 Canonical coframe of a surface unit tangent bundle

On an oriented Riemannian surface, set $\alpha (\xi )=⟨v,d\pi \xi ⟩$, $\theta (\xi )=⟨iv,d\pi \xi ⟩$, and let $\omega$ be the Levi-Civita connection form with $\omega (V)=1$. The dual frame is $(X,H,V)$ andIn 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 $[V,X]=H$, $[V,H]=-X$, $[X,H]=KV$.