The flat torus metric from periodic euclidean coordinates

The map $q$ is open since $q^{-1}q(U)={\bigcup }_{m\in {\mathbb{Z}}^n}(U+m)$ is open when $U$ is. Its restriction to a box of side lengths less than $1$ is injective and hence a homeomorphism onto its open image. On overlap components the inverse charts differ by a constant integer translation. These smooth maps have derivative $I_n$.

For two distinct orbits represented by $x,y$, their displacement vectors $x-y-m$ never vanish. Only finitely many $m\in {\mathbb{Z}}^n$ have $\mathrm{\parallel }x-y-m\mathrm{\parallel }\le 1$, because every coordinate of such an integer vector is bounded. Taking the minimum of $1$ and these finitely many positive lengths gives $\delta >0$. Images of radius-$\delta \mathrm{/}3$ balls around $x,y$ are disjoint, proving Hausdorffness. The images of rational boxes form a countable basis. Thus the charts in step 1.1 define a smooth manifold.

Since all transition derivatives are $I_n$, the local tensors ${\sum }_{i}d{x}_i^2$ agree on overlaps. They glue to a smooth tensor with positive-definite identity matrix in every quotient chart. The coordinate criterion proves it is Riemannian. For example in dimension two the local vector $2{\mathrm{\partial }}_1+3{\mathrm{\partial }}_2$ has squared norm $4+9=13$, independent of the integer translate chosen for the chart. Local equality to the Euclidean metric is the flatness meant here.