You have a closed curve on a Euclidean manifold $c: S^1 \to M$ and a tangent vector $v \in T_{x_0} M$ which you parallel translate around $c$ using the LeviCivita connection. The monodromy representative of this curve should be a rotation on $T_{x_0}M$ (which is a plane in our case). Could you have gotten this angle by integrating the curvature over the area enclosed by the curve? What is the name of that result?
