这本《微分流形导论(第2版)》由美国Serge Lang所著，内容是：  The book gives an introduction to the basic concepts which are used indifferential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps andthe possibility of finding suitable differentiable maps in them (immersions,embeddings, isomorphisms, etc.). One may also use differentiable structureson topological manifolds to determine the topological structure of themanifold (for example, a la Smale [Sm 67]). In differential geometry, oneputs an additional structure on the differentiable manifold (a vector field, aspray, a 2-form, a Riemannian metric, ad lib.) and studies properties con-nected especially with these objects.

Foroword

Acknowledgments

CHAPTER Ⅰ Differential Calculus

 §1. Categories

 §2. Finite Dimensional 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 end 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 I-Forms Under Self Duality

 §7. The Canonical 2-Form

 §8. Darboux's Theorem

CHAPTER Ⅵ The Theorem of Frobenlus

 §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

CHAPTER Ⅶ Metrics

 §1. Definition and Functoriality

 §2. The Metric Group

 §3. Reduction to the Metric Group

 §4. Metric Tubular Neighborhoods

 §5, The Morse Lemma

 §6. The Riemannian Distance

 §7. The Canonical Spray

CHAPTER Ⅷ Integration of Differential Forms

 §1. Sets of Measure 0

 §2, Change of Variables Formula

 §3. Orientation

 §4. The Measure Associated with a Differential Form

CHAPTER Ⅸ Stokes' Theorem,

 §1. Stokes' Theorem for a Rectangular Simplex

 §2. Stokes' Theorem on a Manifold

 §3. Stokes' Theorem with Singularities

CHAPTER Ⅹ Applications of Stokes' Theorem

 §1. The Maximal de Rham Cohomology

 §2. Volume forms and the Divergence

 §3. The Divergence Theorem

 §4. Cauchy's Theorem

 §5. The Residue Theorem

Bibliography

Index