A study of the concircular curvature tensor and its interactions with other tensors under the Lie derivative in \(GBK- 5RF_n\)

This research paper delves into a comprehensive analysis of the concircular curvature tensor and its intricate relationships with other tensors under the Lie derivative. The concircular curvature tensor, a fundamental geometric invariant, plays a pivotal role in characterizing the local geometry of Riemannian manifolds. By employing the powerful tool of the Lie derivative, we explore how the concircular curvature tensor transforms under infinitesimal transformations of the underlying manifold. Our study uncovers novel connections between the concircular curvature tensor and other significant tensors, such as the Ricci tensor and Weyl tensor, providing deeper insights into the geometric structure of Riemannian spaces. The results obtained in this paper not only contribute to the advancement of differential geometry but also have potential applications in various fields, including general relativity and theoretical physics.This research expands the definition of concircular curvature tensor within the context of generalized fifth recurrent Finsler space for Cartan's fourth curvature tensor \(K_{jkh}^i\) in sense of Berwald. By employing the Lie-derivative, we delve into the various connections between concircular, conformal, conharmonice curvature tensors and the Cartan’s third curvature tensor \(R_{jkh}^i\).