Ebook: Riemannian Geometry and Geometric Analysis

This textbook introduces techniques from nonlinear analysis at an early stage. Such techniques have recently become an indispensable tool in research in geometry, and they are treated here for the first time in a textbook. Topics treated include: Differentiable and Riemannian manifolds, metric properties, tensor calculus, vector bundles; the Hodge Theorem for de Rham cohomology; connections and curvature, the Yang-Mills functional; geodesics and Jacobi fields, Rauch comparison theorem and applications; Morse theory (including an introduction to algebraic topology), applications to the existence of closed geodesics; symmetric spaces and K?hler manifolds; the Palais-Smale condition and closed geodesics; Harmonic maps, minimal surfaces.

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This textbook introduces techniques from nonlinear analysis at an early stage. Such techniques have recently become an indispensable tool in research in geometry, and they are treated here for the first time in a textbook. Topics treated include: Differentiable and Riemannian manifolds, metric properties, tensor calculus, vector bundles; the Hodge Theorem for de Rham cohomology; connections and curvature, the Yang-Mills functional; geodesics and Jacobi fields, Rauch comparison theorem and applications; Morse theory (including an introduction to algebraic topology), applications to the existence of closed geodesics; symmetric spaces and K?hler manifolds; the Palais-Smale condition and closed geodesics; Harmonic maps, minimal surfaces.
Content:
Front Matter....Pages I-XI
Foundational Material....Pages 1-54
De Rham Cohomology and Harmonic Differential Forms....Pages 55-75
Parallel Transport, Connections, and Covariant Derivatives....Pages 77-123
Geodesics and Jacobi Fields....Pages 125-163
A Short Survey on Curvature and Topology....Pages 165-171
Morse Theory and Closed Geodesics....Pages 173-210
Symmetric Spaces and K?hler Manifolds....Pages 211-261
The Palais-Smale Condition and Closed Geodesics....Pages 263-276
Harmonic Maps....Pages 277-384
Back Matter....Pages 385-404

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This textbook introduces techniques from nonlinear analysis at an early stage. Such techniques have recently become an indispensable tool in research in geometry, and they are treated here for the first time in a textbook. Topics treated include: Differentiable and Riemannian manifolds, metric properties, tensor calculus, vector bundles; the Hodge Theorem for de Rham cohomology; connections and curvature, the Yang-Mills functional; geodesics and Jacobi fields, Rauch comparison theorem and applications; Morse theory (including an introduction to algebraic topology), applications to the existence of closed geodesics; symmetric spaces and K?hler manifolds; the Palais-Smale condition and closed geodesics; Harmonic maps, minimal surfaces.
Content:
Front Matter....Pages I-XI
Foundational Material....Pages 1-54
De Rham Cohomology and Harmonic Differential Forms....Pages 55-75
Parallel Transport, Connections, and Covariant Derivatives....Pages 77-123
Geodesics and Jacobi Fields....Pages 125-163
A Short Survey on Curvature and Topology....Pages 165-171
Morse Theory and Closed Geodesics....Pages 173-210
Symmetric Spaces and K?hler Manifolds....Pages 211-261
The Palais-Smale Condition and Closed Geodesics....Pages 263-276
Harmonic Maps....Pages 277-384
Back Matter....Pages 385-404
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