The left-invariant Sasakian structure on the group model of the real extension of the Lobachevsky plane

It has been proved that there is left-invariant contact metric structure $(\eta,\xi,\varphi, g)$ whose Riemannian metric is different from the metric of the direct product on the group model of the real extension of the Lobachevsky plane $\mathbb{H}^2\times\mathbb{R}$. The restriction of the metric $g$ to the contact distribution is the metric of the Lobachevsky plane and, together with a completely nonholonomic contact distribution, defines a sub-Riemann structure on $\mathbb{H}^2\times\mathbb{R}$. The found almost contact metric structure is normal and therefore Sasakian. The lie group of automorphisms of this structure has maximum dimension. The basis vector fields of its Lie algebra are found. In addition to the Levi-Civita connection $\nabla$, we consider a contact metric connection $\tilde{\nabla}$ with skew-symmetric torsion, which, like the Levi-Civita connection, is also invariant under the automorphism group. The structure tensors $\eta,\xi,\varphi, g$, the torsion tensor $\tilde{S}$ and the curvature tensor $\tilde{R}$ of a given connection are covariantly constant. The curvature tensor $\tilde{R}$ of the connection $\tilde{\nabla}$ has the necessary properties to introduce the concept of sectional curvature. It is established that the sectional curvature $\tilde{k}$ belongs to the numerical segment $[-2,0]$. Using the field of orthonormal frames adapted to the contact distribution, the coefficients of the truncated connection and the differential equations of its geodesics are found. It has been proved that the contact geodesics of the connections $\nabla$ and $\tilde{\nabla}$ coincide with the geodesics of truncated connection, that is, both connections are compatible with the contact distribution. This means that there is only one contact geodesic through each point in each contact direction.