Hypersurfaces of the homogeneous nearly Kahler S6 and S3 x S3 with anticommutative structure tensors

Each hypersurface of a nearly Kahler manifold is naturally equipped with two tensor fields of (1,1)-type, namely the shape operator A and the induced almost contact structure phi. In this paper, we show that, in the homogeneous nearly Kahler S-6 a hypersurface satisfies the condition A phi + phi A = 0 if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly Kahler manifold S-3 x S-3 does not admit a hypersurface that satisfies the condition A phi + phi A = 0.