M\"{o}bius structures and Weingarten endomorphisms of spacelike submanifolds

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 and it has been inspired by the Fefferman and Graham ambient metric for conformal structures. Every metric in the conformal class corresponds to the induced metric on 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 which Weingarten endomorphisms provide a M\"{o}bius structure on the Riemannian conformal structure. Conversely, every M\"{o}bius structure on a Riemannian conformal structure arises in this way. The Ricci-flatness condition along the scale bundle as a lightlike hypersurface into these Lorentzian manifolds is studied by means of the initial velocity of the $1$-parameter family of self-adjoint tensor fields. Finally, flat M\"{o}bius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces