Foreword
Acknowledgments
PART Ⅰ General Differential Theory
CHAPTER Ⅱ 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
PART Ⅲ Volume Forms and Integration
Index
