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现代黎曼几何导论PDF|Epub|txt|kindle电子书版本网盘下载
- ISAAC CHAVEL著 著
- 出版社： 世界图书出版公司北京公司
- ISBN：750624702X
- 出版时间：2000
- 标注页数：386页
- 文件大小：37MB
- 文件页数：399页
- 主题词：
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图书目录
1 Riemannian manifolds1
1.1 Connections3
1.2 Parallel translation of vector fields7
1.3 Geodesics and the exponential map8
1.4 The torsion and curvature tensors12
1.5 Riemannian metrics14
1.6 The metric space structure17
1.7 Geodesics and completeness24
1.8 Calculations with moving frames27
1.9 Notes and exercises30
2 Riemannian curvature49
2.1 The Riemann sectional curvature51
2.2 Riemannian submanifolds53
2.3 Spaces of constant sectional curvature57
2.4 First and second variation of arc length66
2.5 Jacobi's equation and criteria70
2.6 Elementary comparison theorems77
2.7 Jacobi fields and the exponential map81
2.8 Riemann normal coordinates82
2.9 Notes and exercises86
3 Riemannian volume101
3.1 Geodesic spherical coordinates102
3.2 The conjugate and cut loci104
3.3 Riemannian measure110
3.4 Volume comparison theorems117
3.5 The area of spheres126
3.6 Fermi coordinates128
3.7 Integration of differential forms136
3.8 Notes and exercises144
3.9 Appendix:Eigenvalue comparison theorems155
4 Riemannian coverings172
4.1 Riemannian coverings173
4.2 The fundamental group178
4.3 Volume growth of Riemannian coverings181
4.4 Discretization of Riemannian manifolds189
4.5 The free homotopy classes199
4.6 Gauss-Bonnet theory of surfaces202
4.7 Notes and exercises210
5 The kinematic density219
5.1 The differential geometry of analytical dynamics220
5.2 Santalo's formula231
5.3 The Berger-Kazdan inequalities235
5.4 On manifolds with no conjugate points245
5.5 Notes and exercises254
6 Isoperimetric inequalities262
6.1 Isoperimetric constants264
6.2 The isoperimetric inequality in Euclidean space276
6.3 The isoperimetric inequality on spheres280
6.4 Symmetrization and isoperimetric inequalities283
6.5 Buser's isoperimetric inequality288
6.6 Croke's isoperimetric inequality295
6.7 Discretizations and isoperimetry298
6.8 Notes and exercises305
7 Comparison and finiteness theorems315
7.1 Preliminaries316
7.2 H.E.Rauch's comparison theorem317
7.3 Comparison theorems with initial submanifolds320
7.4 Refinements of the Rauch theorem326
7.5 Triangle comparison theorems329
7.6 Convexity332
7.7 Center of mass336
7.8 Cheeger's finiteness theorem338
7.9 Notes and exercises349
Hints and sketches of solutions355
Bibliography371
Index383