芬斯勒几何就是没有二次型限制的黎曼几何。作为重要的几何不变量，体积在整体微分几何中扮演了关键的角色，它与微分流形的曲率与拓扑密切相关。必须指出的是，对于给定的黎曼度量，体积形式被唯一确定；但对确定的芬斯勒度量，有不同的体积形式可供选择。因此在芬斯勒几何的研究中选择合适的体积形显得十分重要。吴炳烨著的《芬斯勒几何中的比较定理与子流形(英文版)》以体积形式为主线，介绍整体芬斯勒几何研究前沿的若干课题，并系统反映作者本人的研究成果。本书可作为数学专业研究生教材或教学参考书，也可供相关研究人员参考。

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