A note on Blaschke isoparametric hypersurfaces

In this paper, we prove that a Möbius isoparametric hypersurface is a Blaschke isoparametric hypersurface, and a Blaschke isoparametric hypersurface is a Möbius isoparametric hypersurface provided that the Blaschke tensor has more than two distinct eigenvalues.

PARA-BLASCHKE ISOPARAMETRIC HYPERSURFACES IN A UNIT SPHERE Sn + 1(1)

Let A = ρ2∑i,jAijθi ⊗ θj and B = ρ2∑i,jBij θi ⊗ θj be the Blaschke tensor and the Möbius second fundamental form of the immersion x. Let D = A + λB be the para-Blaschke tensor of x, where λ is a constant. If x: Mn ↦ Sn + 1(1) is an n-dimensional para-Blaschke isoparametric hypersurface in a unit sphere Sn + 1(1) and x has three distinct Blaschke eigenvalues one of which is simple or has three distinct Möbius principal curvatures one of which is simple, we obtain the full classification theorems of the hypersurface.