Periodic orbits of non-degenerate lacunary contact forms on prequantization bundles

Abstract

A non-degenerate contact form is lacunary if the indexes of every contractible periodic Reeb orbit have the same parity. To the best of our knowledge, every contact form with finitely many periodic orbits known so far is non-degenerate and lacunary. We show that every non-degenerate lacunary contact form on a suitable prequantization of a closed symplectic manifold $B$ has precisely $r_B$ contractible closed orbits, where $r_B=\dim \mathrm{H}_*(B;{\mathbb Q})$. Examples of such prequantizations include the standard contact sphere, the unit cosphere bundle of a compact rank one symmetric space (CROSS) and many others. We also consider some prequantizations of orbifolds, like lens spaces and the unit cosphere bundle of lens spaces, and obtain multiplicity results for these prequantizations.