Geodesic
The contact metric connection on the Heisenberg group
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2018), pp. 51-59

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We prove that there is only one contact metric connection with skew-torsion on the Heisenberg group endowed with a left-invariant Sasakian structure. We obtain the expression of this connection through the contact form and the metric tensor and show that the torsion tensor and the curvature tensor are constant and the sectional curvature varies between $-1$ and $0$.
Keywords: Heisenberg group, Sasakian structure, connection with skew-torsion, sectional curvature. 
@article{IVM_2018_11_a4,
     author = {V. I. Panzhenskii and T. R. Klimova},
     title = {The contact metric connection on the {Heisenberg} group},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {51--59},
     publisher = {mathdoc},
     number = {11},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2018_11_a4/}
}
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V. I. Panzhenskii; T. R. Klimova. The contact metric connection on the Heisenberg group. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2018), pp. 51-59. http://geodesic.mathdoc.fr/item/IVM_2018_11_a4/