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?
 A: Let $S_r$ be the circle of radius $r$ in $\mathbb R^2$. Let $C$ be a curve inside this $S_r$ and have length equals that of $S_r$. Then $S_r\times \mathbb R$ is isometric to $C\times \mathbb R$: Let $i: S_r \to C$ be the unit length parametrization of $C$, then 
$$ \phi : S_r\times \mathbb R \to C\times \mathbb R, \ \ \ \phi(s, t) = (i(s), t)$$
is an isometry. Thus your argument is not rigorous, that one surface is "inside" the other one does not mean that they are not isometric. 
However, your idea is definitely a good one. Mathematically, you need to know that if 
$$\phi: C_{r_1} \to C_{r_2}$$
is an isometry, then $r_1=r_2$. Using your observation, consider the geodesic $ S_{r_1} \times \{0\}\subset C_{r_1}$. The image of this geodesic under $\phi$ is also a closed geodesic in $C_{r_2}$. Can you show that this geodesic is also of the form $S_{r_2} \times \{t\}$ for some $t$? If yes, then as isometry preserves length, one has 
$$ 2\pi r_1 = 2\pi r_2 \Rightarrow r_1 = r_2.$$
So it really spoils down to this question: 

Are all closed geodesics in $C_r$ of the form $S_r \times \{t\}$ for some $t$? 

A: 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 $r>0$?
A: The intersection of a cylinder and plane is never a geodesic. (except there are 2 inflections points) The development is a sine curve which is not a straight line. 
To visualize and model a cylinder containing a bent helical line,
take a rectangular $xz$ plane between $ x=\pm a, z=\pm b $ containing a transverse line. Imagine plane, portion of interest to be a thin flexible plastic sheet. Bend the sheet/plane to make bent edge to be a circle segment  of a cylinder tangent to the plane. Somewhat as you imagined, the $ r$ in $(x-r)^2 +y^2 =r^2$ is a variable in bending.The series of circular segments which are boundary of bent cylinder are bent circumferentials. The bent circular sheet/segment contains your helix line whose original geodesic length is always preserved.
