微分几何中的度量结构_PDF下载[129MB-百度云]沃尔斯齐普

内容简介

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　　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.

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PrefaceChapter 1.Differentiable Manifolds1.Basic Definitions2.Differentiable Maps3.Tangent Vectors4.The Derivative5.The Inverse and Implicit Function Theorems6.Submanifolds7.Vector Fields8.The Lie Bracket9.Distributions and Frobenius Theorem10.Multilinear Algebra and Tensors11.Tensor Fields and Differential Forms12.Integration on Chains13.The Local Version of Stokes’ Theorem14.Orientation and the Global Version of Stokes’ Theorem15.Some Applications of Stokes’ TheoremChapter 2.Fiber Bundles1.Basic Definitions and Examples2.Principal and Associated Bundles3.The Tangent Bundle of Sn4.Cross—Sections of Bundles5.Pullback and Normal Bundles6.Fibrations and the Homotopy Lifting／Covering Properties7.Grassmannians and Universal BundlesChapter 3.Homotopy Groups and Bundles Over Spheres1.Differentiable Approximations2.Homotopy Groups3.The Homotopy Sequence of a Fibration4.Bundles Over Spheres5.The Vector Bundles Over Low—Dimensional SpheresChapter 4.Connections and Curvature1.Connections on Vector Bundles2.Covariant Derivatives3.The Curvature Tensor of a Connection4.Connections on Manifolds5.Connections on Principal BundlesChapter 5.Metric Structures1.Euclidean Bundles and Riemannian Manifolds2.Riemannian Connections3.Curvature Quantifiers4.Isometric Immersions5.Riemannian Submersions6.The Gauss Lemma7.Length—Minimizing Properties of Geodesics8.First and Second Variation of Arc—Length9.Curvature and Topology10.Actions of Compact Lie GroupsChapter 6.Characteristic Classes1.The Weil Homomorphism2.Pontrjagin Classes3.The Euler Class4.The Whitney Sum Formula for Pontrjagin and Euler Classes5.Some Examples6.The Unit Sphere Bundle and the Euler Class7.The Generalized Gauss—Bonnet Theorem8.Complex and Symplectic Vector Spaces9.Chern ClassesBibliographyIndex