Introduction to Riemannian manifolds / John M. Lee.
Editor: New York, NY : Springer Berlin Heidelberg, 2018Edición: 2nd edDescripción: xiii, 437 p. il. 24 cmTipo de contenido: text Tipo de medio: unmediated Tipo de portador: volumeISBN: 9783319917542Otra clasificación: 53-01 (53B20 53B30 53C20 53C21)1 What Is Curvature? [1] The Euclidean Plane [1] Surfaces in Space [4] Curvature in Higher Dimensions [7] 2 Riemannian Metrics [9] Definitions [9] Methods for Constructing Riemannian Metrics [15] Basic Constructions on Riemannian Manifolds [25] Lengths and Distances [33] Pseudo-Riemannian Metrics [40] Other Generalizations of Riemannian Metrics [46] Problems [47] 3 Model Riemannian Manifolds [55] Symmetries of Riemannian Manifolds [55] Euclidean Spaces [57] Spheres [58] Hyperbolic Spaces [62] Invariant Metrics on Lie Groups [67] Other Homogeneous Riemannian Manifolds [72] Model Pseudo-Riemannian Manifolds [79] Problems [80] 4 Connections [85] The Problem of Differentiating Vector Fields [85] Connections [88] Covariant Derivatives of Tensor Fields [95] Vector and Tensor Fields Along Curves [100] Geodesics [103] Parallel Transport [105] Pullback Connections [110] Problems [111] 5 The Levi-Civita Connection [115] The Tangential Connection Revisited [115] Connections on Abstract Riemannian Manifolds [117] The Exponential Map [126] Normal Neighborhoods and Normal Coordinates [131] Tubular Neighborhoods and Fermi Coordinates [133] Geodesics of the Model Spaces [136] Euclidean and Non-Euclidean Geometries [142] Problems [145] 6 Geodesics and Distance [151] Geodesics and Minimizing Curves [151] Uniformly Normal Neighborhoods [163] Completeness [166] Distance Functions [174] Semigeodesic Coordinates [181] Problems [185] 7 Curvature [193] Local Invariants [193] The Curvature Tensor [196] Flat Manifolds [199] Symmetries of the Curvature Tensor [202] The Ricci Identities [205] Ricci and Scalar Curvatures [207] The Weyl Tensor [212] Curvatures of Conformally Related Metrics [216] Problems [222] 8 Riemannian Submanifolds [225] The Second Fundamental Form [225] Hypersurfaces [234] Hypersurfaces in Euclidean Space [244] Sectional Curvatures [250] Problems [255] 9 The Gauss-Bonnet Theorem [263] Some Plane Geometry [263] The Gauss-Bonnet Formula [271] The Gauss-Bonnet Theorem [276] Problems [281] 10 Jacobi Fields [283] The Jacobi Equation [284] Basic Computations with Jacobi Fields [287] Conjugate Points [297] The Second Variation Formula [300] Cut Points [307] Problems [313] 11 Comparison Theory [319] Jacobi Fields, Hessians, and Riccati Equations [320] Comparisons Based on Sectional Curvature [327] Comparisons Based on Ricci Curvature [336] Problems [342] 12 Curvature and Topology [345] Manifolds of Constant Curvature [345] Manifolds of Nonpositive Curvature [352] Manifolds of Positive Curvature [361] Problems [368] Appendix A: Review of Smooth Manifolds [371] Topological Preliminaries [371] Smooth Manifolds and Smooth Maps [374] Tangent Vectors [376] Submanifolds [378] Vector Bundles [382] The Tangent Bundle and Vector Fields [384] Smooth Covering Maps [388] Appendix B: Review of Tensors [391] Tensors on a Vector Space [391] Tensor Bundles and Tensor Fields [396] Differential Forms and Integration [400] Densities [405] Appendix C: Review of Lie Groups [407] Definitions and Properties [407] The Lie Algebra of a Lie Group [408] Group Actions on Manifolds [411] References [415] Notation Index [419] Subject Index [423]
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Originally published with the title 'Riemannian manifolds: an introduction to curvature' in 1997.
1 What Is Curvature? [1] --
The Euclidean Plane [1] --
Surfaces in Space [4] --
Curvature in Higher Dimensions [7] --
2 Riemannian Metrics [9] --
Definitions [9] --
Methods for Constructing Riemannian Metrics [15] --
Basic Constructions on Riemannian Manifolds [25] --
Lengths and Distances [33] --
Pseudo-Riemannian Metrics [40] --
Other Generalizations of Riemannian Metrics [46] --
Problems [47] --
3 Model Riemannian Manifolds [55] --
Symmetries of Riemannian Manifolds [55] --
Euclidean Spaces [57] --
Spheres [58] --
Hyperbolic Spaces [62] --
Invariant Metrics on Lie Groups [67] --
Other Homogeneous Riemannian Manifolds [72] --
Model Pseudo-Riemannian Manifolds [79] --
Problems [80] --
4 Connections [85] --
The Problem of Differentiating Vector Fields [85] --
Connections [88] --
Covariant Derivatives of Tensor Fields [95] --
Vector and Tensor Fields Along Curves [100] --
Geodesics [103] --
Parallel Transport [105] --
Pullback Connections [110] --
Problems [111] --
5 The Levi-Civita Connection [115] --
The Tangential Connection Revisited [115] --
Connections on Abstract Riemannian Manifolds [117] --
The Exponential Map [126] --
Normal Neighborhoods and Normal Coordinates [131] --
Tubular Neighborhoods and Fermi Coordinates [133] --
Geodesics of the Model Spaces [136] --
Euclidean and Non-Euclidean Geometries [142] --
Problems [145] --
6 Geodesics and Distance [151] --
Geodesics and Minimizing Curves [151] --
Uniformly Normal Neighborhoods [163] --
Completeness [166] --
Distance Functions [174] --
Semigeodesic Coordinates [181] --
Problems [185] --
7 Curvature [193] --
Local Invariants [193] --
The Curvature Tensor [196] --
Flat Manifolds [199] --
Symmetries of the Curvature Tensor [202] --
The Ricci Identities [205] --
Ricci and Scalar Curvatures [207] --
The Weyl Tensor [212] --
Curvatures of Conformally Related Metrics [216] --
Problems [222] --
8 Riemannian Submanifolds [225] --
The Second Fundamental Form [225] --
Hypersurfaces [234] --
Hypersurfaces in Euclidean Space [244] --
Sectional Curvatures [250] --
Problems [255] --
9 The Gauss-Bonnet Theorem [263] --
Some Plane Geometry [263] --
The Gauss-Bonnet Formula [271] --
The Gauss-Bonnet Theorem [276] --
Problems [281] --
10 Jacobi Fields [283] --
The Jacobi Equation [284] --
Basic Computations with Jacobi Fields [287] --
Conjugate Points [297] --
The Second Variation Formula [300] --
Cut Points [307] --
Problems [313] --
11 Comparison Theory [319] --
Jacobi Fields, Hessians, and Riccati Equations [320] --
Comparisons Based on Sectional Curvature [327] --
Comparisons Based on Ricci Curvature [336] --
Problems [342] --
12 Curvature and Topology [345] --
Manifolds of Constant Curvature [345] --
Manifolds of Nonpositive Curvature [352] --
Manifolds of Positive Curvature [361] --
Problems [368] --
Appendix A: Review of Smooth Manifolds [371] --
Topological Preliminaries [371] --
Smooth Manifolds and Smooth Maps [374] --
Tangent Vectors [376] --
Submanifolds [378] --
Vector Bundles [382] --
The Tangent Bundle and Vector Fields [384] --
Smooth Covering Maps [388] --
Appendix B: Review of Tensors [391] --
Tensors on a Vector Space [391] --
Tensor Bundles and Tensor Fields [396] --
Differential Forms and Integration [400] --
Densities [405] --
Appendix C: Review of Lie Groups [407] --
Definitions and Properties [407] --
The Lie Algebra of a Lie Group [408] --
Group Actions on Manifolds [411] --
References [415] --
Notation Index [419] --
Subject Index [423] --
MR, MR3887684
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