Old and New Aspects in Spectral Geometry

by Craioveanu, Mircea; Puta, Mircea; Rassias, Themistocles M.

• ISBN13: 9781402000522
• ISBN10: 1402000529
• Format: Hardcover
• Copyright: 2002-02-01
• Publisher: Kluwer Academic Pub

Summary

This work presents some classical as well as some very recent results and techniques concerning the spectral geometry corresponding to the Laplace-Beltrami operator and the Hodge-de Rham operators. It treats many topics that are not usually dealt with in this field, such as the continuous dependence of the eigenvalues with respect to the Riemannian metric in the CINFINITY-topology, and some of their consequences, such as Uhlenbeck's genericity theorem; examples of non-isometric flat tori in all dimensions greater than or equal to four; Gordon's classical technique for constructing isospectral closed Riemannian manifolds; a detailed presentation of Sunada's technique and Pesce's approach to isospectrality; Gordon and Webb's example of non-isometric convex domains in Rn (n>=4) that are isospectral for both Dirichlet and Neumann boundary conditions; the Chanillo-Tréves estimate for the first positive eigenvalue of the Hodge-de Rham operator, etc. Significant applications are developed, and many open problems, references and suggestions for further reading are given. Several themes for additional research are pointed out. Audience: This volume is designed as an introductory text for mathematicians and physicists interested in global analysis, analysis on manifolds, differential geometry, linear and multilinear algebra, and matrix theory. It is accessible to readers whose background includes basic Riemannian geometry and functional analysis. These mathematical prerequisites are covered in the first two chapters, thus making the book largely self-contained.

Table of Contents

Preface	vii
Introduction to Riemannian Manifolds	1	(74)
Tensor Fields on Smooth Differential Manifolds	1	(4)
Riemannian Structures. Examples	5	(22)
The Levi-Civita Connection	27	(9)
The Curvature of a Riemannian Manifold	36	(26)
Geodesics and the Exponential Map	62	(13)
References	72	(3)
Canonical Differential Operators Associated to a Riemannian Manifold	75	(44)
Hilbert Spaces Associated to a Compact Riemannian Manifold	75	(14)
Some Canonical Differential Operators on a Riemannian Manifold	89	(30)
References	116	(3)
Spectral Properties of the Laplace-Beltrami Operator and Applications	119	(94)
The Fundamental Solution of the Heat Equation on Riemannian Manifolds	120	(28)
Examples of Explicit Spectra	148	(14)
Characterizing Eigenvalues of the Laplace-Beltrami Operator	162	(5)
Generic Properties of the Riemannian Metrics on Closed Smooth Manifolds	167	(14)
Estimates of the Eigenvalues through Geometric Data	181	(32)
References	207	(6)
Isospectral Closed Riemannian Manifolds	213	(60)
Asymptotic Expansion for the Trace of the Heat Kernel and Consequences	213	(17)
Isospectral Flat Tori	230	(13)
Sunada's Theorem and Pesce's Approach to Isospectrality	243	(30)
References	265	(8)
Spectral Properties of the Laplacians for the de Rham Complex	273	(54)
The Heat Equation Associated to a Hodge-de Rham Operator	273	(11)
Characterizing Eigenvalues of Δ(p)	284	(12)
A Continuity Property of the Eigenvalues of the Hodge-de Rham Operators	296	(6)
Asymptotic Expansion for the Trace of the Heat p-Kernel and Spectral Geometry	302	(16)
Lower Bounds for the Smallest Positive Eigenvalue of the Hodge-de Rham Operator	318	(9)
References	322	(5)
Applications to Geometry and Topology	327	(28)
The Hodge-de Rham Decomposition Theorem	327	(6)
Vanishing Theorems for the Real Cohomology of Closed Riemannian Manifolds	333	(4)
Lefschetz Fixed Point Theorem	337	(6)
Chern-Gauss-Bonnet Theorem	343	(12)
References	352	(3)
An Introduction to Witten-Helffer-Sjostrand Theory	355	(38)
Introduction	355	(1)
Analytic Preliminaries	356	(8)
Morse Inequalities	364	(3)
Generalized Triangulations	367	(4)
Witten's Deformation	371	(4)
The Main Results of the Witten-Helffer-Sjostrand Theory	375	(12)
Strong Morse Inequalities	387	(6)
References	389	(4)
Open Problems and Comments	393	(16)
References	402	(7)
APPENDIX	409	(32)
1. Review of Matrix Algebra	409	(1)
2. Eigenvectors and Eigenvalues	410	(8)
3. Diagonalizable Matrices. Triangularizable Matrices. Jordan Canonical Form	418	(12)
4. Eigenvalues and Eigenvectors of Real Symmetric and Hermitian Matrices	430	(11)
References	438	(3)
Subject Index	441