黎曼幾何 [Riemannian Geometry] 下載 mobi epub pdf 電子書 2025

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內容簡介

　　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。

內頁插圖

目錄

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

黎曼幾何 [Riemannian Geometry] 下載 mobi epub pdf txt 電子書 格式

評分☆☆☆☆☆

微分幾何是數學的一個重要分支，它起源於微積分在幾何上的應用，並與微分方程，復分析，代數，拓撲以及理論物理等相互滲透成為推動這些理論發展的一項重要工具。此外，微分幾何在機械工程，力學等領域有廣泛應用。[1]

評分☆☆☆☆☆

微分幾何學的産生和發展是和數學分析密切相連的。在這方麵第一個做齣貢獻的是瑞士數學傢歐拉。1736年他首先引進瞭平麵麯綫的內在坐標這一概念，即以麯綫弧長這以幾何量作為麯綫上點的坐標，從而開始瞭麯綫的內在幾何的研究。

評分☆☆☆☆☆

書是正版，快遞很快，價格實惠，值得購買

評分☆☆☆☆☆

List of books Edit

評分☆☆☆☆☆

任一仿緊微分流形總具有黎曼度量，這種黎曼度量的數目是非常繁多的，但也不是完全任意的。微分流形的度量結構是受它的拓撲結構所製約的，而這種製約關係正是黎曼幾何研究的一個重要內容，還存在許多沒有解決的問題。

評分☆☆☆☆☆

微分幾何學的産生和發展是和數學分析密切相連的。在這方麵第一個做齣貢獻的是瑞士數學傢歐拉。1736年他首先引進瞭平麵麯綫的內在坐標這一概念，即以麯綫弧長這以幾何量作為麯綫上點的坐標，從而開始瞭麯綫的內在幾何的研究。

評分☆☆☆☆☆

一本不錯的書，是正版

評分☆☆☆☆☆

不錯就是全英文的

評分☆☆☆☆☆

好

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