瀋一兵、瀋忠民編著的《現代芬斯勒幾何初步》
介紹：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.

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