Cyclic conformally flat hypersurfaces revisited

TL;DR

This paper classifies certain conformally flat hypersurfaces in Euclidean and product spaces, providing new insights into their geometric properties and a simplified proof of their classification based on principal curvature conditions.

Contribution

It offers a new classification of conformally flat hypersurfaces with three distinct principal curvatures in specific spaces, and simplifies the proof of cyclic conformally flat hypersurfaces in 4.

Findings

Classification of conformally flat hypersurfaces in 4, 32, and 32 with a principal direction property.

Simplified proof of the classification of cyclic conformally flat hypersurfaces in 4.

Characterization of cyclic conformally flat hypersurfaces via conformal Killing vector fields.

Abstract

In this article we classify the conformally flat Euclidean hypersurfaces of dimension three with three distinct principal curvatures of $\mathbb{R}^4$, $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$ with the property that the tangent component of the vector field $\partial/\partial t$ is a principal direction at any point. Here $\partial/\partial t$ stands for either a constant unit vector field in $\mathbb{R}^4$ or the unit vector field tangent to the factor $\mathbb{R}$ in the product spaces $\mathbb{S}^3\times \mathbb{R}$ and $\mathbb{H}^3\times \mathbb{R}$, respectively. Then we use this result to give a simple proof of an alternative classification of the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$, that is, the conformally flat hypersurfaces of $\mathbb{R}^4$ with three distinct principal curvatures such that the curvature lines correspondent to one of its principal curvatures are extrinsic circles. We also characterize the cyclic conformally flat hypersurfaces of $\mathbb{R}^4$ as those conformally flat hypersurfaces of dimension three with three distinct principal curvatures for which there exists a conformal Killing vector field of $\mathbb{R}^4$ whose tangent component is an eigenvector field correspondent to one of its principal curvatures.