现代Finsler几何初步 (英文版)

作者: 沈一兵著 , 沈忠民著

出版社: 高等教育出版社

出版时间: 2016-01

版次: 1

ISBN: 9787040444247

定价: 98.00

装帧: 精装

开本: 16开

纸张: 胶版纸

页数: 393页

字数: 530千字

正文语种: 英语

原版书名: Introduction to Modern Finsler Geometry

分类: 自然科学

5人买过

　　This comprehensive book is an introduction to the basics of Finsler geometry with recent developments in its area. It includes local geometry as well as global geometry of Finsler manifolds. In Part Ⅰ， the authors discuss differential manifolds， Finsler metrics， the Chern connection， Riemannian and non- Riemannian quantities. Part Ⅱ is written for readers who would like to further their studies in Finsler geometry. It covers projective transformations，comparison theorems， fundamental group， minimal immersions，harmonic maps， Einstein metrics， conformal transformations，amongst other related topics.The authors made great efforts to ensure that the contents are accessible to senior undergraduate students， graduate students， mathematicians and scientists. 沈一兵，浙江大学数学系教授，研究方向：现代微分几何与几何分析。著有《解析几何》、《微分几何》、《整体微分几何》、《黎曼几何》等。
沈忠民，知名华人学者，美国Indiana University-Purdue University数学系主任，曾任美国国家自然科学基金委几何与分析学科主管。著有Riemann-Finsler Geometry, Differential Geometry of Spray and Finsler Spaces等。 Preface
Foundations
1． Differentiable Manifolds
1．1 Differentiable manifolds
1．1．1 Differentiable manifolds
1．1．2 Examples of differentiable manifolds
1．2 Vector fields and tensor fields
1．2．1 Vector bundles
1．2．2 Tensor fields
1．3 Exterior forms and exterior differentials
1．3．1 Exterior differential operators
1．3．2 de Rham theorem
1．4 Vector bundles and connections
1．4．1 Connection of the vector bundle
1．4．2 Curvature of a connection
Exercises
2． Finsler Metrics
2．1 Finsler metrics
2．1．1 Finsler metrics
2．1．2 Examples of Finsler metrics
2．2 Cartan torsion
2．2．1 Cartan torsion
2．2．2 Deicke theorem
2．3 Hilbert form and sprays
2．3．1 Hilbert form
2．3．2 Sprays
2．4 Geodesics
2．4．1 Geodesics
2．4．2 Geodesic coefficients
2．4．3 Geodesic completeness
Exercises
3． Connections and Curvatures
3．1 Connections
3．1．1 Chern connection
3．1．2 Berwald metrics and Landsberg metrics
3．2 Curvatures
3．2．1 Curvature form of the Chern connection
3．2．2 Flag curvature and Ricci curvature
3．3 Bianchi identities
3．3．1 Covariant differentiation
3．3．2 Bianchi identities
3．3．3 Other formulas
3．4 Legendre transformation
3．4．1 The dual norm in the dual space
3．4．2 Legendre transformation
3．4．3 Example
Exercises
4． S-Curvature
4．1 Volume measures
4．1．1 Busemann-Hausdorff volume element
4．1．2 The volume element induced from SM
4．2 S-curvature
4．2．1 Distortion
4．2．2 S-curvature and E-curvature
4．3 Isotropic S-curvature
4．3．1 Isotropic S-curvature and isotropic E-curvature
4．3．2 Randers metrics of isotropic S-curvature
4．3．3 Geodesic flow
Exercises
5． Riemann Curvature
5．1 The second variation of arc length
5．1．1 The second variation of length
5．1．2 Elements of curvature and topology
5．2 Scalar flag curvature
5．2．1 Schur theorem
5．2．2 Constant flag curvature
5．3 Global rigidity results
5．3．1 Flag curvature with special conditions
5．3．2 Manifolds with non-positive flag curvature
5．4 Navigation
5．4．1 Navigation problem
5．4．2 Randers metrics and navigation
5．4．3 Ricci curvature and Einstein metrics
Exercises
Further Studies
6． Projective Changes
6．1 The projective equivalence
6．1．1 Projective equivalence
6．1．2 Projective invariants
6．2 Projectively flat metrics
6．2．1 Projectively flat metrics
6．2．2 Projectively fiat metrics with constant flag curvature
6．3 Projectively fiat metrics with almost isotropic S-curvature
6．3．1 Randers metrics with almost isotropic S-curvature
6．3．2 Projectively flat metrics with almost isotropic
S-curvature
6．4 Some special projectively equivalent Finsler metrics
6．4．1 Projectively equivalent Randers metrics
6．4．2 The projective equivalence of （α， β）-metrics
6．4．3 The projective equivalence of quadratic （α， β）
metrics
Exercises
7． Comparison Theorems
7．1 Volume comparison theorems for Finsler manifolds
7．1．1 The Jacobian of the exponential map
7．1．2 Distance function and comparison theorems
7．1．3 Volume comparison theorems
7．2 Berger-Kazdan comparison theorems
7．2．1 The Kazdan inequality
7．2．2 The rigidity of reversible Finsler manifolds
7．2．3 The Berger-Kazdan comparison theorem
Exercises
8． Fundamental Groups of Finsler Manifolds
8．1 Fundamental groups of Finsler manifolds
8．1．1 Fundamental groups and covering spaces
8．1．2 Algebraic norms and geometric norms
8．1．3 Growth of fundamental groups
8．2 Entropy and finiteness of fundamental group
8．2．1 Entropy of fundamental group
8．2．2 The first Betti number
8．2．3 Finiteness of fundamental group
8．3 Gromov pre-compactness theorems
8．3．1 General metric spaces
8．3．2 δ-Gromov-Hausdorff convergence
8．3．3 Pre-compactness of Finsler manifolds
8．3．4 On the Gauss-Bonnet-Chern theorem
Exercises
9． Minimal Immersions and Harmonic Maps
9．1 Isometric immersions
9．1．1 Finsler submanifolds
9．1．2 The variation of the volume
9．1．3 Non-existence of compact minimal submanifolds
9．2 Rigidity of minimal submanifolds
9．2．1 Minimal surfaces in Minkowski spaces
9．2．2 Minimal surfaces in （α， β）-spaces
9．2．3 Minimal surfaces in special Minkowskian （α， β）
spaces
9．3 Harmonic maps
9．3．1 A divergence formula
9．3．2 Harmonic maps
9．3．3 Composition maps
9．4 Second variation of harmonic maps
9．4．1 The second variation
9．4．2 Stress-energy tensor
9．5 Harmonic maps between complex Finsler manifolds
9．5．1 Complex Finsler manifolds
9．5．2 Harmonic maps between complex Finsler manifolds
9．5．3 Holomorphic maps
Exercises
10． Einstein Metrics
10．1 Projective rigidity and m-th root metrics
10．1．1 Projective rigidity of Einstein metrics
10．1．2 m-th root Einstein metrics
10．2 The Ricci rigidity and Douglas-Einstein metrics
10．2．1 The Ricci rigidity
10．2．2 Douglas （α， β）-metrics
10．3 Einstein （α， β）-metrics
10．3．1 Polynomial （α， β）-metrics
10．3．2 Kropina metrics
Exercises
11． Miscellaneous Topics
11．1 Conformal changes
11．1．1 Conformal changes
11．1．2 Conformally flat metrics
11．1．3 Conformally flat （α， β）-metrics
11．2 Conformal vector fields
11．2．1 Conformal vector fields
11．2．2 Conformal vector fields on a Randers manifold
11．3 A class of critical Finsler metrics
11．3．1 The Einstein-Hilbert functional
11．3．2 Some special g-critical metrics
11．4 The first eigenvalue of Finsler Laplacian and the generalized maximal principle
11．4．1 Finsler Laplacian and weighted Ricci curvature
11．4．2 Lichnerowicz-Obata estimates
11．4．3 Li-Yau-Zhong-Yang type estimates
11．4．4 Mckean type estimates
Exercises
Appendix A Maple Program
A．1 Spray coefficients of two-dimensional Finsler metrics
A．2 Gauss curvature
A．3 Spray coefficients of （α， β）-metrics
Bibliography
Index
内容简介:
　　This comprehensive book is an introduction to the basics of Finsler geometry with recent developments in its area. It includes local geometry as well as global geometry of Finsler manifolds. In Part Ⅰ， the authors discuss differential manifolds， Finsler metrics， the Chern connection， Riemannian and non- Riemannian quantities. Part Ⅱ is written for readers who would like to further their studies in Finsler geometry. It covers projective transformations，comparison theorems， fundamental group， minimal immersions，harmonic maps， Einstein metrics， conformal transformations，amongst other related topics.The authors made great efforts to ensure that the contents are accessible to senior undergraduate students， graduate students， mathematicians and scientists.
作者简介:
沈一兵，浙江大学数学系教授，研究方向：现代微分几何与几何分析。著有《解析几何》、《微分几何》、《整体微分几何》、《黎曼几何》等。
沈忠民，知名华人学者，美国Indiana University-Purdue University数学系主任，曾任美国国家自然科学基金委几何与分析学科主管。著有Riemann-Finsler Geometry, Differential Geometry of Spray and Finsler Spaces等。
目录:
Preface
Foundations
1． Differentiable Manifolds
1．1 Differentiable manifolds
1．1．1 Differentiable manifolds
1．1．2 Examples of differentiable manifolds
1．2 Vector fields and tensor fields
1．2．1 Vector bundles
1．2．2 Tensor fields
1．3 Exterior forms and exterior differentials
1．3．1 Exterior differential operators
1．3．2 de Rham theorem
1．4 Vector bundles and connections
1．4．1 Connection of the vector bundle
1．4．2 Curvature of a connection
Exercises
2． Finsler Metrics
2．1 Finsler metrics
2．1．1 Finsler metrics
2．1．2 Examples of Finsler metrics
2．2 Cartan torsion
2．2．1 Cartan torsion
2．2．2 Deicke theorem
2．3 Hilbert form and sprays
2．3．1 Hilbert form
2．3．2 Sprays
2．4 Geodesics
2．4．1 Geodesics
2．4．2 Geodesic coefficients
2．4．3 Geodesic completeness
Exercises
3． Connections and Curvatures
3．1 Connections
3．1．1 Chern connection
3．1．2 Berwald metrics and Landsberg metrics
3．2 Curvatures
3．2．1 Curvature form of the Chern connection
3．2．2 Flag curvature and Ricci curvature
3．3 Bianchi identities
3．3．1 Covariant differentiation
3．3．2 Bianchi identities
3．3．3 Other formulas
3．4 Legendre transformation
3．4．1 The dual norm in the dual space
3．4．2 Legendre transformation
3．4．3 Example
Exercises
4． S-Curvature
4．1 Volume measures
4．1．1 Busemann-Hausdorff volume element
4．1．2 The volume element induced from SM
4．2 S-curvature
4．2．1 Distortion
4．2．2 S-curvature and E-curvature
4．3 Isotropic S-curvature
4．3．1 Isotropic S-curvature and isotropic E-curvature
4．3．2 Randers metrics of isotropic S-curvature
4．3．3 Geodesic flow
Exercises
5． Riemann Curvature
5．1 The second variation of arc length
5．1．1 The second variation of length
5．1．2 Elements of curvature and topology
5．2 Scalar flag curvature
5．2．1 Schur theorem
5．2．2 Constant flag curvature
5．3 Global rigidity results
5．3．1 Flag curvature with special conditions
5．3．2 Manifolds with non-positive flag curvature
5．4 Navigation
5．4．1 Navigation problem
5．4．2 Randers metrics and navigation
5．4．3 Ricci curvature and Einstein metrics
Exercises
Further Studies
6． Projective Changes
6．1 The projective equivalence
6．1．1 Projective equivalence
6．1．2 Projective invariants
6．2 Projectively flat metrics
6．2．1 Projectively flat metrics
6．2．2 Projectively fiat metrics with constant flag curvature
6．3 Projectively fiat metrics with almost isotropic S-curvature
6．3．1 Randers metrics with almost isotropic S-curvature
6．3．2 Projectively flat metrics with almost isotropic
S-curvature
6．4 Some special projectively equivalent Finsler metrics
6．4．1 Projectively equivalent Randers metrics
6．4．2 The projective equivalence of （α， β）-metrics
6．4．3 The projective equivalence of quadratic （α， β）
metrics
Exercises
7． Comparison Theorems
7．1 Volume comparison theorems for Finsler manifolds
7．1．1 The Jacobian of the exponential map
7．1．2 Distance function and comparison theorems
7．1．3 Volume comparison theorems
7．2 Berger-Kazdan comparison theorems
7．2．1 The Kazdan inequality
7．2．2 The rigidity of reversible Finsler manifolds
7．2．3 The Berger-Kazdan comparison theorem
Exercises
8． Fundamental Groups of Finsler Manifolds
8．1 Fundamental groups of Finsler manifolds
8．1．1 Fundamental groups and covering spaces
8．1．2 Algebraic norms and geometric norms
8．1．3 Growth of fundamental groups
8．2 Entropy and finiteness of fundamental group
8．2．1 Entropy of fundamental group
8．2．2 The first Betti number
8．2．3 Finiteness of fundamental group
8．3 Gromov pre-compactness theorems
8．3．1 General metric spaces
8．3．2 δ-Gromov-Hausdorff convergence
8．3．3 Pre-compactness of Finsler manifolds
8．3．4 On the Gauss-Bonnet-Chern theorem
Exercises
9． Minimal Immersions and Harmonic Maps
9．1 Isometric immersions
9．1．1 Finsler submanifolds
9．1．2 The variation of the volume
9．1．3 Non-existence of compact minimal submanifolds
9．2 Rigidity of minimal submanifolds
9．2．1 Minimal surfaces in Minkowski spaces
9．2．2 Minimal surfaces in （α， β）-spaces
9．2．3 Minimal surfaces in special Minkowskian （α， β）
spaces
9．3 Harmonic maps
9．3．1 A divergence formula
9．3．2 Harmonic maps
9．3．3 Composition maps
9．4 Second variation of harmonic maps
9．4．1 The second variation
9．4．2 Stress-energy tensor
9．5 Harmonic maps between complex Finsler manifolds
9．5．1 Complex Finsler manifolds
9．5．2 Harmonic maps between complex Finsler manifolds
9．5．3 Holomorphic maps
Exercises
10． Einstein Metrics
10．1 Projective rigidity and m-th root metrics
10．1．1 Projective rigidity of Einstein metrics
10．1．2 m-th root Einstein metrics
10．2 The Ricci rigidity and Douglas-Einstein metrics
10．2．1 The Ricci rigidity
10．2．2 Douglas （α， β）-metrics
10．3 Einstein （α， β）-metrics
10．3．1 Polynomial （α， β）-metrics
10．3．2 Kropina metrics
Exercises
11． Miscellaneous Topics
11．1 Conformal changes
11．1．1 Conformal changes
11．1．2 Conformally flat metrics
11．1．3 Conformally flat （α， β）-metrics
11．2 Conformal vector fields
11．2．1 Conformal vector fields
11．2．2 Conformal vector fields on a Randers manifold
11．3 A class of critical Finsler metrics
11．3．1 The Einstein-Hilbert functional
11．3．2 Some special g-critical metrics
11．4 The first eigenvalue of Finsler Laplacian and the generalized maximal principle
11．4．1 Finsler Laplacian and weighted Ricci curvature
11．4．2 Lichnerowicz-Obata estimates
11．4．3 Li-Yau-Zhong-Yang type estimates
11．4．4 Mckean type estimates
Exercises
Appendix A Maple Program
A．1 Spray coefficients of two-dimensional Finsler metrics
A．2 Gauss curvature
A．3 Spray coefficients of （α， β）-metrics
Bibliography
Index

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现代Finsler几何初步 (英文版)

内容简介:

作者简介:

目录: