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Metrizability of Holonomy Invariant Projective Deformation of Sprays

Part of: Variational problems in infinite-dimensional spaces Global differential geometry Local differential geometry Miscellaneous topics in calculus of variations and optimal control Infinite-dimensional manifolds

Published online by Cambridge University Press:  23 January 2020

S. G. Elgendi  and
Zoltán Muzsnay
S. G. Elgendi
Affiliation:
Department of Mathematics, Faculty of Science, Benha University, Egypt e-mail: salah.ali@fsci.bu.edu.eg
Zoltán Muzsnay
Affiliation:
Institute of Mathematics, University of Debrecen, Debrecen, Hungary e-mail: muzsnay@science.unideb.hu URL: http://math.unideb.hu/muzsnay-zoltan
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Abstract

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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.

Keywords

sprayprojective deformationmetrizability problemholonomy invariant functionholonomy distribution

MSC classification

Primary: 53C60: Finsler spaces and generalizations (areal metrics)
Secondary: 53B40: Finsler spaces and generalizations (areal metrics) 58B20: Riemannian, Finsler and other geometric structures 49N45: Inverse problems 58E30: Variational principles
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 3 , September 2023 , pp. 701 - 714
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Copyright
© Canadian Mathematical Society 2020

Footnotes

This work is partially supported by the EFOP-3.6.2-16-2017-00015 and EFOP-3.6.1-16-2016-00022 projects and the 307818 TKA-DAAD exchange project.

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