On Einstein hypersurfaces of $I\times_{f}\mathbb{Q}^{n}(c)$

TL;DR

This paper classifies Einstein hypersurfaces in warped products of an interval and a space form, revealing their principal curvature structure and conditions for constant sectional curvature.

Contribution

It provides a classification of Einstein hypersurfaces with up to three principal curvatures in warped product spaces, including conditions for constant curvature and new structural insights.

Findings

At most three principal curvatures on Einstein hypersurfaces.

Hypersurfaces with one or two principal curvatures have constant sectional curvature.

Classification of hypersurfaces with three principal curvatures under specific conditions.

Abstract

In this paper, we investigate Einstein hypersurfaces of the warped product $I\times_{f}\mathbb{Q}^{n}(c)$, where $\mathbb{Q}^{n}(c)$ is a space form of curvature $c$. We prove that $M$ has at most three distinct principal curvatures and that it is locally a multiply warped product with at most two fibers. We also show that exactly one or two principal curvatures on an open set imply constant sectional curvature on that set. For exactly three distinct principal curvatures this is no longer true, and we classify such hypersurfaces provided it does not have constant sectional curvature and a certain principal curvature vanishes identically.