Almost Ricci Solitons on Class $\mathcal A$ Hypersurfaces of Product Spaces

TL;DR

This paper classifies hypersurfaces in product spaces with a principal direction related to the Ricci soliton structure, showing that only rotational hypersurfaces admit such solitons.

Contribution

It provides a local classification of hypersurfaces with three principal curvatures in product spaces and characterizes those admitting almost Ricci solitons, proving non-existence for certain examples.

Findings

Hypersurfaces with three principal curvatures are classified locally.

Necessary and sufficient conditions for almost Ricci solitons are established.

Only rotational hypersurfaces admit the almost Ricci soliton structure.

Abstract

In this paper, we study hypersurfaces in the product spaces $\mathbb{Q}_{\epsilon}^3 \times \mathbb{R}$ for which the tangential component $T$ of the vector field $\frac{\partial}{\partial t}$ is a principal direction, where $\mathbb{Q}_{\epsilon}^3$ denotes the three-dimensional non-flat Riemannian space form with sectional curvature $\epsilon = \pm 1$, and $\frac{\partial}{\partial t}$ is the unit vector field tangent to the $\mathbb{R}$-factor. We obtain a local classification of hypersurfaces with three distinct principal curvatures satisfying specific functional relations. Then, we determine the necessary and sufficient conditions for such hypersurfaces to admit an almost Ricci soliton structure with potential vector field $T$. Finally, we prove that the only hypersurfaces admitting such solitons are rotational, by showing that the constructed examples with three distinct principal curvatures do not admit almost Ricci solitons.