On the Berwald–Weyl curvature

In this paper, we study the Berwald–Weyl curvature which is defined for a spray/Finsler metric with a volume form. We obtain some expressions for the Berwald–Weyl curvature. This quantity is a projective invariant with respect to a fixed volume form. We prove that for any spray of scalar curvature on a manifold of dimension [Formula: see text], the Berwald–Weyl curvature vanishes with respect to any volume form. We also show that for any Finsler metric of constant Ricci curvature and constant S-curvature, the Berwald–Weyl curvature vanishes with respect to the Busemann–Hausdorff volume form. This study leads to a new notion of BWeyl-flat sprays/Finsler metrics.