Isoparametric hypersurfaces induced by navigation in Lorentz Finsler geometry

TL;DR

This paper explores how a navigation process with a Lorentz Finsler metric induces isoparametric hypersurfaces, establishing local correspondences and providing examples in Funk spaces of Lorentz Randers type.

Contribution

It introduces a method to relate isoparametric hypersurfaces before and after navigation in Lorentz Finsler geometry, with new examples in Funk spaces.

Findings

Established local correspondences between isoparametric functions and hypersurfaces.

Constructed examples of isoparametric hypersurfaces in Funk spaces.

Extended the theory of isoparametric hypersurfaces to Lorentz Finsler settings.

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.