Invariant almost contact structure on the real extension of a sphere

The existence of contact and almost contact metric structures invariant under the group of motions on the real extension of a two-dimensional sphere with a Riemannian direct product metric was examined. The basis vector fields of the Lie algebra associated with the Lie group of motions were found. The results obtained show that invariant contact structures do not exist, but there is an almost contact metric structure, which is integrable, normal, and has a closed fundamental form, thus making it quasi-Sasakian. The Lie group of automorphisms of this structure coincides with the group of motions and has the maximum possible dimension. All linear connections were found that are invariant under the automorphism group and in which the structural tensors of the quasi-Sasakian structure are covariantly constant. Each such connection is uniquely determined by the quasi-Sasakian structure and by fixing one constant. It was established that the contact distribution of the almost contact structure is completely geodesic. Therefore, the derived connections are consistent with this distribution.