I am looking for references on contact structures in the framework of pseudoriemannian manifolds. For instance on the LorentzMinkowski 3space (resp. (n+1)space). Denoting by L (resp. H) the LorentzMinkowski 3space (resp. the hyperbolic plane regarded as pseudosphere in L), it seems to me that w= udx+vdywdz, (x,y,z,u,v,w) in LxH, defines a contact structure on LxH. It is as the classical example of the unit tangent bundle of 3dimensional Euclidean space E but replacing E by L and the Euclidean scalar product by the Lorentzian one in <(dx,dy,dz),(u,v,w)>.
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Yes, this is a contact structure. The point is that, using the pseudoRiemannian metric, you can identify its futuredirected 'unit' tangent bundle with an open set in the projectivized cotangent bundle. Since the projectivized cotangent bundle of any manifold carries a canonical contact structure, this identification induces a canonical contact structure on the futuredirected 'unit' tangent bundle. In your case, $L\times H$ is exactly the futuredirected unit tangent bundle. As you note, it corresponds to the unit sphere bundle of a Riemannian manifold (which also inherits a contact structure in exactly the same way). 

