On Akbar-Zadeh's Theorem on a Finsler Space of Constant Curvature

On the gradient flows on Finsler manifolds

The purpose of this article is to provide a general overview of curvature functional in Finsler geometry and use its information to introduce the gradient flow on Finsler manifolds. For this purpose, we first prove that the space of Finslerian metrics is a Riemannian manifold. Then it is given a decomposition for the tangent space of this Riemannian manifold by means of Riemannian metric and the Berger-Ebin theorem. Next, Finslerian functional is introduced and show that Akbar-Zadeh curvature functional is the example of Finslerian functional. After that, the critical points of Akbar-Zadeh fun…

关于芬斯勒空间上的射影对称性

On projective symmetries on Finsler Spaces

本文证明了Akbar-Zadeh的两种Einstein-Finsler空间定义在$I$-不变射影向量场积分曲线上等价，揭示了闭Finsler空间中射影对称性退化为Killing向量场的条件，并完成了紧致各向同性平均Landsberg流形的分类。

On k-nullity foliations in Finsler geometry and completeness

Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant then the distribution is involutive and each maximal integral manifold is totally geodesic. Characterization of the k-nullity foliation is given, as well as some results concerning constancy of the flag curvature, and completeness of their integral…