GENERALIZATION OF WELY'S PROJECTIVE CURVATURE TENSOR IN FINSLER SPACES: A STUDY ON GENERALIZED W-RECURRENT, BIRECURRENT, AND RICCI TENSORS

Abstract

In this study, we explore the generalization of Wely’s projective curvature tensor \(W_{jkh}^i\) satisfying a specific recurrence relation (3.1), which defines a generalized W-recurrent space, denoted as G^2nd W_|h - BRF_n . By investigating the conditions under which this tensor satisfies the generalized recurrence relations, we define and analyze the properties of generalized W-recurrent, birecurrent, and Ricci tensors in the context of second-order covariant derivatives. We derive a series of equations that describe the behavior of these tensors, including the covariant derivative expressions and their relationships with scalar curvatures and deviation tensors. A key contribution is the proof of various theorems that relate the generalized recurrence conditions to torsion tensors and curvature properties. Specifically, we show that the Ricci tensor and associated tensors exhibit generalized birecurrent Finsler space behavior under certain conditions. The results further provide insights into the torsion and deviation tensors, highlighting their role in the overall curvature structure of Finsler spaces. These findings extend the theory of recurrent spaces, offering a comprehensive framework for understanding higher-order curvature phenomena in Finsler geometry.