Generalized \(R^h\)-Recurrent Spaces in Finsler Geometry: Properties, Identities, and Characterizations

In this paper, we investigate a class of Finsler spaces, termed generalized Rh-recurrent spaces (denoted by GRh-RFn), where the Cartan's third curvature tensor satisfies a specific condition involving non-null covariant vector fields. We provide several characterizations and properties of these spaces, starting with the equivalence of two distinct forms of the curvature condition, and prove key theorems about the non-vanishing of essential tensors like the Ricci tensor, curvature vector, deviation tensor, and the curvature scalar. Furthermore, we derive important identities associated with the covariant differentiation of tensors in these spaces, such as the h-covariant derivative of the h(v)-torsion tensor and deviation tensor. These identities play a crucial role in understanding the geometric structure of GRh-RFn​, and the results contribute to the broader theory of recurrent spaces in Finsler geometry. The paper concludes by presenting several new identities, which hold in GRh-RFn, thus advancing the theoretical framework of generalized recurrent spaces in differential geometry.