Removing coordinate singularities

Question

Consider the Riemannian metric given in this picture. Wikipedia claims that this Riemannian manifold has the topology $\mathbb{R}^2\times S^2$, but the coordinate expression given in the picture seems to be only smooth and non-singular for $r>r_0$ and $0<\theta<\pi$, giving a Riemannian metric on $$(r_0,\infty)\times S^1\times(S^2\setminus\{N,S\})\cong(\mathbb{R}^2\setminus\{0\})\times(S^2\setminus\{N,S\}),$$ where $N$ and $S$ are the north and south poles on $S^2$. Since Wikipedia claims the topology is $\mathbb{R}^2\times S^2$, my guess is that the singularities at $\theta=0$, $\theta=\pi$ and $r=r_0$ are just coordinate singularities. I want to show this more rigorously. In other words, I want to show that the metric in the picture extends smoothly to a Riemannian metric on all of $\mathbb{R}^2\times S^2$, and my guess is that one would have to come up with a clever choice of new coordinates in which these singularities vanish. But how does one approach the problem of coming up with such coordinates?