Suppose that $m = 1$ and $n=2$, hence you are in $\mathbb{R}^3$. You should see that the space $X \subseteq \mathbb{R}^3$ is the unitary sphere $S^2 \subseteq \mathbb{R}^3$ minus the two antipodal points $(-1,0,0)$ and $(1,0,0)$. It should be clear that $X$ is homeomorphic to the meridian $\mathbb{R}$ times the equator $S^2 \cap \{ x = 0 \} \simeq S^1$, i.e. $\mathbb{R} \times S^1$ as wanted. This is a sort of Mercator projection of the earth planet without the two poles onto an infinite cylinder which is tangent to the equator. Can you write down formulas in this particular case?
Try to generalize the formulas found in the particular case.