Show that the cylinder $C^{n-1}$ is diffeomorphic to $S^{n-m-1}\times \mathbb{R}^{m}.$

Question

Show that the cylinder $C^{n-1}$ is diffeomorphic to $S^{n-m-1}\times \mathbb{R}^{m}.$

Here, diffeomorphic is manifold sense and we caratherize $C^{n-1}$ by

$$C^{n-1}=\{p \in \mathbb{R}^{n} ; |p-p_{0}|^{2}-\langle p-p_{0},a\rangle^{2}=r^2\},$$ where $a\in\mathbb{R}^{n}, |a|=1$ and $r>0.$

My thought was consider $m=1,$ $p_{0}=0$ and $a=(0,0,\dots,1)$ because, the other case is just a rotation and translation. So, we have

$$C^{n-1}=\{p=(p_{1},\dots,p_{n}) \in \mathbb{R}^{n} ; p_{1}^{2}+\dots+p_{n-1}^{2}=r^2\}.$$

And now look trivial that $C^{n-1}$ is diffeomorphic to $S^{n-2}\times \mathbb{R}.$

My question is: is simples as I write above or I need to work with charts?