On Möbius form and Möbius isoparametric hypersurfaces

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