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OverviewThis 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-Treves 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. Full Product DetailsAuthor: M.-E. Craioveanu , Mircea Puta , Themistocles RASSIASPublisher: Springer Imprint: Springer Edition: Softcover reprint of hardcover 1st ed. 2001 Volume: 534 Dimensions: Width: 16.00cm , Height: 2.30cm , Length: 24.00cm Weight: 0.700kg ISBN: 9789048158379ISBN 10: 9048158370 Pages: 446 Publication Date: 06 December 2010 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Out of Print Availability: Out of stock  Table of ContentsPreface. 1. Introduction to Riemannian Manifolds. 2. Canonical Differential Operators Associated to a Riemannian Manifold. 3. Spectral Properties of the Laplace-Beltrami Operator and Applications. 4. Isospectral Closed Riemannian Manifolds. 5. Spectral Properties of the Laplacians for the De Rham Complex. 6. Applications to Geometry and Topology. 7. An Introduction to Witten-Helffer-Sjoestrand Theory. 8. Open Problems and Comments. Appendix. Subject Index.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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