Cartan Connections Associated to a $\beta$-Conformal Change in Finsler Geometry

TL;DR

This paper studies a generalized $eta$-conformal change in Finsler geometry, deriving explicit relations between the original and transformed Cartan connections and geometric objects, unifying many known results as special cases.

Contribution

It provides explicit formulas relating Cartan connections and geometric objects under a broad class of $eta$-conformal changes in Finsler geometry, extending previous results.

Findings

Derived explicit relations between original and transformed Cartan connections.

Expressed fundamental geometric objects of the transformed Finsler structure.

Characterized $eta$-homothetic changes and conditions for vanishing difference tensors.

Abstract

On a Finsler manifold $(M,L)$, we consider the change $L\longrightarrow\bar{L}(x,y)=e^{\sigma(x)}L(x,y)+\beta (x,y)$, which we call a $\beta$-conformal change. This change generalizes various types of changes in Finsler geometry: conformal, $C$-conformal, $h$-conformal, Randers and generalized Randers changes. Under this change, we obtain an explicit expression relating the Cartan connection associated to $(M,L)$ and the transformed Cartan connection associated to $(M,\bar{L})$. We also express some of the fundamental geometric objects (canonical spray, nonlinear connection, torsion tensors, ...etc.) of $(M,\bar{L})$ in terms of the corresponding objects of $(M,L)$. We characterize the $\beta$-homothetic change and give necessary and sufficient conditions for the vanishing of the difference tensor in certain cases. It is to be noted that many known results of Shibata, Matsumoto, Hashiguchi and others are retrieved as special cases from this work.