The action spectrum characterizes closed contact 3-manifolds all of whose Reeb orbits are closed
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.10617
- arXiv:
- arXiv:1901.10617
- Bibcode:
- 2019arXiv190110617C
- Keywords:
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- Mathematics - Symplectic Geometry;
- Mathematics - Differential Geometry;
- Mathematics - Dynamical Systems;
- 53C22;
- 58E10
- E-Print:
- 18 pages