几何．3，曲面理论

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. Weyls 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 Weyls 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. Minkowskis 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. Efimovs 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. Gausss 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

《国外数学名著系列(续1)(影印版)57：几何3(曲面理论)》由俄罗斯作家阿诺德编写。The theory of surfaces in Euclidean spaces is remarkably rich in deep results and applications.This volume of the Encyclopaedia is concerned mainly with the connection between the theory of embedded surfaces and Riemannian geometry and with the geometry of surfaces as influenced by intrinsic metrics.

《国外数学名著系列(续1)(影印版)57：几何3(曲面理论)》为《国外数学名著系列》丛书之一。该丛书是科学出版社组织学术界多位知名院士、专家精心筛选出来的一批基础理论类数学著作，读者对象面向数学系高年级本科生、研究生及从事数学专业理论研究的科研工作者。 本册为《几何(Ⅲ曲面理论影印版)57》，《国外数学名著系列(续1)(影印版)57：几何3(曲面理论)》包含了欧几里德的几何曲面理论。