Generalized Bi-Recurrent Structures in \(G^{2nd} C_{|h} -RF_n\) Spaces via the Weyl Conformal Curvature Tensor

Abstract

In this paper, we investigate the properties of the Weyl conformal curvature tensor \(C_{jkh}^i\) in the context of n=4 Riemannian and Finslerian spaces, with a particular focus on generalized recurrent and birecurrent structures. We derive several equivalent forms of the conformal curvature tensor under various covariant derivatives, revealing deep interrelations between curvature tensors, Ricci tensors, scalar curvature, and their derivatives. By transvecting the conformal curvature expressions with vectors such as yⁱ, y^k, and tensors such as g_ij we deduce necessary and sufficient conditions for the conformal curvature tensor, torsion tensor, Ricci tensor, and projective deviation tensor to represent generalized recurrent and birecurrent Finsler spaces. The results culminate in a sequence of theorems (Theorems 3.1 to 3.8), offering a comprehensive characterization of G^2nd C_|h-RF_n spaces and G^2nd C_|h-BRF_n spaces. These findings contribute to the geometric understanding of recurrence structures in differential geometry and extend the theoretical framework of Finsler geometry.