Left-invariant para-Sasakian structure on the group model of the real extension of the de Sitter plane

In this paper, a group model for a real extension of the de Sitter plane is proposed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.