= Liouville volume of a surface geodesic flow
{c}
{title2=$\mu=\alpha\wedge\theta\wedge\omega$}
The positive Liouville volume for the base-orientation/positive-fibre orientation is $\mu=\pi^*\Omega_a\wedge\omega$. Its fibre integral is $2\pi\Omega_a$, so its total volume is $2\pi\operatorname{Area}(M)$. 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.
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