= Solution
An orientation of $S$ chooses a unit normal at every point. The <Gauss map> is
$$
N:S\longrightarrow S^2,
\qquad p\longmapsto N(p).
$$
For the unit tangent $T=\dot\alpha$, the <surface covariant derivative> is the tangential projection
$$
D_sT=(\ddot\alpha)^{\top}.
$$
The signed <geodesic curvature> is
$$
\boxed{\kappa_g=\langle D_sT,N\times T\rangle}.
$$
Because $|T|=1$, $D_sT$ is perpendicular to $T$ and hence
$$
D_sT=\kappa_g(N\times T).
$$
\b[Consequently $\kappa_g=0$ means $D_sT=0$: the tangent is parallel along $\alpha$, so $\alpha$ is a <geodesic>.]
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