Geometry III: Theory of Surfaces几何III
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作者： Yu.D. Burago等著
出 版 社： 新世纪出版社
出版时间： 1992-9-1字数：版次： 1页数： 256印刷时间： 1992/09/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540533771包装： 精装内容简介
The theory of two-dimensional surfaces in Euclidean spaces is remarkably rich in deep results and applications, for example in the theory of non-linear partial differential equations, physics and mechanics. This theory has great clarity and intrinsic beauty, and differs in many respects from the theory of multidimensional submanifolds. A separate volume of the Encyclopaedia is therefore devoted to surfaces. It is concerned mainly with the connection between the theory of embedded surfaces and two-dimensional Riemannian geometry (and its generalizations), and, above all, with the question of the influence of properties of intrinsic metrics on the geometry of surfaces. In the first article Yu.D.Burago and S.Z.Shefel' give an extended survey of surfaces from a non-traditional viewpoint stressing the connection between classes of metrics and classes of surfaces in En. A number of conjectures are included. The article of E.R.Rozendorn considers the state of the art of the still incomplete theory of the geometry of surfaces of negative curvature in three-dimensional Euclidean space, and the article of I.Kh.Sabitov considers subtle questions of local bendability and rigidity ofsurfaces. These articles reflect the development of the results of N.V.Efimov and also include statements of unsolved problems.
目录
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
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