On the Blaschke isoparametric hypersurfaces in the unit sphere

Given an immersed submanifold x: M m . S n in the unit sphere S n without umbilics, a Blaschke eigenvalue of x is by definition an eigenvalue of the Blaschke tensor of x. x is called Blaschke isoparametric if its Möbius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the Möbius geometry of submanifolds. When n = 4, the corresponding classification theorem was given by the authors. In this paper, we are able to complete the corresponding classification for n = 5. In particular, we shall prove that all the Blaschke isoparametric hypersurfaces in S5 with more than two distinct Blaschke eigenvalues are necessarily Möbius isoparametric.