Let $\Lambda\subset\mathbb{R}^n$ a lattice, i.e., a discrete subgroup that spans $\mathbb{R}^n$. Now we can look at the torus $T=\mathbb{R}^n/\Lambda$ which naturally carries the metric $d\_T$ induced by the euclidean metric $d$ on $\mathbb{R}^n$: $$d\_T(x+\Lambda,y+\Lambda):=\min\_{a,b\in\Lambda}(d(x+a,y+b)).$$ Now my question is the following: What information about $\Lambda$ can be recovered from the metric space $T$? Can we completely recover $\Lambda$ (up to some orthogonal equivalence)?



What information about the lattice $\Lambda$ can be recovered from the metric space $\mathbb{R}^n/\Lambda$?
Yes, we can. Consider the universal cover $U$ of your torus $T$. One can easily show that $U=R^n$ equipped with a Euclidean metric.
So we have $f:R^n\to T$,


The $f$-preimage of a point is your lattice, up to the shift of the origin (to one point of this preimage) and an orthogonal transformation (an isometry of the Euclidean metric).


