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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 topologiesS¹×S²,$S^1\tilde{\times }{S}^2$,$S^1\times \mathbb{R}{\mathbb{P}}^2$,$\mathbb{R}{\mathbb{P}}^3\#\mathbb{R}{\mathbb{P}}^3$, or covered by the sphereS³or torusT³. 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 three-manifolds, by applying a generalization of the almost splitting theorem together with a curvature formula of Ehlers and Ellis.

We prove positive mass theorems for asymptotically hyperbolic (AH) and asymptotically locally hyperbolic Riemannian manifolds with black-hole-type boundaries.