An Introduction to
                Riemannian Geometry
              with Applications to Mechanics and Relativity
                  Leonor Godinho and Jos´e Nat´ario
                          Lisbon, 2004
                                                       Contents
                          Chapter 1.  Differentiable Manifolds                                   3
                            1.  Topological Manifolds                                           3
                            2.  Differentiable Manifolds                                         9
                            3.  Differentiable Maps                                             13
                            4.  Tangent Space                                                  15
                            5.  Immersions and Embeddings                                      22
                            6.  Vector Fields                                                  26
                            7.  Lie Groups                                                     33
                            8.  Orientability                                                  45
                            9.  Manifolds with Boundary                                        48
                            10.  Notes on Chapter 1                                            51
                          Chapter 2.  Differential Forms                                        57
                            1.  Tensors                                                        57
                            2.  Tensor Fields                                                  64
                            3.  Differential Forms                                              66
                            4.  Integration on Manifolds                                       72
                            5.  Stokes Theorem                                                 75
                            6.  Orientation and Volume Forms                                   78
                            7.  Notes on Chapter 2                                             80
                          Chapter 3.  Riemannian Manifolds                                     87
                            1.  Riemannian Manifolds                                           87
                            2.  Affine Connections                                               94
                            3.  Levi-Civita Connection                                         98
                            4.  Minimizing Properties of Geodesics                            104
                            5.  Hopf-Rinow Theorem                                            111
                            6.  Notes on Chapter 3                                            114
                          Chapter 4.  Curvature                                               115
                            1.  Curvature                                                     115
                            2.  Cartan’s Structure Equations                                  122
                            3.  Gauss-Bonnet Theorem                                          131
                            4.  Manifolds of Constant Curvature                               137
                            5.  Isometric Immersions                                          144
                            6.  Notes on Chapter 4                                            150
                                                             1
                          2                             CONTENTS
                          Chapter 5.  Geometric Mechanics                                     151
                            1.  Mechanical Systems                                            151
                            2.  Holonomic Constraints                                         160
                            3.  Rigid Body                                                    164
                            4.  Non-Holonomic Constraints                                     177
                            5.  Lagrangian Mechanics                                          186
                            6.  Hamiltonian Mechanics                                         194
                            7.  Completely Integrable Systems                                 203
                            8.  Notes on Chapter 5                                            209
                          Chapter 6.  Relativity                                              211
                            1.  Galileo Spacetime                                             211
                            2.  Special Relativity                                            213
                            3.  The Cartan Connection                                         223
                            4.  General Relativity                                            224
                            5.  The Schwarzschild Solution                                    229
                            6.  Cosmology                                                     240
                            7.  Causality                                                     245
                            8.  Singularity Theorem                                           253
                            9.  Notes on Chapter 6                                            263
                          Bibliography                                                        265
                          Index                                                               267
                                      CHAPTER 1
                               Differentiable Manifolds
                    This chapter introduces the basic notions of differential geometry.
                    The first section studies topological manifolds of dimension n, which
                  is the rigorous mathematical concept corresponding to the intuitive notion
                  of “continuous n-dimensional spaces”. Several examples are studied, partic-
                  ularly in dimension 2 (surfaces).
                    Section 2 specializes to differentiable manifolds, on which one can
                  definedifferentiable functions (Section 3) and tangent vectors (Section
                  4). Important examples of differentiable maps, namely immersions and
                  embeddings, are examined in Section 5.
                    Vector fields and their flows are the main topic of Section 6. It is
                  shown that there is a natural differential operation between vector fields,
                  called the Lie bracket, which produces a new vector field.
                    Section 7 is devoted to the important class of differentiable manifolds
                  which are also groups, the so-called Lie groups. It is shown that to each
                  Lie group one can associate a Lie algebra, i.e. a vector space equipped with
                  a Lie bracket, and the exponential map, which maps the Lie algebra to
                  the Lie group.
                    The notion of orientability of a manifold (which generalizes the intu-
                  itive notion of “having two sides”) is discussed in Section 8.
                    Finally, manifolds with boundary are studied in Section 9.
                                 1. Topological Manifolds
                    We will begin this section by studying spaces that are locally like Rn,
                  meaning that there exists a neighborhood around each point which is home-
                  omorphic to an open subset of Rn.
                    Definition 1.1. A topological manifold M of dimension n is a topo-
                  logical space with the following properties:
                     (i) M is Hausdorff, that is, for each pair p ,p of distinct points of
                                                   1 2
                        M,there exist neighborhoods V ,V of p and p such that V !V =
                                            1 2  1    2       1  2
                        !.
                     (ii) Each point p ∈ M possesses a neighborhood V homeomorphic to an
                        open subset U of Rn.
                    (iii) M satisfies the second countability axiom, that is, M has a
                        countable basis for its topology.
                                          3