Table Of ContentOld 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
