In an article called synthetic geometry in Riemannian manifolds M. Gromov says that

tori (and in general flat manifolds) must be seen as

local geometric objects.

He does so after making an example where he shows that, fixed a scale at which to look at a manifold, i.e. the radius $\epsilon$ of the neigbourbood of a point $U_a$, then its topology is quite free. In particular it can be produced a flat $n$-dimensional manifold where such a neigbourhood looks like the product of an $k$-torus by a $(n-k)$-ball, for any $k$.

My understanding of this observation comes from comparing this example with the case of negative curvature where once fixed the (bounds on the) curvature we have estimates on the systole, or the injectivity radius. It follows that below a given scale the topology of a negatively curved manifold has some restrictions, if it is not trivial. Hence the geometry of such objects happens at bigger scales (or even global ones, as the boundary of the fundamental groups of negatively curved spaces carry a lot of information).

I'm not quite satisfied with this point of view and I would like to recieve further comments on this point.

He also talks about *nontrivial local objects* and *nontrivial local geometry* but I'm not sure about what he means with these expressions