Isoparameteric hypersurfaces in a Randers sphere of constant flag curvature

TL;DR

This paper classifies isoparametric hypersurfaces in Randers spheres of constant flag curvature, showing that those tangent to the navigation vector field W form a special class with limited principal curvatures, and all homogeneous hypersurfaces belong to this class.

Contribution

It provides a classification of isoparametric hypersurfaces in Randers spheres with constant flag curvature, extending known results and identifying a special class related to the navigation data.

Findings

Isoparametric hypersurfaces tangent to W remain isoparametric after navigation.

The number of principal curvatures is limited to 1, 2, or 4.

All homogeneous hypersurfaces are part of the classified special isoparametric hypersurfaces.

Abstract

In this paper, I study the isoparametric hypersurfaces in a Randers sphere $(S^n,F)$ of constant flag curvature, with the navigation datum $(h,W)$. I prove that an isoparametric hypersurface $M$ for the standard round sphere $(S^n,h)$ which is tangent to $W$ remains isoparametric for $(S^n,F)$ after the navigation process. This observation provides a special class of isoparametric hypersurfaces in $(S^n,F)$, which can be equivalently described as the regular level sets of isoparametric functions $f$ satisfying $-f$ is transnormal. I provide a classification for these special isoparametric hypersurfaces $M$, together with their ambient metric $F$ on $S^n$, except the case that $M$ is of the OT-FKM type with the multiplicities $(m_1,m_2)=(8,7)$. I also give a complete classificatoin for all homogeneous hypersurfaces in $(S^n,F)$. They all belong to these special isoparametric hypersurfaces. Because of the extra $W$, the number of distinct principal curvature can only be 1,2 or 4, i.e. there are less homogeneous hypersurfaces for $(S^n,F)$ than those for $(S^n,h)$.