On Generalized BW-Four-Recurrent Finsler Spaces and Their Role in the Geometric Modelling of Intelligent Robotic Systems

Abstract

This paper presents and examines a new class of Finsler spaces known as Generalized BW-Four-Recurrent Finsler spaces -   , in which a particular recurrence relation involving non-null covariant vector fields a_lmns and b_lmns is satisfied by the fourth-order Berwald covariant derivative of the projective curvature tensor . We prove that all generalized BW-recurrent spaces are special cases of GBW-FR  , and we determine the necessary and sufficient conditions for the non-vanishing of several important curvature tensors, including the scalar curvature W, the curvature vector , and the projective deviation tensor . In the setting of -  geometry, the work also provides recurrence conditions for Cartan's third and fourth curvature tensors and the h(v)-torsion tensor. A detailed analysis is conducted of the interactions between several higher-order curvature tensors under repeated Berwald differentiation. Significantly, the paper shows how -  structures can be used as sophisticated mathematical models for intelligent robotic systems that operate in non-Euclidean, anisotropic situations. In robots operating in dynamically complicated or curved settings, such geometric frameworks can improve motion planning, adaptive control, and sensor-based navigation. The theoretical findings provide a basis for interdisciplinary applications of higher-order Finsler geometry in autonomous systems and advanced robotics.