= Solution
Parametrize a regular curve by arclength $s$, write its unit tangent as $T$, and choose an oriented surface unit normal $N$. Its signed <geodesic curvature> is
$$
\boxed{k_g=\langle dT/ds,N\times T\rangle.}
$$
It vanishes exactly when the acceleration of the unit-speed curve is normal to the surface, equivalently when the curve is a geodesic with arclength parametrization. For an arbitrary regular parametrization, this means the curve becomes a geodesic after reparametrization.
The global <Gauss-Bonnet theorem> for a compact oriented surface with smooth boundary, positively oriented with the surface on the left, is
$$
\boxed{\int_SK\,dA+\int_{\partial S}k_g\,ds=2\pi\chi(S).}
$$
For piecewise smooth boundary, add the signed exterior turning angles at its corners to the left side.
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