微分几何初步

1 Basic Objects and Notation 1.1 Sets 1.2 Functions 2 Linear Algebra Essentials 2.1 Vector Spaces 2.2 Subspaces 2.3 Constructing Subspaces I: Spanning Sets 2.4 Linear Independence, Basis, and Dimension 2.5 Linear Transformations 2.6 Constructing Linear Transformations 2.7 Constructing Subspaces II: Subspaces and Linear Transformations 2.8 The Dual of a Vector Space, Forms, and Pullbacks 2.9 Geometric Structures I: Inner Products 2.10 Geometric Structures II" Linear Symplectic Forms 2.11 For Further Reading 2.12 Exercises 3 Advanced Calculus 3.1 The Derivative and Linear Approximation 3.2 The Tangent Space I" A Geometric Definition 3.3 Geometric Sets and Subspaces of Tp(Rn) 3.4 The Tangent Space II: An Analytic Definition 3.5 The Derivative as a Linear Map Between Tangent Spaces 3.6 Diffeomorphisms 3.7 Vector Fields: From Local to Global 3.8 Integral Curves 3.9 Diffeomorphisms Generated by Vector Fields 3.10 ForFurther Reading 3.11 Exercises 4 Differential Forms and Tensors 4.1 The Algebra of Alternating Linear Forms 4.2 Operations on Linear Forms 4.3 Differential Forms 4.4 Operations on Differential Forms 4.5 Integrating Differential Forms 4.6 Tensors 4.7 The Lie Derivative 4.8 For Further Reading 4.9 Exercises 5 Rlemannlan Geometry 5.1 Basic Concepts 5.2 Constructing Metrics; Metrics on Geometric Sets 5.3 The Riemannian Connection 5.4 Parallelism and Geodesics 5.5 Curvature 5.6 Isometrics 5.7 For Further Reading 5.8 Exercises 6 Contact Geometry 6.1 Motivation I: Huygens' Principle and Contact Elements ... 6.2 Motivation II: Differential Equations and Contact Elements 6.3 Basic Concepts 6.4 Contact Diffeomorphisms 6.5 Contact Vector Fields 6.6 Darboux's Theorem 6.7 Higher Dimensions 6.8 For Further Reading 6.9 Exercises 7 Symplectic Geometry 7.1 Motivation: Hamiltonian Mechanics and Phase Space 7.2 Basic Concepts 7.3 Symplectic Diffeomorphisms 7.4 Symplectic and Hamiitonian Vector Fields 7.5 Geometric Sets in Symplectic Spaces 7.6 Hypersurfaces of Contact Type 7.7 Symplectic Invariants 7.8 For Further Reading 7.9 Exercises References Index