@article{Elgendi2025, 
author = {Salah G. Elgendi and Zoltán Muzsnay},
title = {Metrizability of 1-form projective deformation of sprays},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {12},
pages = {28815-28828},
keywords = {sprays, metrizability, projective deformation, Hilbert's fourth problem, projectively flat},
url = {https://www.sciopen.com/article/10.3934/math.20251268},
doi = {10.3934/math.20251268},
abstract = {In this paper, we consider a special case of the inverse problem of calculus of variations. For a given spray    S on a manifold    M, we investigate the projective deformation of    S by a 1-form    β  ∈      Λ    1    (  M  ) on    M, precisely,        S                                            β                      =  S  −  2  β      C  . We show that, in general, the spray        S                                            β                     is not Finsler metrizable. Moreover, the metrizability of        S                                            β                    , when the background spray is flat, is characterized. In this case, we establish an explicit formula for the Finsler function whose geodesic spray is        S                                            β                    . We conclude that this metric is a projectively flat metric of nonzero constant flag curvature; that is, the obtained metric is a solution for Hilbert's fourth problem.}
}