A characterization of hypersurfaces of nearly K\"ahler $\mathbb{S}^3\times\mathbb{S}^3$ with $\mathcal{P}$-principal normal

In this article, we continue the study of hypersurfaces of homogeneous nearly K"ahler ${\ensuremath{\mathbb{S}^3\times\mathbb{S}^3}}$ with a $\mathcal{P}$-principal normal vector field $\xi$. After proving that the smooth angle function $\theta$, from the defining relation $P\xi=\cos\theta\xi+\sin\theta J\xi$, is constant, with the only possibilities being $\{-\frac{\pi}{3},\frac{\pi}{3},\pi\}$, we prove that all such hypersurfaces are Hopf, with either $3$ or $5$ distinct principal curvatures. The main results are: the complete classification of hypersurfaces of ${\ensuremath{\mathbb{S}^3\times\mathbb{S}^3}}$ with $\mathcal{P}$-principal normal and with $3$ distinct principal curvatures, as well as the general form of the immersion of such hypersurfaces with $5$ distinct principal curvatures. If these $5$ curvatures are all additionally constant, we give the explicit classification. We also provide new examples of hypersurfaces of ${\ensuremath{\mathbb{S}^3\times\mathbb{S}^3}}$ with $\mathcal{P}$-principal normal, such that certain of them have constant mean curvature.