Codimension two spacelike submanifolds in Lorentzian manifolds and conformal structures

TL;DR

This paper introduces a method to construct Lorentzian manifolds from Riemannian conformal structures, linking spacelike submanifolds and Mobius structures through extrinsic geometry and tensor fields.

Contribution

It provides a novel construction connecting Riemannian conformal structures with Lorentzian geometry and Mobius structures via spacelike submanifolds and tensor fields.

Findings

Construction of Lorentzian manifolds from conformal structures

Characterization of Mobius structures via extrinsic geometry

Conditions for flat Mobius 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 Mobius structure on the Riemannian conformal structure. Conversely, every Mobius structure on a Riemannian conformal structure arises in this way. Flat Mobius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces.