Sprays of weakly isotropic curvature

Abstract

Spray geometry and Finsler geometry are related to each other because every Finsler metric on a manifold induces a spray on it. In the past decades, some sprays of scalar curvature and isotropic curvature are studied to tell the relations with Finsler metrics of scalar flag curvature and isotropic flag curvature. It is a natural way to study more new sprays with special curvature to find more interesting properties of sprays and their relations with Finsler metrics. In this paper, we introduce a new notion of curvature named weakly isotropic curvature which is more general than isotropic curvature for sprays. The relationship between this new curvature and the existed curvatures (Riemann curvature, χ-curvature and H-curvature) is revealed. Then we prove this curvature notion is a natural extension of weakly isotropic flag curvature for Finsler metrics. Two new groups of sprays of weakly isotropic curvature not induced by any Finsler metric are constructed based on our result.