Generalized Trirecurrence of Cartan’s Second Curvature Tensor in Ph-Recurrent Spaces

Abstract

This paper explores Cartan's second curvature tensor  within the framework of generalized recurrent spaces. We derive and analyze the generalized recurrence conditions, and show that  satisfies the recurrence relations of the first, second, and third orders. Specifically, we investigate the conditions that lead to the tensor’s behavior in generalized -recurrent spaces, denoted as  -G-TR , and establish several theorems regarding the curvature tensor’s covariant derivatives and its relationship with other curvature tensors. Furthermore, we connect these recurrence conditions to the P-Ricci tensor and vector fields in higher-dimensional spaces. We demonstrate the interrelations between these tensors through a series of covariant derivative computations and conclude that the curvature tensor cannot vanish under specific geometric conditions. This study advances the field of differential geometry and its applications in physics by offering a thorough foundation for comprehending the characteristics of Cartan's curvature tensor in generalized recurrent spaces.