Hypersurfaces of the sphere S6(1) with four-dimensional nullity distribution

The sphere $S^6\left(1\right)$ is one of the four homogeneous, six-dimensional nearly Kähler manifolds, and the only one where the nearly Kähler structure is given with the standard metric. A nullity distribution of a submanifold consists of the vector fields X such that the second fundamental form h satisfies $h(X,.)=0$. The totally geodesic sphere $S^5$ trivially admits a five-dimensional nullity distribution. In this paper, we investigate non totally geodesic hypersurfaces of the nearly Kähler sphere $S^6\left(1\right)$, that admit nullity distribution of the maximal possible dimension, i.e. with nullity distribution of the dimension four and classify them.