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-Trèves 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.
Old and New Aspects in Spectral Geometry
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Let M (resp. N) be a connected, smooth (= Cx) n-dimensional manifold without boundary. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
principal isotropy group, isospectral flat tori, generalized triangulation, one gets successively, same theta series, curvature tensor field, real analytic family, orthogonal automorphism, geodesic fibers, smooth tensor field, isospectral domains, metric tensor field, isospectral manifolds, isospectral deformations, closed smooth manifold, cocompact discrete subgroup, constant sectional curvature, smooth differential forms, spectral geometry, isometry classes, complete orthonormal sequence, flat torus, elliptic complex, mannian manifolds, canonical inner product Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hodge-de Rham, New York, Academic Press, Lecture Notes, Colin de Verdière, Duke Math, Pure Math, Riemannian Heisenberg, Pure Appl, Springer Verlag, Cambridge University Press, Reine Angew, Annals of Math, Birkhäuser Verlag, G-invariant Riemannian, Geometry of Manifolds, Let Mbe, London Mathematical Society, Michigan Math, Princeton Univ, Rhode Island, Second Edition, Supplemento al Vol, Tôhoku Math, Translations of Mathematical Monographs New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Let M (resp. N) be a connected, smooth (= Cx) n-dimensional manifold without boundary. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
principal isotropy group, isospectral flat tori, generalized triangulation, one gets successively, same theta series, curvature tensor field, real analytic family, orthogonal automorphism, geodesic fibers, smooth tensor field, isospectral domains, metric tensor field, isospectral manifolds, isospectral deformations, closed smooth manifold, cocompact discrete subgroup, constant sectional curvature, smooth differential forms, spectral geometry, isometry classes, complete orthonormal sequence, flat torus, elliptic complex, mannian manifolds, canonical inner product Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hodge-de Rham, New York, Academic Press, Lecture Notes, Colin de Verdière, Duke Math, Pure Math, Riemannian Heisenberg, Pure Appl, Springer Verlag, Cambridge University Press, Reine Angew, Annals of Math, Birkhäuser Verlag, G-invariant Riemannian, Geometry of Manifolds, Let Mbe, London Mathematical Society, Michigan Math, Princeton Univ, Rhode Island, Second Edition, Supplemento al Vol, Tôhoku Math, Translations of Mathematical Monographs New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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