CHAINS AND LORENTZ GEOMETRY (CR MANIFOLDS)

Fefferman and others have shown that if M('2n+1) is a pseudo-convex CR manifold (of hypersurface type) then a certain circle bundle over M carries a conformal class of Lorentz metrics invariant under CR diffeomorphisms of M. The conformal geometry of these Fefferman metrics gives rise to the CR geometry of M. In particular, the null geodesics of these metrics project to an invariant system of curves called chains, originally defined by Cartan, Chern, and Moser.;More generally, this work is a study of conformal Lorentz manifolds L('2n+2) which admit a geodesic null conformal Killing field K such that curl (K) is nondegenerate and = 0 for X (PERP) K (where C is the Weyl conformal curvature tensor of L), and of the quotients M('2n+1) of such manifolds by their null congruence. These Lorentz manifolds are shown to include all Fefferman manifolds. All such L induce a CR structure on M related to the geometry of L. The system of curves on M obtained by projecting to M the null geodesics of L is studied; such systems of curves are shown to connect pairs of nearby points of M, even if L is not Fefferman.;An example is given of a four-dimensional Lorentz manifold with the properties described above, but which is not Fefferman. Its quotient CR manifold is identified, and the projections of its null geodesics are shown to be qualitatively different from the standard chains.;Jacobowitz has recently proved, by analytic methods, that any two sufficiently nearby points of a CR manifold are connected by a smooth chain. This dissertation provides a geometric proof, exploiting the Fefferman construction, of Jacobowitz' Theorem.