Isoparametric functions and mean curvature in manifolds with Zermelo navigation

TL;DR

This paper explores the relationship between isoparametric functions and mean curvature in Finsler manifolds affected by wind or current, extending classical geometric concepts to environments modeled by Zermelo navigation.

Contribution

It introduces a coordinate-free analysis of isoparametric functions and mean curvature in Finsler manifolds with vector fields, linking navigation problems with differential geometry.

Findings

Relationship between isoparametric functions with and without vector field W

Comparison of mean curvatures in the presence of W for positive-definite cases

Extension of geometric analysis to Zermelo navigation environments

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. For the positive-definite cases, we also compare the mean curvatures in the manifold. Overall, we follow a coordinate-free approach.