Isoparametric 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 classification 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 $$(S^n,h)$$ .