On Mbius Form and Mbius Isoparametric Hypersurfaces

An umbilic-free hypersurface in the unit sphere is called Mbius isoparametric if it satisfiestwo conditions, namely, it has vanishing Mbius form and has constant Mbius principal curvatures.In this paper, under the condition of having constant Mbius principal curvatures, we show that thehypersurfaee is of vanishing Mbius form if and only if its Mbius form is parallel with respect to theLevi-Civita connection of its Mbius metric. Moreover, typical examples are constructed to show thatthe condition of having constant Mbius principal curvatures and that of having vanishing Mbius formare independent of each other.