Chains on CR manifolds and Lorentz geometry

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. 0. 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. 1. CR manifolds and chains. An abstract almost CR manifold of hypersur-face 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, N(X, Y) = [JX, JY] + J2[X, Y]-J([X, JY] + [JX, Y]), 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._

Robinson manifolds as the Lorentzian analogs of Hermite manifolds

“…where a is a real function, the b s are complex and a det b = 0, where b = (b α β ). An almost CR manifold can be defined as an odd-dimensional manifold with an atlas of compatible CR charts, their compatibility being defined by (13). The (n + 1)-form…”

“…where the dots stand for exterior products of pairs of the local basis 1-forms other than the products µ α ∧μ β , 1 α, β n. The transformation (13) (13). The almost CR structure is said to be non-degenerate if det h = 0; it is called pseudo-convex (sometimes: strongly pseudo-convex) if the associated Hermitean form is definite.…”

“…Several solutions of Einstein's equations admit this congruence. As an example, we show this for the Gödel universe [13]. Take its metric in the form given in [28],…”

Chains in CR geometry as geodesics of a Kropina metric

“…The first assertion does hold for strictly pseudoconvex CR manifolds: nearby points are connected by chains. See [17], p. 185, and the original references therein, including [18,20]. Surprisingly, the second assertion is false, even if the compact manifold is locally CR equivalent to the standard model, S 3 with its canonical strictly pseudoconvex structure.…”

“…The fact that chains are geodesics of a Kropina metric allows us to employ variational methods and techniques of metric geometry to investigate chains. We will use these methods to reprove and generalize the famous result of [18,20] on local chain connectivity. Theorem 1.5.…”