 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 isIt 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, isFor piecewise smooth boundary, add the signed exterior turning angles at its corners to the left side.