Isoparameteric hypersurfaces in a Randers sphere of constant flag curvature

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 $(S^n,h)$.