Isoparametric functions and mean curvature in manifolds with Zermelo navigation. (English) Zbl 07864266
Summary: 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.
MSC:
| 53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |
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