# The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed

## TL;DR

This paper characterizes closed contact 3-manifolds with all Reeb orbits closed by showing equivalence of conditions involving common periods and the rank of the action spectrum, providing a classification framework.

## Contribution

It establishes the equivalence of conditions for Reeb flows on 3-manifolds and characterizes contact forms with rank 1 action spectrum up to diffeomorphism.

## Key findings

- All Reeb orbits are closed if and only if the action spectrum has rank 1.
- A contact form with rank 1 action spectrum is uniquely determined by minimal periods.
- The conditions for closed Reeb orbits are equivalent on closed connected 3-manifolds.

## Abstract

A classical theorem due to Wadsley implies that, on a connected contact manifold all of whose Reeb orbits are closed, there is a common period for the Reeb orbits. In this paper we show that, for any Reeb flow on a closed connected 3-manifold, the following conditions are actually equivalent: (1) every Reeb orbit is closed; (2) all closed Reeb orbits have a common period; (3) the action spectrum has rank 1. We also show that, on a fixed closed connected 3-manifold, a contact form with an action spectrum of rank 1 is determined (up to pull-back by diffeomorphisms) by the set of minimal periods of its closed Reeb orbits.