夏普编著的《微分几何（英文影印版）》系统并深入地阐述了微分几何的概念及性质，并且全面研究了cartan联络，很适合研究生一年级学习理解。内容体系脉络清晰，完整。前四章全面讲述了微分拓扑、叶状结构，李群和齐次空间；第五章讲述了产生齐次空间的cartan几何，和黎曼几何产生欧几里得几何很相似。cartan几何的一个很漂亮的方面将曲率看作是破缺对称性的精确局部度量；最后三章研究了黎曼几何、共形几何和射影几何；五个附录中介绍了cartan和ehresmann联络和从对称的角度考虑散度和旋度算子的不同特性。目次：微分拓扑；叶状结构；微积分基本定理；klein几何；cartan几何；黎曼几何；mobius几何；射影几何。附录：ehresmann联络；不滑动和扭曲的滚动；一维有效klein对的分类；来自对称的微分算子；主丛的分类。

foreword

note on the second printing

preface

1 in the ashes of the ether: differential topology

 1. smooth manifolds

 2. submanifolds

 3. fiber bundles

 4. tangent vectors, bundles, and fields

 5. differential forms

2 looking for the forest-in the leaves: foliations

 1. integral curves

 2. distributions

 3. integrability conditions

 4. the frobenius theorem

 5. the frobenius theorem in terms of differential forms

 6. foliations

 7. leaf holonomy

 8. simple foliations

3 the fundamental theorem of calculus

 1. the maurer-cartan form

 2. lie algebras

 3. structural equation

 4. adjoint action

 5. the darboux derivative

 6. the fundamental theorem: local version

 7. the fundamental theorem: global version

 8. monodromy and completeness

4 shapes fantastic: klein geometries

 1. examples of planar klein geometries

 2. principal bundles: characterization and reduction

 3. klein geometries

 4. a fundamental property

 5. the tangent bundle of a klein geometry

 6. the meteor tracking problem

 7. the gauge view of klein geometries

5 shapes high fantastical: cartan geometries

 1. the base definition of cartan geometries

 2. the principal bundle hidden in a cartan geometry

 3. the bundle definition of a cartan geometry

 4. development, geometric orientation, and holonomy

 5. flat cartan geometries and uniformization

 6. cartan space forms

 7. symmetric spaces

6 riemannian geometry

 1. the model euclidean space

 2. euclidean and riemannian geometry

 3. the equivalence problem for riemannian metrics

 4. riemannian space forms

 5. subgeometry of a riemannian geometry

 6. isoparametric submanifolds

7 msbius geometry

 1. the msbius and weyl models

 2. msbius and weyl geometries

 3. equivalence problems for a conformal metric

 4. submanifolds of msbius geometry

 5. immersed curves

 6. immersed surfaces

8 projective geometry

 1. the projective model

 2. projective cartan geometries

 3. the geometry of geodesics

 4. the projective connection in a riemannian geometry

 5. a brief tour of projective geometry

a ehresmann connections

 1. the geometric origin of ehresmann connections

 2. the reductive case

 3. ehresmann connections generalize cartan connections

 4. covariant derivative

b rolling without slipping or twisting

 1. rolling maps

 2. the existence and uniqueness of rolling maps

 3. relation to levi-civita and normal connections

 4. transitivity of rolling without slipping or twisting

c classification of one-dimensional effective klein pairs

 1. classification of one-dimensional effective klein pairs

d differential operators obtained from symmetry

 1. real representations of so2(r)

 2. operators on riemannian surfaces

e characterization of principal bundles

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