Universe with zero curvature
In a universe with zero curvature, the local geometry is flat . The most obvious global structure is that of Euclidean space , which is infinite in extent. Flat universes that are finite in extent include the torus and Klein bottle . Moreover, in three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable. These are the Bieberbach manifolds . The most familiar is the aforementioned 3-torus universe . In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching zero. With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases. The ultimate fate of the universe is the same as that of an open universe. A flat universe can have zero total energy .