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

TL;DR

This paper proves that non-degenerate lacunary contact forms on certain prequantizations have a number of contractible closed orbits equal to the Betti number of the base manifold, extending known results to orbifolds.

Contribution

It establishes a precise count of contractible periodic Reeb orbits for lacunary contact forms on prequantizations, including orbifold cases, based on topological invariants.

Findings

Number of contractible orbits equals Betti number of base manifold

Results apply to standard contact sphere and cosphere bundles of CROSS

Includes multiplicity results for orbifold prequantizations

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.