One form deformation of sprays

TL;DR

This paper introduces the concept of one form deformation of sprays, characterizes the metrizability of the deformed spray, and constructs new solutions to Hilbert's fourth problem with novel metrics.

Contribution

It presents a new deformation method for sprays, explores their metrizability, and provides fresh solutions to Hilbert's fourth problem with non-isometric metrics.

Findings

New projectively flat metrics of constant flag curvature 1 are constructed.

The deformed sprays are generally not isometric to Klein metrics.

Several examples illustrating the deformation and metric properties are discussed.

Abstract

In this paper, we introduce the notion of one form deformation of sprays. The metrizability of the new spray, when the background spray is flat, is characterized. Therefore, we obtain new projectively flat metrics of constant flag curvature $1$. Moreover, these new metrics are not, generally, isometric to the Klein metric via affine transformations. New solutions for Hilbert's fourth problem are obtained and constructed. Various examples are discussed and studied.