Real hypersurfaces of a complex projective space

Abstract

Let M be a real hypersurface of a Kähler manifold $$(\overline M, J)$$ and let $$\xi$$ be its unit normal vector field. Then M is a CR submanifold of maximal CR dimension and $$\xi$$ is the distinguished normal vector field, used to define the almost contact structure F on M, induced from the almost complex structure J of $$\overline M$$ . Moreover, since a real hypersurface M of a Kähler manifold $$\overline M$$ has two geometric structures: an almost contact structure F and a submanifold structure represented by the shape operator A with respect to $$\xi$$ , in this section we study the commutativity condition of A and F and we present its geometric meaning.