TEXTS IN MATHEMATICS
CONTENTS CHAPTER 1 RIEMANNIAN METRICS 1 RIEMANNIAN MANIFOLDS AND MAPS 2 GROUPS AND RIEMANNIAN MANIFOLDS 3 LOCAL REPRESENTATIONS OF METRICS 4 DOUBLY WARPED PRODUCTS 5 EXERCISES CHAPTER 2 CURVATURE 1 CONNECTIONS 2 THE CONNECTION IN LOCAL COORDINATES 3 CURVATURE 4 THE FUNDAMENTAL CURVATURE EQUATIONS 5 THE EQUATIONS OF RIEMANNIAN GEOMETRY 6 SOME TENSOR CONCEPTS 7 FURTHER STUDY 8 EXERCISES CHAPTER 3 EXAMPLES 1 COMPUTATIONAL SIMPLI?CATIONS 2 WARPED PRODUCTS 3 HYPERBOLIC SPACE 4 METRICS ON LIE GROUPS 5 RIEMANNIAN SUBMERSIONS 6 FURTHER STUDY 7 EXERCISES CHAPTER 4 HYPERSURFACES 1 THE GAUSS MAP 2 EXISTENCE OF HYPERSURFACES 3 THE GAUSS-BONNET THEOREM 4 FURTHER STUDY CHAPTER 5 GEODESICS AND DISTANCE 1 MIXED PARTIALS 2 GEODESICS 3 THE METRIC STRUCTURE OF A RIEMANNIAN MANIFOLD 4 FIRST VARIATION OF ENERGY 5 THE EXPONENTIAL MAP 6 WHY SHORT GEODESICS ARE SEGMENTS 7 LOCAL GEOMETRY IN CONSTANT CURVATURE 8 COMPLETENESS 9 CHARACTERIZATION OF SEGMENTS 10 RIEMANNIAN ISOMETRIES 11 FURTHER STUDY 12 EXERCISES CHAPTER 6 SECTIONAL CURVATURE COMPARISON I 1 THE CONNECTION ALONG CURVES 2 SECOND VARIATION OF ENERGY 3 NONPOSITIVE SECTIONAL CURVATURE 4 POSITIVE CURVATURE 5 BASIC COMPARISON ESTIMATES 6 MORE ON POSITIVE CURVATURE CHAPTER 7 THE BOCHNER TECHNIQUE 1 KILLING FIELDS 2 HODGE THEORY 3 HARMONIC FORMS 4 CLI?ORD MULTIPLICATION ON FORMS 5 THE CURVATURE TENSOR CHAPTER 8 SYMMETRIC SPACES AND HOLONOMY 1 SYMMETRIC SPACES 2 EXAMPLES OF SYMMETRIC SPACES 3 HOLONOMY 4 CURVATURE AND HOLONOMY 5 FURTHER STUDY 6 EXERCISES CHAPTER 9 RICCI CURVATURE COMPARISON 1 VOLUME COMPARISON 2 FUNDAMENTAL GROUPS AND RICCI CURVATURE 3 MANIFOLDS OF NONNEGATIVE RICCI CURVATURE CHAPTER 10 CONVERGENCE 1 GROMOV-HAUSDOR?CONVERGENCE 2 H¨OLDER SPACES AND SCHAUDER ESTIMATES 3 NORMS AND CONVERGENCE OF MANIFOLDS 4 GEOMETRIC APPLICATIONS 5 HARMONIC NORMS AND RICCI CURVATURE CHAPTER 11 SECTIONAL CURVATURE COMPARISON II 1 CRITICAL POINT THEORY 2 DISTANCE COMPARISON 3 SPHERE THEOREMS 4 THE SOUL THEOREM 5 FINITENESS OF BETTI NUMBERS 6 HOMOTOPY FINITENESS