On Hopf hypersurfaces of the homogeneous nearly Kähler ${\mathbf{S}}^3\times {\mathbf{S}}^3$

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 Kähler) 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 ${\mathbf{S}}^3\times {\mathbf{S}}^3$.

26 pages