definition of geodesic cuvature, $k_g$. Projection of the Normal vector of a curve into the Tangent space of a surface at the point where we are evaluating the normal vector of the curve.

It represents the acceleration of the curve inside the surface, that is projected to the surface, locally. So one can see that a curve with $0$ geodesic everywhere curvature is a Geodesic!

Important theorem: Let $\gamma$ be a smooth closed simple Curve on a patch of a Surface $S$, enclosing a region $R$. Then:

$\int_\gamma k_g ds = 2\pi - \int_R K dA$

where $k_g$ is the geodesic curvature along the curve, $K$ is the Gaussian curvature, and $dA$ is the measure of Area induced on the surface by the First fundamental form. – Picture

Proof