CR-submanifolds of the nearly Kähler 6-sphere

Abstract

There is an almost complex structure J on the sphere S6(1) defined by multiplication of the Cayley numbers. This structure is nearly Kähler. A submanifold of a manifold with an almost complex structure is CR, by Bejancu, if it has a differentiable holomorphic distribution H such that its orthogonal complement H⊥⊂TM is a totally real distribution. A CR-submanifolds of S6(1) has to be at least three-dimensional, so with disregarding the hypersurfaces which are trivially CR in the focus of investigation are three and four dimensional submanifolds. We give examples of such submanifolds, show the existence and uniqueness theorem for the three dimensional case, and present the results concerning H and H⊥ totally geodesic submanifolds. We also give examples obtained from the almost contact manifolds. In the four dimensional case, we show the classification of CR minimal submanifolds that satisfy Chen’s basic equality and of those that are not linearly full in S6(1).