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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。

内页插图

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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　　 加点八卦，其实梁老师是这样说的，他说S.Hawking当年的那本《大尺度》是本天书，R.Wald（梁老师的导师）为了让更多的学者能够理解 Hawking的天书，写了本《General relativity》，后来这本书也成为一代经典，梁老师说他为了让更多的学生（甚至本科生）理解Wald的书，才写了他本人这套书，这套书本来是英文版的，后来适合国情才改成中文版的。还有一位Wald的学生Sean Carroll也写了一本Wald书的过渡版《spacetime and geometry》。如果你读梁老师书，这本也可以看看。呵呵。

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