Contact foliations and generalised Weinstein conjectures

Abstract

We consider contact foliations: objects which generalise to higher dimensions the flow of the Reeb vector field on contact manifolds. We list a number of properties of such foliations, and propose two conjectures about the topological types of their leaves, both of which coincide with the classical Weinstein conjecture in the case of contact flows. We give positive partial results for our conjectures in particular cases -when the holonomy of the contact foliation preserves a Riemannian metric, for instance -extending already established results in the field of Contact Dynamics.

Compact leaves of the foliation defined by the kernel of a <i>T</i>²-invariant presymplectic form

“…, R r ∈ R. On the other hand, in Theorem 7, M does not always satisfy that Z 1 ∈ ker ω, as (8) in Example 5 shows. Thus, [3,Theorem 3.23] and Theorem 7 are independent results.…”

“…For instance, Viterbo [8] proved it for compact contact manifolds which are hypersurfaces of contact type in (R 2n , ω 0 ), where ω 0 = n j=1 dy j ∧ dx j is the standard symplectic form. Hofer [4] proved this conjecture for S 3 and for overtwisted 3-dimensional contact manifolds, and after that Taubes [7] solved it affirmatively in dimension 3.…”

“…However, they did not mention the case where dim F ω ≥ 2. For such a direction, it seems that there are few works which assume that (M, ω) is r-contact (see Remark in Section 3, and [3] for details).…”

“…, α r with a splitting T M = R ⊕ (∩ r i ker α i ) satisfying that dα i | ∩ker αi is nondegenerate and ker dα i = R for every i. In our context, R corresponds to ker ω. Finamore [3,Theorem 3.23] proved that if a closed presymplectic manifold (M, ω) is r-contact with a special metric, then the r-dimensional foliation defined by R has at least two leaves which are homeomorphic to an r-dimensional torus. In the case where M is r-contact, by definition, M admits a locally free R r -action with the infinitesimal generators R 1 , .…”

Multicontact formulation for non-conservative field theories

“…This was developed by some of the authors in [46,48,61,83]. Other less-general approaches to these kinds of geometric structures were also introduced in [8,19,44,79,87]. (See also [3,5] for other generalizations of contact geometry related with polysymplectic geometry and other geometric structures).…”