Generalized βP-Recurrent Finsler Spaces: Characterization, Properties, and Covariant Derivatives of Weyl's Projective Tensors

This paper investigates the properties and behaviour of generalized βP-recurrent Finsler spaces with an emphasis on the covariant derivatives of Weyl's projective tensors, which include torsion, curvature, and deviation tensors. We define a GβP-recurrent space as a Finsler space where the second curvature tensor satisfies a specific recurrence condition involving non-zero covariant vector fields. Through a series of mathematical derivations, the paper explores the equivalence of different characterizations of the GβP-recurrent space, proving that the torsion tensor, its associate tensor, the P-Ricci tensor, and the curvature vector are non-vanishing. Further, we analyze the relationship between the covariant vector fields showing their dependence or independence on the direction argument. The study also includes the application of Berwald’s covariant derivative to the projective curvature tensor, concluding that these tensors exhibit generalized recurrence under certain conditions. The results are presented in a series of theorems that contribute to the deeper understanding of the geometric structure of -  spaces.