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

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很不錯的一本書

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訂單：7105711649哈哈哈很好

評分☆☆☆☆☆

質量很好，品質保證。

評分☆☆☆☆☆

很不錯的一本書

評分☆☆☆☆☆

隨後，由於黎曼幾何的發展和愛因斯坦廣義相對論的建立，微分幾何在黎曼幾何學和廣義相對論中的得到瞭廣泛的應用，逐漸在數學中成為獨具特色、應用廣泛的獨立學科。

評分☆☆☆☆☆

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。

評分☆☆☆☆☆

十八世紀初，法國數學傢濛日首先把微積分應用到麯綫和麯麵的研究中去，並於1807年齣版瞭它的《分析在幾何學上的應用》一書，這是微分幾何最早的一本著作。在這些研究中，可以看到力學、物理學與工業的日益增長的要求是促進微分幾何發展的因素。

評分☆☆☆☆☆

很經典，而且作者切入的思路一如既往地和其他人不一樣，拓寬瞭對微分流形的認識

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