Local geometry of levi-forms associated with the existence of complex submanifolds and the minimality of generic CR manifolds

Let M be a generic CR manifold in C ^m+d of codimension d, locally given as the common zero set of real-valued functions r ¹,..., r ^d. Given an integer δ = 1,..., d, we find a necessary and sufficient condition for M to contain a real submanifold of codimension δ with the same CR structure. We also find a necessary and sufficient condition and several sufficient conditions for M to admit a complex submanifold of complex dimension n, for any n = 1,..., m. We use the method of prolongation of an exterior differential system. The conditions are systems of partial differential equations on r ¹,..., r ^d of third order.