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Space can have a hyperbolic geometry but spacetime cannot. Hyperbolic geometry always implies a positive-definite signature. The geometry relevant to cosmology and general relativity is a not a Riemannian geometry, but a Lorentzian one (with an indefinite signature). Hyperbolic geometry is related to Lorentzian geometry (hyperbolic space can be embedded in Minkowski space) but it is not the same thing. The closest analog of hyperbolic space in the Lorentzian setting is anti de Sitter space but I don't think that should be categorized here. -- Fropuff 08:19, 2 February 2006 (UTC)[reply]
Yeah, well, depending on how much hand-waving one is willing to put up with, Carrionluggae has a point, sort of. The Lorentz group is SL(2,C); this group is central to GR and strings and etc. It's discrete subgroups are the Kleinian and Fuchsian groups, which are deeply implicated in all things hyperbolic. String theory scattering ampltudes are Riemann surfaces; monstrous moonshine caused a stir for a reason. Although this handwaving may be too blatent, it does seem sometimes to be important to the beginner. (its an endless source of amazement to me) linas 01:21, 3 February 2006 (UTC)[reply]
