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
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