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Isoparametric functions and mean curvature in manifolds with Zermelo navigation

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Abstract

The generalized Zermelo navigation problem looks for the shortest time paths in an environment, modeled by a Finsler manifold (M, F), under the influence of wind or current, represented by a vector field W. The main objective of this paper is to investigate the relationship between the isoparametric functions on the manifold M with and without the presence of the vector field W. Our work generalizes results in (Dong and He in Differ Geom Appl 68:101581, 2020; He et al. in Acta Math Sinica Engl Ser 36:1049–1060, 2020; He et al. in Differ Geom Appl 84:101937, 2022; Ming et al. in Pub Math Debr 97:449–474, 2020; Xu et al. in Isoparametric hypersurfaces induced by navigation in Lorentz Finsler geometry, 2021). For the positive-definite cases, we also compare the mean curvatures in the manifold. Overall, we follow a coordinate-free approach.

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  1. Instituto de Matemática e Estatística, Universidade Federal da Bahia, Rua Barão de Jeremoabo, 40170-115, Salvador, BA, Brazil

    Benigno Oliveira Alves

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão, 05508-090, São Paulo, SP, Brazil

    Patrícia Marçal

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  1. Benigno Oliveira Alves
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Alves, B.O., Marçal, P. Isoparametric functions and mean curvature in manifolds with Zermelo navigation. Annali di Matematica 203, 1285–1310 (2024). https://doi.org/10.1007/s10231-023-01402-2

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