Investigating the Relationships Between Weyl's and Cartan's \(2^{th}\) Curvature Tensors in Finsler Spaces

Abstract

This study investigates the interconnection between Weyl's curvature tensor \(W_{jkh}^i\) and Cartan's second curvature tensor \(P_{jkh}^i\) in the frame of Finsler geometry (or \(F\)-geometry), a broader framework that generalizes Riemannian geometry(or \(R\)-geometry). When describing the curvature characteristics of \(F\)-space which are crucial for simulating a variety of physical events both tensors are crucial. Even though the geometric meanings and physical consequences of these tensors have been thoroughly investigated, their interconnection remains an open area for research. In the present work, we demonstrate that the Weyl's and Cartan's second curvature tensors are connected by a novel set of identities and inequalities that we deduce by examining their algebraic and geometric characteristics. A series of theorems that outline particular circumstances in which the tensors exhibit generalized birecurrent behavior in Finsler spaces (or \(F\)-spaces) are presented. In addition to offering further insight into how these notions are applied in physics, especially in the gravitational field and cosmology, these results are anticipated to improve our knowledge of the curvature structure in \(F\)-spaces and yield interesting findings in the frame of differential geometry and its physical applications.