Contents Preface ix A Brief History xi Part 1. Isometric Embedding of Riemannian Manifolds 1 Chapter 1. Fundamental Theorems 3 1.1. Local Isometric Embedding of Analytic Metrics 4 1.2. Local Isometric Embedding of Smooth Metrics 9 1.3. Global Isometric Embedding of Smooth Metrics 16 Notes 31 Chapter 2. Surfaces in Low Dimensional Euclidean Spaces 33 2.1. Surfaces in R3 33 2.2. Isometric Immersions of Surfaces of Constant Curvature 38 2.3. Isometric Immersions of Surfaces in M4 40 Notes 42 Part 2. Local Isometric Embedding of Surfaces in R3 43 Chapter 3. Basic Equations 45 3.1. The Darboux Equation 45 3.2. The Gauss-Codazzi System 47 3.3. The Rozhdestvenskii-Poznyak System 51 Notes 53 Chapter 4. Nonzero Gauss Curvature 55 4.1. Positive Gauss Curvature 56 4.2. Negative Gauss Curvature 58 4.3. Appendix: Sobolev Spaces 63 4.4. Appendix: Some Lemmas 64 4.5. Appendix: Symmetric Hyperbolic Linear Differential Systems 68 Notes 70 Chapter 5. Gauss Curvature Changing Sign Cleanly 71 5.1. The Setting 71 5.2. Iterations 74 5.3. Symmetric Positive Linear Differential Systems 79 Notes 86
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