On Some Properties of Finsler Manifolds of Stretch Curvature

Abstract

Finsler metrics with relatively non-negative (non-positive, respectively), constant and isotropic stretch curvatures are investigated in this paper. In particular, it is proved that every non-Riemannian $(\alpha, \beta)$-metric with a nonzero constant flag curvature and a non-zero relatively isotropic stretch curvature over a manifold of dimension $n\geq 3$ is of a characteristic scalar constant over the Finsler geodesics. It is also shown that every compact Finsler manifold with a relatively non-negative (non-positive, respectively) stretch curvature is a Landsberg metric. Finsler manifolds with $2$-dimensional relative stretch curvature are also investigated.