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內容簡介

　　This text is an elementary introduction to differential geometry. Although it was written for a graduate-level audience, the only requisite is a solid back-ground in calculus, linear algebra, and basic point-set topology.
　　The first chapter covers the fundamentals of differentiable manifolds that are the bread and butter of differential geometry. All the usual topics are covered, culnunating in Stokes' theorem together with some applications. The stu dents' first contact with the subject can be overwhelming because of the wealth of abstract definitions involved, so examples have been stressed throughout. One concept, for instance, that students often find confusing is the definition of tangent vectors. They are first told that these are derivations on certain equiv-alence classes of functions, but later that the tangent space of Rl is "the same" as Rn. We have tried to keep these spaces separate and to carefully explain how a vector space E is canonically isomorphic to its tangent space at a point. This subtle distinction becomes essential when later discussing the vertical bundle of a given vector bundle.

內頁插圖

目錄

Preface
Chapter 1.Differentiable Manifolds
1.Basic Definitions
2.Differentiable Maps
3.Tangent Vectors
4.The Derivative
5.The Inverse and Implicit Function Theorems
6.Submanifolds
7.Vector Fields
8.The Lie Bracket
9.Distributions and Frobenius Theorem
10.Multilinear Algebra and Tensors
11.Tensor Fields and Differential Forms
12.Integration on Chains
13.The Local Version of Stokes' Theorem
14.Orientation and the Global Version of Stokes' Theorem
15.Some Applications of Stokes' Theorem

Chapter 2.Fiber Bundles
1.Basic Definitions and Examples
2.Principal and Associated Bundles
3.The Tangent Bundle of Sn
4.Cross—Sections of Bundles
5.Pullback and Normal Bundles
6.Fibrations and the Homotopy Lifting／Covering Properties
7.Grassmannians and Universal Bundles

Chapter 3.Homotopy Groups and Bundles Over Spheres
1.Differentiable Approximations
2.Homotopy Groups
3.The Homotopy Sequence of a Fibration
4.Bundles Over Spheres
5.The Vector Bundles Over Low—Dimensional Spheres

Chapter 4.Connections and Curvature
1.Connections on Vector Bundles
2.Covariant Derivatives
3.The Curvature Tensor of a Connection
4.Connections on Manifolds
5.Connections on Principal Bundles

Chapter 5.Metric Structures
1.Euclidean Bundles and Riemannian Manifolds
2.Riemannian Connections
3.Curvature Quantifiers
4.Isometric Immersions
5.Riemannian Submersions
6.The Gauss Lemma
7.Length—Minimizing Properties of Geodesics
8.First and Second Variation of Arc—Length
9.Curvature and Topology
10.Actions of Compact Lie Groups

Chapter 6.Characteristic Classes
1.The Weil Homomorphism
2.Pontrjagin Classes
3.The Euler Class
4.The Whitney Sum Formula for Pontrjagin and Euler Classes
5.Some Examples
6.The Unit Sphere Bundle and the Euler Class
7.The Generalized Gauss—Bonnet Theorem
8.Complex and Symplectic Vector Spaces
9.Chern Classes
Bibliography
Index

前言/序言

微分幾何中的度量結構 [Metric Structures in Differential Geometry] 下載 mobi epub pdf txt 電子書

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數學專業人士

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書沒看，是替人買的，這個係列是專業必備書

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本書是 2004 年由斯普林格齣版社推齣，今年初由世界圖書做瞭麵前的這個影印版。本書的內容實質是微分流形和縴維叢的很基本的理論。這本書是根據作者給研究生上課的講義修改完成的，適閤高年級的本科生和研究生閱讀，是比本的微分幾何著作，寫的也清晰。本書有165個習題，不容易，有點費勁

評分☆☆☆☆☆

數學專業人士

評分☆☆☆☆☆

很好很適閤學生老師閱讀

評分☆☆☆☆☆

經典專業書籍，多次購買，質量一如既往。

評分☆☆☆☆☆

GTM係列的書，很棒！

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