On deformations of a codimension-1 foliation into contact structures (Q352775)

The aim of the paper under review is to study certain special deformations of 1-codimensional foliations into contact structures on an odd-dimensional manifold \(V\). Namely, by definition, a foliation defined by a non-singular 1-form \(V\) can be deformed into contact structures if there exists a non-singular 1-form \(\beta\) on \(V\) such that for every \(t\in(0,\infty)\) the 1-form \(\alpha_t=C(t)\alpha+B(t)\beta\) is a contact form, where \(C:[0,\infty)\to(0,\infty)\) is a suitable function continuous at \(0\) with \(C(0)=1\) and \(B:[0,\infty)\to[0,\infty)\) is a suitable continuous increasing function with \(B(0)=0\). For \(C(t)=1\) and \(B(t)=t\) one recovers the linear deformations investigated by the first author and \textit{P. Rukimbira} [Adv. Geom. 4, No. 1, 75--81 (2004; Zbl 1042.53058)]. The affine deformations allow one to provide a condition which is necessary and sufficient for the deformation of any integrable 1-form into contact structures.