In this paper we study generalized symmetric Berwald spaces. We show that if a Berwald space $(M,F)$ admits a parallel $s-$structure then it is locally symmetric. For a complete Berwald space which admits a parallel s-structure we show that if the flag curvature of $(M,F)$ is everywhere nonzero, then $F$ is Riemannian.

This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. The first remark is that there is a canonical Kähler structure on the space of geodesics of such a manifold. The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out ...

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...

This is a survey article about the recent developments in classifying Randers metrics of scalar flag curvature under an additional condition on the isotropic S-curvature. The authors give an outline of the proof for the classification theorem.

In this paper, we establish a closer relation between the Berwald scalar curvature and [Formula: see text]-curvature. fact, prove that Finsler metric has isotropic if only it weakly For metrics of flag text]-curvature, they have almost text]-curvature is isotropic.

The notion of holomorphic bisectional curvature for a complex Finsler space (M, F ) is defined with respect to the Chern complex linear connection on the pull-back tangent bundle. By means of holomorphic curvature and holomorphic flag curvature of a complex Finsler space, a special approach is emloyed to obtain the characterizations of the holomorphic bisectional curvature. For the class of gen...

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 a...

In this paper, we define a new projective invariant and call it W̃ -curvature. We prove that a Finsler manifold with dimension n ≥ 3 is of constant flag curvature if and only if its W̃ -curvature vanishes. Various kinds of projectively flatness of Finsler metrics and their equivalency on Riemannian metrics are also studied. M.S.C. 2010: 53B40, 53C60.