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Let M4 be a closed minimal hypersurface in $$\mathbb{S}^5$$ S 5 with constant nonnegative scalar curvature. Denote by f3 the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f3 and g are constant, then M4 is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form...
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