Invariant almost contact structures and connections on the Lobachevsky space

It has been proved that there is left-invariant normal almost contact metric structure on the group model of the Lobachevsky space. All left-invariant linear connections compatible with this structure have been found and connections with a zero curvature tensor have been distinguished among them. On the Lobachevsky space, in addition to the Levi-Civita connection, there is a 1-parameter family of metric connections with skew-torsion that is invariant with respect to the complete six-dimensional group of motions. Also, there is only one semi symmetric almost contact metric connection that is invariant with respect to a 4-dimensional subgroup of the group of motions.