Codimension two spacelike submanifolds in Lorentzian manifolds and conformal structures

Abstract

Starting from a Riemannian conformal structure on a manifold M, we provide a method to construct a family of Lorentzian manifolds. The construction relies on the choice of a metric in the conformal class and a smooth 1-parameter family of self-adjoint tensor fields. Then, every metric in the conformal class corresponds to the induced metric on M seen as a codimension two spacelike submanifold into these Lorentzian manifolds. Under suitable choices of the 1-parameter family of tensor fields, there exists a lightlike normal vector field along such spacelike submanifolds whose Weingarten endomorphism provide a Möbius structure on the Riemannian conformal structure. Conversely, every Möbius structure on a Riemannian conformal structure arises in this way. Flat Möbius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces.