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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Bbb{C}^{m+d}$\end{document} of codimension d, locally given as the common zero set of real-valued functions r1,…,rd. 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 r1,…,rd of third order.