On the Berwald-Weyl Curvature

TL;DR

This paper investigates the Berwald-Weyl curvature in Finsler geometry, deriving expressions, establishing invariance properties, and identifying conditions under which it vanishes, leading to a new concept of BWeyl-flat metrics.

Contribution

It introduces new expressions for the Berwald-Weyl curvature, proves its invariance under projective transformations, and characterizes conditions for its vanishing, including scalar curvature and constant Ricci and S-curvatures.

Findings

Berwald-Weyl curvature expressions derived

Invariance under projective transformations proven

Conditions for vanishing curvature established

Abstract

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 $n\geq 3$, 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.