| Preface Chapter 1. The Geometry of Two-Dimensional Manifolds and Surfaces in En 1. Statement of the Problem 1.1. Classes of Metrics and Classes of Surfaces. Geometric Groups and Geometric Properties 2. Smooth Surfaces 2.1. Types of Points 2.2. Classes of Surfaces 2.3. Classes of Metrics 2.4. G-Connectedness 2.5. Results and Conjectures 2.6. The Conformal Group 3. Convex, Saddle and Developable Surfaces with No Smoothness Requirement 3.1. Classes of Non-Smooth Surfaces and Metrics 3.2. Questions of Approximation 3.3. Results and Conjectures 4. Surfaces and Metrics of Bounded Curvature 4.1. Manifolds of Bounded Curvature 4.2. Surfaces of Bounded Extrinsic Curvature Chapter 2. Convex Surfaces 1. Weyl's Problem 1.1. Statement of the Problem 1.2. Historical Remarks 1.3. Outline of One of the Proofs 2. The Intrinsic Geometry of Convex Surfaces. The Generalized Weyl Problem 2.1. Manifolds of Non-Negative Curvature in the Sense of Aleksandrov 2.2. Solution of the Generalized Weyl Problem 2.3. The Gluing Theorem 3. Smoothness of Convex Surfaces 3.1. Smoothness of Convex Immersions 3.2. The Advantage of Isothermal Coordinates 3.3. Consequences of the Smoothness Theorems 4. Bendings of Convex Surfaces 4.1. Basic Concepts 4.2. Smoothness of Bendings 4.3. The Existence of Bendings 4.4. Connection Between Different Forms of Bendings 5. Unbendability of Closed Convex Surfaces 5.1. Unique Determination 5.2. Stability in Weyrs Problem 5.3. Use of the Bending Field 6. Infinite Convex Surfaces 6.1. Non-Compact Surfaces 6.2. Description of Bendings 7. Convex Surfaces with Given Curvatures 7.1. Hypersurfaces 7.2. Minkowski's Problem 7.3. Stability 7.4. Curvature Functions and Analogues of the Minkowski Problem 7.5. Connection with the Monge-Amprre Equations 8. Individual Questions of the Connection Between the Intrinsic and Extrinsic Geometry of Convex Surfaces 8.1. Properties of Surfaces 8.2. Properties of Curves 8.3. The Spherical Image of a Shortest Curve 8.4. The Possibility of Certain Singularities Vanishing Under Bendings Chapter 3. Saddle Surfaces 1. Efimov's Theorem and Conjectures Associated with It 1.1. Sufficient Criteria for Non-Immersibility in E3 1.2. Sufficient Criteria for Immersibility in E3 1.3. Conjecture About a Saddle Immersion in En . 1A. The Possibility of Non-Immersibility when the Manifold is Not Simply-Connected …… Chapter 4. Surfaces of Bounded Extrinsic Curvature Comments on the References References |
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