On vector fields having properties of Reeb fields. (English) Zbl 1317.53113

Hajduk, Bogusław; Walczak, Rafał

Topol. Methods Nonlinear Anal. 41, No. 2, 401-408 (2013).

The aim of this paper is to study vector fields with properties similar to those of the Reeb vector field on a contact manifold. Given a non-vanishing vector field \(V\), a 1-form \(\eta\) is said to be a connection form if \(\iota_Vd\eta=0\) and \(\eta(V)=1\). D. Sullivan [J. Pure Appl. Algebra 13, 101–104 (1978; Zbl 0402.57015)] proved that \(V\) admits a connection form if and only if it is geodesible, i.e., the manifold admits a Riemannian metric with respect to which the orbits of \(V\) are geodesics.
The authors first prove that a closed oriented odd-dimensional manifold \(M\) admits a geodesible vector field \(V\). The proof is based on a study of the open book decomposition of the manifold. Moreover the connection form \(\eta\) of \(V\) may be chosen such that \([d\eta]_b\) is a non-zero class in \(H^2_b(M,{\mathfrak V})\), the basic cohomology group of the foliation \(\mathfrak V\) induced by \(V\).
A closed non-degenerate 2-form \(\omega\) is said to be presymplectic; here non-degeneracy means that the rank of \(\omega\) is maximal at every point. Thus it is \(2n\) if \(\text{dim}\,M=2n+1\). In this case we have the Reeb foliation \(\mathfrak R\) defined by \(\iota_{\mathfrak R}\omega=0\). The \(2n\)-form \(\omega^n\) defines an orientation of any subbundle transverse to \(\mathfrak R\) and hence non-vanishing vector fields. Such vector fields will be called Reeb vector fields.
A 1-form \(\alpha\) is a presymplectic confoliation form if \(\alpha\) is a (positive) confoliation form (i.e., \(\alpha\wedge(d\alpha)^n\geq 0\)) and \(d\alpha\) is presymplectic. The authors prove that if \(\alpha\) is such a form and the Reeb vector of \(d\alpha\) has a connection form \(\eta\), then for sufficiently small \(\varepsilon\), \(\alpha+\varepsilon\eta\) is a contact form.

Reviewer: David E. Blair (East Lansing)

MSC:

53D35	Global theory of symplectic and contact manifolds
53D10	Contact manifolds (general theory)
53D15	Almost contact and almost symplectic manifolds
57R25	Vector fields, frame fields in differential topology

Keywords:

Reeb field; presymplectic form; contact form; geodesible vector field

Citations:

Zbl 0402.57015