Generalized Birecurrent Finsler Geometry: Curvature Identities and Mathematical Foundations for Mechanical and Renewable Energy Systems

Abstract

This paper investigates a class of generalized birecurrent Finsler spaces characterized by a generalized recurrence condition imposed on Cartan’s third curvature tensor with respect to Cartan’s connection. The geometric framework is formulated using non-null covariant vector fields and higher-order covariant tensor fields. Fundamental consequences of the generalized birecurrence condition are derived, including identities involving the -torsion tensor, deviation tensor, Ricci tensor, curvature vector, and curvature scalar. It is shown that the Ricci tensor, curvature vector, deviation tensor, and curvature scalar are non-vanishing in the considered class of Finsler spaces. Several curvature identities are established through covariant differentiation, tensor contraction, and transvection of the fundamental geometric relations. The developed framework provides a mathematical basis for studying direction-dependent and curvature-constrained dynamical structures. It also suggests potential extensions to mechanical systems and renewable energy technologies involving directional states, nonlinear motion, and geometric constraints. These results establish a theoretical foundation for developing Finsler-geometric models of mechanical dynamics, energy conversion, and renewable energy systems.