Complex vs. convex Morse functions and geodesic open books

Abstract

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