Canonical coframe of a surface unit tangent bundle ($(\alpha ,\theta ,\omega )$)

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 $[V,X]=H$, $[V,H]=-X$, $[X,H]=KV$.

On an oriented Riemannian surface, the vertical vector field generates positive rotation in each fibre of its unit tangent bundle: $Vf(x,v)={{\partial }_{t}f(x,\cos tv+\sin tiv)|}_{t=0}$. Hence $V(v)=iv$ and $V(iv)=-v$. In the canonical frame, $[V,X]=H$, $[V,H]=-X$ and $[X,H]=KV$.

The positive Liouville volume for the base-orientation/positive-fibre orientation is $\mu ={\pi }^{\ast }{\Omega }_a\wedge \omega$. Its fibre integral is $2\pi {\Omega }_a$, so its total volume is $2\pi \mathrm{Area}(M)$. With the stated coframe convention, $\alpha \wedge d\alpha =-\mu$; the signed contact volume and the positive measure should not be confused. The fields $X,H,V$ preserve this volume by their structure equations.