微分几何（英文版）
作者：（加）夏普 著
出版时间：2011年版
内容简介
This book is a study of an aspect of Elie Cartan's contribution to thequestion "What is geometry？"In the last century two great generalizations of Euclidean geometry ap-peared. The first was the discovery of the non-Euclidean geometries. Thesewere organized into a coherent whole by Felix Klein， who recognized themas various examples of coset spaces G/H of Lie groups. In this book we refer to these latter as Klein geometries. The second generalization was Georg Riemann's discovery of what we now call Riemannian geometry. These two theories seemed largely incompatible with one other.1In the early 1920s Elie Cartan， one of the pioneers of the theory of Lie groups， found that it was possible to obtain a common generalization of these theories， which he called espaces generalizes and we call Cartan geometries （see diagram）.
目录
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-cartanform
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
bibliography
index
