# How many unequal metrics can one define on a torus such that the projection map from $\mathbb R^n$ to the torus is a local isometry?

In how many ways can one define a Riemannian metric on $$T^n$$ such that the projection map $$\pi:\mathbb R^n\rightarrow T^n$$ defined by $$\pi(x_1,...,x_n)=(e^{ix_1},...,e^{ix_n})$$ is a local isometry?

If you have a covering projection $$\pi\colon M\to N$$ and $$M$$ is a Riemannian manifold whose Riemannian metric $$g$$ is invariant under the group of deck (covering) transformations, then the only metric on $$N$$ which makes $$\pi$$ a local isometry is the obvious one, namely $$(\pi^{-1})^*g$$. This is computed locally (since $$\pi$$ has smooth local inverses), but gives a well-defined metric on $$N$$ because of the assumption on the invariance of $$g$$. To emphasize: There is a unique metric on $$N$$ so that $$\pi$$ is a local isometry.