Geodesic
The contact metric connection with skew torsion
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2019), pp. 54-63

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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. The expression of this connection through the contact form and the metric tensor is received. It is shown that the torsion tensor and the curvature tensor are constant and the sectional curvature varies between $-1$ and $0$. It is proved that the obtained connection is the contact metric connection for all $k$-contact metric structures, therefore it is the contact metric connection for all Sasakian structures.
Keywords: Heisenberg group, contact metric structure, connection with skew-torsion, sectional curvature. 
@article{IVM_2019_11_a6,
     author = {V. I. Panzhenskii and T. R. Klimova},
     title = {The contact metric connection with skew torsion},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {54--63},
     publisher = {mathdoc},
     number = {11},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2019_11_a6/}
}
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V. I. Panzhenskii; T. R. Klimova. The contact metric connection with skew torsion. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2019), pp. 54-63. http://geodesic.mathdoc.fr/item/IVM_2019_11_a6/