微分幾何基礎

前言

　　The present book aims to give a fairly comprehensive account of thefundamentals of differential manifolds and differential geometry. The sizeof the book influenced where to stop， and there would be enough materialfor a second volume （this is not a threat）.　　At the most basic level， the book gives an introduction to the basicconcepts which are used in differential topology， differential geometry， anddifferential equations.　In differential topology， one studies for instancehomotopy classes of maps and the possibility of finding suitable differen-tiable maps in them （immersions， embeddings， isomorphisms， etc.）. Onemay also use differentiable structures on topological manifolds to deter-mine the topological structure of the manifold （for example， h ia Smale[Sin 67]）. In differential geometry， one puts an additional structure on thedifferentiable manifold （a vector field， a spray， a 2-form， a Riemannianmetric， ad lib.） and studies properties connected especially with theseobjects. Formally， one may say that one studies properties invariant underthe group of differentiable automorphisms which preserve the additionalstructure. In differential equations， one studies vector fields and their in-tegral curves， singular points， stable and unstable manifolds， etc. A certainnumber of concepts are essential for all three， and are so basic and elementarythat it is worthwhile to collect them together so that more advanced expositionscan be given without having to start from the very beginnings.　　Those interested in a brief introduction could run through Chapters II，III， IV， V， VII， and most of Part III on volume forms， Stokes theorem，and integration. They may also assume all manifolds finite dimensional.

內容概要

本書介紹了微分拓撲、微分幾何以及微分方程的基本概念。本書的基本思想源于作者早期的《微分和黎曼流形》，但重點卻從流形的一般理論轉移到微分幾何，增加了不少新的章節(jié)。這些新的知識為Banach和Hilbert空間上的無限維流形做準備，但一點都不覺得多余，而優(yōu)美的證明也讓讀者受益不淺。在有限維的例子中，討論了高維微分形式，繼而介紹了Stokes定理和一些在微分和黎曼情形下的應用。給出了Laplacian基本公式，展示了其在浸入和浸沒中的特征。書中講述了該領域的一些主要基本理論，如：微分方程的存在定理、唯一性、光滑定理和向量域流，包括子流形管狀鄰域的存在性的向量叢基本理論，微積分形式，包括經(jīng)典2-形式的辛流形基本觀點，黎曼和偽黎曼流形協(xié)變導數(shù)以及其在指數(shù)映射中的應用，Cartan-Hadamard定理和變分微積分第一基本定理。目次：（第一部分）一般微分方程；微積分；流形；向量叢；向量域和微分方程；向量域和微分形式運算；Frobenius定理；（第二部分）矩陣、協(xié)變導數(shù)和黎曼幾何：矩陣；協(xié)變導數(shù)和測地線；曲率；二重切線叢的張量分裂；曲率和變分公式；半負曲率例子；自同構和對稱；浸入和浸沒；（第三部分）體積形式和積分：體積形式；微分形式的積分；Stokes定理；Stokes定理的應用；譜理論。

書籍目錄

Foreword Acknowledgments PART Ⅰ General Differential Theory 　CHAPTER Ⅱ Differential Calculus 　　1.Categories 　　2.Topological Vector Spaces 　　3.Derivatives and Composition of Maps 　　4.Integration and Taylor's Formula 　　5.The Inverse Mapping Theorem 　CHAPTER Ⅱ Manifolds 　　1.Atlases, Charts, Morphisms 　　2.Submanifolds, Immersions, Submersions 　　3.Partitions of Unity 　　4.Manifolds with Boundary 　CHAPTER Ⅲ Vector Bundles 　　1.Definition, Pull Backs 　　2.The Tangent Bundle 　　3.Exact Sequences of Bundles 　　4.Operations on Vector Bundles 　　5.Splitting of Vector Bundles 　CHAPTER Ⅳ Vector Fields and Differential Equations 　　1.Existence Theorem for Differential Equations 　　2.Vector Fields, Curves, and Flows 　　3.Sprays 　　4.The Flow of a Spray and the Exponential Map 　　5.Existence of Tubular Neighborhoods 　　6.Uniqueness of Tubular Neighborhoods 　CHAPTER Ⅴ Operations on Vector Fields and Differential Forms 　　1.Vector Fields, Differential Operators, Brackets 　　2.Lie Derivative 　　3.Exterior Derivative 　　4.The Poincare Lemma. 　　5.Contractions and Lie Derivative 　　6.Vector Fields and l-Forms Under Self Duality 　　7.The Canonical 2-Form 　　8.Darboux's Theorem 　CHAPTER Ⅵ The Theorem ol Frobenius 　　1.Statement of the Theorem 　　2.Differential Equations Depending on a Parameter 　　3.Proof of the Theorem 　　4.The Global Formulation 　　5.Lie Groups and Subgroups PART Ⅱ Metrics, Covariant Derivatives, and Riemannian Geometry 　CHAPTER Ⅶ Metrics 　　1.Definition and Functoriality 　　2.The Hilbert Group 　　3.Reduction to the Hiibert Group 　　4.Hilbertian Tubular Neighborhoods 　　5.The Morse-Palais Lemma 　　6.The Riemannian Distance 　　7.The Canonical Spray 　CHAPTER Ⅷ Covarlent Derivatives and Geodesics 　　1.Basic Properties 　　2.Sprays and Covariant Derivatives 　　3.Derivative Along a Curve and Parallelism 　　4.The Metric Derivative 　　5.More Local Results on the Exponential Map 　　6.Riemannian Geodesic Length and Completeness 　CHAPTER Ⅸ curvature 　　1.The Riemann Tensor 　　2.Jacobi Lifts. 　　3.Application of Jacobi Lifts to Texp 　　4.Convexity Theorems. 　　5.Taylor ExpansionsPART Ⅲ Volume Forms and IntegrationIndex

章節(jié)摘錄

　　We shall recall briefly the notion of derivative and some of its usefulproperties. As mentioned in the foreword, Chapter VIII of Dieudonn6sbook or my books on analysis [La 83], [La 93] give a self-contained andcomplete treatment for Banach spaces.　We summarize certain factsconcerning their properties as topological vector spaces, and then wesummarize differential calculus. The reader can actually skip this chapterand start immediately with Chapter II if the reader is accustomed tothinking about the derivative of a map as a linear transformation. （In thefinite dimensional case, when bases have been selected, the entries in thematrix of this transformation are the partial derivatives of the map.） Wehave repeated the proofs for the more important theorems, for the ease ofthe reader.　　It is convenient to use throughout the language of categories. Thenotion of category and morphism （whose definitions we recall in 1） isdesigned to abstract what is common to certain collections of objects andmaps between them.　For instance, topological vector spaces and continuous linear maps, open subsets of Banach spaces and differentiablemaps, differentiable manifolds and differentiable maps, vector bundles andvector bundle maps, topological spaces and continuous maps, sets and justplain maps. In an arbitrary category, maps are called morphisms, and infact the category of differentiable manifolds is of such importance in thisbook that from Chapter II on, we use the word morphism synonymouslywith differentiable map （or p-times differentiable map, to be precise）. Allother morphisms in other categories will be qualified by a prefix to in-dicate the category to which they belong.

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