  Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar
curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor
field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and
$\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=1,\cdots, n-1$. We show that
all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of
de Almeida and Brito \cite{dB90} to higher dimensions.
  As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric
if one takes $\mathfrak{a}$ above to be the second fundamental form, giving
affirmative evidence to Chern's conjecture.
