| Preface to the first edition Preface to the second edition Preface to the English edition How to use this book CHAPTER 0-DIFFERENTIABLE MANIFOLDS 1. Introduction 2. Differentiable manifolds；tangent space 3. Immersions and embeddings；examples 4. Other examples of manifolds，Orientation 5. Vector fields； brackets，Topology of manifolds CHAPTER 1-RIEMANNIAN METRICS 1. Introduction 2. Riemannian Metrics CHAPTER 2-AFFINE CONNECTIONS；RIEMANNIAN CONNECTIONS 1. Introduction 2. Affine connections 3. Riemannian connections CHAPTER 3-GEODESICS；CONVEX NEIGHBORHOODS 1．Introduction 2．The geodesic flow 3．Minimizing properties ofgeodesics 4．Convex neighborhoods CHAPTER 4-CURVATURE 1．Introduction 2．Curvature 3．Sectional curvature 4．Ricci curvature and 8calar curvature 5．Tensors 0n Riemannian manifoids CHAPTER 5-JACOBI FIELDS 1．Introduction 2．The Jacobi equation 3．Conjugate points CHAPTER 6-ISOMETRIC IMMERSl0NS 1．Introduction． 2．The second fundamental form 3．The fundarnental equations CHAPTER 7-COMPLETE MANIFoLDS；HOPF-RINOW AND HADAMARD THEOREMS 1．Introduction． 2．Complete manifolds；Hopf-Rinow Theorem． 3．The Theorem of Hadamazd． CHAPTER 8-SPACES 0F CONSTANT CURVATURE 1．Introduction 2．Theorem of Cartan on the determination ofthe metric by mebns of the curvature． 3．Hyperbolic space 4．Space forms 5．Isometries ofthe hyperbolic space；Theorem ofLiouville CHAPTER 9一VARIATl0NS 0F ENERGY 1．Introduction． 2．Formulas for the first and second variations of enezgy 3．The theorems of Bonnet—Myers and of Synge-WeipJtein CHAPTER 10-THE RAUCH COMPARISON THEOREM 1．Introduction 2．Ttle Theorem of Rauch． 3．Applications of the Index Lemma to immersions 4．Focal points and an extension of Rauch’s Theorem CHAPTER 11—THE MORSE lNDEX THEOREM 1．Introduction 2.The Index Theorem CHAPTER 12-THE FUNDAMENTAL GROUP OF MANIFOLDS 0F NEGATIVE CURVATURE 1．Introduction 2．Existence of closed geodesics CHAPTER 13-THE SPHERE THEOREM References Index |
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