The object of this book is to familiarize the reader with the basic language of and some fundamental theorems in Riemannian Geometry.To avoid referring to previous knowledge of differentiable manifolds, we include Chapter 0, which contains those concepts and results on differentiable manifolds which are used in an essential way in the rest of the book.

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.

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