黎曼几何 [Riemannian Geometry] 下载 mobi epub pdf 电子书 2024

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　　《黎曼几何》非常值得一读。

内容简介

　　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。

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目录

Preface to the first edition
Preface 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 neighborhoods

CHAPTER 4-CURVATURE
1．Introduction
2．Curvature
3．Sectional curvature
4．Ricci curvature and 8calar curvature
5．Tensors 0n Riemannian manifoids

CHAPTER 5-JACOBI FIELDS
1．Introduction
2．The Jacobi equation
3．Conjugate points

CHAPTER 6-ISOMETRIC IMMERSl0NS
1．Introduction．
2．The second fundamental form
3．The fundarnental equations

CHAPTER 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 ofLiouville

CHAPTER 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-WeipJtein

CHAPTER 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 Theorem

CHAPTER 11—THE MORSE lNDEX THEOREM
1．Introduction
2.The Index Theorem

CHAPTER 12-THE FUNDAMENTAL GROUP OF MANIFOLDS 0F NEGATIVE CURVATURE
1．Introduction
2．Existence of closed geodesics
CHAPTER 13-THE SPHERE THEOREM
References
Index

前言/序言

黎曼几何 [Riemannian Geometry] 下载 mobi epub pdf txt 电子书 格式

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微分几何学的产生和发展是和数学分析密切相连的。在这方面第一个做出贡献的是瑞士数学家欧拉。1736年他首先引进了平面曲线的内在坐标这一概念，即以曲线弧长这以几何量作为曲线上点的坐标，从而开始了曲线的内在几何的研究。

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还可以，还可以！

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1827年，高斯发表了《关于曲面的一般研究》的著作，这在微分几何的历史上有重大的意义，它的理论奠定了现代形式曲面论的基础。微分几何发展经历了150年之后，高斯抓住了微分几何中最重要的概念和带根本性的内容，建立了曲面的内在几何学。其主要思想是强调了曲面上只依赖于第一基本形式的一些性质，例如曲面上曲面的长度、两条曲线的夹角、曲面上的一区域的面积、测地线、测地线曲率和总曲率等等。他的理论奠定了近代形式曲面论的基础。

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　　二十七岁的李敖曾在《十三年和十三月》中说过一段话：

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好好好……拿来做广义相对论的数学参考的

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