Structure Properties and Fundamental Identities of Generalized R-Recurrent Finsler Manifolds

In this paper, we introduce and investigate a new class of Finsler spaces, termed generalized  \(R^h\)-recurrent Finsler spaces, denoted by \(F_n\) G \(R^h\) - R . These are defined via a generalized recurrence condition imposed on Cartan’s third curvature tensor, involving three non-null covariant vector fields. We derive the fundamental characterizations of such spaces and establish their equivalence through multiple tensorial identities. The behavior of related geometric objects such as the h(v)-torsion tensor, Ricci tensor, curvature vector, deviation tensor, and scalar curvature is analyzed. Furthermore, several non-trivial identities involving covariant derivatives and contractions are proven, demonstrating rich internal symmetries. The study concludes with a series of structural theorems that extend classical recurrence concepts in Finsler geometry.