Hopf hypersurfaces

Abstract

This thesis is concerned with Hopf hypersurfaces of Kähler and nearly Kähler manifolds and gives special emphasis to the cases of hypersurfaces of complex projective spaces and of the 6-sphere endowed with its nearly Kähler almost complex structure. Although there is already a wealth of investigations done in the case of complex space forms and the 6-sphere, a full classification of these hypersurfaces in the former spaces was done under assumption of constancy of the rank of its focal map. Here, the classification is revisited and this assumption is removed although a complete classification is still not obtained. The characterization of the Hopf hypersurfaces of the 6-sphere as tubular hypersurfaces around almost complex curves is used to determine among these hypersurfaces special examples which have constant mean curvature or are Einstein hypersurfaces. The invariants needed to decide when a pair of hypersurfaces of S(^6) and CP(^n) are respectively G(_2)-congruent and holomorphically congruent are determined and this result is applied to characterize the hypersurfaces of these spaces whose Hopf vector fields are also Killing field. Finally, the linearly full almost complex 2-spheres of S(^6) with at most two singularities are determined up to G(^C)(_2)-congruence of their directrix curves and this is used to determine the space of linearly full almost complex 2-spheres of S(^6) with suitably small induced area.