A sufficient condition for a hypersurface to be isoparametric

TL;DR

This paper establishes a sufficient condition involving scalar curvature and tensor eigenvalues for a hypersurface to be isoparametric, extending previous results and supporting Chern's conjecture.

Contribution

It generalizes a theorem on eigenvalues of symmetric tensors to higher dimensions and confirms a condition under which hypersurfaces are isoparametric.

Findings

Eigenvalues of the tensor are constant under the given conditions

A hypersurface in a sphere is isoparametric if the second fundamental form meets the criteria

Supports Chern's conjecture on hypersurface geometry

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 $k=1,\cdots, n-1$. 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.