Abstract
On a hypersurface of a unit sphere without umbilical points, we know that three Mobius invariants can be defined and analogous to Euclidean case, we have the concepts of Mobius isoparametric and isotropic hypersurfaces. In this paper, we study the relationship between Euclidean geometry and Mobius geometry, and prove that a hypersurface in a sphere with constant length of the second fundamental form is Euclidean isoparametric if and only if it is Mobius isoparametric. When restricting to the case of three distinct principal curvatures, we show that such a hypersurface is either Mobius isoparametric or isotropic if the Blaschke tensor has constant eigenvalues.
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