= Solution
For local surface coordinates $u^1,u^2$ and metric coefficients $g_{ij}$, affinely parametrized <geodesics> satisfy
$$
\boxed{\ddot u^k+\Gamma^k_{ij}\dot u^i\dot u^j=0,\qquad
\Gamma^k_{ij}=\frac12g^{k\ell}(\partial_i g_{j\ell}+\partial_jg_{i\ell}-\partial_\ell g_{ij}).}
$$
These are the <geodesic equation> and <Christoffel symbols>. They follow by applying the <Euler-Lagrange equation> to the energy $\tfrac12g_{ij}\dot u^i\dot u^j$. 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 $a^2d\theta^2+dz^2$, with constant coefficients, so all <Christoffel symbols> vanish in this chart. Thus
$$
\boxed{\theta(s)=\theta_0+cs,\qquad z(s)=z_0+ds.}
$$
These <geodesics on a circular cylinder> are straight generators when $c=0$, circles around the cylinder when $d=0$, and <helices> when $cd\ne0$. Constant curves arise if both vanish. For unit-speed nonconstant curves, $a^2c^2+d^2=1$. This description is global after allowing the angle to run through arbitrary multiples of $2\pi$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2009/ib/paper-3-cylinder-geodesics.png]
{title=Generators, circular parallels and helices on a cylinder become straight lines when the cylinder is unrolled}
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.
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