Jvr E2 , Jvr f (jxr γ) = jfr (x) (f ◦

γ ◦ f −1 ), jxr γ ∈ Jvr E1 . Another example is the so called vertical Weil functor V A : ABm → F M corresponding to a Weil algebra A, where for a ABm -object p : E → M S we have V A E = x∈M T A (Ex ) and for a ABm -map f : E1 → E2 we have S V A f = x∈M1 T A (fx ) : V A E1 → V A E2 . In general, the composition of a fiber product preserving bundle functor T (A,H,t) on F Mm for some triple (A, H, t), see Ref. 4, with the forgetting functor ABm → F Mm we obtain fiber product preserving gauge bundle functor T (A,H,t) : ABm → F M. The fiber product F1 ×BABm F2 : ABm → F M of fiber product preserving gauge bundle functors F1 , F2 : ABm → F M is again a fiber product preserving gauge bundle functor. We recall that (F1 ×BABm F2 )(E) = F1 E ×M F2 E for any ABm -object E → M and (F1 ×BABm F2 )(f )(v1 , v2 ) = (F1 f (v1 ), F2 f (v2 )) for any ABm -map f : E → G and any (v1 , v2 ) ∈ F1 E ×M F2 E. The composition of some fiber product preserving gauge bundle functors on ABm is again a fiber product preserving gauge bundle functor on ABm . (In Proposition 5.2, it will be proved that every fiber product preserving gauge bundle functor has values in ABm . So, the composition is possible.) The first main result in this paper is that all fiber product preserving gauge bundle functors F on ABm of finite order r are in bijection with so called admissible systems, i.e. systems (V, H, t, 1), where V is a finite dimensional vector space over R, 1 ∈ V is an element, H : Grm → GL(V ) is a smooth group homomorphism from Grm = invJ0r (Rm , Rm )0 into GL(V ) r → gl(V ) is a Grm -equivariant with H(ξ)(1) = 1 for any ξ ∈ Grm , t : Dm r unity preserving associative algebra homomorphism from Dm = J0r (Rm , R) into gl(V ). The second main result is that natural transformations between two fiber product preserving gauge bundle functors on ABm of order r are in

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bijection with the morphisms between corresponding admissible systems. The third main result is that any fiber product preserving gauge bundle functor on ABm is of finite order. All manifolds are assumed to be finite dimensional. All manifolds and maps are assumed to be smooth, i.e. of class C ∞ . 2. Fiber product preserving gauge bundle functors on ABm corresponding to admissible systems Definition 2.1. An admissible system of order r and dimension m is a system (V, H, t, 1), where V is a finite dimensional vector space over R, 1 ∈ V is an element, H : Grm → GL(V ) is a smooth group homomorphism from the Lie group Grm = invJ0r (Rm , Rm )0 of invertible r-jets at 0 ∈ Rm of diffeomorphisms Rm → Rm preserving 0 into the group GL(V ) of linear r isomorphisms of V with H(ξ)(1) = 1 for any ξ ∈ Grm , and t : Dm → gl(V ) is r a Gm -equivariant unity preserving algebra homomorphism from the algebra r Dm = J0r (Rm , R) of r-jets at 0 ∈ Rm of maps Rm → R into the associative algebra gl(V ) of linear endomorphisms of V . r by j0r ϕ.j0r γ = j0r (γ ◦ ϕ−1 ), j0r ϕ ∈ Grm , We recall that Grm acts on Dm r r . We also recall that Grm acts ∈ Dm . This action will be denoted by Hm −1 on gl(V ) by ξ.A = H(ξ) ◦ A ◦ H(ξ ), ξ ∈ Grm , A ∈ gl(V ). These actions are by unity preserving algebra isomorphisms. Let (V, H, t, 1) be an admissible system of order r and dimension m. We are going to construct a fiber product preserving gauge bundle functor on ABm corresponding to (V, H, t, 1).

j0r γ

Example 2.1. For an ABm -object p : E → M we put [ T (V,H,t,1)E = {Φ ∈ Homtx (J r F IBAF Fx (E), V˜x M ) | Φ(jxr 1) = 1x }. x∈M

Here V˜ : Mfm → VB is the vector natural bundle corresponding to the Grm space V , i.e. V˜ M = P r M [V, H] (the associated bundle) for any m-manifold M and V˜ ϕ = P r ϕ[idV ] : V˜ M1 → V˜ M2 for any embedding ϕ : M1 → M2 between m-manifolds, 1x ∈ V˜x M is the canonical element induced by 1, 1x := V˜ ϕ(1) for any local diffeomorphism ϕ : Rm → M with ϕ(0) = x (the definition of 1x is independent of the choice of ϕ), Homtx (J r F IBAF Fx (E), V˜x M ) is the space of module homomorphisms over tx : Jxr (M, R) → gl(V˜x M ) from the (free) Jxr (M, R)-module J r F IBAF Fx (E) of r-jets at x ∈ M of germs at x of fiber affine maps

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E → R into the gl(V˜x M )-module V˜x M , where tx : Jxr (M, R) → gl(V˜x M ) is the induced by t unity preserving algebra homomorphism such that tx (jxr γ) = V˜0 ϕ ◦ t(j0r (γ ◦ ϕ)) ◦ (V˜0 ϕ)−1 for any γ : M → R and any embedding ϕ : Rm → M with ϕ(0) = x (tx is well-defined because of the Grm -equivariance of t). Given an affine bundle trivialization (x1 ◦ p, ..., xm ◦ p, y 1 , ..., y n ) : E|U → Rm × Rn we have an induced fiber bundle trivialization (˜ x1 , ..., x ˜m , y˜1 , ..., y˜n ) : T (V,H,t,1)E|U → Rm × V n such that x ˜i (Φ) = (V,H,t,1) i i j r ˜ x (xo ) ∈ R and y˜ (Φ) = Φ(jxo (y )) ∈ Vxo M =V E, i = ˜ for any Φ ∈ Txo ˜ V˜ ((x1 , ..., xm )−1 ◦ τ(xi (xo )) )(v), 1, ..., m, j = 1, ..., n, where V = ˜ V˜xo M by v = v ∈ V , τy : Rm → Rm is the translation by y ∈ Rm . Then T (V,H,t,1)E with obvious projection is a fiber bundle over M . Every ABm -map f : E1 → E2 covering f : M1 → M2 induces a fibred map T (V,H,t,1)E1 → T (V,H,t,1)E2 covering f such that T (V,H,t,1)f (Φ)(jfr (x) ξ) = V˜ f ◦ Φ(jxr (ξ ◦ f ))) (V,H,t,1)

for any Φ ∈ Tx E1 , x ∈ M1 , and any fiber affine map ξ : E2 → R. The correspondence T (V,H,t,1) : ABm → F M is a fiber product preserving gauge bundle functor of order r. Definition 2.2. We call T (V,H,t,1) : ABm → F M the fiber product preserving gauge bundle functor corresponding to admissible system (V, H, t, 1). We have the following fact. Proposition 2.1. (i) Given an ABm -object E, T (V,H,t,1)E is an affine bundle with the corresponding vector bundle T (V,H,t,0)E. (ii) Given an ABm -morphism f : E → G, T (V,H,t,1)f : T (V,H,t,1)E → T (V,H,t,1)G is an affine bundle map with the corresponding vector bundle map T (V,H,t,0)f : T (V,H,t,0)E → T (V,H,t,0)G. (iii) The vector bundle T (V,H,t,0)E is canonically isomorphic (by vector bundle isomorphism) with T (V,H,t)(E → ) (see Ref. 5 for the definition of T (V,H,t) ). Proof. The proof of Parts (i) and (ii) is standard. More precisely, given (V,H,t,0) E for z ∈ M and α ∈ R we have standardly defined Φ1 , Φ2 ∈ T z (V,H,t,0) (V,H,t,0) Φ 1 + Φ2 ∈ T z E and αΦ1 ∈ Tz E. That is why, T (V,H,t,0) E (V,H,t,1) is a vector bundle over M . Similarly, given Φ1 ∈ Tz E and Φ2 ∈

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(V,H,t,0)

E. E, z ∈ M , we have standardly defined Φ1 + Φ2 ∈ Tz Tz (V,H,t,1) That is why, T E is an affine bundle with the corresponding vector bundle T (V,H,t,0) E. Part (iii) will be clear after Section 10 because T (V,H,t,0) and T (V,H,t) ◦ → ( ) have isomorphic the corresponding admissible systems. 3. Admissible systems corresponding to fiber product preserving gauge bundle functors on ABm . Let F : ABm → F M be a f.p.p.g.b. functor of order r. We are going to construct an admissible system corresponding to F . Example 3.1. We put V F := (F0 (Rm × R), F0 (+), F0 λa , F0 0) , where F0 (Rm × R) is the fiber of F (Rm × R) over 0 ∈ Rm and Rm × R is the trivial affine bundle over Rm with fiber R with the corresponding vector bundle Rm × R, and where the fiber sum map +, the fiber scalar multiplications λa , a ∈ R, and the zero map 0 of the vector bundle Rm × R are treated as ABm -morphisms. Then V F is a finite dimensional vector space over R. We define H F : Grm → GL(V F ) by

H F (ξ)(v) := F0 (ϕ × idR )(v) , v ∈ V F , ξ = j0r ϕ ∈ Grm .

H F (ξ)(v) is well defined because of F is of order r. By the definition of V F , H F (ξ) ∈ GL(V F ). By the functoriality of F , H F is a group homomorphism. By the regularity of F , H F is smooth. r We define tF : Dm → gl(V F ) by r tF (η)(v) = F0 (˜ γ )(v) , v ∈ V F η = j0r γ ∈ Dm

where γ˜ : Rm × R → Rm × R is an ABm -map such that γ˜ (x, y) = (x, γ(x)y), x ∈ Rm , y ∈ R. tF (η)(v) is well defined because of F is of order r. By the definition of V F , tF (η) ∈ gl(V F ). By the functoriality of F and the definitions of the actions one can standardly verify that tF is a Grm -equivariant unity preserving algebra homomorphism. We define 1F ∈ V F by {1F } = the image of F0 (idRm , 1) ,

where (idRm , 1) : Rm → Rm × R is the ABm -morphism. It is easy to see that H F (ξ)(1F ) = 1F for any ξ ∈ Grm .

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Then (V F , H F , tF , 1F ) is an admissible system of order r and dimension m. Definition 3.1. We call (V F , H F , tF , 1F ) the admissible system corresponding to F . 4. Admissible systems corresponding to some fiber product preserving gauge bundle functors on ABm . In this section we present admissible systems corresponding to the presented in Introduction fiber product preserving gauge bundle functors on ABm . The results of this section will not be used to prove the main result. Fact 4.1. The admissible system corresponding to ( )→ : ABm → F M is r r , 0), where IdR : G (R, IdR , κDm m → GL(R) is the trivial group homomorr r phism and κDm : Dm → R = gl(R) is the trivial unity preserving algebra homomorphism. Fact 4.2. The admissible system corresponding to T (V,H,t) ◦ ( )→ is (V, H, t, 0),where T (A,H,t) : VBm → F M is described in Ref. 5. Fact 4.3. The admissible system corresponding to the r-jet prolongation r r r gauge bundle functor J r : ABm → F M is (Dm , Hm , trm , j0r 1), where Hm : r r r r r Gm → Aut(Dm ) is defined after Definition 2.1 and tm : Dm → gl(Dm ) is r given by trm (η)(ρ) = ηρ,where η, ρ ∈ Dm . Fact 4.4. The admissible system corresponding to the vertical r-jet prolonr r gation gauge bundle functor Jvr : ABm → F M is (Dm , Hm , trm ◦ ǫrm , j0r 1), r r r r where Hm : Gm → Aut(Dm ) is defined after Definition 2.1, trm : Dm → r r r r gl(Dm ) is defined above and ǫm : Dm → R ⊂ Dm is the algebra homomorphism. Fact 4.5. The admissible system corresponding to the vertical Weil gauge bundle functor V A : ABm → F M corresponding to a Weil algebra A is (A, idA , ǫA , 1), where idA : Grm → {idA } ⊂ GL(A) is the trivial group r homomorphism and ǫA : Dm → gl(A) is given by ǫA (η)(a) = γ(0)a, η = r r j0 γ ∈ Dm , a ∈ A. Fact 4.6. The admissible triple corresponding to T (A,H,t) : ABm → F M (obtained from T (A,H,t) : F Mm → F M described in Ref. 4), where A is a Weil algebra of order r, H : Grm → Aut(A) is a smooth group homor morphism and t : Dm → A is a Grm -equivariant unity preserving algebra

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˜ H, ˜ t˜, 1), where A˜ is A considered as the vector space, homomorphism, is (A, r ˜ : G → GL(A) ˜ is H : Gr → Aut(A) ⊂ GL(A), ˜ t˜ : Dr → gl(A) ˜ is given H m m m by t˜(η)(a) = t(η)a for a ∈ A˜ and η ∈ Dr , and 1 ∈ A is the unity of the m

Weil algebra A.

Fact 4.7. The admissible system corresponding to F1 ×BABm F2 is (V F1 ⊕ V F2 , H F1 ⊕ H F2 , tF1 ⊕ tF2 , 1F1 ⊕ 1F2 ). An open problem. By Proposition 5.2., any fiber product preserving gauge bundle functor on ABm has values in ABm . So, we can compose fiber product preserving gauge bundle functors on ABm . Compute the admissible system corresponding to the composition F1 ◦ F2 of two fiber product preserving gauge bundle functors F1 and F2 on ABm in terms of admissible systems corresponding to F1 and F2 . 5. Classification of fiber product preserving gauge bundle functors on ABm of order r in terms of admissible systems of order r and dimension m The following classification proposition shows that any fiber product preserving gauge bundle functor on ABm of order r is equivalent to some fiber product preserving gauge bundle functor as in Example 2.1. Proposition 5.1. Let F : ABm → F M be a fiber product preserving gauge bundle functor of order r. Let (V F , H F , tF , 1F ) be the admissible system (of order r and dimension m) corresponding to F . Then we have a natural F F F F equivalence ΘF : F =T ˜ (V ,H ,t ,1 ) . Proof. Let p : E → M be an ABm -object. We construct canonically a F F F F diffeomorphism ΘF : F E → T (V ,H ,t ,1 ) E as follows. Given a point y ∈ Fx E, x ∈ M , we define ΘF (y) : J r F IBAF Fx (E) → V˜F x M by ΘF (y)(ξ) = Fx (f )(y) ∈ Fx (M × R)= ˜ V˜F x M, ξ = jxr f ∈ J r F IBAF Fx (E), where a fiber affine map f : E → R is (in obvious way) considered as the ABm -map f : E → M × R covering the identity of M and where the identification Fx (M × R)= ˜ V˜F x M is given by Fx (M × R) ∋ F (ϕ × idR )(v)= ˜ < j0r ϕ, v >∈ V˜F x M , v ∈ V F = F0 (Rm × R), ϕ : Rm → M is an embedding with ϕ(0) = x. ΘF (y) is well defined because F is of order r. Recalling the definition of (V F , H F , tF , 1F ) (see Example 3.1) and using the functoriality of F one can standardly verify that ΘF (y) is a module

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r F r F ˜F homomorphism over tF x : Jx (M, R) → gl(V x M ) with Θ (y)(jx 1) = 1x , (V F ,H F ,tF ,1F )

E. i.e. ΘF (y) ∈ Tx F F F F It remains to show that ΘF : F E → T (V ,H ,t ,1 ) E is a diffeomorphism. F F F F Because of ΘF : F → T (V ,H ,t ,1 ) is natural with respect to ABm F F F F maps and F and T (V ,H ,t ,1 ) preserve fiber product and E is locally a (multi) fiber product of Rm ×R we may assume that E = Rm ×R, the trivial affine bundle over Rm with fiber R. But for E = Rm × R transformation ΘF F F F F is the composition F (Rm × R)=R ˜ m × V F =T ˜ (V ,H ,t ,1 ) (Rm × R), where the first identification is given by Fx (Rm × R) ∋ v = (x, F (τ−x × idR )(v)) ∈ {x} × V F , x ∈ Rm , and where the second trivialization is induced (see Example 2.1.) by the obvious trivialization of Rm × R. From Propositions 2.1 and 5.1 we obtain Proposition 5.2. Any fiber product preserving gauge bundle functor F on ABm of finite order has values in ABm . 6. Classification of admissible systems of order r and dimension m in terms of fiber product preserving gauge bundle functors on ABm of order r The following classification proposition shows that any admissible system of order r and dimension m is isomorphic to some admissible system as in Example 3.1. Proposition 6.1. Let (V, H, t, 1) be an admissible system of order r and dimension m. Let F = T (V,H,t,1). Then we have an isomorphism O(V,H,t,1) : (V, H, t, 1)=(V ˜ F , H F , tF , 1F ) of admissible systems. We recall that a morphism (V1 , H1 , t1 , 11 ) → (V2 , H2 , t2 , 12 ) of admissible systems is a linear map O : V1 → V2 such that H2 (ξ) ◦ O = O ◦ H1 (ξ) r for any ξ ∈ Grm , t2 (η) ◦ O = O ◦ t1 (η) for any η ∈ Dm and O(11 ) = 12 . Proof. We have O(V,H,t,1) : V → V F that the composition V → V F = (V F ,H F ,tF ,1F ) (Rm × R)={0} ˜ × V of O(V,H,t,1) with the isomorphism inT0 duced (see Example 2.1) by the usual trivialization of Rm ×R is the (almost) identity map. One can show standardly (but long) that O(V,H,t,1) is a morphism (V, H, t, 1) → (V F , H F , tF , 1F ) of admissible systems.

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7. Natural transformations of fiber product preserving gauge bundle functors on ABm of order r and induced morphisms between corresponding admissible systems Let F1 , F2 : ABm → F M be fiber product preserving gauge bundle functors of order r. Let (V F1 , H F1 , tF1 , 1F1 ) and (V F2 , H F2 , tF2 , 1F2 ) be the corresponding admissible systems of order r and dimension m. Let µ : F1 → F2 be a natural transformation. Example 7.1. Define ν µ : V F1 → V F2 to be the restriction of µ : F1 (Rm × R) → F2 (Rm × R) to V F1 = (F1 )0 (Rm × R) and V F2 = (F2 )0 (Rm × R). Then ν µ : (V F1 , H F1 , tF1 , 1F1 ) → (V F2 , H F2 , tF2 , 1F2 ) is a morphism of admissible systems. If µ is an isomorphism, then so is ν µ . Definition 7.1. We call ν µ the morphism corresponding to µ. 8. Morphisms between admissible systems of order r and dimension m and induced natural transformations between corresponding fiber product preserving gauge bundle functors Let (V1 , H1 , t1 , 11 ) and (V2 , H2 , t2 , 12 ) be admissible systems of order r and dimension m. Let ν : (V1 , H1 , t1 , 11 ) → (V2 , H2 , t2 , 12 ) be a morphism of admissible systems. Example 8.1. Given an ABm -object p : E → M define a base preserving fibred map µν : T (V1 ,H1 ,t1 ,11 ) E → T (V2 ,H2 ,t2 ,12 ) E as follows. Let (V ,H ,t ,1 ) Φ ∈ Tx 1 1 1 1 E, x ∈ M . Put µν (Φ) = ν˜x ◦ Φ : J r F IBAF Fx (E) → (V˜2 )x M , where ν˜x : (V˜1 )x M → (V˜2 )x M , ν˜x (< jxr ϕ, v >) =< jxr ϕ, ν(v) >, v ∈ V1 , ϕ : Rm → M is an embedding with ϕ(0) = x. We see that (V ,H ,t ,1 ) µν (Φ) ∈ Tx 2 2 2 2 E and that µν : T (V1 ,H1 ,t1 ,11 ) → T (V2 ,H2 ,t2 ,12 ) is a natural transformation. If ν is an isomorphism, then so is µν . Definition 8.1. We call µν the natural transformation corresponding to ν. 9. Object Classification Theorem The first main result in this paper is the following theorem. Theorem 9.1. The correspondence ”F → (V F , H F , tF , 1F )” induces a bijective correspondence between the equivalence classes of fiber product preserving gauge bundle functors F on ABm of order r and the equivalence classes of admissible systems (V, H, t, 1) of order r and dimension m.

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The inverse correspondence is induced by the correspondence ”(V, H, t, 1) → T (V,H,t,1)”. Proof. The correspondence “[F ] → [(V F , H F , tF , 1F )]” is well-defined. For, if µ : F1 → F2 is an isomorphism, then so is ν µ : (V F1 , H F1 , tF1 , 1F1 ) → (V F2 , H F2 , tF2 , 1F2 ). The correspondence “[(V, H, t, 1)] → [T (V,H,t,1)]” is well-defined. For, if ν : (V1 , H1 , t1 , 11 ) → (V2 , H2 , t2 , 12 ) is an isomorphism, then so is µν : T (V1 ,H1 ,t1 ,11 ) → T (V2 ,H2 ,t2 ,12 ) . F F F F From Proposition 5.1 it follows that [F ] = [T (V ,H ,t ,1 ) ]. From Proposition 6.1. it follows that [(V, H, t, 1)] = [(V F , H F , tF , 1F )] if F = T (V,H,t,1). 10. Morphism Classification Theorem Let F1 and F2 be two fiber product preserving gauge bundle functors on ABm of order r. Let (V F1 , H F1 , tF1 , 1F1 ) and (V F2 , H F2 , tF2 , 1F2 ) be the corresponding admissible systems of order r and dimension m. Lemma 10.1. Let ν : (V F1 , H F1 , tF1 , 1F1 ) → (V F2 , H F2 , tF2 , 1F2 ) be a morphism of admissible systems. Let µ[ν] : F1 → F2 be a natural transformation given by the composition Θ F1

→ T (V F1 −−−

F1

ν ,H F1 ,tF1 ,1F1 ) µ

− −→ T (V

F2

F2 −1 ,H F2 ,tF2 ,1F2 ) (Θ )

−−−−−→ F2 ,

where ΘF is as in Proposition 5.1. and µν is described in Example 8.1. Then µ = µ[ν] is the unique natural transformation F1 → F2 such that ν µ = ν, where ν µ is as in Example 7.1. Proof. Suppose µ : F1 → F2 is another natural transformation such that ν µ = ν. Then µ coincides with µ on the affine bundle Rm × R. Hence µ = µ because of the same argument as in the proof of Proposition 5.1. Now, the following second main result in this paper is clear. Theorem 10.1. Let F1 and F2 be two fiber product preserving gauge bundle functors on ABm of order r. The correspondence “µ → ν µ ” is a bijection between natural transformations F1 → F2 and morphisms (V F1 , H F1 , tF1 , 1F1 ) → (V F2 , H F2 , tF2 , 1F2 ) between corresponding admissible systems. The inverse correspondence is “ν → µ[ν] ”.

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11. Finite order theorem Theorem 11.1. Any fiber product preserving gauge bundle functor F : ABm → F M is of finite order. Proof. Define AF : F IBAF F (Rm × R) → C ∞ (F (Rm × R), F (Rm × R)) by AF (f ) = F f , where a fiber affine map f : Rm × R → R is considered as a base preserving ABm -map Rm × R → Rm × R in obvious way. Clearly, AF is π-local, where π : F (Rm × R) → Rm is the projection. The zero map O ∈ F IBAF F (Rm × R) is invariant with respect to the translations of Rm . Clearly AF (tf ) = tAF (f ) for any f . Here F (Rm × R) = Rm × V F is considered as the trivial vector bundle. Then by the non -linear Peetre theorem3 and the homogeneous function theorem3 we see that AF is linear. Then we can assume that f is fiber linear or f is fibre constant. Moreover, given f in question, v ∈ F0 (Rm × R), a neighbourhood W of j0∞ (0) and a neighbourhood U of 0 ∈ F0 (Rm × R), there is t ∈ R+ such that j0∞ (tf ) ∈ W and F λt (v) ∈ U , where λt is the fiber homothety by t. Then standardly by the non-linear (or classical) Peetre theorem we deduce that the operator AF is of finite order r1 . (More precisely, there exists finite r1 , a neighbourhood W of j0∞ (0), a neighbourhood U of 0 ∈ F0 (Rm ×R) such that for fiber affine f1 , f2 : Rm × R → R with j0∞ (f1 ), j0∞ (f2 ) ∈ W, v ∈ U , from j0r1 f1 = j0r1 f2 it follows F f1 (v) = F f2 (v). Now, let f1 , f2 : Rm × R → R be fiber linear ( resp. fiber constant), v ∈ F0 (Rm × R), j0r1 f1 = j0r1 f2 . Then there exists t ∈ R+ such that j0∞ (tf1 ), j0∞ (tf2 ) ∈ W ,j0r1 (tf1 ) = j0r1 (tf2 ) and F λt (v) ∈ U . Then F (tf1 )(F λt (v)) = F (tf2 )(F λt (v)). Then t2 F f1 (v) = t2 F f2 (v) (resp. tF f1 (v) = tF f2 (v)). Then F f1 (v) = F f2 (v) as well.) We have the bundle functor GF : Mfm → F M such that GF M = F (M × R) for any m-manifold M and GF ϕ = F (ϕ × idR ) : GF M → GF N for any embedding ϕ : M → N between m-manifolds. By the Palais-Terng theorem, see Ref. 3, GF has finite order r2 . We prove that F is of order r = max(r1 , r2 ). We consider an ABm -map f : E1 → E2 and a point x ∈ M1 . It remains to show that Fx f depends on jxr f . Using ABm -trivialization we can assume that E1 = Rm ×Rn , E2 = Rm × q R and x = 0 ∈ Rm . Since F preserves fiber product, we can assume that q = 1. Then we can write f = f˜◦(ϕ×idRn ), where f˜ : Rm ×Rn → Rm ×R is base preserving ABm -map and ϕ : Rm → Rm is a 0-preserving embedding. Pn Pn If f˜(x, y) = (x, i=1 (ai (x)y i +h(x))), we have F f˜(v) = i=1 F (x, ai (x)y + h(x))(vi ) for v = (vi ) ∈ F0 (Rm × Rn ) = ×n F0 (Rm × R), where the sum is the one of the vector space V F = F0 (Rm × R) and where y : Rm × R → R

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r1 ˜ is the usual fiber coordinate. Then F0 f = F0 f˜ ◦ (×n GF 0 ϕ) depends on j0 f r2 and j0 ϕ.

References 1. M. Doupovec and I. Kol´ aˇr, Iteration of fiber product preserving bundle functors, Monatsh. Math. 134 (2001) 39–50. 2. I. Kol´ aˇr, Bundle functors of jet type, In: Differential Geom. and Appl. (Proc. of the 7th International Conf. Brno, 1988) 231–237. 3. I. Kol´ aˇr and P. Michor and J. Slov´ ak, Natural Operations in Differential Geometry (Springer-Verlag, 1993). 4. I. Kol´ aˇr and W.M. Mikulski, On the fiber product preserving bundle functors, Differential Geometry and Its Appl. 11 (1999) 105–115. 5. W.M. Mikulski, On the fiber product preserving gauge bundle functors on vector bundles, Ann. Polon. Math. 82 (3) (2003) 251–264.

Differential Geometry and its Applications Proc. Conf., in Honour of Leonhard Euler, Olomouc, August 2007 c 2008 World Scientific Publishing Company, pp. 489–500

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Two analytical Goodwillie’s theorems R´ emi L´ eandre Institut de Math´ ematiques, Universit´ e de Bourgogne, 21000, Dijon, France E-mail: [email protected] We define two analytical Goodwillie’s theorem, either by using an Hida type Fock space or by using an holomorphic type Fock space. Keywords: Goodwillie theorem, Fock space. MS classification: 60H40.

1. Introduction Cyclic homology is a purely algebraic object. Several works in order to give an analytical meaning to cyclic cohomology were done in the works of Connes,2 Jaffe-Lesniewski-Osterwalder,9 Cuntz3 and Meyer.15 In these works, cyclic complex is endowed with some topology or some bornologies. L´eandre-Ouerdiane14 have introduced in this context the interacting Fock space of Accardi-Bozejko and have defined a cyclic complex in Hida sense. Let us remark that L´eandre11 instead of using the Fock space, has defined Hochschild homology with some auxiliary operators and tensor product of Hilbert Sobolev spaces. Jaffe8 has done similar works, but with the cyclic homology. In the algebraic context, people associate to an algebra A the model of T (A), the tensor algebra, given by non-commutative algebraic differential forms of even degree. Cuntz-Quillen consider the X-complex of T (A), and the cyclic complex associated to the tensor algebra T (A). A particular case of Goodwillie’s theorem says that the cohomology of the X-complex of T (A) is equal to the cyclic cohomology associated to T (A). Goodwillie considered only finite sums. Perrot,16 by using some bornologies, shows that this theorem remains true in the bornological Calculus.

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The goal of this paper is to give an analytical version of Goodwillie’s theorem, when A is an algebra of functions on a manifold. In the first part, we introduce the analogous of some Fock space, and we state a Goodwillie’s theorem in Hida sense. In the second part, we consider the case of a group. We deduce a Goodwillie’s theorem, when there are some conditions of analyticity in the derivatives for the elements in A. We thank D. Arnal for helpful discussions. 2. Goodwillie’s theorem and Hida Calculus Let M be a compact Riemannian manifold. We consider the Hilbert space H = L2 (M ) ⊗ C and we define the Hilbert Sobolev space Hk of φ ∈ H such that (∆ + 2)k/2 φ belongs to H. Hk+1 is included in Hk and the intersections of Hk is by the Sobolev imbedding theorem nothing else than the algebra of smooth functions on the manifold M . We consider a > 1 fixed in this part and Sk,C (M ) the Hilbert space of P formal sums σ ˜ = σ2n where σ2n belongs to Hk ⊗ Hk⊗2n where we take the Hilbert norm X Cn kσ2n k2k < ∞ (1) k˜ σ k2k,C = (an)!

If k > k ′ , k˜ σ kk,C ≥ k˜ σ kk′ ,C and if C > C ′ , k˜ σ kk,C ≥ k˜ σ kk,C ′ . We denote by ∩k,C Sk,C (M ) = S∞− (M ). This space is called the space of Hida functionals. We write X X λJ φiJ0 dφiJ1 ..dφiJ = λJ σ ˜J (2) σ2n = |J|

|J|=2n+1

where φi constitute an orthonormal basis of the Hilbert space H inherited by diagonalizing the Laplacian ∆M associated to the eigenvalues Ci of ∆M . We adjoint the formal unity 1 to H. We write d(φφ′ ) = (dφ)φ′ + φdφ′ and d2 φ = 0. d1 = 0 if 1 is the formal unit in H ⊕ C. If σ ˜ and σ ˜1 are Hida test functionals, we consider the Fedosov product (See Refs. 4,5,16) σ ˜◦σ ˜1 = σ ˜σ ˜1 − d˜ σd˜ σ1

(3)

Theorem 2.1. The Fedosov product is continuous from S∞− (M ) × S∞− (M ) into S∞− (M ). Proof. Let us write: σ ˜=

X

λJ σ ˜J

(4)

Two analytical Goodwillie’s theorems

X

σ ˜1 = such that σ ˜σ ˜1 = We have k˜ σσ ˜1 kk,C ≤

X

X

λ1J σ ˜J

491

(5)

λJ λ1J ′ σ ˜J σ ˜J ′

(6)

|λJ ||λ1J ′ |k˜ σJ σ ˜J ′ kk,C

(7)

But (dφ)φ′ = d(φφ′ ) − φdφ′ . By Sobolev imbedding theorem, we deduce that Y ′ ′Y |J|+|J ′ | σJ kk1 ,C1 (Cj+2)−k (Cj ′+2)−k (8) |J|k˜ σJ ′ kk1 ,C1 k˜ k˜ σJ σ ˜J ′ kk,C ≤ C2 j ′ ∈J ′

j∈J

for some small C2 , some big k1 , C1 and k ′ . We apply Cauchy-Schwartz inequality in (7) and the bound (8) in order to show that σ1 kk1 ,C1 k˜ σσ ˜1 kk,C ≤ Kk˜ σkk1 ,C1 k˜ where K=

X J,J ′

2|J|+|J ′ |

C2

Y

(Cj + 2)−2k

′

Y

(Cj ′ + 2)−2k

(9) ′

But if we fix |J| = n, we have: Y X ′ ′ |J| sum (Cj + 2)−2k = ( (Cj + 2)−2k )|J| = C3 < ∞ j∈J

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j ′ ∈J ′

j∈J

(11)

j

It remains to choose C1 big enough in order that C22 C3 < 1 in order to conclude that K < ∞. σ1 kk1 ,C1 The proof that kd˜ σd˜ σ1 kk,C can be estimated by Kk˜ σkk1 ,C1 k˜ for some big k1 and some big C1 is simpler because we we don’t have to multiply dφiJ by φiJ1 but we have to multiply it by dφiJ1 and the formula |J| 0 0 is simpler. Therefore the result. We follow the indications of Cuntz-Quillen,4,5 Meyer15 and Perrot16 in order to choose as model of the universal tensor algebra associated to ∩Hk = A, S∞− (M ) where we apply Ψ: φ0 dφ1 ..dφ2n → φ0 ⊗ (φ1 φ2 − φ1 ⊗ φ2 ).. ⊗ (φ2n−1 φ2n − φ2n−1 ⊗ φ2n ) (12) Ψ map the Fedosov product on the traditional tensor product in the tensor algebra. S∞− (M ) is therefore a topological algebra for the Fedosov product ◦. There is the same system of orthogonal basis σ ˜J on all Sk,C (M ).

492

´ R. Leandre

Let us consider Ωk (Sk,C (M )). There is a basis ˜σJ I ...d˜ ˜σJ I α ˜I = σ ˜J0I d˜ 1

(13)

|I|

P I we denote by |I| = k and by the total weight kIk the quantity |Ji | + |I|. We consider the Hilbert space Ωk,C (Sk,C (M )) as the space of α ˜ = P λI σ ˜I such that Y X C kIk kφj k2k < ∞ (14) kα ˜ kk,C = |λI |2 (akIk)! j j∈Ji ∈I

for some fixed a > 0. Definition 2.1. We put: Ω∞− (S∞− (M )) = ∩k,C Ωk,C (Sk,C (M ))

(15)

Let us recall that the Hochschild boundary ˜b is given (See Refs. 3,15,16) by ˜σ ) = (−1)n [˜ ˜b(α˜n d˜ αn , σ ˜]

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Theorem 2.2. ˜b is continuous on Ω∞− (S∞− (M )). Proof. Let α ˜=

P

λI α ˜ I . We get: ˜bα ˜=

X

˜I λI ˜bα

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In ˜bα ˜ I , there are |I| + 1 terms which appear, whose norms can be bounded by increasing C. The product of some φi in some σ ˜JrI by some φj in some σ ˜J I′ can be estimated as it was done before. We deduce that for some big r k1 , some big C1 and some big k ′ and some small C3 that X (18) k˜bαk ˜ k,C ≤ |λI |KI kα ˜ I kk1 ,C1

where

kIk

KI = C3

Y

(Cj + 2)−k

′

j∈JiI ∈I

By using the same arguments than in theorem 1, we deduce that ∞. We conclude by using the Cauchy-Schwartz inequality. ˜ be the Connes boundary: Let B X ˜σ1 ..d˜ ˜σi−1 ˜σ0 d˜ ˜σ1 ...d˜ ˜σn ) = ˜ i ...d˜ ˜σn d˜ ˜ σ0 d˜ sign dσ B(˜

We get easily:

˜ is continuous on Ω∞− (S∞− (M )). Theorem 2.3. B

(19) P

|KI |2
3kIk. In Φ αI ), there are at most the product of two φj belonging to the same σ ˜JiI . The Sobolev norm of each such product can be estimated, by Sobolev imbedding theorem, by the product of some Sobolev norms. If a term which appears in ˜ l (α˜I ) as a total weight k, when we consider Φ ˜ l+1 (˜ Φ αI ), there are deduced from this term bk terms which appears with total weight k+ or k + 2. Therefore, we deduce: X kIk X C2 φ˜l α ˜ I kk,C ≤ k l≤3kIk

X

l1