幾何III

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，引進國外的先進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。這次科學(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)與拓撲三本，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《拓撲學(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。當(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數(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

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.

作者簡介

作者：(俄羅斯)布拉格 (Burago.Y.D.)

書籍目錄

PrefaceChapter 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 CurvatureChapter 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 Weyl's 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-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 BendingsChapter 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 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-ImmersibilityChapter 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. Gauss's 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 ClassesComments on the ReferencesReferences

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