Metrizability of 1-form projective deformation of sprays

Salah G. Elgendi; Zoltán Muzsnay; Salah G. Elgendi; Zoltán Muzsnay

doi:10.3934/math.20251268

AIMS Mathematics

2025, Volume 10, Issue 12: 28815-28828. doi: 10.3934/math.20251268

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Research article

Metrizability of 1-form projective deformation of sprays

Salah G. Elgendi ^{1
,
,},
Zoltán Muzsnay ²

1.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, KSA
2.
Institute of Mathematics, University of Debrecen, Debrecen, Hungary

Received: 13 August 2025 Revised: 25 November 2025 Accepted: 01 December 2025 Published: 09 December 2025
MSC : 53B40, 53C60

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 $ \beta \in \Lambda^1(M) $ on $ M $, precisely, $ S_{\! {_\beta}} = S-2\beta \mathcal{C} $. We show that, in general, the spray $ S_{\! {_\beta}} $ is not Finsler metrizable. Moreover, the metrizability of $ S_{\! {_\beta}} $, 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_{\! {_\beta}} $. 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.
- sprays,
- metrizability,
- projective deformation,
- Hilbert's fourth problem,
- projectively flat
Citation: Salah G. Elgendi, Zoltán Muzsnay. Metrizability of 1-form projective deformation of sprays[J]. AIMS Mathematics, 2025, 10(12): 28815-28828. doi: 10.3934/math.20251268

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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 $ \beta \in \Lambda^1(M) $ on $ M $, precisely, $ S_{\! {_\beta}} = S-2\beta \mathcal{C} $. We show that, in general, the spray $ S_{\! {_\beta}} $ is not Finsler metrizable. Moreover, the metrizability of $ S_{\! {_\beta}} $, 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_{\! {_\beta}} $. 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.

References

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