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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 9Ò»VARIATl0NS 0F ENERGY
1.Introduction.
2.Formulas for the first and second variations of enezgy
3.The theorems of Bonnet¡ªMyers 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¡ªTHE 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