Isoparametric functions on $\mathbb{R}^n\times\mathbb{M}^m$

TL;DR

This paper classifies isoparametric functions with compact level sets on product manifolds and characterizes hypersurfaces with constant principal curvatures in a specific product space, advancing understanding of geometric structures.

Contribution

It provides a complete classification of isoparametric functions on certain product manifolds and describes isoparametric hypersurfaces in 2 with constant principal curvatures.

Findings

Classified isoparametric functions on ^m with compact level sets.

Characterized isoparametric hypersurfaces in ^2 with constant principal curvatures.

Abstract

We classify the isoparametric functions on $\mathbb{R}^n\times\mathbb{M}^m$, $n, m\geq2$, with compact level sets, where $\mathbb{M}^m$ is a connected, closed Riemannian manifold of dimension $m$. Also, we classify the isoparametric hypersurfaces in $\mathbb{S}^2\times\mathbb{R}^2$ with constant principal curvatures.