Real Hypersurfaces in Nearly Kaehler 6-Sphere

In this paper we characterize Hopf hypersurfaces in the nearly Kaehler 6-Sphere S^6 using some restrictions on the characteristic vector field xi = -JN, where J is the almost complex structure on S^6 and N is the unit normal to the hypersurface. It is shown that if the characteristic vector field xi of a compact and connected real hypersurface M of the nearly Kaehler sphere S^6 is harmonic and the Ricci curvature in the direction of xi is nonnegative, then M is a Hopf hypersurface and therefore congruent to either a totally geodesic hypersphere or a tube over almost complex curve on S^6. It is also observed that similar result holds if xi is Jacobi-type vector field (a notion similar to Jacobi fields along geodesics). We also show that if a connected real hypersurface M is a Ricci soliton with potential vector field xi , then M is congruent to an open piece of either a totally geodesic hypersphere or a tube over an almost complex curve in S^6.