Holomorphic Curves in Lorentzian CR-Manifolds

A CR-manifold is said to be Lorentzian if its Levi form has one negative eigenvalue and the rest positive. In this case, it is possible that the CR-manifold contains holomorphic curves. In this paper, necessary and sufficient conditions are derived (in terms of the "derivatives" of the CR-structure) in order that holomorphic curves exist. A "flatness" theorem is proven characterizing the real Lorentzian hyperquadric ${Q_5} \subseteq {\mathbf {C}}{P^3}$, and examples are given showing that nonflat Lorentzian hyperquadrics can have a richer family of holomorphic curves than the flat ones.