黎曼几何概论

出版时间：2011-4  出版社：世界图书出版公司  作者：Marcel Berger  页数：824  
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内容概要

　　《黎曼几何概论》既是一本学习黎曼几何发展参考书，也是一本很好的教程，包括了学习现代微分几何研究生需要了解的方方面面。黎曼几何变得越来越重要，《黎曼几何概论》中着手本领域比较熟悉的话题，并且尽快过渡到最新科研成果。这些结果并没有给出详细的证明，但一些的重要的结果仍然描述的十分详细生动，使得读者对该领域有详细深刻的理解。然而，黎曼流形作为一个很小的科目，概念的建立需要一个过程，前三章侧重于通过常规的方法引入各种概念和黎曼几何的工具，紧接着详细讲述gauss和riemann。

作者简介

作者：（法国）贝格（Marcel Berger）

书籍目录

1　Euclidean Geometry
　1.1　Preliminaries
　1.2　Distance Geometry
　　1.2.1 A Basic Formula
　　1.2.2 The Length of a Path
　　1.2.3 The First Variation Formulaand Application to
Billiards
　1.3　Plane Curves
　　1.3.1 Length
　　1.3.2 Curvature
　1.4　Global Theory of Closed Plane Curves
　　 1.4.1 "Obvious" Truths About Curves Which are Hard
to Prove
　　 1.4.2 The Four Vertex Theorem
　　 1.4.3 Convexity with Respect to Arc Length
　　 1.4.4 Umlaufsatz with Corners
　　 1.4.5 Heat Shrinking of Plane Curves
　　 1.4.6 ArnoFd's Revolution in Plane Curve
Theory
　1.5　The Isoperimetric Inequality for Curves
　1.6　The Geometry of Surfaces Before and After Gaufl
　　 1.6.1 Inner Geometry:. a First Attempt
　　 1.6.2 Looking for Shortest Curves: Geodesics
　　 1.6.3 The Second Fundamental Formand Principal
Curvatures
　　 1.6.4 The Meaning of the Sign of K
　　 1.6.5 Global Surface Geometry
　　 1.6.6 Minimal Surfaces
　　 1.6.7 The Hartman-Nirenberg Theoremfor Inner Flat
Surfaces
　　 1.6.8 The Isoperimetric Inequality in E3 ~ la
Gromov
　　1.6.8.1 Notes
　1.7　Generic Surfaces
　1.8　Heat and Wave Analysis in
　　 1.8.1 Planar Physics
　　1.8.1.1 Bibliographical Note
　　……

章节摘录

版权页：   插图：   This would describe the motion of the sea,if there were nocontinents on the planet.It might not be a bad approximation,since oceansare the bulk of the Earth's surface.His first result backs intuition.Disturb the water with a big shock（a Dirac distribution,if you prefer）somewhere,say at the north pole.Then you will always find another big shock at the south pole after a time T which is no larger than 2π.But Meyer's second series of results oppose intuition: the big shock can move from the north to the south pole while remaining extremely small everywhere for all time between O and T,as in figure 1.102 on the next page.Worse some apparently moderate shocks can,with larger and larger time,be very small almost everywhere and almost all the time,but at some times and some places can be as big as desired（everything there and then will break down）.Meyer's results explained the following strange observation: quite recently a tidal wave was observed at Martinique,and came back there later,this being completely unnoticed eyerywhere else in between times. 1.9.3 Billiards in Higher Dimensions  The Faber-Krahn inequality extends to any dimension.Again,the proof involves the isoperimetric inequality（in any dimension）.Contrarily,billiards in dimensions larger than two are still to be fully explored.There is one ex-ception: billiards in concave regions,for which we refer to Katok,Strelcyn,Ledrappier & Przytycki 1986（788）.Such regions have the best possible ergodic behavior.But for nonconcave regions,even for polyhedra in three dimensional space,almost nothing is known concerning periodic motions or ergodicity,except for rectangular parallelepipeds,balls（of course）and the Berard examples mentioned above.There is an unpublished result of Katok asserting that the length counting function is subexponential.But,even in the simplest possi-ble cases,things are either not finished,or completely open.Let us mention only two cases.The first is that of the cube.Dynamics in a cube are compa-rable to dynamics in a square: a trajectory is periodic or everywhere dense.

编辑推荐

《黎曼几何概论》既是一本学习黎曼几何发展参考书，也是一本很好的教程，包括了学习现代微分几何研究生需要了解的方方面面。黎曼几何变得越来越重要，《黎曼几何概论》中着手本领域比较熟悉的话题，并且尽快过渡到最新科研成果。这些结果并没有给出详细的证明，但一些的重要的结果仍然描述的十分详细生动，使得读者对该领域有详细深刻的理解。

用户评论 (总计7条)

•   本书作者Bege是法国的大几何学家，本书堪称巨著，有824页。该书对黎曼几何做了全景式的描述，不是一本教材。希望对数学的其它分支也能有人写出这样全面深入的书籍。
•   领域内最全的黎曼几何百科全书了吧。。
•   这本书的确出自大人物之手，配的插图很多，但有很多式子、手把手教你并不是这本书的任务，而更多具体的内容比如算本书提供了索引。我或许买了本大字典，但不适合用来初学。
•   经典，值得看！黎曼几何概论
•   书还不错，但是英文而不是中文。卖方写的是中文
•   能使你很快的了解黎曼几何的前沿，不错的书。
•   这是法国伟大数学家Berger的著作，很全面，一般黎曼几何的书没有这么全的题材。感觉非常好！