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Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere withconstant nonnegative scalar curvature. We prove that, if the sum of the cubesof all principal curvatures and the number of distinct principal curvatures areconstant, then M is isoparametric. Moreover, We give all possible values forsquared length of the second fundamental form of M. This result providesanother piece of supporting evidence to the Chern conjecture.

An intrinsic rigidity theorem for closed minimal hypersurfaces in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature