內容大鋼
本書介紹了微分拓撲、微分幾何以及微分方程的基本概念。本書的基本思想源於作者早期的《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。這些新的知識為Banach和Hilbert空間上的無限維流形做準備，但一點都不覺得多餘，而優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徵。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，包括經典2-形式的辛流形基本觀點，黎曼和偽黎曼流形協變導數以及其在指數映射中的應用，Cartan-Hadamard定理和變分微積分第一基本定理。
目錄
Foreword
Acknowledgments
PART Ⅰ General Differential Theory
CHAPTER Ⅰ 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 Expansions
CHAPTER Ⅹ Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle
§1.Convexity of Jacobi Lifts
§2.Global Tubular Neighborhood of a Totally Geodesic Submanifold
§3.More Convexity and Comparison Results
§4.Splitting of the Double Tangent Bundle
§5.Tensorial Derivative of a Curve in TX and of the Exponential Map
§6.The Flow and the Tensorial Derivative
CHAPTER ? Curvature and the Variation Formula
§1.The Index Form, Variations, and the Second Variation Formula
§2.Growth of a Jacobi Lift.
§3.The Semi Parallelogram Law and Negative Curvature
§4.Totally Geodesic Submanifolds
§5.Rauch Comparison Theorem
CHAPTER ? An Example of Seminegative Curvature
§1.Posn(R) as a Riemannian Manifold
§2.The Metric Increasing Property of the Exponential Map
§3.Totally Geodesic and Symmetric Submanifolds
CHAPTER ⅩⅢ Automorphisms and Symmetries
§1.The Tensorial Second Derivative
§2.Alternative Definitions of Killing Fields
§3.Metric Killing Fields
§4.Lie Algebra Properties of Killing Fields
§5.Symmetric Spaces
§6.Parallelism and the Riemann Tensor
CHAPTER ⅩⅣ Immersions and Submersions
§1.The Covariant Derivative on a Submanifold.
§2.The Hessian and Laplacian on a Submanifold
§3.The Covariant Derivative on a Riemannian Submersion
§4.The Hessian and Laplacian on a Riemannian Submersion
§5.The Riemann Tensor on Submanifolds
§6.The Riemann Tensor on a Riemannian Submersion
PART Ⅲ Volume Forms and Integration
CHAPTER ⅩⅤ Volume Forms
§1.Volume Forms and the Divergence
§2.Covariant Derivatives
§3.The Jacobian Determinant of the Exponential Map
§4.The Hodge Star on Forms
§5.Hodge Decomposition of Differential Forms
§6.Volume Forms in a Submersion
§7.Volume Forms on Lie Groups and Homogeneous Spaces
§8.Homogeneously Fibered Submersions
CHAPTER ⅩⅥ Integration of Differential Forms
§1.Sets of Measure O
§2.Change of Variables Formula
§3.Orientation
§4.The Measure Associated with a Differential Form
§5.Homogeneous Spaces
CHAPTER ⅩⅦ Stokes' Theorem
§1.Stokes' Theorem for a Rectangular Simplex
§2.Stokes' Theorem on a Manifold
§3.Stokes' Theorem with Singularities
CHAPTER ⅩⅧ Applications of Stokes' Theorem
§1.The Maximal de Rham Cohomology
§2.Moser's Theorem
§3.The Divergence Theorem
§4.The Adjoint of d for Higher Degree Forms
§5.Cauchy's Theorem
§6.The Residue Theorem
APPENDIX The Spectral Theorem
§1.Hilbert Space
§2.Functionals and Operators
§3.Hermitian Operators
Bibliography
Index
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