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

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☆☆☆☆☆

[葡] 卡莫 著

齣版社： 世界圖書齣版公司

ISBN：9787506292184

版次：1

商品編碼：10096470

包裝：平裝

外文名稱：Riemannian Geometry

開本：24開

齣版時間：2008-05-01

用紙：膠版紙

頁數：300

正文語種：英語

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　　《黎曼幾何》非常值得一讀。

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

評分☆☆☆☆☆

4 The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Russell A. Gordon (1994, ISBN 978-0-8218-3805-1)

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書是正版，快遞很快，價格實惠，值得購買

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數學係的同學你懂的，好書一本。

評分☆☆☆☆☆

　　希望自己不做俎肉，而是一條活生生的遊魂！

評分☆☆☆☆☆

卡莫的黎曼幾何寫的不錯，內容比較詳細簡單，講得很細膩，很清楚，入門比較好，思維方式的不同吧。外國的書很多都是寫的比較人性化，就是想一個老師在給你講解一樣。較重引導個人的思考，內容講得全麵一點。當然要學好一個理論當然對這個理論的各種不同的見解、認識、解釋、說明都要有所瞭解，綜閤各方麵的思想，重要是自己思考的過程。這本書比較適閤自學，實際上，自學是獲得知識的很重要的方麵。

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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]

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近代微分幾何的創始人是黎曼，他在1854年創立瞭黎曼幾何（實際上黎曼提齣的是芬斯勒幾何），這成為近代微分幾何的主要內容，並在相對論有極為重要的作用。埃利·嘉當和陳省身等人曾在微分幾何領域做齣極為傑齣的貢獻。

評分☆☆☆☆☆

問題明確，解決問題直接瞭當，用極少的概念引理命題把要說的東西講明白，絕對好書