Author: Bingye-Wu
Language: English
ISBN/ISSN: 9787030434364
Published on: 2015-03
Paperback
Language: English
ISBN/ISSN: 9787030434364
Published on: 2015-03
Paperback
《芬斯勒几何中的比较定理与子流形(英文版)》对芬斯勒几何的比较定理作了全面系统的研究，改进已有的结果并得到许多全新的结论，极大地丰富了现代微分几何理论的内容，尤其在整体芬斯勒几何的若干基础性工作，如比较定理、子流形几何学等，具有重要的学术价值，已成为相关研究重要成果之一，也为进一步研究整体芬斯勒几何以及芬斯勒子流形几何研究打下坚实的基础。本成果具有很高的原创性与先进性，后续研究前景广阔。
Chapter 1 Basics on Finsler Geometry
1.1 Minkowski Space
1.1.1 Definition and Examples
1.1,2 Legendre Transformation
1.1.3 Cartan Tensor
1.2 Finsler Manifold
1.2.1 The Definition of Finsler Manifold
1.2.2 Connection and Curvature
1.3 Geodesic
1.3.1 Geodesic and Exponential Map
1.3.2 The First Variation of Arc Length
1.3.3 The Second Variation of Arc Length
1.4 Jacobi Fields and Conjugate Points
1.4.1 Jacobi Fields
1.4.2 Conjugate Points
1.5 Basic Index Lemma
Chapter 2 Comparison Theorems in Finsler Geometry
2.1 Rauch Comparison Theorem
2.2 Volume Form
2.2.1 Definition and Examples
2.2.2 Distortion and S-Curvature
2.3 Hessian Comparison Theorem and Laplacian Comparison Theorem
2.3.1 Polar Coordinates
2.3.2 Hessian Comparison Theorem
2.3.3 Laplacian Comparison Theorem
2.4 Volume Comparison Theorems (Ⅰ): Pointwise Curvature Bounds
2.5 Volume Comparison Theorems (Ⅱ): Integral Curvature Bouads
2.6 Volume Comparison Theorems (Ⅲ): Tubular Neighborhoods
2.6.1 Fermi Coordinates for Minkowski Space
2.6.2 Jacobi Fields with Initial Sublnanifolds
2.6.3 Fermi Coordinates and Focal Cut Locus
2.6.4 Volume Comparison Theorem for Tubular Neighborhoods of Submanifolds
2.7 Comparison Theorems with Weighted Curvature Bounds
2.8 Toponogov Type Comparison Theorem
Chapter 3 Applications of Comparison Theorems
3.1 Generalized Myers Theorem and Linearly Growth Theorem of Volume
3.1.1 Generalized Myers Theorem
3.1.2 Linearly Growth Theorem of Volume
3.2 McKean Type Inequalities for the First Eigenvalue
3.2.1 The Divergence Lemma
3.2.2 The Mckean Type Inequalities
3.3 Gromov Pre-Compactness Theorem
3.4 The First Betti Number
3.5 Curvature and Fundamental Group
3.5.1 Universal Covering Space and Fundamental Group
3.5.2 Growth of Fundamental Group
3.5.3 Finiteness of Fundamental Group
3.5.4 Results Related to Milnor's Conjecture
3.6 A Lower Bound of Injectivity Radius
3.7 Finite Topological Type
Chapter 4 Geometry of Finsler Submanifolds
4.1 Mean Curvature
4.1.1 Projection in a Minkowski Space
4.1.2 The Mean Curvature for Finsler Submanifolds
4.2 Some Results on Submanifolds in Minkowski Space
4.3 Volmne Growth of Submanifolds in Minkowski Space
4.4 Rigidity of Minimal Surfaces in Randers-Minkowski 3-Space
4.4.1 The Mean Curvature of a Graph in
4.4.2 The Rigidity Results
Bibliography
Index
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