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Isoparametric Hypersurfaces Induced by Navigation in Lorentz Finsler Geometry

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Abstract

Using a navigation process with the datum (F, V), in which F is a Finsler metric and the smooth tangent vector field V satisfies F(−V(x)) > 1 everywhere, a Lorentz Finsler metric \(\tilde{F}\) can be induced. Isoparametric functions and isoparametric hypersurfaces with or without involving a smooth measure can be defined for \(\tilde{F}\). When the vector field V in the navigation datum is homothetic, we prove the local correspondences between isoparametric functions and isoparametric hypersurfaces before and after this navigation process. Using these correspondences, we provide some examples of isoparametric functions and isoparametric hypersurfaces on a Funk space of Lorentz Randers type.

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Acknowledgements

We thank the referees for their time and comments. We thank Qun He and Miguel Angel Javaloyes for helpful discussions and suggestions. The first author thanks Anhui University and Sichuan University for hospitality during the preparation of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, P. R. China

    Ming Xu

  2. School of Microelectronics and Data Science, Anhui University of Technology, Maanshan, 243032, P. R. China

    Ju Tan & Na Xu

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  1. Ming Xu
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  2. Ju Tan
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  3. Na Xu
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Correspondence to Ju Tan.

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Supported by Beijing Natural Science Foundation (Grant No. 1222003), National Natural Science Foundation of China (Grant Nos. 12131012, 11821101 and 12001007), Natural Science Foundation of Anhui province (Grant Nos. 2008085QA03 and 1908085QA03)

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Xu, M., Tan, J. & Xu, N. Isoparametric Hypersurfaces Induced by Navigation in Lorentz Finsler Geometry. Acta. Math. Sin.-English Ser. 39, 1547–1564 (2023). https://doi.org/10.1007/s10114-023-1187-x

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