Contact and almost contact structures on the real extension of the Lobachevsky plane

In this article, we propose a group model $\mathbb{G}$ of a real extension of the Lobachevsky plane $\mathbb{H}^2 \times \mathbb{R}$. The group $\mathbb{G}$ is a Lie group of special-form matrices and a subgroup of the general linear group $GL(3, \mathbb{R})$. It is proved that, on the group model of the real extension of the Lobachevsky plane, there is a unique left-invariant almost contact metric structure with the Riemannian metric of the direct product that is invariant with respect to the isometry group. The concept of a linear connection compatible with the distribution is introduced. All left-invariant linear connections for which the tensors of the almost contact metric structure $(\eta, \xi, \varphi, g)$ are covariantly constant are found. Among the left-invariant differential $1$-forms, a canonical form defining a contact structure on $\mathbb{G}$ is distinguished. The left-invariant contact metric connections are found. There is a unique left-invariant connection for which all tensors of the almost contact metric structure and the canonical contact form are covariantly constant. It is proved that this connection is compatible with the contact distribution in the sense that a single geodesic tangent to the contact distribution passes through each point in each contact direction. Parametric equations of geodesics of the given connection are found. It is also established that the Levi-Civita connection of the Riemannian metric of the direct product is not a connection compatible with the contact distribution.