New classes of Finsler metrics: The birth of new projective invatiant

Abstract

This paper presents a pioneering projective invariant in Finsler geometry, introducing a new class of Finsler metrics that are preserved under projective transformations. The newly formulated weakly generalized Douglas-Weyl $(W-G D W)$ equation facilitates the generalization of generalized Douglas-Weyl $(G D W)$-metrics into the broader category of $W-G D W$-metrics, which encompasses all $G D W$-metrics. Within this class, there are also two additional subclasses: generalized weakly-Weyl metrics, characterized by a milder form of Weyl curvature, and generalized $\tilde{D}$-metrics, defined by a less strict version of Douglas curvature. The paper provides a comprehensive overview of these generalized class of Finsler metrics and elucidates their properties, supported by detailed examples.