$\varepsilon\,$-contact structures and six-dimensional supergravity

We introduce the concept of $\varepsilon\,$-contact metric structures on oriented (pseudo-)Riemannian three-manifolds, which encompasses the usual Riemannian contact metric, Lorentzian contact metric and para-contact metric structures, but which also allows the possibility for the Reeb vector field to be null. We investigate in more detail this latter case, which we call null contact structure. We observe that it is possible to extend in a natural and meaningful way both the Sasaki and K-contact conditions for null-contact structures, but we find that they are not equivalent conditions, in contradistinction to the situation for non-lightlike Reeb vector fields. Finally, we define the notion of $\varepsilon\eta\,$-Einstein structures and we discover that appropriate direct products of these structures produce solutions of six-dimensional minimal supergravity coupled to a tensor multiplet with constant dilaton.