Chapter 1Basics on Finsler Geometry1 1.1Minkowski Space1 1.1.1Definition and Examples1 1.1.2Legendre Transformation6 1.1.3Cartan Tensor10 1.2Finsler Manifold12 1.2.1The Definition of Finsler Manifold12 1.2.2Connection and Curvature 13 1.3Geodesic16 1.3.1Geodesic and Exponential Map16 1.3.2The First Variation ofArc Length18 1.3.3The Second Variation ofArc Length25 1.4Jacobi Fields and ConjugatePoints28 1.4.1Jacobi Fields28 1.4.2Conjugate Points32 1.5Basic Index Lemma33 Chapter 2Comparison Theorems in Finsler Geometry39 2.1Rauch Comparison Theorem39 2.2VolumeForm43 2.2.1Definition and Examples43 2.2.2Distortion and S-Curvature 47 2.3Hessian Comparison Theorem and Laplacian Comparison Theorem48 2.3.1Polar Coordinates 48 2.3.2Hessian Comparison Theorem51 2.3.3Laplacian Comparison Theorem53 2.4VolumeComparison Theorems(I): Pointwise Curvature Bounds56 2.5VolumeComparison Theorems(II): Integral Curvature Bounds62 2.6VolumeComparison Theorems(III): Tubular Neighborhoods72 2.6.1Fermi Coordinates for Minkowski Space73 2.6.2Jacobi Fields with Initial Submanifolds75 2.6.3Fermi Coordinates and Focal Cut Locus80 2.6.4Volume Comparison Theorem for Tubular Neighborhoods of Submanifolds82 2.7Comparison Theorems with Weighted Curvature Bounds85 2.8Toponogov Type Comparison Theorem94 Chapter 3Applications of Comparison Theorems98 3.1Generalized Myers Theorem and Linearly Growth Theorem of Volume98 3.1.1Generalized Myers Theorem98 3.1.2Linearly Growth Theorem of Volume100 3.2McKean Type Inequalities for the First Eigenvalue101 3.2.1The Divergence Lemma101 3.2.2The Mckean Type Inequalities103 3.3Gromov Pre-Compactness Theorem107 3.4The First Betti Number109 3.5Curvature and Fundamental Group114 3.5.1Universal Covering Space and Fundamental Group114 3.5.2Growth of Fundamental Group118 3.5.3Finiteness of Fundamental Group124 3.5.4Results Related to Milnor’s Conjecture125 3.6A Lower Bound of Injectivity Radius128 3.7Finite Topological Type131 Chapter 4Geometry of Finsler Submanifolds135 4.1Mean Curvature135 4.1.1Projection in a Minkowski Space135 4.1.2The Mean Curvature for Finsler Submanifolds137 4.2Some Results on Submanifolds in Minkowski Space139 4.3Volume Growth of Submanifolds in Minkowski Space145 4.4Rigidity of Minimal Surfaces in Randers-Minkowski 3-Space149 4.4.1The Mean Curvature of a Graph in (Rn+1,Fb)149 4.4.2The Rigidity Results153 Bibliography156 Index160
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