Consider a right circular cylinder of radius a centred on the z axis. Show that in cylindrical polar coordinates that an element of arc length on the surface of a cylinder is given by $$ds^2 = a^2dφ^2 + dz^2$$
Following on from this, the next part asks: Find the equation giving φ as a function of z for the geodesic (shortest path) on the cylinder between two points with cylindrical polar coordinates (a, φ1, z1) and (a, φ2, z2). I think i have done it but wondered how you'd go about this problem.
The parametric equation of the cylinder is $$ r=\{a\cos\varphi,a\sin\varphi,z\}. $$ So $$ ds^2=dr\cdot dr=\{-a\sin\varphi d\varphi,a\cos\varphi d\varphi,dz\}\cdot \{-a\sin\varphi d\varphi,a\cos\varphi d\varphi,dz\}=a^2(\sin^2\varphi+\cos^2\varphi)d\varphi^2+dz^2=a^2d\varphi^2+dz^2. $$