1 Riemannian manifolds 1
1.1 Connections 3
1.2 Parallel translation of vector fields 7
1.3 Geodesics and the exponential map 8
1.4 The torsion and curvature tensors 12
1.5 Riemannian metrics 14
1.6 The metric space structure 17
1.7 Geodesics and completeness 24
1.8 Calculations with moving frames 27
1.9 Notes and exercises 30
2 Riemannian curvature 49
2.1 The Riemann sectional curvature 51
2.2 Riemannian submanifolds 53
2.3 Spaces of constant sectional curvature 57
2.4 First and second variation of arc length 66
2.5 Jacobi's equation and criteria 70
2.6 Elementary comparison theorems 77
2.7 Jacobi fields and the exponential map 81
2.8 Riemann normal coordinates 82
2.9 Notes and exercises 86
3 Riemannian volume 101
3.1 Geodesic spherical coordinates 102
3.2 The conjugate and cut loci 104
3.3 Riemannian measure 110
3.4 Volume comparison theorems 117
3.5 The area of spheres 126
3.6 Fermi coordinates 128
3.7 Integration of differential forms 136
3.8 Notes and exercises 144
3.9 Appendix:Eigenvalue comparison theorems 155
4 Riemannian coverings 172
4.1 Riemannian coverings 173
4.2 The fundamental group 178
4.3 Volume growth of Riemannian coverings 181
4.4 Discretization of Riemannian manifolds 189
4.5 The free homotopy classes 199
4.6 Gauss-Bonnet theory of surfaces 202
4.7 Notes and exercises 210
5 The kinematic density 219
5.1 The differential geometry of analytical dynamics 220
5.2 Santalo's formula 231
5.3 The Berger-Kazdan inequalities 235
5.4 On manifolds with no conjugate points 245
5.5 Notes and exercises 254
6 Isoperimetric inequalities 262
6.1 Isoperimetric constants 264
6.2 The isoperimetric inequality in Euclidean space 276
6.3 The isoperimetric inequality on spheres 280
6.4 Symmetrization and isoperimetric inequalities 283
6.5 Buser's isoperimetric inequality 288
6.6 Croke's isoperimetric inequality 295
6.7 Discretizations and isoperimetry 298
6.8 Notes and exercises 305
7 Comparison and finiteness theorems 315
7.1 Preliminaries 316
7.2 H.E.Rauch's comparison theorem 317
7.3 Comparison theorems with initial submanifolds 320
7.4 Refinements of the Rauch theorem 326
7.5 Triangle comparison theorems 329
7.6 Convexity 332
7.7 Center of mass 336
7.8 Cheeger's finiteness theorem 338
7.9 Notes and exercises 349
Hints and sketches of solutions 355
Bibliography 371
Index 383