  For a closed hypersurface $M^n\subset S^{n+1}(1)$ with constant mean
curvature and constant non-negative scalar curvature, the present paper shows
that if $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=3,\ldots, n-1$ for
shape operator $\mathcal{A}$, then $M$ is isoparametric. The result generalizes
the theorem of de Almeida and Brito \cite{dB90} for $n=3$ to any dimension $n$,
strongly supporting Chern's conjecture.
