Old and New Aspects in Spectral Geometry:
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
2001
|
| Series: | Mathematics and Its Applications
534 |
| Subjects: | |
| Online Access: | Volltext |
| Item Description: | 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 |
| Physical Description: | 1 Online-Ressource (X, 446 p) |
| ISBN: | 9789401724753 9789048158379 |
| DOI: | 10.1007/978-94-017-2475-3 |
Staff View
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| 500 | |a 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 | ||
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| id | DE-604.BV042424291 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:15Z |
| institution | BVB |
| isbn | 9789401724753 9789048158379 |
| language | English |
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| spelling | Craioveanu, Mircea Verfasser aut Old and New Aspects in Spectral Geometry by Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias Dordrecht Springer Netherlands 2001 1 Online-Ressource (X, 446 p) txt rdacontent c rdamedia cr rdacarrier Mathematics and Its Applications 534 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 Mathematics Matrix theory Global analysis Global differential geometry Global Analysis and Analysis on Manifolds Differential Geometry Applications of Mathematics Linear and Multilinear Algebras, Matrix Theory Mathematik Puta, Mircea Sonstige oth Rassias, Themistocles M. Sonstige oth https://doi.org/10.1007/978-94-017-2475-3 Verlag Volltext |
| spellingShingle | Craioveanu, Mircea Old and New Aspects in Spectral Geometry Mathematics Matrix theory Global analysis Global differential geometry Global Analysis and Analysis on Manifolds Differential Geometry Applications of Mathematics Linear and Multilinear Algebras, Matrix Theory Mathematik |
| title | Old and New Aspects in Spectral Geometry |
| title_auth | Old and New Aspects in Spectral Geometry |
| title_exact_search | Old and New Aspects in Spectral Geometry |
| title_full | Old and New Aspects in Spectral Geometry by Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias |
| title_fullStr | Old and New Aspects in Spectral Geometry by Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias |
| title_full_unstemmed | Old and New Aspects in Spectral Geometry by Mircea Craioveanu, Mircea Puta, Themistocles M. Rassias |
| title_short | Old and New Aspects in Spectral Geometry |
| title_sort | old and new aspects in spectral geometry |
| topic | Mathematics Matrix theory Global analysis Global differential geometry Global Analysis and Analysis on Manifolds Differential Geometry Applications of Mathematics Linear and Multilinear Algebras, Matrix Theory Mathematik |
| topic_facet | Mathematics Matrix theory Global analysis Global differential geometry Global Analysis and Analysis on Manifolds Differential Geometry Applications of Mathematics Linear and Multilinear Algebras, Matrix Theory Mathematik |
| url | https://doi.org/10.1007/978-94-017-2475-3 |
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