The left-invariant contact metric structure on the Sol manifold

Among the known eight-dimensional Thurston geometries, there is a geometry of the Sol manifold – a Lie group consisting of real special matrices. For a left-invariant Riemannian metric on the Sol manifold, the left shift group is a maximal simple transitive group of isometry. In this paper, we found all left-invariant differential 1-forms and proved that on the oriented Sol manifold there is only one left-invariant differential 1-form, such that this form and the left-invariant Riemannian metric together define the contact metric structure on the Sol manifold. We identified all left-invariant contact metric connections and distinguished flat connections among them. A completely non-holonomic contact distribution along with the restriction of a Riemannian metric to this distribution define the contact metric structure on the Sol manifold, and an orthogonal projection of the Levi-Chivita connection is a truncated connection. We obtained geodesic parameter equations of the truncated connection, which are the sub-geodesic equations, using a non-holonomic field of frames adapted to the contact metric structure. We revealed that these geodesics are a part of the geodesics of the flat contact metric connection.