幾何IV

前言

　　要使我國的數(shù)學(xué)事業(yè)更好地發(fā)展起來，需要數(shù)學(xué)家淡泊名利并付出更艱苦地努力。另一方面，我們也要從客觀上為數(shù)學(xué)家創(chuàng)造更有利的發(fā)展數(shù)學(xué)事業(yè)的外部環(huán)境，這主要是加強對數(shù)學(xué)事業(yè)的支持與投資力度，使數(shù)學(xué)家有較好的工作與生活條件，其中也包括改善與加強數(shù)學(xué)的出版工作。　　從出版方面來講，除了較好較快地出版我們自己的成果外，引進(jìn)國外的先進(jìn)出版物無疑也是十分重要與必不可少的。從數(shù)學(xué)來說，施普林格（springer）出版社至今仍然是世界上最具權(quán)威的出版社?？茖W(xué)出版社影印一批他們出版的好的新書，使我國廣大數(shù)學(xué)家能以較低的價格購買，特別是在邊遠(yuǎn)地區(qū)工作的數(shù)學(xué)家能普遍見到這些書，無疑是對推動我國數(shù)學(xué)的科研與教學(xué)十分有益的事?！　∵@次科學(xué)出版社購買了版權(quán)，一次影印了23本施普林格出版社出版的數(shù)學(xué)書，就是一件好事，也是值得繼續(xù)做下去的事情。大體上分一下，這23本書中，包括基礎(chǔ)數(shù)學(xué)書5本，應(yīng)用數(shù)學(xué)書6本與計算數(shù)學(xué)書12本，其中有些書也具有交叉性質(zhì)。這些書都是很新的，2000年以后出版的占絕大部分，共計16本，其余的也是1990年以后出版的。這些書可以使讀者較快地了解數(shù)學(xué)某方面的前沿，例如基礎(chǔ)數(shù)學(xué)中的數(shù)論、代數(shù)與拓?fù)淙?，都是由該領(lǐng)域大數(shù)學(xué)家編著的“數(shù)學(xué)百科全書”的分冊。對從事這方面研究的數(shù)學(xué)家了解該領(lǐng)域的前沿與全貌很有幫助。按照學(xué)科的特點，基礎(chǔ)數(shù)學(xué)類的書以“經(jīng)典”為主，應(yīng)用和計算數(shù)學(xué)類的書以“前沿”為主。這些書的作者多數(shù)是國際知名的大數(shù)學(xué)家，例如《拓?fù)鋵W(xué)》一書的作者諾維科夫是俄羅斯科學(xué)院的院士，曾獲“菲爾茲獎”和“沃爾夫數(shù)學(xué)獎”。這些大數(shù)學(xué)家的著作無疑將會對我國的科研人員起到非常好的指導(dǎo)作用?！　‘?dāng)然，23本書只能涵蓋數(shù)學(xué)的一部分，所以，這項工作還應(yīng)該繼續(xù)做下去。更進(jìn)一步，有些讀者面較廣的好書還應(yīng)該翻譯成中文出版，使之有更大的讀者群?！　】傊?，我對科學(xué)出版社影印施普林格出版社的部分?jǐn)?shù)學(xué)著作這一舉措表示熱烈的支持，并盼望這一工作取得更大的成績。

內(nèi)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．

書籍目錄

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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