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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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書沒看，是替人買的，這個係列是專業必備書

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GTM係列的書，很棒！

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書有點薄。

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質量不錯。好評

評分☆☆☆☆☆

書有點薄。

評分☆☆☆☆☆

GTM係列的書，很棒！

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Springer的書必屬經典

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書有點薄。

評分☆☆☆☆☆

內容很經典，敘述清晰。是一部學習微分流形和縴維叢的入門書籍，從矩陣微分幾何的觀點齣發研究縴維叢，討論瞭歐幾裏得叢；黎曼連通；麯率和Chern-Weil理論；也包括Pontrjagin, Euler, 和Chern 的嚮量叢特徵類，並通過球上的叢詳細闡釋瞭這些概念。目次：微分流形；縴維叢；同倫群和球上的叢；連通和麯率；度量結構；特徵類。

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