Left-invariant para-Sasakian structure on the Heisenberg group

Among the eight three-dimensional Thurston geometries, there is the Heisenberg group, the nilpotent Lie group of real 3$\times$3 matrices of a special form. It is known that this group has a left-invariant Sasakian structure. This article proves that there is also a paracontact metric structure on the Heisenberg group, which is also Sasakian. This group has a unique contact metric connection with skew-symmetric torsion, which is invariant under the group of automorphisms of the para-Sasakian structure. The discovered connection is proved to be a contact metric connection for any para-Sasakian structure. The concept of a connection compatible with the distribution is introduced. It is found that the Levi-Civita connection and the contact metric connection on the Heisenberg group endowed with a para-Sasakian structure are compatible with the contact distribution. Their orthogonal projections on this distribution determine the same truncated connection. It is proved that Levi-Civita contact geodesics and truncated geodesics coincide. It is found that contact geodesics are either straight lines lying in the contact planes or parabolas the orthogonal projections of which on the contact planes are straight lines. The results obtained in this article are also valid for the multidimensional Heisenberg group.