THE CALABI (VERONESE) IMBEDDINGS AS INTEGRAL SUBMANIFOLDS OF ℂP^{2n+1}

Considering odd-dimensional complex projective space as a complex contact manifold, one may ask which of the Calabi (Veronese) imbeddings can be positioned by a holomorphic congruence as integral submanifolds of the complex contact structure. It is first shown that when the first normal space is the whole normal space, this is impossible. It is also shown to be impossibile for a Calabi surface (complex dimension 2) in complex projective space of dimension 9 where one has both a first and second normal space. However when the complex dimension of the submanifold is odd and the whole normal space consists of the first and second normal spaces, then there is a holomorphic congruence positioning the Calabi imbedding as an integral submanifold of the complex contact structure.