  Is the universe finite or infinite, and what shape does it have? These
fundamental questions, of which relatively little is known, are typically
studied within the context of the standard model of cosmology where the
universe is assumed to be homogeneous and isotropic. Here we address the above
questions in highly general cosmological models, with the only assumption being
that the average flow of matter is irrotational. Using techniques from
differential geometry, specifically extensions of the Bonnet-Myers theorem, we
derive a condition which implies a finite universe and yields a bound for its
diameter. Furthermore, under a weaker condition involving the interplay between
curvature and diameter, together with the assumption that the universe is
finite (i.e., has closed spatial slices), we provide a concise list of possible
topologies. Namely, the spatial sections then would be either the ring
topologies $S^1 \times S^2$, $S^1\tilde{\times}S^2$, $S^1\times\mathbb{RP}^2$,
$\mathbb{RP}^3 \# \mathbb{RP}^3$, or covered by the sphere $S^3$ or torus
$T^3$. In particular, under this condition the basic construction of connected
sums would be ruled out (save for one), along with the plethora of topologies
associated with negative curvature. These results are obtained from
consequences of the geometrization of 3-manifolds, by applying a generalization
of the almost splitting theorem together with a curvature formula of Ehlers and
Ellis.
