Old and New Aspects in Spectral Geometry Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science. Amsterdam. The Netherlands Volume 534 Old and New Aspects in Spectral Geometry by Mircea Craioveanu Mircea Puta Facultatea de Matematicii, Universitatea de Vest din Timi/$oara, Timi/$oara, Romania and Themistocles M. Rassias Department of Mathematics, National Technical University ofA thens, Athens, Greece SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5837-9 ISBN 978-94-017-2475-3 (eBook) DOI 10.1007/978-94-017-2475-3 Printed on acidlree paper AII Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1s t edition 2001 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. CONTENTS PREFACE ................................................................................................................. vii CHAPTER 1. INTRODUCTION TO RIEMANNIAN MANIFOLDS ....................... 1 1. Tensor Fields on Smooth Differential Manifolds ................................................. 1 2. Riemannian Structures. Examples ....................................................................... 5 3. The Levi-Civita Connection ................................................................................ 27 4. The Curvature of a Riemannian Manifold .......................................................... 36 5. Geodesics and the Exponential Map ................................................................... 62 References ........................................................................................................... 72 CHAPTER 2. CANONICAL DIFFERENTIAL OPERA TORS ASSO· CIATED TO A RIEMANNIAN MANIFOLD .................................... 75 1. Hilbert Spaces Associated to a Compact Riemannian Manifold ........................ 75 2. Some Canonical Differential Operators on a Riemannian Manifold .................. 89 References ......................................................................................................... 116 CHAPTER 3. SPECTRAL PROPERTIES OF THE LAPLACE· BELTRAMI OPERA TOR AND APPLICA TIONS ......................... 119 1. The Fundamental Solution of the Heat Equation on Riemannian Manifolds .......................................................................................................... 120 2. Examples of Explicit Spectra ............................................................................ 148 3. Characterizing Eigenvalues of the Laplace-Beltrami Operator ......................... 162 4. Generic Properties of the Riemannian Metrics on Closed Smooth Manifolds .......................................................................................................... 167 5. Estimates of the Eigenvalues through Geometric Data ..................................... 181 References ......................................................................................................... 207 CHAPTER 4. ISOSPECTRAL CLOSED RIEMANNIAN MANIFOLDS ............ 213 1. Asymptotic Expansion for the Trace of the Heat Kernel and Consequences .................................................................................................... 213 2. Isospectral Flat Tori .......................................................................................... 230 3. Sunada's Theorem and Pesce's Approach to Isospectrality .............................. 243 References ......................................................................................................... 265 CHAPTER 5. SPECTRAL PROPERTIES OF THE LAPLACIANS FOR THE DE RHAM COMPLEX ................................................... 273 1. The Heat Equation Associated to a Hodge-de Rham Operator. ........................ 273 2. Characterizing Eigenvalues of 284 L1(P) •••••••••••.•••••.••••..•••••••••••••••••••••••••••••••••••••••••• 3. A Continuity Property of the Eigenvalues of the Hodge-de Rham Operators ........................................................................................................... 296 4. Asymptotic Expansion for the Trace of the Heat p-Kernel and Spectral Geometry ............................................................................................. 302 5. Lower Bounds for the Smallest Positive Eigenvalue of the Hodge-de Rham Operator ................................................................................. 318 v vi Contents References ......................................................................................................... 322 CHAPTER 6. APPLICATIONS TO GEOMETRY AND TOPOLOGy ............... 327 1. The Hodge-de Rham Decomposition Theorem ................................................ 327 2. Vanishing Theorems for the Real Cohomology of Closed Riemannian Manifolds ...................................................................................... 333 3. Lefschetz Fixed Point Theorem ........................................................................ 337 4. Chem-Gauss-Bonnet Theorem .......................................................................... 343 References ......................................................................................................... 352 CHAPTER 7. AN INTRODUCTION TO WITTEN-HELFFER- SJOSTRAND THEORY ..................................................................... 355 1. Introduction ....................................................................................................... 355 2. Analytic Preliminaries ....................................................................................... 356 3. Morse Inequalities ............................................................................................. 364 4. Generalized Triangulations ............................................................................... 367 5. Witten's Deformation ....................................................................................... 371 6. The Main Results of the Witten-Helffer-Sjostrand Theory .............................. 375 7. Strong Morse Inequalities ................................................................................. 387 References ......................................................................................................... 389 CHAPTER 8. OPEN PROBLEMS AND COMMENTS .......................................... 393 References ......................................................................................................... 402 APPENDIX ............................................................................................................... 409 1. Review of Matrix Algebra ................................................................................ .409 2. . Eigenvectors and Eigenvalues .......................................................................... .41 0 3. Diagonalizable Matrices. Triangularizable Matrices. Jordan Canonical Form ................................................................................................ .418 4. Eigenvalues and Eigenvectors of Real Symmetric and Hermitian Matrices ............................................................................................................. 430 References .......................................................................................................... 438 Subject Index ............................................................................................................... 441 Preface It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva lent. Thus, one of the main tasks in this subject is to find methods allowing to extract geometric and topological information from spectral data. On the other hand, the only way to identify specific geometric invariants, which are not spectrally determined, is through explicit constructions of isospectral closed Riemannian manifolds. This text is designed to introduce spectral geometry to graduate students and inter ested mathematicians and physicists. It is readily accessible to anyone whose background includes introductory Riemannian geometry and functional analysis. However, the book is largely self-contained, the prerequisite in mathematics being contained in the first two chapters. Many significant applications are developed, making obvious the ways in which the spectrum of a closed Riemannian manifold (M, g) influences the geometry of (M, g) and the topology of M, as well as pointing out the fascinating interplay between analysis, topology and geometry. This work includes an extensive bibliography of books and pa pers of both current and historical interest. The references appear at the end of each chap ter. At the end of the book a symbol index is also included. After reviewing some basic facts on smooth differential manifolds, Chapter I pro vides some fundamental concepts and results of Riemannian geometry. In particular, we explain the notions of curvature, Singer-Thorpe irreducible decomposition of the alge braic curvature tensor space, and geodesic, with many examples. The second chapter is devoted to canonical differential operators (gradient, divergence, the Laplace-Beltrami operator and the Hodge-de Rham operators on forms) in the context of Riemannian mani folds. The third chapter covers spectral properties of the Laplace-Beltrami operator asso ciated to a compact Riemannian manifold. Among them included are Courant's principle, the mini-max characterization of the eigenvalues, the continuous dependence of the ei genvalues with respect to the Riemannian metric in the ex -topology and some of their consequences such as Uhlenbeck's theorem on the genericity of the Riemannian metrics with the property that the eigenvalues of the corresponding Laplace-Beltrami operators have multiplicity one. We also treat Ebin's theorem on the genericity of the Riemannian vii viii Preface metrics with finite isometry groups and estimation of various geometric features of a compact Riemannian manifold in terms of spectral data, usually the dimension, the vol ume and the eigenvalues of the Laplace-Beltrami operator as well as of geometric data such as the diameter and curvature bounds. Some basic examples concerning the explicit computation of spectra for some standard closed Riemannian manifolds are presented. In Chapter 4 the method of heat asymptotics is discussed and geometric information is de rived (which may be inferred from the heat coefficients and thus from the spectrum on functions). For example, it is proved that the round sphere and the real projective space with the standard Riemannian metric are spectrally determined in all dimensions less than or equal to six. Examples are constructed of non-isometric isospectral flat tori in all di mensions greater than or equal to four. The classical technique of Gordon as well as other ones for constructing isospectral closed Riemannian manifolds are discussed in this chap ter. Among them Sunada's technique and Pesce's approach to isospectrality are presented in detail. The example of Gordon and Webb of non-isometric convex domains in )R" (n 2: 4) that are isospectral for both Dirichlet and Neumann boundary conditions is described. The subject matter of Chapter 3 and Chapter 4, § I is extended in Chapter 5, in which we study spectral properties of the Hodge-de Rham operators (Laplacians on forms) associated to a compact Riemannian manifold. Thus, we present various charac terizations of the eigenvalues and the continuity property of the eigenvalues with respect to the Riemannian metric in the C' -topology. The corresponding heat flow as well as the corresponding heat asymptotics are described in a detailed way. In particular, one derives Patodi's classical result showing that from the spectra of the Laplace-Beltrami operator acting on functions and of the Hodge-de Rham operators acting on I-forms and 2-forms one can tell whether the closed oriented Riemannian manifold has constant scalar curva ture (resp. is Einsteinian) in all dimensions greater than or equal to four. Using an argu ment of Chanillo and Treves, one can show that the first positive eigenvalue of the Hodge-de Rham operator on the regular level sets of a real analytic functionJ on S" (n 2: 2) .i s bounded below by a constant times a power of the distance between the value ofJand the set of critical values off Some aspects of the interplay between analy sis, topology and geometry are also discussed in Chapter 6. The main tool in the subject is the heat flow associated to a Hodge-de Rham operator of a closed Riemannian mani fold. Accordingly, the Hodge-de Rham decomposition theorem, vanishing theorems for the real cohomology of closed Riemannian manifolds, the Lefschetz fixed point theorem and the Chern-Gauss-Bonnet theorem are treated with complete proofs. Chapter 7 pro vides an introduction to the Witten-Helffer-Sjostrand theory, which allows one to relate some of the spectral properties of the Hodge-de Rham operators associated to a Rieman nian metric g on a closed smooth manifold M to the combinatorial Laplacians associated to a certain smooth triangulation defined by a pair (J, g), where J: M --> IR is a Morse function. In particular, this can be used in order to obtain a short proof of the weak and strong Morse inequalities - a very useful result in topology. In writing this chapter we have greatly profited by Burghelea's lecture notes on Witten-He If fer-Sj ostrand theory de livered at Ohio State University as well as at the Third International Workshop of Differ ential Geometry and its Applications (Sibiu, September 18-23, 1997). Finally, the last chapter contains open problems as well as related comments. The Appendix presents spectral properties of square matrices with entries in IR or C. These include the exis tence and uniqueness of the Jordan canonical form, density theorems of the set of matri ces, whose eigenvalues have certain properties, in the space of all square matrices, the ex- Preface ix istence of Hermitian (symmetric) matrices with prescribed eigenvalues as well as the theorem of Rayleigh-Ritz. One should mention that some of the topics presented in this text-book are still in rapid progress and are in the front line of current research. The writing of this book itself has been under way for at least five years, and has been the subject of extensive correspondence and discussions with several mathemati cians, who provided us with valuable information. new ideas, and the necessary encour agement during the preparation of this manuscript. We are sincerely grateful to the fol lowing colleagues: D. Alekseevski, A. Bacopoulos, M. Berger, P. Berard, G. Besson, M. Bordoni, D. Burghelea, J. Dodziuk, H. Donnelly, J. Eichhorn, Th. Friedrich, P.S. Gilkey, L. Gligor, St. Ianu~, R. Iordanescu. E. Macias-Virgos, St. Marchiafava, P. Michor, P.T. Nagy, D.I. Papuc, A. Savo, R. Schimming, D. Schuth. Z.L Szabo. A. Torok, Gr. Tsagas, L. Vanhecke, C. Vizman, S. Zelditch, and others. Special thanks go to C.S. Gordon for her many useful comments and remarks on different topics of the book. She and Z.I. Szabo have also provided us with their recent unpublished results for Section 3 in Chapter 4. Some of the above topics have been presented in courses given by M. Craioveanu in 199511996 and 1997-2000 at the West University of Timi~oara, the University of Craiova, the University of Dortmund and the University ofUdine, as well as by Th.M. Rassias in the period 1995-1999 at the National Technical University of Athens. We wish to thank several former students of ours who have read and commented on earlier versions of various chapters of the manuscript: S. Benczik, P. Birtea, O. Bodro gean, D. Boros, M. Buligii. L. David. V.-D. Lalescu. N. Laos, F. Sakellaridis, V. Slesar and R. Tudoran. Work on the book has been much aided by visits ofM. Craioveanu and M. Puta to the Erwin Schrodinger Institute of Mathematical Physics in Vienna, to the Mathematical Institutes of the Humboldt University of Berlin, to the University of Debrecen, the Uni versity of Dortmund, the Technical University of Lausanne. the Catholic University of Leuven, the Technical University of Munich. the University of Rome "La Sapienza" and to the University of Santiago de Compostela. Also. Th.M. Rassias expresses his thanks to the Departments of Mathematics of the following Universities: Harvard University, M.LT., Oxford University, The Technion (Israel Institute of Technology) and the Univer sity of Rome "La Sapienza", which provided facilities during the preparation of part of the manuscript. We thank our colleagues in these Institutes for making these visits possible, productive, memorable and enjoyable. The preparation of this book has been partially supported by travel and other ex penses awarded to M. Craioveanu by the West University of Timisoara and the companies S.c. Argirom International SA, Bucharest, and Petrom S.A .. Timi~oara Branch. Th.M. Rassias also thanks the Department of Mathematics of the National Technical University of Athens, for granting him partial financial support during this project. We would also like to express our thanks to X. Hadjiliadis for the time and energy devoted to the preparation of the camera ready manuscript. Last, but not least, we want to thank Kluwer Academic Publishers for their generous co-operation. Timi~oaral Athens Mircea Craioveanu June 2001 Mircea Pula Themistocles M. Rassias Chapter 1 INTRODUCTION TO RIEMANNIAN MANIFOLDS 1. Tensor Fields on Smooth Differential Manifolds Let M (resp. N) be a connected. smooth (= (' '- ) n-dimensional manifold without boundary, We denote by C' (M) the ring of smooth real valued functions on M and by X (M) the Lie-algebra of all smooth vector fields on JI. Recall that X E X (M) is a smooth map X : M -> TM = LJ T,M r":.\1 such that X (x) = X, E T,M (= the tangent space of M at x) for each x EM. TxM may be characterized as the space of all derivations of the algebra of smooth real valued functions defined on neighborhoods ofx. Note that a vector fieldXand a function f E ex (M) give rise to a new function X (f) E C' (M) defined by X(f)(x)=X,(f). xE.\1. Therefore a smooth vector field X E X(M) leads to an IR-Iinear map, denoted also by X. namely which is obviously a derivation, i.e. Conversely, a mapping X: C' (M) -> C' (.\1) which is IR-linear and a derivation defines an element ofX(M). Indeed. one defines X, E T,JI. XE.\1 . as follows. First note that if .r; E ex (M) has the property .r; I, = O. where C is a neighborhood of x, then (X(J;))(x)=O. In fact, choose a .f; EC'(M) such that .I; (x)=O and .1;1111 =1. For the existence of such a .I; assume that M satisfies the second countability axiom. so that in particular Mis paracompact and admits partitions of unity. Then one gets i; = hi; , and consequently which implies that (X (i;))(x) = O. Now for a smooth real valued function J defined on a neighborhood of x, let us define X, (1) = (X(f))(x). where f E ex (M) is an extension of ]. This definition does not depend on the choice M. Craioveanu et al., Old and New Aspects in Spectral Geometry © Springer Science+Business Media Dordrecht 2001