Let $(\Sigma,g)$ be an oriented Riemannian surface, let $U\subset\Sigma$ be an [open set](/page/Open%20Set) with local oriented orthonormal frame $(e_1,e_2)$, and let $\alpha\in\Omega^1(U)$ be the connection form. Under the standard identification $\mathfrak{so}(2)\cong \mathbb{R}$ given by the generator $J_{11}=0$, $J_{12}=-1$, $J_{21}=1$, and $J_{22}=0$, the scalar coefficient $F$ of the associated principal $SO(2)$-connection curvature is locally represented by $F=d\alpha$. With the convention $\nabla_Xe_1=\alpha(X)e_2$ and $R(X,Y)=\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]}$, one has $d\alpha= -K\,\theta_1\wedge \theta_2$, where $K$ is the Gaussian curvature and $(\theta_1,\theta_2)$ is the dual coframe.