The Laplacian on a Riemannian Manifold : An Introduction to Analysis on Manifolds
Steven. Rosenberg
Bok Engelsk 1997 · Electronic books.
| Annen tittel | |
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| Utgitt | Cambridge : Cambridge University Press , c1997
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| Omfang | 1 online resource (186 p.)
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| Opplysninger | Description based upon print version of record.. - Cover; Title; Copyright; Contents; Introduction; 1 The Laplacian on aRiemannian Manifold; 1.1 Basic Examples; 1.1.1 The Laplacian on S1 and R; 1.1.2 Heat Flow on S1 and R; 1.2 The Laplacian on a Riemannian Manifold; 1.2.1 Riemannian Metrics; 1.2.2 L2 Spaces of Functions and Forms; 1.2.3 The Laplacian on Functions; 1.3 Hodge Theory for Functions and Forms; 1.3.1 Analytic Preliminaries; 1.3.2 The Heat Equation Proof of the Hodge Theorem forFunctions; 1.3.3 The Hodge Theorem for Differential Forms; 1.3.4 Regularity Results; 1.4 De Rham Cohomology; 1.5 The Kernel of the Laplacian on Forms. - 2 Elements of DifferentialGeometry2.1 Curvature; 2.2 The Levi-Civita Connection and BochnerFormula; 2.2.1 The Levi-Civita Connection; 2.2.2 Weitzenböck Formulas and Gårding's Inequality; 2.3 Geodesies; 2.4. The Laplacian in Exponential Coordinates; 3 The Construction of theHeat Kernel; 3.1 Preliminary Results for the Heat Kernel; 3.2 Construction of the Heat Kernel; 3.2.1 Construction of the Parametrix; 3.2.2 The Heat Kernel for Functions; 3.3. The Asymptotics of the Heat Kernel; 3.4 Positivity of the Heat Kernel; 4 The Heat EquationApproach to theAtiyah-Singer IndexTheorem. - 4.1 The Chern-Gauss-Bonnet Theorem4.1.1 The Heat Equation Approach; 4.1.2 Proof of the Chern-Gauss-Bonnet Theorem; 4.2 The Hirzebruch Signature Theorem and the Atiyah-Singer Index Theorem; 4.2.1 A Survey of Characteristic Forms; 4.2.2 The Hirzebruch Signature Theorem; 4.2.3 The Atiyah-Singer Index Theorem; 5 Zeta Functions ofLaplacians; 5.1 The Zeta Function of a Laplacian; 5.2 Isospectral Manifolds; 5.3 Reidemeister Torsion and Analytic Torsion; 5.3.1 Reidemeister Torsion; 5.3.2 Analytic Torsion; 5.3.3 The Families Index Theorem and Analytic Torsion; Bibliography; Index. - This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.
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| Emner | |
| Sjanger | |
| Dewey | |
| ISBN | 0521463009. - 0521468310
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