  We study the non-relativistic (NR) limit of relativistic spacetimes in
relation with the topology of the Universe. We first show that the NR limit of
the Einstein equation is only possible in Euclidean topologies, i.e. for which
the covering space is $\mathbb{E}^3$. We interpret this result as an
inconsistency of general relativity in non-Euclidean topologies and propose a
modification of that theory which allows for the limit to be performed in any
topology. For this, a second reference non-dynamical connection is introduced
in addition to the physical spacetime connection. The choice of reference
connection is related to the covering space of the spacetime topology. Instead
of featuring only the physical spacetime Ricci tensor, the modified Einstein
equation features the difference between the physical and the reference Ricci
tensors. This theory should be considered instead of general relativity if one
wants to study a model universe with a non-Euclidean topology and admitting a
non-relativistic limit.
