Left-invariant paracontact metric structure on a group Sol

Abstract

Among Thurston's famous list of eight three-dimensional geometries is the geometry of the manifold Sol. The variety Sol is a connected simply connected Lie group of real matrices of a special form. The manifold Sol has a left-invariant pseudo-Riemannian metric for which the group of left shifts is the maximal simply transitive isometry group. In this paper, we prove that on the manifold Sol there exists a left-invariant differential 1-form, which, together with the left-invariant pseudo-Riemannian metric, defines a paracontact metric structure on Sol. A three-parameter family of left-invariant paracontact metric connections is found, that is, linear con­nec­tions invariant under left shifts, in which the structure tensors of the par­acontact structure are covariantly constant. Among these connections, a flat connection is distinguished. It has been established that some geo­desics of a flat connection are geodesics of a truncated connection, which is an orthogonal projection of the original connection onto a 2n-dimen­sional con­tact distribution. This means that this connection is con­sistent with the con­tact distribution. Thus, the manifold Sol has a pseudo-sub-Riemannian structure determined by a completely non-holonomic contact distribution and the restriction of the original pseudo-Riemannian metric to it.

Keywords

• sol group
• paracontact metric structure
• paracontact metric connection
• truncated connection