# A nonisoparametric hypersurface with constant principal curvatures

## TL;DR

This paper constructs a specific example of a conformally flat manifold with a totally geodesic foliation that challenges the assumption that all hypersurfaces with constant principal curvatures are isoparametric, providing a negative answer.

## Contribution

It provides the first explicit example of a nonisoparametric hypersurface with constant principal curvatures in a conformally flat manifold.

## Key findings

- Existence of a conformally flat manifold with a totally geodesic foliation of codimension one
- Counterexample to the hypothesis that all hypersurfaces with constant principal curvatures are isoparametric
- Clarification of the relationship between constant principal curvatures and isoparametric hypersurfaces

## Abstract

In this note we construct an explicit example of a (compact) conformally flat Riemannian manifold which admits a totally geodesic foliation of codimension one with no isoparametric leaves. This answers negatively the question: is every hypersurface with constant principal curvatures isoparametric?