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《微分几何基础(英文版)》介绍了微分拓扑、微分几何以及微分方程的基本概念。《微分几何基础(英文版)》的基本思想源于作者早期的《微分和黎曼流形》，但重点却从流形的一般理论转移到微分几何，增加了不少新的章节。这些新的知识为Banach和Hilbert空间上的无限维流形做准备，但一点都不觉得多余，而优美的证明也让读者受益不浅。在有限维的例子中，讨论了高维微分形式，继而介绍了Stokes定理和一些在微分和黎曼情形下的应用。给出了Laplacian基本公式，展示了其在浸入和浸没中的特征。书中讲述了该领域的一些主要基本理论，如：微分方程的存在定理、少数性、光滑定理和向量域流，包括子流形管状邻域的存在性的向量丛基本理论，微积分形式，包括经典2-形式的辛流形基本观点，黎曼和伪黎曼流形协变导数以及其在指数映射中的应用，Cartan-Hadamard定理和变分微积分一基本定理。目次：（一部分）一般微分方程；微积分；流形；向量丛；向量域和微分方程；向量域和微分形式运算；Frobenius定理；（第二部分）矩阵、协变导数和黎曼几何：矩阵；协变导数和测地线；曲率；二重切线丛的张量分裂；曲率和变分公式；半负曲率例子；自同构和对称；浸入和浸没；（第三部分）体积形式和积分：体积形式；微分形式的积分；Stokes定理；Stokes定理的应用；谱理论。

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目录

Foreword
Acknowledgments
PART Ⅰ
General Differential Theory，
CHAPTER Ⅰ
Oifferenlial Calculus
Categories
Topological Vector Spaces
Derivatives and Composition of Maps
Integration and Taylors Formula
The Inverse Mapping Theorem

CHAPTER Ⅱ
Manifolds
Atlases， Charts， Morphisms
Submanifolds， Immersions， Submersions
Partitions of Unity
Manifolds with Boundary

CHAPTER Ⅲ
Vector Bundles
Definition， Pull Backs
The Tangent Bundle
Exact Sequences of Bundles
Operations on Vector Bundles
Splitting of Vector Bundles

CHAPTER Ⅳ
Vector Fields and Differential Equations
Existence Theorem for Differential Equations .
Vector Fields， Curves， and Flows
Sprays
The Flow of a Spray and the Exponential Map
Existence of Tubular Neighborhoods
Uniqueness of Tubular Neighborhoods

CHAPTER Ⅴ
Operations on Vector Fields and Differential Forms
Vector Fields， Differential Operators， Brackets
Lie Derivative
Exterior Derivative
The Poincar Lemma
Contractions and Lie Derivative
Vector Fields and l-Forms Under Self Duality
The Canonical 2-Form
Darbouxs Theorem

CHAPTER Ⅵ
The Theorem of Frobenius
Statement of the Theorem
Differential Equations Depending on a Parameter
Proof of the Theorem
The Global Formulation
Lie Groups and Subgroups

PART Ⅱ
Metrics， Covariant Derivatives， and Riemannian Geometry

CHAPTER Ⅶ
Metrics
Definition and Functoriality
The Hilbert Group
Reduction to the Hilbert Group
Hilbertian Tubular Neighborhoods
The Morse-Palais Lemma
The Riemannian Distance
The Canonical Spray

CHAPTER Ⅷ
Covariant Derivatives and Geodesics.
Basic Properties
Sprays and Covariant Derivatives
Derivative Along a Curve and Parallelism
The Metric Derivative
More Local Results on the Exponential Map
Riemannian Geodesic Length and Completeness

CHAPTER Ⅸ
Curvature
The Riemann Tensor
Jacobi Lifts
Application of Jacobi Lifts to Texpx
Convexity Theorems
Taylor Expansions

CHAPTER Ⅹ
Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle
Convexity of Jacobi Lifts
Global Tubular Neighborhood of a Totally Geodesic Submanifold.
More Convexity and Comparison Results
Splitting of the Double Tangent Bundle
Tensorial Derivative of a Curve in TX and of the Exponential Map
The Flow and the Tensorial Derivative

CHAPTER XI
Curvature and the Variation Formula
The Index Form， Variations， and the Second Variation Formula
Growth of a Jacobi Lift
The Semi Parallelogram Law and Negative Curvature
Totally Geodesic Submanifolds
Rauch Comparison Theorem
CHAPTER XII
An Example of Seminegative Curvature
Pos，，(R) as a Riemannian Manifold
The Metric Increasing Property of the Exponential Map
Totally Geodesic and Symmetric Submanifolds

CHAPTER XIII
Automorphisms and Symmetries.，
The Tensorial Second Derivative
Alternative Definitions of Killing Fields
Metric Killing Fields
Lie Algebra Properties of Killing Fields
Symmetric Spaces
Parallelism and the Riemann Tensor
CHAPTER XlV
Immersions and Submersions .
The Covariant Derivative on a Submanifoid
The Hessian and Laplacian on a Submanifold
The Covariant Derivative on a Riemhnnian Submersion .
The Hessian and Laplacian on a Riemannian Submersion
The Riemann Tensor on Submanifolds
The Riemann Tensor on a Riemannian Submersion

PART III
Volume Forms and Integration
CHAPTER XV
Volume Forms
Volume Forms and the Divergence
Covariant Derivatives
The Jacobian Determinant of the Exponential Map
The Hodge Star on Forms
Hodge Decomposition of Differential Forms
Volume Forms in a Submersion
Volume Forms on Lie Groups and Homogeneous Spaces
Homogeneously Fibered Submersions

CHAPTER XVI
Integration of Differential Forms
Sets of Measure 0
Change of Variables Formula
Orientation
The Measure Associated with a Differential Form
Homogeneous Spaces

CHAPTER XVII
Stokes Theorem
Stokes Theorem for a Rectangular Simplex
Stokes Theorem on a Manifold
Stokes Theorem with Singularities

CHAPTER XVIII
Applications of Stokes Theorem
The Maximal de Rham Cohomology
Mosers Theorem
The Divergence Theorem
The Adjoint of d for Higher Degree Forms
Cauchys Theorem
The Residue Theorem

APPENDIX
The Spectral Theorem，
Hilbert Space
Functionals and Operators
Hermitian Operators
Bibliography
Index

精彩书摘

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.

前言/序言

　　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.

微分几何基础（英文版） [Fundamentals of Differential Geometry] 电子书 下载 mobi epub pdf txt

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这系列真心不错，会努力去读的

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很好的书，京东性价比高，喜欢！

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据说很不错的。同学帮忙推荐的

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书的微分几何的学习很有帮助。本书介绍曲线和曲面几何的入门知识，主要内容包括欧氏空间上的积分、帧场、欧氏几何、曲面积分、形状算子、曲面几何、黎曼几何、曲面上的球面结构等。修订版扩展了一些主题，更加强调拓扑性质、测地线的性质、向量场的奇异性等。更为重要的是，修订版增加了计算机建模的内容，提供了Mathematica和Maple程序。此外，还增加了相应的计算机习题，补充了奇数号码习题的答案，更便于教学。《微分几何基础(英文版·第2版修订版)》适合作为高等院校本科生相关课程的教材，也适合作为相关专业研究生和科研人员的参考书。也不失为学习英语的好方法。书的编排很好，正版

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一本较老的讲诉微分几何的教材，牛人之作必属精品。

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　　 再说广义相对论，这本书是不可多得的一本好书。当然会有些人说过于数学。我这里只引用W.Thirring的一句话——许多看似神秘的物理原理，结论或者现象往往是一些高等的数学结构的中的平庸的结果。这本书中关于广义相对论的内容及时与国际上的名著比较也是难得的并极具价值的。现在国际广义相对论的书籍情况跟国内科技书籍的行市一样，就是初等的书一大堆，就像国内书店里高数，线数，概率的书塞满书架，内容大同小异（就那么点东西写这么多书，唉），真正高等点儿的书就凤毛麟角了。广义相对论的书一般来说，先写点儿张量分析，线性联络，好点儿的用坐标不依赖的观点写。然后介绍爱因斯坦方程，再写点儿物理，也就是这个方程的意义（很少有人能能说清楚）以及怎么退化到牛顿理论。好，然后再写几个对称解，球对称解讨论讨论黑洞，均匀各向同性解讨论讨论宇宙学，然后就完了。呵呵，国际上这样的书也多得让人烦了，这样的书差不多谁都可以写。真正包含一些时空的更深刻的结构和认识（比如，初值问题，因果结构与奇点定理，N-P形式，引力的辛结构，引力的规范结构，准局域能与正能定理等等，实际上场方程的发现已经快一个世纪了，但是广义相对论揭示的全新的物理是人们经过了很长时间的努力才逐渐了解的）的书就只剩下MTW，Wald的书，Hawking & Ellis的书以及梁老师的书了（当然还有一些其他专著）。梁老师的书与大师们的书相比其严谨性也并不逊色（严谨的读起来如同嚼腊，顿悟之后又倍感欣悦）。然而梁老师的更新，包含更多新的结果。这使梁老师的这本书成为一本极具价值的参考书。

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希望能够了解物理学的基本理论，所以买了……

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这本书算是经典了，过段时间再看

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书的微分几何的学习很有帮助。本书介绍曲线和曲面几何的入门知识，主要内容包括欧氏空间上的积分、帧场、欧氏几何、曲面积分、形状算子、曲面几何、黎曼几何、曲面上的球面结构等。修订版扩展了一些主题，更加强调拓扑性质、测地线的性质、向量场的奇异性等。更为重要的是，修订版增加了计算机建模的内容，提供了Mathematica和Maple程序。此外，还增加了相应的计算机习题，补充了奇数号码习题的答案，更便于教学。《微分几何基础(英文版·第2版修订版)》适合作为高等院校本科生相关课程的教材，也适合作为相关专业研究生和科研人员的参考书。也不失为学习英语的好方法。书的编排很好，正版

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