Contact Geometry and Lens Spaces

There exists a canonical symplectic structure on the cotangent bundle of a differentiable manifold and similarly a canonical contact structure on the positive projectivization of the cotangent bundle under the scaling R action. Given a Riemannian (or Finsler) metric on the original manifold, this contact structure is contactomorphic to the contact structure determined by the metric on the unit tangent bundle. In symplectic topology it is a general question to ask to what extent the symplectic or contact geometry of these structures determines the smooth structure of the smooth underlying manifold. For example one might expect symplectic invariants to give interesting new smooth invariants. In this paper we study the canonical contact structures associated to 3 dimensional Lens spaces L(r, s). These are the first examples of manifolds which may be homotopic but not diffeomorphic. We will see that the contact structures on their unit tangent bundles do indeed distinguish the smooth structures on the Lens spaces. We note that homotopic but non diffeomorphic Lens spaces still have diffeomorphic tangent bundles. Therefore we do need contact methods to distinguish these structures and in particular we discover examples of different 5 dimensional contact structures on the same manifold (with the same Chern class). Such examples are still quite rare (but see [6]). One natural approach to this problem is to compute the contact homology of our manifolds (see [3]). This was the approach of Ustilovsky. However the calculations in sections 2 and 3 show that this fails to separate the contact structures. Contact homology is a homology theory with chain groups generated by certain periodic orbits of a Reeb vector field on our contact manifold and with the differential defined by counting holomorphic curves. We will see that nondegenerate Reeb vector fields on our contact manifolds have isomorphic periodic orbits. Moreover these orbits are such that the necessary differentials vanish in all cases. Motivated by work of Bourgeois [1] we study holomorphic curves corresponding to natural Morse-Bott contact forms in section 4. Relative to these contact forms we can describe fairly explicitly various moduli spaces of holomorphic curves. They do not appear in the differential defining contact homology although would certainly influence the more subtle invariants coming from Symplectic Field Theory (see [2]). In any case, in section 4 we show directly that properties of these moduli spaces of holomorphic curves are enough to distinguish our contact manifolds.