Topology of Dupin hypersurfaces with six distinct principal curvatures

Abstract

Let M be a Dupin hypersurface in the unit sphere $S^{n+1}$ with six distinct principal curvatures. We will prove in the present paper that M is either diffeomorphic to $SU(2)\times SU(2)/Q_8 $ or homeomorphic to a tube around an embedded 5-dimensional complex Fermat hypersurface $X_5(2)$ in $S^{13}$ , where $ Q_8\subset SU(2)=Sp(1)$ denotes the subgroup $\{ \pm 1, \pm i, \pm j,\pm k\}$ and $X_5(2)= \{ [z_0, z_1,\cdots z_{6}] \in CP^{6}\vert z_0^2 +z_1^2+ \cdots +z_{6}^2=0\} $ . Moreover, in the former case, all of the focal manifolds are diffeomorphic to $S^3\times RP^2$ ; In the latter case, one of the focal manifolds is homeomorphic to $X_5(2)$ .