几何IV

出版时间：2009-1  出版社：科学  作者：列舍特尼亚克  页数：250  
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前言

　　要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。　　从出版方面来讲，除了较好较快地出版我们自己的成果外，引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说，施普林格（springer）出版社至今仍然是世界上最具权威的出版社。科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。　　这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。　　当然，23本书只能涵盖数学的一部分，所以，这项工作还应该继续做下去。更进一步，有些读者面较广的好书还应该翻译成中文出版，使之有更大的读者群。　　总之，我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持，并盼望这一工作取得更大的成绩。

内容概要

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 and integral curvature of a set and the length and integral curvature of a curve are also defined for these manifolds．Some fundamental results of Riemannian geometry 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 giyen constants．The main result iS that these spaces are in fact Riemannian． This result has important applications in global Riemannian geometry． 　 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．

书籍目录

Chapter 1.Preliminary Information　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 of a 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 of a 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 of a 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.Differentiable 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 of a Manifold with Polyhedral Metric　  5.3 Curvature of a Set on a Polyhedron.Turn of the Boundary. The Gauss-Bonnet Theorem..　  5.4.A Turn of a 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 ofa K.Polyhedron.Chapter 2.Different Ways of Defining Two.Dimensional 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 Two.Dimensional 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 Characterization 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 ofTheorems 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

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