微分几何基础（英文版） [Fundamentals of Differential Geometry] 下载 mobi epub pdf 电子书 2024

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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] 下载 mobi epub pdf txt 电子书 格式

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最近这段时间，“向党中央看齐”成为时政新闻中的高频词。一个多月来，习近平先后4次讲话

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很不错的一本书

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不过也不是盖的

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不错的书，比较适合自己，内容很详细

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1827年，德国数学家高斯发表了《关于曲面的一般研究》的著作，这在微分几何的历史上有重大的意义，它的理论奠定了曲面论的基础。高斯抓住了微分几何中最重要的概念和根本性的内容，建立了曲面的内蕴几何学。其主要思想是强调了曲面上只依赖于第一基本形式的一些性质，例如曲面上曲线的长度、两条曲线的夹角、曲面上的某一区域的面积、测地线、测地曲率和总曲率等等。

评分☆☆☆☆☆

在曲面上有两条重要概念，就是曲面上的距离和角。比如，在曲面上由一点到另一点的路径是无数的，但这两点间最短的路径只有一条，叫做从一点到另一点的测地线。在微分几何里，要讨论怎样判定曲面上一条曲线是这个曲面的一条测地线，还要讨论测地线的性质等。另外，讨论曲面在每一点的曲率也是微分几何的重要内容。

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还可以的呀还可以的呀

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1870年德国数学家克莱因（Felix Klein）在德国埃尔朗根大学作就职演讲时，阐述了他的《埃尔朗根纲领》，用变换群对已有的几何学进行了分类。在《埃尔朗根纲领》发表后的半个世纪内，它成了几何学的指导原理，推动了几何学的发展，导致了射影微分几何、仿射微分几何、共形微分几何的建立。特别是射影微分几何起始于1878年阿尔方的学位论文，后来1906年起经以威尔辛斯基为代表的美国学派所发展，1916年起又经以富比尼为首的意大利学派所发展。在仿射微分几何方面，布拉施克（W. Blaschke）也做出了决定性的工作。

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速度很快，非常满意~！！！

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