Calculating geodesic curvature for a general curve

Question

A cylinder of radius $R$ can be parameterized by $X(\theta, z) = [R\cos\theta, \sin\theta, z]$, where $-\pi < \theta < \pi$ and $\infty < z < \infty$.

Part b of a question I'm working on (studying for an exam) asks me to calculate the geodesic curvature for a general curve - I am stuck on this. Part a asks to find the metric and the normal to the surface, so I assume those quantities are useful in the part I am stuck on.

If anyone could give me guidance on how to calculate geodesic curvature for a general curve on the above surface, that would be great.

Thanks.

Answer 1

On the cylinder $C$ consider the curve $\gamma(t)=(\cos t,\sin t, h(t))$, where $h:\mathbb{R}\to\mathbb{R}$ is some smooth function.

the geodesic curvature of $\gamma$ is $$κ_g=\dfrac{h''(t)}{(1+h'(t)^2)^{3/2}}$$

The geodesic curvature of $\gamma$ vanishes if and only if $h(t)=at+b$ for certain constants a and b (line a plane).

Use the isometry between the cylinder and plane to argue that the geodesic curvature of the curve $\gamma$ on the cylinder must be the same as that of the graph of $u= h(v)$ in the plane.