On Curvature Inheritance in Generalized Fifth Recurrent Finsler Space

This paper builds upon a new class of Finsler space, generalized fifth recurrent Finsler space. For Cartan’s fourth curvature tensor in the sense of Berwald, we denote such space by Finsler space. Finsler geometry is a kind of differential geometry. It is an extension of Riemannian geometry; Riemann studied the distance between points in n-dimensions using only positional coordinates, while Finsler generalized Riemann's idea and studied the distance between points in n-dimensions by using two coordinates: positional and directional. The key idea revolves around a mathematical object called the curvature inheritance, which is defined in terms of the Lie-derivative, where curviture is a scalar function. In this work, we introduce new relationships for certain curvature inheritance, with the curvature tensor's components related to the components of a vector field during the application of a force at an arbitrary point by the Lie-derivative, and which it inherits a new tensor at a flow line of via transformation mapping. We prove that the Lie-derivative of K-curvature inheritance and the Berwald covariant derivative of fifth order are commutative under certain conditions. Also we prove that the scalar function of the Cartan’s fourth curvature tensor, the Berwald’s curvature tensor, the H-Ricci tensor and the K-Ricci tensor all have values of zero under the same certain conditions. Further, an equality relationship is established between Cartan’s fourth curvature tensor and Berwald’s curvature tensor if their Berwald covariant derivatives of fifth order are equal. Finally, we derived a mathematical formula for the change occurring in the vector field of the fifth order during its flow through a smooth vector field under the influence of an applied force and we proved that the direction of this force is opposite to the direction of the vector field of the fifth order itself within this space.