Geodesic
Metrizability of Holonomy Invariant Projective Deformation of Sprays
Canadian mathematical bulletin, Tome 66 (2023) no. 3, pp. 701-714

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DOI

In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions. Starting with a Finsler spray $S$ and a holonomy invariant function ${\mathcal{P}}$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\unicode[STIX]{x1D706}{\mathcal{P}}{\mathcal{C}}$. We prove that for any holonomy invariant nontrivial function ${\mathcal{P}}$ and for almost every value $\unicode[STIX]{x1D706}\in \mathbb{R}$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray. In these cases, the holonomy invariant function ${\mathcal{P}}$ is necessarily one of the principal curvatures of the geodesic structure.
DOI : 10.4153/S0008439520000016
Mots-clés : spray, projective deformation, metrizability problem, holonomy invariant function, holonomy distribution 
Elgendi, S. G.; Muzsnay, Zoltán. Metrizability of Holonomy Invariant Projective Deformation of Sprays. Canadian mathematical bulletin, Tome 66 (2023) no. 3, pp. 701-714. doi: 10.4153/S0008439520000016
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     author = {Elgendi, S. G. and Muzsnay, Zolt\'an},
     title = {Metrizability of {Holonomy} {Invariant} {Projective} {Deformation} of {Sprays}},
     journal = {Canadian mathematical bulletin},
     pages = {701--714},
     year = {2023},
     volume = {66},
     number = {3},
     doi = {10.4153/S0008439520000016},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439520000016/}
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