Complex vs Convex Morse Functions and Geodesic Open Books

Abstract

Suppose that $\Sigma$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are four seemingly distinct constructions of open books on the unit (co)tangent bundle of $\Sigma$, having complex, symplectic, contact, and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $\Sigma$. We show that the resulting open books are pairwise isomorphic provided that the ordered Morse function is adapted to the admissible divide on $\Sigma$. Moreover, we observe that if $\Sigma$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.