现代黎曼几何导论
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作者I.Chavel 编
出版社世界图书出版公司
ISBN9787506247023
出版时间2000-06
版次1
装帧平装
开本16开
纸张胶版纸
页数386页
定价59元
上书时间2024-08-03
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基本信息
书名:现代黎曼几何导论
定价：59元
作者:I.Chavel 编
出版社：世界图书出版公司
出版日期：2000-06-01
ISBN：9787506247023
字数：
页码：386
版次：1
装帧：平装
开本：其他
商品重量：
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内容提要
My goals in thiook on Riemannian geometry are essentially the same as those which guided me in my Eigenvalues in Riemannian Geometry [69], to introduce the subject, to coherently present a number of itasic techniques and results with a mind to future work, and to present some of the results that are attractive in their own right. Thiook differs from Eigenvalues in that it starts at a more basic level,and therefore, it must present a broader view of the ideas from which all the various directions emerge. At the same time, other treatments of Riemannian geometry are available at varying levels and interests,so I need not introduce everything. I have, therefore, attempted a viable introduction to Riemannian geometry for a very broad group of students, with emphases and developments in areas not covered by other books.
目录
Preface1 Riemannian manifolds 1.1 Connections 1.2 Parallel translation of vector fields 1.3 Geodesics and the exponential map 1.4 The torsion and curvature tensors 1.5 Riemannian metrics 1.6 The metric space structure 1.7 Geodesics and completeness 1.8 Calculations with moving frames 1.9 Notes and exercises2 Riemannian curvature 2.1 The Riemann sectional curvature 2.2 Riemannian submanifolds 2.3 Spaces of constant sectional curvature 2.4 First and second variation of arc length 2.5 Jacobi''s equation and criteria 2.6 Elementary comparison theorems 2.7 Jacobi fields and the exponential map 2.8 Riemann normal coordinates 2.9 Notes and exercises3 Riemannian volume 3.1 Geodesic spherical coordinates 3.2 The conjugate and cut loci 3.3 Riemannian measure 3.4 Volume comparison theorems 3.5 The area of spheres 3.6 Fermi coordinates 3.7 Integration of differential forms 3.8 Notes and exercises 3.9 Appendix: Eigenvalue comparison theorems4 Riemannian coverings 4.1 Riemannian coverings 4.2 The fundamental group 4.3 Volume growth Of Riemannian coverings 4.4 Discretization of Riemannian manifolds 4.5 The free homotopy classes 4.6 Gauss-Bonnet theory of surfaces 4.7 Notes and exercises5 The kinematic density 5.1 The differential geometry of analytical dynamics 5.2 Santalo''s formula 5.3 The Berger-Kazdan inequalities 5.4 On manifolds with no conjugate points 5.5 Notes and exercises6 Isoperimetric inequalities 6.1 Isoperimetric constants 6.2 The isoperimetric inequality in Euclidean space 6.3 The isoperimetric inequality on spheres 6.4 Symmetrization and isoperimetric inequalities 6.5 Buser''s isoperimetric inequality 6.6 Croke''s isoperimetric inequality 6.7 Discretizations and isoperimetry 6.8 Notes and exercises7 Comparison and finiteness theorems 7.1 Preliminaries 7.2 H.E. Rauch''s comparison theorem 7.3 Comparison theorems with initial submanifolds 7.4 Refinements of the Rauch theorem 7.5 Triangle comparison theorems 7.6 Convexity 7.7 Center of mass 7.8 Cheeger''s finiteness theorem 7.9 Notes and exercisesHints and sketches of solutionsBibliographyIndex
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