As in the title I want to prove that the $n$-dimensional puntunctured euclidean space: $$(\mathbb{R^n}\setminus\{0\}, g_\mathcal{E})$$
where $g_\mathcal{E}$ is the canonical metric induced by the euclidean on $\mathbb{R}^n$
and the $n-1$-dimensional cylinder: $$((0,+\infty)\times S^{n-1}, dr^2+r^2g_\mathcal{E})$$ where $dr^2$is the canonical euclidean metric on $(0,+\infty)$ and $g_\mathcal{E}$ the canonical one induced by $\mathbb{R}^n$ on $S^{n-1}$.
Given the specific metric on the cylinder (which is not the product one) I am trying to find an explicit isometry, but cannot succed.
I would be very grateful for any help.