Natural almost contact structures and their 3D homogeneous models (Q2825988)

The authors study \textit{natural} almost contact structures (i.e., almost contact structures \((\varphi, \xi, \eta)\) such that \(\xi \in \ker(d\eta)\)), with focus on the \(3\)-dimensional Lie groups. A metric \(g\) compatible with a natural almost contact structure gives a quadruple \((\varphi, \xi, \eta, g)\) called a natural almost contact metric structure. This class of structures shares several properties with contact metric manifolds, and as is pointed out by the authors it is a very large class (e.g., contact metric structures and normal almost contact metric structures belong to the natural almost contact metric class).NEWLINENEWLINEAfter recalling some basic information and general existence results on natural almost contact structures, the authors classify the left-invariant normal almost contact Riemannian structures on \(3\)-dimensional Lie groups. This classification, which includes also the almost contact Lorentzian case, proves that any simply connected homogeneous Riemannian \(3\)-manifold \((M,g)\) admits a natural almost contact structure having \(g\) as a compatible metric. Note that this classification improves and corrects some results obtained in [the first author, J. Geom. Phys. 69, 60--73, (2013; Zbl 1282.53063)].NEWLINENEWLINEFinally, relations between left-invariant CR structures and left-invariant normal almost contact structures on \(3\)-dimensional Lie groups are investigated, and the totally geodesic and minimal examples are completely characterized.NEWLINENEWLINEThe paper includes the following sections: Introduction; Preliminaries; Natural almost contact metric structures; 3D left-invariant natural almost contact metric structures; Geometry of CR structures.