Solution
= Solution
Tangency along $\gamma$ means
$$
T_{\gamma(t)}\Sigma_1=T_{\gamma(t)}\Sigma_2
$$
for every $t$. Consequently $\ddot\gamma(t)$ is perpendicular to the first tangent plane exactly when it is perpendicular to the second. The ambient characterization of a geodesic therefore proves the <tangency invariance of an ambient-surface geodesic>:
$$
\boxed{\gamma\text{ is geodesic in }\Sigma_1
\Longleftrightarrow
\gamma\text{ is geodesic in }\Sigma_2}.
$$