Decomposition of Generalized Recurrent Tensor Fields of R^h-nth Order in Finsler Manifolds

In this paper, we investigate the decomposition of Cartan's third curvature tensor in the framework of generalized -nth recurrent Finsler spaces. We examine three distinct decompositions of the curvature tensor and analyze the resulting equations under covariant derivatives of the decomposed tensor fields. The decompositions are expressed in terms of independent tensor fields, and we explore the recurrence properties of the resulting decomposition tensors. We show that under certain conditions, these decomposition tensors exhibit generalized nth-recurrence properties, which are crucial for understanding the geometric behavior of these tensors in Finsler geometry. The results provide a deeper understanding of the curvature properties in higher-dimensional recurrent Finsler spaces, with implications for both theoretical and applied geometry.