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._
Abstract.A system of distinguished curves distinct from chains is defined on indefinite nondegenerate CR hypersurfaces; the new curves are called nullchains. The properties of these curves are explored, and it is shown that two sufficiently nearby points of any nondegenerate CR hypersurface can be connected by either a chain or a null-chain.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.