= Canonical coframe of a surface unit tangent bundle
{title2=$(\alpha,\theta,\omega)$}
On an oriented Riemannian surface, set $\alpha(\xi)=\langle v,d\pi\xi\rangle$, $\theta(\xi)=\langle iv,d\pi\xi\rangle$, and let $\omega$ be the Levi-Civita connection form with $\omega(V)=1$. The dual frame is $(X,H,V)$ and
$$
d\alpha=\omega\wedge\theta,\qquad d\theta=\alpha\wedge\omega,\qquad d\omega=-K\alpha\wedge\theta.
$$
In a local oriented frame, $\omega=d\vartheta+\omega_0$ and $\alpha,\theta$ are the base coframe rotated through angle $\vartheta$; the torsion-free and curvature equations give these identities. Evaluating exterior derivatives on dual fields yields $[V,X]=H$, $[V,H]=-X$, $[X,H]=KV$.
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