My calculations tell me an empty universe has hyperbolic curvature. Is this correct? If it is, can anyone help me understand why this is intuitively?
I remember being confused by this, and thanks to help from this site I think I understand the problem (though I probably don't! :-).
If you take the FLRW metric and extrapolate to zero density you get the Milne metric, which is hyperbolic and maximally curved. However the Milne metric is equivalent to the Minkowski metric with a co-ordinate transformation, and the Minkowski metric is obviously also a solution to the vacuum eqution. So the two are the same space descibed by different co-ordinates. The hyperbolicity of the Milne universe is just down to taking different spatial slices, and it's Riemann tensor is everywhere zero like the Minkowski space. A quick Google found this article that goes into more detail.