On characteristics of circle invariant presymplectic forms

Abstract

Abstract. We prove that a circle-invariant exact 2-form of rank 2n on a compact (2n+ 1 )-dimensional manifold admits two closed cha^cteristics. This solves a particular case of a generalized Weinstein conjecture.

Contact foliations and generalised Weinstein conjectures

“…Section 3 is heavily inspired by the work of Rukimbira and Banyaga (cf. [41,7]), and culminates in the proof of Theorem 1.1. We show more, proving that an isometric contact foliation on a closed manifold must have at least two closed orbits and that the set of closed orbits decomposes into a disjoint union of invariant even-codimensional submanifolds (cf.…”

“…It follows then, from classical results from Myers-Steenrod [34] and Kobayashi [28], that for Riemannian foliations the leaf closures have the necessary properties: they are submanifolds of even codimension. Following the steps laid down by Banyaga and Rukimbira in [7], we can show that Riemannian contact foliations satisfy the WGWC. Going even further, a reduction argument can be used to prove Theorem 1.1.…”

“…Though not yet proven in its full generality, the conjecture is known to hold in several particular cases: in dimension three [44,26]; for PS-overtwisted contact forms [1] and, more generally, for overtwisted contact manifolds [14]; and finally, when the Reeb vector field is Killing for some metric on M [41,7]. Given the analogy between contact foliations and Reeb flows, it is reasonable that one asks oneself whether or not some Weinstein-like results exist for contact foliations as well.…”

Geodesic and Conformally Reeb Vector Fields on Flat 3-Manifolds

“…If c < 0, then |λ|, |µ| ≤ |c|. In the latter two cases, the conditions are clearly necessary by (2).…”

“…The closure H := G of G is abelian and hence isomorphic to a torus. Note that H acts on (M, α) by strict contactomorphisms: Indeed, since H is abelian, we have that φ t • h = h • φ t for every h ∈ H, where φ t denotes the time-t flow of R. Differentiating this equation with respect to t yields h * R = R. Using the fact that α = i R g, a simple computation then shows that h * α = α (see [2,Proposition 1]). Via the identification of K(M ) with isom(M ), the vector field R corresponds to an element r ∈ h ⊂ isom(M ), where h denotes the Lie algebra of H. Choose an element r ′ ∈ B 1/2 (r) ∩ h corresponding to a periodic vector field R ′ , where B 1/2 (r) denotes the open ball of radius 1/2 with respect to the norm || • || ∞ on isom(M ).…”