Study on generalized BK-5th recurrent Finsler space

In this paper, we present a novel new class and investigate the connection between the K-projective curvature tensor and other tensors of Finsler space Fn, this space is characterized by the property for Cartan’s 4th curvature tensor satisfies the certain relationship with the given covariant vectors field, we define this space as a generalized BK-5th recurrent space and denote it briefly by GBK-5RFn. This paper aims to derive the fifth-order Berwald covariant derivatives of the torsion tensor and the deviation tensor . Additionally, it demonstrates that the curvature vector Kj, the curvature vector Hk, and the curvature scalar H are all non-vanishing within the considered space. We have identified tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal fifth-order Berwald covariant derivatives with their lower-order counterparts.