Riemannian Geometry
                                                          Contents
                   Chapter 1.  Smooth manifolds                                                              5
                      1. Tangent vectors, cotangent vectors and tensors                                      5
                      2. The tangent bundle of a smooth manifold                                             5
                      3. Vector fields, covector fields, tensor fields, n-forms                                 5
                   Chapter 2.  Riemannian manifolds                                                          7
                      1. Riemannian metric                                                                   7
                      2. The three model geometries                                                          9
                      3. Connections                                                                        13
                      4. Geodesics and parallel translation along curves                                    16
                      5. The Riemannian connection                                                          17
                      6. Connections on submanifolds and pull-back connections                              19
                      7. Geodesics in the three geometries                                                  20
                      8. The exponential map and normal coordinates                                         21
                      9. The Riemann distance function                                                      25
                   Chapter 3.  Curvature                                                                    29
                      1. The Riemann curvature tensor                                                       29
                      2. Ricci curvature, scalar curvature, and Einstein metrics                            31
                      3. Riemannian submanifolds                                                            33
                      4. Sectional curvature                                                                36
                      5. Jacobi fields                                                                       38
                      6. Comparison theorems                                                                44
                   Chapter 4.  Space-times                                                                  47
                   Chapter 5.  Multilinear Algebra                                                          49
                      1. Tensors                                                                            49
                      2. Tensors of inner product spaces                                                    51
                      3. Coordinate expressions                                                             52
                   Chapter 6.  Non-euclidean geometry                                                       55
                      1. The hyperbolic plane                                                               55
                   Bibliography                                                                             59
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