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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/}
}
TY  - JOUR
AU  - V. I. Panzhenskii
AU  - T. R. Klimova
TI  - The contact metric connection with skew torsion
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2019
SP  - 54
EP  - 63
IS  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2019_11_a6/
LA  - ru
ID  - IVM_2019_11_a6
ER  - 
%0 Journal Article
%A V. I. Panzhenskii
%A T. R. Klimova
%T The contact metric connection with skew torsion
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2019
%P 54-63
%N 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2019_11_a6/
%G ru
%F IVM_2019_11_a6
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/

[1] Tanno S., “The automorphism groups of almost contact Riemannian manifolds”, Tôhoku Math. J. (2), 21:1 (1969), 21–38 | DOI | MR | Zbl

[2] Gonzalez J. C. and Chinea D., “Quasi-Sasakian homogeneous structures on the generalized Heisenberg group $H(p,1)$”, Proc. Amer. Math. Soc., 105:1 (1989), 173–184 | MR | Zbl

[3] Binz E., Pods S., Geometry of Heisenberg Groups, Amer. Math. Soc., Providence, R.I., 2008 | MR | Zbl

[4] Boyer S. P., “The Sasakian geometry of the Heisenberg group”, Bull. Math. Soc. Sci. Math. Roumanie, 52:3 (2009), 251–262 | MR | Zbl

[5] Vershik A. M., Gershkovich V. Ya., “Negolonomnye dinamicheskie sistemy. Geometriya raspredelenii i variatsionnye zadachi”, Dinam. sist. – 7, Itogi nauki i tekhn. Ser. Sovremen. probl. matem. Fundament. napravleniya, 16, VINITI, M., 1987, 5–85

[6] Agrachev A., Barilari D., Boscain U., “Introduction to Riemannian and sub-Riemannian geometry”, SISSA (Italy, 2012)

[7] Sachkov Yu. L., “Teoriya upravleniya na gruppakh Li”, Sovremen. matem. Fundament. napravleniya, 2008, 5–59

[8] Agrachev A. A., “Nekotorye voprosy subrimanovoi geometrii”, Usp. matem. nauk, 71:6 (2016), 3–36 | DOI | MR | Zbl

[9] Gorbatsevich V. V., “Subrimanovy geometrii na kompaktnykh odnorodnykh prostranstvakh”, Izv. RAN. Ser. matem., 78:3 (2014), 35–52 | DOI | MR | Zbl

[10] Blair D. E., Contact manifolds in Riemannian geometry, Lect. Notes Math., 509, Springer-Verlag, Berlin–New York, 1976 | DOI | MR | Zbl

[11] Panzhenskii V. I., Klimova T. R., “Kontaktnaya metricheskaya svyaznost na gruppe Geizenberga”, Izv. vuzov. Matem., 2018, no. 11, 51–59 | Zbl

[12] Kirichenko V. F., Differentsialno-geometricheskie struktury na mnogoobraziyakh, “Pechatnyi Dom”, Odessa, 2013

[13] Sasaki S., “On differentiable manifolds with certain structures which are closely related to almost contact structure. I”, Tohoku Math. J. (2), 12:3 (1960), 459–476 | DOI | MR | Zbl

[14] Gromol D., Klingenberg V., Maier V., Rimanova geometriya v tselom, Mir, M., 1971