Concurrent $\pi$-vector fields and energy beta-change

TL;DR

This paper investigates the role of concurrent π-vector fields in Finsler geometry, examining their impact on special Finsler spaces and analyzing the effects of energy β-change on various fundamental connections and curvature tensors in a coordinate-free manner.

Contribution

It introduces an intrinsic study of concurrent π-vector fields and their influence on energy β-change, connecting different Finsler connections and their curvature tensors.

Findings

Relation between Barthel connections before and after energy β-change

Impact of concurrent π-vector fields on Finsler space structures

Change in curvature tensors of fundamental Finsler connections

Abstract

The present paper deals with an \emph{intrinsic} investigation of the notion of a concurrent $\pi$-vector field on the pullback bundle of a Finsler manifold $(M,L)$. The effect of the existence of a concurrent $\pi$-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular $\beta$-change, namely the energy $\beta$-change ($\widetilde{L}^{2}(x,y)=L^{2}(x,y)+ B^{2}(x,y) with \ B:=g(\bar{\zeta},\bar{\eta})$; $\bar{\zeta} $ being a concurrent $\pi$-vector field), is established. The relation between the two Barthel connections $\Gamma$ and $\widetilde{\Gamma}$, corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy $\beta$-change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.