  Constructing an extension of Newton's theory which is defined on a
non-Euclidean topology (in the sense of Thurston's decomposition), called a
non-Euclidean Newtonian theory, corresponding to the zeroth order of a
non-relativistic limit of general relativity is an important step in the study
of the backreaction problem in cosmology and might be a powerful tool to study
the influence of global topology on structure formation. After giving a precise
mathematical definition of such a theory, based on the concept of Galilean
manifolds, we propose two such extensions, for spherical or hyperbolic
topologies, using a minimal modification of the Newton-Cartan equations.
However as for now we do not seek to justify this modification from general
relativity. The first proposition features a non-zero cosmological
backreaction, but the presence of gravitomagnetism and the impossibility of
performing exact $N$-body calculations make this theory difficult to be
interpreted as a Newtonian-like theory. The second proposition features no
backreaction, exact $N$-body calculation is possible and no gravitomagnetism
appears. In absence of a justification from general relativity, we argue that
this non-Euclidean Newtonian theory should be the one to be considered, and
could be used to study the influence of topology on structure formation via
$N$-body simulations. For this purpose we give the mass point gravitational
field in $\mathbb{S}^3$.
