# On Hopf hypersurfaces of the homogeneous nearly K\"ahler $\mathbf{S}^3\times\mathbf{S}^3$

## TL;DR

This paper studies Hopf hypersurfaces in the homogeneous nearly K"ahler manifold S^3×S^3, proving they cannot have two distinct principal curvatures and classifying those with three principal curvatures under specific geometric conditions.

## Contribution

It extends previous work by classifying Hopf hypersurfaces with three principal curvatures in S^3×S^3 under a preservation condition of the holomorphic distribution.

## Key findings

- Hopf hypersurfaces in S^3×S^3 cannot have two distinct principal curvatures.
- Complete classification of Hopf hypersurfaces with three principal curvatures under the preservation condition.

## Abstract

In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343--373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly K\"ahler) manifold $\mathbf{S}^3\times\mathbf{S}^3$. First, we show that any Hopf hypersurface of the homogeneous NK $\mathbf{S}^3\times\mathbf{S}^3$ does not admit two distinct principal curvatures. Then, for the important class of Hopf hypersurfaces with three distinct principal curvatures, we establish a complete classification under the additional condition that their holomorphic distributions $\{U\}^\perp$ are preserved by the almost product structure $P$ of the homogeneous NK $\mathbf{S}^3\times\mathbf{S}^3$.