Geodesics on Cylinders I have a question about Geodesics on Cylinders and think I have the right answer but am unsure. The question reads:

Let $C_r:=[(x,y,z)\in\mathbb{R}^3: x^2+y^2=r]$ be the infinite cylinder of radius $r$. Show that $C_{r_1}$ is isometric to $C_{r_2}$ iff $r_1=r_2$.

Now I understand the logic behind this question I think. An isometry preserves geodesics, and because if you intersect a plane parallel to the axis of the cylinder with this cylinder, you get a curve $C$, which is just a circle, that is a geodesic. Now, if the radius between the two cylinders are different, the smaller circle would lie inside of the bigger cylinder, thus not lying on the surface and definitely not a geodesic.
Is this okay to write? Or do I have to explain it mathematically?

Another possible way to proceed.
Hint: Isometries preserve the length of geodesics. What is the minimal length of a closed, simple geodesic in a cylinder of radius $r>0$?
