Generalized Berwald Projective Weyl metrics
Nasrin Sadeghzadeh
TL;DR
The paper defines a new C-projective invariant, the generalized Berwald projective Weyl ($GB\\widetilde{W}$) metric, and situates it within the landscape of Finsler projective invariants by proving it is a proper subset of generalized Douglas–Weyl ($GDW$) metrics. It establishes that $GB\\widetilde{W}$ includes all Berwald and $\\widetilde{W}$ metrics and all constant-flag-curvature metrics ($n>2$), and proves that any non-Riemannian $GDW$ metric with vanishing Landsberg curvature is $R$-quadratic, while $GDW$ metrics encompass all scalar-flag-curvature metrics (extending Numata's theorem). The results further show that $GB\\widetilde{W}$ is invariant under $C$-projective changes and that every $GB\\widetilde{W}$ metric is a $GDW$ metric, with no nonconstant scalar-flag-curvature $GB\\widetilde{W}$ metrics. Numata-type conclusions link Landsberg, scalar curvature, and $R$-quadraticity, highlighting a structured hierarchy: $GB\\widetilde{W} \\subset GDW \\subset$ broader Finsler projective invariants. The work thus clarifies the relationships among key curvature notions in Finsler geometry and extends classical results.
Abstract
This paper introduces a new quantity in Finsler geometry, called the generalized Berwald projective Weyl ($GB\widetilde{W}$) metric. The $C$-projective invariance of these metrics is demonstrated, and it is shown that they constitute a proper subset of the class of generalized Douglas ($GDW$) metrics. The paper also proves that all $GDW$ metrics with vanishing Landsberg curvature are of R-quadratic type. The class of $GDW$ metrics contains all Finsler metrics of scalar curvature, which provides an extension of the well-known Numata's theorem.