黎曼几何

出版时间:2008-5  出版社:世界图书出版公司  作者:Manfredo Perdigao do Carmo  页数:300  
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内容概要

The object of this book is to familiarize the reader with the basic language of and some fundamental theorems in Riemannian Geometry。To avoid referring to previous knowledge of differentiable manifolds, we include Chapter 0, which contains those concepts and results on differentiable manifolds which are used in an essential way in the rest of the book。  The first four chapters of the book present the basic concepts of Riemannian Geometry (Riemannian metrics, Riemannian connections, geodesics and curvature)。A good part of the study of Riemannian Geometry consists of understanding the relationship between geodesics and curvature。Jacobi fields, an essential tool for this understanding, are introduced in Chapter 5。In Chapter 6 we introduce the second fundamental form associated with an isometric immersion, and prove a generalization of the Theorem Egregium of Gauss。This allows us to relate the notion of curvature in Riemannian manifolds to the classical concept of Gaussian curvature for surfaces。

作者简介

作者:(葡萄牙)卡莫(Carmo M.p.)

书籍目录

Preface to the first editionPreface to the second edition Preface to the English edition How to use this book CHAPTER 0-DIFFERENTIABLE MANIFOLDS  1. Introduction  2. Differentiable manifolds;tangent space  3. Immersions and embeddings;examples  4. Other examples of manifolds,Orientation  5. Vector fields; brackets,Topology of manifolds CHAPTER 1-RIEMANNIAN METRICS  1. Introduction  2. Riemannian Metrics CHAPTER 2-AFFINE CONNECTIONS;RIEMANNIAN CONNECTIONS  1. Introduction  2. Affine connections  3. Riemannian connections CHAPTER 3-GEODESICS;CONVEX NEIGHBORHOODS  1.Introduction  2.The geodesic flow  3.Minimizing properties ofgeodesics  4.Convex neighborhoodsCHAPTER 4-CURVATURE    1.Introduction  2.Curvature  3.Sectional curvature  4.Ricci curvature and 8calar curvature  5.Tensors 0n Riemannian manifoidsCHAPTER 5-JACOBI FIELDS  1.Introduction  2.The Jacobi equation  3.Conjugate pointsCHAPTER 6-ISOMETRIC IMMERSl0NS    1.Introduction.  2.The second fundamental form  3.The fundarnental equationsCHAPTER 7-COMPLETE MANIFoLDS;HOPF-RINOW AND HADAMARD THEOREMS  1.Introduction.  2.Complete manifolds;Hopf-Rinow Theorem.  3.The Theorem of Hadamazd.CHAPTER 8-SPACES 0F CONSTANT CURVATURE  1.Introduction  2.Theorem of Cartan on the determination ofthe metric by mebns of the curvature.  3.Hyperbolic space  4.Space forms  5.Isometries ofthe hyperbolic space;Theorem ofLiouvilleCHAPTER 9一VARIATl0NS 0F ENERGY    1.Introduction.  2.Formulas for the first and second variations of enezgy  3.The theorems of Bonnet—Myers and of Synge-WeipJteinCHAPTER 10-THE RAUCH COMPARISON THEOREM    1.Introduction  2.Ttle Theorem of Rauch.  3.Applications of the Index Lemma to immersions  4.Focal points and an extension of Rauch’s TheoremCHAPTER 11—THE MORSE lNDEX THEOREM    1.Introduction  2.The Index TheoremCHAPTER 12-THE FUNDAMENTAL GROUP OF MANIFOLDS 0F NEGATIVE CURVATURE    1.Introduction  2.Existence of closed geodesicsCHAPTER 13-THE SPHERE THEOREMReferencesIndex

编辑推荐

《黎曼几何》非常值得一读。

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用户评论 (总计25条)

 
 

  •     这本书是doCamo教授的名著,他的另一本名著是《曲线和曲面的微分几何》,已经翻译成了中文。可惜英文原版为1992年,近20年的研究没有得到整理。
  •     几何味道很浓,叙述生动,证明简练,习题也很好。
  •     此乃广义相对论两大神器之一,是上乘武功秘籍,你值得拥有!
  •     都不错相当好
  •     一定要看的,作者太强了
  •     当当对书的保护还不错。这本书比较小,没什么问题。
  •     绝对的经典好书!
  •     原版教材,很好的书
  •     要求有相当的数学基础
  •     经典之作,值得收藏!印刷很清晰
  •     书皮比较脏,其他还好!
  •     内容比较充实全面,易于理解。值得推荐。
  •     好书,快读完了!题目还没有做完!
  •     很好的书,经典之作,不容错过
  •     拿到时发现是影印版顿时=。=至于内容还没看
  •     内容基本,是很好的入门教材
  •     很不错呀!送货也快。
  •     这本书应该是微分几何的经典
  •     符号用的有些杂,觉得不太适合自学!
  •     这本书首先给出了微分流形的基础,其定义与一般的微分流形的定义不同,但却是最自然的定义方式。其次,关于黎曼几何的内容,讲解的非常细致,章节和条理非常清晰。书后的习题不多,却极其精炼。作为黎曼几何的入门书,是相当好的。
  •     经典中的经典。强烈推荐。
  •     作者写作思路非常清晰,在简短的介绍了微分流形,矢量场等概念后很快就进入黎曼几何的主题了。
  •     内容精简易懂,讲解教详细,印刷不好。
  •     黎曼几何入门看这本书非常好,此书不像一般的黎曼几何书一开始就充斥了许多张量分析的记号,内容几何直观很强,让人能过很快理解到几何的东西,尤其是大范围黎曼几何部分(从第七章起)内容丰富精彩。阅读本书只需要简单的数学分析和线性代数知识。适合高中生和低年级本科生阅读
  •     外观不错,英语不好是看不懂的,本身就难度大,就算懂英语也不一定看得懂
 

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