Past exam of the mathematics course of the University of Cambridge / 2009 / ib / Paper 3 / 2G / Solution

For local surface coordinates $u^1,u^2$ and metric coefficients $g_{ij}$, affinely parametrized geodesics satisfyThese are the geodesic equation and Christoffel symbols. They follow by applying the Euler-Lagrange equation to the energy $\frac{1}{2}{g}_{ij}{\dot{u}}^i{\dot{u}}^j$. Equivalently, in an embedded surface a constant-speed geodesic has acceleration normal to the surface.

Parameterize a circular cylinder by $(a\cos \theta ,a\sin \theta ,z)$, where $a>0$. Its induced metric is $a^{2}d{\theta }^2+d{z}^2$, with constant coefficients, so all Christoffel symbols vanish in this chart. ThusThese geodesics on a circular cylinder are straight generators when $c=0$, circles around the cylinder when $d=0$, and helices when $cd\ne 0$. Constant curves arise if both vanish. For unit-speed nonconstant curves, $a^2{c}^2+d^2=1$. This description is global after allowing the angle to run through arbitrary multiples of $2\pi$.

The unrolled coordinate $u=a\theta$ exhibits the intrinsic flatness of a circular cylinder: every displayed family is a straight line in $(u,z)$. A circle is therefore a geodesic despite bending in ambient three-dimensional space; its acceleration is normal to the cylinder.