Tohoku Mathematical Journal
2020

December
SECOND SERIES VOL. 72, NO. 4

Tohoku Math. J.
72 (2020), 493-505

Title A SUFFICIENT CONDITION FOR A HYPERSURFACE TO BE ISOPARAMETRIC

Author Zizhou Tang, Dongyi Wei and Wenjiao Yan

(Received March 19, 2019, revised May 13, 2019)
Abstract. 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 $k=1,\ldots, n-1$. We show that all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito [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.


Mathematics Subject Classification. Primary 53C12, Secondary 53C20, 53C40.

Key words and phrases. Isoparametric hypersurfaces, scalar curvature, Chern conjecture.

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