The Laplacian on a Riemannian Manifold

The Laplacian on a Riemannian Manifold
Author: Steven Rosenberg
Publsiher: Cambridge University Press
Total Pages: 190
Release: 1997-01-09
Genre: Mathematics
ISBN: 0521468310

Download The Laplacian on a Riemannian Manifold Book in PDF, Epub and Kindle

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Eigenfunctions of the Laplacian on a Riemannian Manifold

Eigenfunctions of the Laplacian on a Riemannian Manifold
Author: Steve Zelditch
Publsiher: American Mathematical Soc.
Total Pages: 394
Release: 2017-12-12
Genre: Eigenfunctions
ISBN: 9781470410377

Download Eigenfunctions of the Laplacian on a Riemannian Manifold Book in PDF, Epub and Kindle

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.

Spectral Geometry Of The Laplacian Spectral Analysis And Differential Geometry Of The Laplacian

Spectral Geometry Of The Laplacian  Spectral Analysis And Differential Geometry Of The Laplacian
Author: Urakawa Hajime
Publsiher: World Scientific
Total Pages: 312
Release: 2017-06-02
Genre: Mathematics
ISBN: 9789813109100

Download Spectral Geometry Of The Laplacian Spectral Analysis And Differential Geometry Of The Laplacian Book in PDF, Epub and Kindle

The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the first eigenvalue, the Lichnerowicz–Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne–Pólya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.

Eigenvalues in Riemannian Geometry

Eigenvalues in Riemannian Geometry
Author: Isaac Chavel
Publsiher: Academic Press
Total Pages: 379
Release: 1984-11-07
Genre: Mathematics
ISBN: 9780080874340

Download Eigenvalues in Riemannian Geometry Book in PDF, Epub and Kindle

The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery.

Dirac Operators in Riemannian Geometry

Dirac Operators in Riemannian Geometry
Author: Thomas Friedrich
Publsiher: American Mathematical Soc.
Total Pages: 213
Release: 2000
Genre: Dirac equation
ISBN: 9780821820551

Download Dirac Operators in Riemannian Geometry Book in PDF, Epub and Kindle

For a Riemannian manifold M, the geometry, topology and analysis are interrelated in ways that have become widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants. In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm{spin}mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag.

The Laplacian on a Riemannian Manifold

The Laplacian on a Riemannian Manifold
Author: Steven Rosenberg
Publsiher: Unknown
Total Pages: 185
Release: 2014-05-14
Genre: MATHEMATICS
ISBN: 1107362067

Download The Laplacian on a Riemannian Manifold Book in PDF, Epub and Kindle

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Riemannian Manifolds

Riemannian Manifolds
Author: John M. Lee
Publsiher: Springer Science & Business Media
Total Pages: 232
Release: 2006-04-06
Genre: Mathematics
ISBN: 9780387227269

Download Riemannian Manifolds Book in PDF, Epub and Kindle

This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

Old and New Aspects in Spectral Geometry

Old and New Aspects in Spectral Geometry
Author: M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS
Publsiher: Springer Science & Business Media
Total Pages: 447
Release: 2013-03-14
Genre: Mathematics
ISBN: 9789401724753

Download Old and New Aspects in Spectral Geometry Book in PDF, Epub and Kindle

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.