Solution
= Solution
A regularly parametrized curve $\gamma$ on an embedded surface $\Sigma$ is a geodesic when its covariant <acceleration> vanishes. Equivalently, in the ambient Euclidean space,
$$
\ddot\gamma(t)\perp T_{\gamma(t)}\Sigma
$$
for an affine geodesic parameter. A <complete geodesic> is one whose maximal affine parameter interval is all of $\mathbb R$.