Let $M=S^1 \times \mathbb{R}^n$ with the metric $g=-d\psi^2+q$, where $d \psi^2$ is the standard metric on $S^1$ and $q$ is the euclidean metric on $\mathbb{R}^n$.
I know that the universal covering of $S^1$ is given by
$$\mathbb{R} \rightarrow S^1, t \mapsto e^{2 \pi i t}$$
Is is then correct that the universal covering of $M$ is given by the Minkowski space-time
$$ \mathbb{R} \times \mathbb{R}^n \rightarrow S^1 \times \mathbb{R}^n, (t,x) \mapsto (e^{2 \pi i t}, x)$$ ?
