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 Weyl's 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-Ampere 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 E"
1.4. The Possibility of Non-Immersibility when the Manifold is Not Simply-Connected
2. On the Extrinsic Geometry of Saddle Surfaces
2.1. The Variety of Saddle Surfaces
2.2. Tapering Surfaces
3. Non-Regular Saddle Surfaces
3.1. Definitions
3.2. Intrinsic Geometry
3.3. Problems of Immersibility
3.4. Problems of Non-Immersibility
Chapter 4. Surfaces of Bounded Extrinsic Curvature
1. Surfaces of Bounded Positive Extrinsic Curvature
1.1. Extrinsic Curvatures of a Smooth Surface
1.2. Extrinsic Curvatures of a General Surface
1.3. Inequalities
2. The Role of the Mean Curvature
2.1. The Mean Curvature of a Non-Smooth Surface
2.2. Surfaces of Bounded Mean Curvature
2.3. Mean Curvature as First Variation of the Area
3. C1-Smooth Surfaces of Bounded Extrinsic Curvature
3.1. The Role of the Condition of Boundedness of the Extrinsic Curvature
3.2. Normal C1-Smooth Surfaces
3.3. The Main Results
3.4. Gauss's Theorem
3.5. Cl-Smooth Surfaces
4. Polyhedra
4.1. The Role of Polyhedra in the General Theory
4.2. Polyhedral Metric and Polyhedral Surface
4.3. Results and Conjectures
5. Appendix. Smoothness Classes
Comments on the References
References
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