Three dimensional homogeneous Finsler manifolds

In this paper we study three dimensional homogeneous Finsler manifolds. We first obtain a complete list of the three‐dimensional homogeneous manifolds which admit invariant Finsler metrics. Then we consider invariant Randers metrics and present the classification of three dimensional homogeneous Randers spaces under isometrics.     

On a class of Douglas Finsler metrics

In this article, we study a class of Finsler metrics called general(α, β)-metrics,which are defined by a Riemannian metric α and a 1-form β. We determine all of Douglas general(α, β)-metrics on a manifold of dimension n 2.     

Nontrivial minimal surfaces in a hyperbolic Randers space

The contribution of this paper is two‐fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing vector field, by considering the Busemann–Hausdorff measure and Holmes–Thompson measure simultaneously. The second one is to obtain the explicit local expressions of two types of nontrivial rotational BH‐minimal surfaces in a Randers domain of constant flag curvature , which are the first examples of BH‐minimal surfaces in the hyperbolic Randers space.     

Some constructions of projectively flat Finsler metrics

In this paper, we find some solutions to a system of partial differential equations that characterize the projectively flat Finsler metrics. Further, we discover that some of these metrics actually have the zero flag curvature.     

射影平坦Finsler度量的解析构造

利用Hamel关于射影平坦的基本方程,我们导出了Randers度量的λ形变保持射影平坦的充分条件.特别,对一类具有特殊旗曲率性质的Randers度量我们证明了这类度量一定存在保持射影平坦性的λ形变.     

Riemann-Finsler geometry

In this paper, a survey on Riemann-Finsler geometry is given. Non-trivial examples of Finsler metrics satisfying different curvature conditions are presented. Local and global results in Finsler geometry are analyzed.     

Comparison theorems in Finsler geometry and their applications

We prove Hessian comparison theorems, Laplacian comparison theorems and volume comparison theorems for Finsler manifolds under various curvature conditions. As applications, we derive McKean type theorems for the first eigenvalue of Finsler manifolds, as well as generalize to Finsler manifolds a result on fundamental groups due to Milnor.The research of the Y. L. Xin was partially supported by NSFC of and SFECC.     

On the non-Riemannian quantity H in Finsler geometry

In this paper, we study a new non-Riemannian quantity H defined by the S-curvature. We find that the non-Riemannian quantity is closely related to S-curvature. We characterize Randers metrics of almost isotropic S-curvature if and only if they have almost vanishing H-curvature. Furthermore, the Randers metrics actually have zero S-curvature if and only if they have vanishing H-curvature.     

K(a)hler Finsler Metrics Are Actually Strongly K(a)hler

In this paper,the K(a)hler conditions of the Chern-Finsler connection in complex Finsler geometry are studied,and it is proved that K(a)hler Finsler metrics are actually strongly K(a)hler.     

On conformal vector fields on Randers manifolds

In this paper,we study conformal vector fields on a Randers manifold with certain curvature properties.In particular,we completely determine conformal vector fields on a Randers manifold of weakly isotropic flag curvature.     

Polyhedral Finsler spaces with locally unique geodesics

We study Finsler PL spaces, that is simplicial complexes glued out of simplices cut off from some normed spaces. We are interested in the class of Finsler PL spaces featuring local uniqueness of geodesics (for complexes made of Euclidean simplices, this property is equivalent to local CAT(0)). Though non-Euclidean normed spaces never satisfy CAT(0), it turns out that they share many common features. In particular, a globalization theorem holds: in a simply-connected Finsler PL space local uniqueness of geodesics implies the global one. However the situation is more delicate here: some basic convexity properties do not extend to the PL Finsler case.     

Quantifier elimination for elementary geometry and elementary affine geometry

We introduce new first‐order languages for the elementary n‐dimensional geometry and elementary n‐dimensional affine geometry (n ≥ 2), based on extending $\mathsf {FO}(\beta ,\equiv )$ and $\mathsf {FO}(\beta )$, respectively, with new function symbols. Here, β stands for the betweenness relation and ≡ for the congruence relation. We show that the associated theories admit effective quantifier elimination.     

Locally symmetric positively curved Finsler spaces

We showed that any compact locally symmetric Finsler metric with positive flag curvature must be Riemannian. Dedicated to Professor Karsten Grove on the occassion of his sixtieth birthday Received: 8 May 2006     

Uniform convexity and smoothness,and their applications in Finsler geometry

We generalize the Alexandrov–Toponogov comparison theorems to Finsler manifolds. Under suitable upper (lower, resp.) bounds on the flag and tangent curvatures together with the 2-uniform convexity (smoothness, resp.) of tangent spaces, we show the 2-uniform convexity (smoothness, resp.) of Finsler manifolds. As applications, we prove the almost everywhere existence of the second order differentials of semi-convex functions and of c-concave functions with the quadratic cost function.     

Completing the conformal boundary of a simply connected Lorentz surface

This paper extends Kulkarni's conformal boundary for a simply connected Lorentz surface to a compact conformal boundary . The procedure used is analogous to Carathéodory's construction (in the definite metric setting) of prime ends from the accessible points of a bounded simply connected planar domain. The space of conformal boundary elements is homeomorphic to the circle, and contains Kulkarni's conformal boundary as a dense subspace.

     

Closed geodesics on positively curved Finsler spheres

In this paper, we prove that for every Finsler n-sphere (Sn,F) for n?3 with reversibility λ and flag curvature K satisfying , either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiμ) with an irrational μ. Furthermore, there always exist three prime closed geodesics on any (S³,F) satisfying the above pinching condition.     

Examples of Finsler metrics with special curvature properties

We give a lot of new Finsler metrics of quadratic Weyl curvature which are non‐trivial in the sense that these metrics are not of Weyl type. We also manufacture new Weyl metrics.