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

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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

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截麵麯率、裏奇麯率以及數量麯率是非常重要的幾何量。研究這些量與黎曼流形的幾何性質以及拓撲性質之間的關係是黎曼幾何的一個重要課題。例如，嘉當－阿達馬定理斷言：若一個n維單連通完備黎曼流形的截麵麯率處處不大於零，那麼它與Rn微分同胚。再如邁爾斯定理斷言：若完備黎曼流形的裏奇麯率處處大於一個正常數h，那麼它必是緊流形而且基本群有限。W.剋林格貝格和M.伯熱證明的球定理斷言：如果完備單連通n維黎曼流形M的截麵麯率KM 滿足，那麼M與n維歐氏球麵Sn同胚。這些結果顯示瞭流形的拓撲性質與度量性質之間有密切的聯係。在這方麵還有許多未解決的問題。

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2 Combinatorial Rigidity, Jack Graver, Brigitte Servatius, Herman Servatius (1993, ISBN 978-0-8218-3801-3)

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此用戶未填寫評價內容

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學習黎曼幾何的好書，正在學習中

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一個方嚮的書要多買幾本不同人寫的，纔能從不同的角度去看！

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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。

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不錯。。。。。。。。。。。。。。。。。

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書包裝很好，服務也很滿意！