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Metrizability of Holonomy Invariant Projective Deformation of Sprays
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 23 January 2020, pp. 701-714
- Print publication:
- September 2023
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- Article
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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.
