On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry

Understand

Using global considerations, Mess proved that the moduli space of globally hyperbolic flat Lorentzian structures on $S\times\mathbb{R}$ is the tangent bundle of the Teichm\"uller space of $S$, if $S$ is a closed surface.

• One of the goals of this paper is to deepen this surprising occurrence and to make explicit the relation between the Mess parameters and the embedding data of any Cauchy surface.
• This relation is pointed out by using some specific properties of Codazzi tensors on hyperbolic surfaces.
• As a by-product we get a new Lorentzian proof of Goldman's celebrated result about the coincidence of the Weil-Petersson symplectic form and the Goldman pairing.