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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

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  • $\begingroup$ It's hard to say since I don't think you've defined $n$, but do you really mean to take the product of an $n$-torus and an ($n - k$)-ball (rather than, say, a $k$-torus and an ($n - k$)-ball)? $\endgroup$
    – LSpice
    Jul 6 at 20:19
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    $\begingroup$ No, in the negative curvature case (even constant $-1$) you do not get bounds on the systole (neither upper nor lower). $\endgroup$ Jul 6 at 23:18
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    $\begingroup$ One should also add the compactness assumption here and many even the condition that dimension is $>3$. $\endgroup$ Jul 6 at 23:53

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Maybe this relates to how being a locally flat space is rather restrictive on the global structure of the space. A locally flat space $(M,g)$ is characterized by the equation $\mathrm{Rm}_{g}=0$ on the curvature tensor $g$, which is a completely local equation. On the other hand, $(M,g)$ is in particularly a locally symmetric space with Einstein constant $\lambda=0$. The classification theorem for locally symmetric spaces states that there exists a simply connected complete manifold $(\tilde{M},\tilde{g})$ and a discrete Lie group $\Gamma$ that acts acts freely, properly, and isometrically on $\tilde{M}$ such that $M=\tilde{M}/{\Gamma}$ and $\tilde{M}$ has the same Einstein constant as $M$. This implies that $\tilde{M}$ is a simply connected flat space, hence must be isometric to $\mathbb{R}^{n}$. This shows that $M$ is virtually the Euclidean product of flat tori and an Euclidean space, hence its global structure is fully characterized.

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