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 Poincaré Lemma

5．Contractions and Lie Derivative

6．Vector Fields and 1-Forms Under Self Duality

7．TheCanonical 2-Form

8．Darboux's Theorem

CHAPTER Ⅵ The Theorem of 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 Hilbert Group

4．Hilbertian Tubular Neighborhoods

5．The Morse-Palais Lemma

6．The Riemannian Distance

7．The Canonical Spray

CHAPTER Ⅷ Covariant 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 Texpx

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 Ⅺ 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 Ⅻ 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 0

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