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 Bounds
2.6 Volume Comparison Theorems III: Tubular Neighborhoods
2.6.1 Fermi Coordinates for Minkowski Space
2.6.2 Jacobi Fields with Initial Submanifolds
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 ~ndamental 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 Volume 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 Rn+1,fb
4.4.2 The Rigidity Results
Bibliography
Index
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