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Randers metrics based on deformations by gradient winds

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Abstract

This paper investigates the deformations of Riemannian metrics, in particular Hessian metrics, by Zermelo’s navigation under the action of the weak gradient winds. Various descriptions of the resulting Randers metrics are given in relation to other special classes of Finsler metrics, e.g., projectively flat, locally dually flat. We prove that the resulting Randers metric obtained from perturbation by a conformal gradient wind is locally dually flat if and only if the background Riemannian metric is homothetic with the Euclidean metric. The inverse problem answers the question, when a given Randers metric comes from a Hessian metric and a gradient vector field through the Zermelo deformation. Some relevant examples are indicated at the end.

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Acknowledgements

The authors wish to express their gratitute to the anonymous reviewers for a careful reading of the manuscript, several helpful and detailed comments.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Braşov, Romania

    Nicoleta Aldea

  2. Faculty of Navigation, Gdynia Maritime University, Gdynia, Poland

    Piotr Kopacz

  3. Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

    Robert Wolak

Authors
  1. Nicoleta Aldea
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  2. Piotr Kopacz
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  3. Robert Wolak
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Correspondence to Piotr Kopacz.

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Aldea, N., Kopacz, P. & Wolak, R. Randers metrics based on deformations by gradient winds. Period Math Hung 86, 266–280 (2023). https://doi.org/10.1007/s10998-022-00464-8

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  • DOI: https://doi.org/10.1007/s10998-022-00464-8

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