Geometry iv : non-regular riemannian geometry几何IV:非正则黎曼代数
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作者： 本社 编
出 版 社： 新世纪出版社
出版时间： 1993-10-1字数：版次： 1页数： 250印刷时间： 1993/10/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540547013包装： 精装内容简介
This volume of the Encyclopaedia contains two articles，which give a survey of modern research into non-regular Riemannian geometry，carried out mostly by Russian mathematicians。The first article written by Reshetnyak is devoted to the theory of two-dimensional Riemannian manifolds of bounded curvature。Concepts of Riemannian geometry，such as the area andintegral curvature of a set，and the length and integral curvature of a curve are also defined for these manifolds。 Some fundamental results of Riemannian goemetry like the Gauss-Bonnet formula are true in the more general case considered in the book。The second article by Berestovskij and Nikolaev is devoted to the theory of metric spaces whose curvature lies between two given constants。The main result is that these spaces are infact Riemannian。This result has important applications in global Riemanniangeometry。Both parts cover topics，which have not yet been treated in monograph form。 Hence the book will be immensely useful to graduate students and researchers in geometry， in particular Riemannian geometry。
目录
Chapter 1.Preliminary lnformation
§1.Introduction
1.1 General Information about the Subject of Research and a Survey of Results
1.2 Some Notation and Terminology
2 The Concept ofa Space with Intrinsic Metric
2.1 The Concept of the Length ofa Parametrized Curve
2.2 A Space with Intrinsic Metric.The Induced Metric
2.3 The Concept ofa Shortest Curve
2.4 The Operation of Cutting of a Space with Intrinsic Metric
3 TWO.Dimensional Manifolds with Intrinsic Metric
3.1 Definition.Triangulation ofa Manifold
3.2 Pasting of TWO.Dimensional Manifolds with Intrinsic Metric
3.3 Cutting of Manifolds
3.4 A Side of a Simple Arc in a TWO.Dimensional Manifold
4 TWO.Dimensional Riemannian Geometry
4.1 Difrerentiable Two.Dimensional Manifolds.
4.2 The Concept of a Two.Dimensional Riemannian Manifold
4.3 The Curvature of a Curve in a Riemannian Manifold Integral Curvature.The GaUSS—Bonnet Formula
4.4 Isothermal Coordinates in TWO.Dimensional Riemannian Manifolds of Bounded Curvature
5 Manifolds with Polyhedral Metric
5.1 Cone and Angular Domain
5.2 Definition ofa Manifold with Polyhedral Metric
5.3 Curvature of a Set on a Polyhedron.Turn of the Boundarv
The GaussBonnet Theorem
5.4 A Turn ofa Polygonal Line on a Polyhedron
5.5 Characterization of the Intrinsic Geometry of Convex Polyhedra
5.6 An Extremal Property of a Convex Cone.The Method of Cutting and Pasting as a Means of Solving Extremal Problems for Polyhedra
5.7 The Concept of a K-Polyhedron
Chapter 2 Different Ways of Defining TwoDimensional Manifolds
of Bounded Curvature
6 Axioms of a Two-Dimensional Manifold of Bounded Curvature
Characterization of such Manifolds by Means of Approximation by Polyhedra
6.1 Axioms of a Two—Dimensional Manifold of Bounded Curvature
6.2 Theorems on the Approximation of TwoDimensional
Manifolds of Bounded Curvature by Manifolds with
Polyhedral and Riemannian Metric
6.3 Proof of the First Theorem on Approximation
6.4 Proof of Lemma 6.3.1
6.5 Proof of the Second Theorem on Approximation
7.Analytic Characterjzation of Two.Dimensional Manifolds of Bounded Curvature
7.1 Theorems on Isothermal Coordinates in a Two.Dimensional Manifold of Bounded Curvature
7.2 Some Information about Curves on a Plane and in a Riemannian manifold
7.3 Proofs of Theorems 7.1.1，7.1.2，7.1.3
7.4.On the Proof ofTheorem 7.3.1
Chapter 3.Basic Facts of the Theory of Manifolds of Bounded Curvature
8.Basic Results of the Theory of Two.Dimensional Manifolds
of Bounded Curvature
8.1 A Turn ofa Curve and the Integral Curvature ofa Set
8.2 A Theorem on the Contraction of a Cone.Angle between Curves.Comparison Theorems
8.3.A Theorem on Pasting Together Two—Dimensional Manifolds of Bounded Curvature
……
References
