Chains on CR Manifolds and Lorentz Geometry

Abstract

ABSTRACT. We show that two nearby points of a strictly pseudoconvex CR manifold are joined by a chain. The proof uses techniques of Lorentzian geometry via a correspondence of Fefferman. The arguments also apply to more general systems of chain-like curves on CR manifolds. Introduction.If M is a nondegenerate CR manifold, its Fefferman metric is a conformal class of pseudo-Riemannian metrics on a circle bundle over M. The various CR invariants of M may be described in terms of the conformal geometry of this metric; the description of chains in this setting is especially appealing. Recently, H. Jacobowitz showed that chains on a strictly pseudo-convex CR manifold connect pairs of nearby points. This paper presents a new proof of Jacobowitz's result which makes use of the Fefferman correspondence. We hope that this approach will yield new insights into the behavior of chains.We begin with a definition of an abstract CR manifold of hypersurface type. We then sketch briefly a construction due to Lee [15] of the pseudo-Riemannian Fefferman manifold associated to a CR manifold, and discuss the special properties of such manifolds. Much of this discussion applies to a slightly more general class of pseudo-Riemannian manifolds.We then proceed with our proof that chains on a strictly pseudoconvex CR manifold connect pairs of sufficiently nearby points. This proof applies also to the "pseudochains" associated to the more general pseudo-Riemannian manifolds discussed above. Finally, we discuss an example of a CR manifold with a system of pseudochains which are not chains. CR manifoldsand chains. An abstract almost CR manifold of hypersurface type is an odd-dimensional orientable manifold M2n+1 (which we shall always take to be smooth) together with a field of tangent hyperplanes 772™ on which a (smooth) complex structure J, J: H -> 77 linear, J2 = -id [h, is given. This J is called a CR structure tensor on M. An almost CR manifold is called a CR manifold if its Nijenhuis torsion tensor,vanishes for X, Y E 77; a CR structure satisfying this condition is said to be formally integrable.Let 9 E T*M be a one-form which annihilates 77. A choice of such a one-form is called a pseudo-Hermitian structure on M. The almost CR structure on M is said to be nondegenerate if 0 A (d0)n ^ 0; this condition is independent of the choice of 6._