Decomposition and Generalized Recurrence of Cartan’s Third Curvature Tensor in Generalized -Recurrent Finsler Spaces

This paper investigates the decomposition properties of Cartan’s third curvature tensor in generalized recurrent Finsler spaces. More precisely, we consider an -dimensional Finsler space equipped with Cartan’s connection and satisfying a generalized recurrence condition for the third curvature tensor. Several decomposable forms of the curvature tensor are examined, including , , and . By applying -covariant differentiation, tensor contraction, and transvection with respect to the supporting element , we derive recurrence relations for the corresponding decomposition tensors. Under suitable constancy and covariant-constancy assumptions on the associated tensor fields, we establish that the resulting decomposed curvature tensors inherit generalized recurrence properties. Further consequences are obtained for the contracted tensors , , and . The obtained results provide a systematic characterization of decomposable Cartan curvature tensors and their recurrence behavior, thereby extending the structural analysis of recurrent curvature tensors in Finsler geometry.