The first four chapters of the book present the basic concepts
of Riemannian Geometry(Riemannian metrics, Riemannian connections,
geodesics and curvature).A good part of the study of Riemannian
Geometry consists of understanding the relationship between
geodesics and curvature.Jacobi fields, an essential tool for this
understanding, are introduced in Chapter 5.In Chapter 6 we
introduce the second fundamental form associated with an isometric
immersion, and prove a generalization of the Theorem Egregium of
Gauss.This allows us to relate the notion of curvature in
Riemannian manifolds to the classical concept of Gaussian
curvature for surfaces.
目录
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
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
3. Preissman's Theorem
CHAPTER 13-THE SPHERE THEOREM
1. Introduction
2. The cut locus
3. The estimate of the injectivity radius
4. The Sphere Theorem
5. Some further developments
References
Index
