I am looking for references on contact structures in the framework of pseudo-riemannian manifolds. For instance on the Lorentz-Minkowski 3-space (resp. (n+1)-space). Denoting by L (resp. H) the Lorentz-Minkowski 3-space (resp. the hyperbolic plane regarded as pseudo-sphere in L), it seems to me that w= udx+vdy-wdz, (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 3-dimensional 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 pseudo-Riemannian metric, you can identify its future-directed '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 future-directed 'unit' tangent bundle. In your case, $L\times H$ is exactly the future-directed 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). |
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