Narrow Results

In this paper, we study the non-Riemannian quantity H in Finsler geometry. We obtain some rigidity theorems of a compact Finsler manifold under some conditions related to H. We also prove that the S-curvature for a Randers metric is almost isotropic if and only if H almost vanishes. In particular, S-curvature is isotropic if and only if H = 0.

Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray $G$ on a manifold is of vanishing $H$-curvature, but $G$ has not isotropic curvature, then $G$ is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain $U\subset {ℝ}^n$ with scalar curvature and vanishing $H$-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

In this paper, we show that a Finsler warped product metric is of almost vanishing $H$-curvature if and only if it is of almost vanishing $\chi$-curvature. Furthermore, the corresponding one form is exact.