Null-geodesics in complex conformal manifolds and the LeBrun correspondence

Abstract

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally flat. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal infinity of a selfdual four-manifolds. We also find a relation between the conformal invariants of the conformal infinity and its ambient.

17 pages, 1 figure, part of the paper is contained in (an old version of) the paper math.DG/0002029