微分幾何

內(nèi)容概要

　　《微分幾何(英文影印版)》系統(tǒng)并深入地闡述了微分幾何的概念及性質(zhì)，并且全面研究了cartan聯(lián)絡(luò)，很適合研究生一年級學(xué)習(xí)理解。內(nèi)容體系脈絡(luò)清晰，完整。前四章全面講述了微分拓撲、葉狀結(jié)構(gòu)，李群和齊次空間；第五章講述了產(chǎn)生齊次空間的cartan幾何，和黎曼幾何產(chǎn)生歐幾里得幾何很相似。cartan幾何的一個很漂亮的方面將曲率看作是破缺對稱性的精確局部度量；最后三章研究了黎曼幾何、共形幾何和射影幾何；五個附錄中介紹了cartan和ehresmann聯(lián)絡(luò)和從對稱的角度考慮散度和旋度算子的不同特性。目次：微分拓撲；葉狀結(jié)構(gòu)；微積分基本定理；klein幾何；cartan幾何；黎曼幾何；mobius幾何；射影幾何。附錄：ehresmann
聯(lián)絡(luò)；不滑動和扭曲的滾動；一維有效klein對的分類；來自對稱的微分算子；主叢的分類。
　　讀者對象：數(shù)學(xué)專業(yè)的高年級本科生、研究生和科研人員。

作者簡介

作者：(加拿大)夏普(Sharpe.R.W.)

書籍目錄

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
bibliography
index

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