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2020 A sufficient condition for a hypersurface to be isoparametric
Zizhou Tang, Dongyi Wei, Wenjiao Yan
Tohoku Math. J. (2) 72(4): 493-505 (2020). DOI: 10.2748/tmj.20190611

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.

Citation

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Zizhou Tang. Dongyi Wei. Wenjiao Yan. "A sufficient condition for a hypersurface to be isoparametric." Tohoku Math. J. (2) 72 (4) 493 - 505, 2020. https://doi.org/10.2748/tmj.20190611

Information

Published: 2020
First available in Project Euclid: 22 December 2020

MathSciNet: MR4194182
Digital Object Identifier: 10.2748/tmj.20190611

Subjects:
Primary: 53C12
Secondary: 53C20, 53C40

Rights: Copyright © 2020 Tohoku University

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Vol.72 • No. 4 • 2020
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