Left-invariant para-Sasakian structure on the group model of the real extension of the de Sitter plane

V. I. Pan'zhenskii ; Yu. V. Dyranova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 82 (2023), pp. 14-27

Résumé

In this paper, a group model for a real extension of the de Sitter plane is proposed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.

Détail

Mots-clés : paracontact structure, automorphism
Keywords: contact metric connection, contact geodesics, truncated connection.

Bibliographie

[1] S. V. Galaev, “Pochti kontaktnye metricheskie prostranstva s $N$-svyaznostyu”, Izvestiya Saratovskogo universiteta. Novaya ser. Ser. Matematika. Mekhanika. Informatika, 15:3 (2015), 258–264 | DOI | Zbl

[2] S. V. Galaev, “$\nabla_N$-Einshteinovy pochti kontaktnye metricheskie mnogoobraziya”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 70 (2021), 5–15 | DOI | MR | Zbl

[3] M. B. Banaru, “O pochti kontaktnykh metricheskikh 1-giperpoverkhnostyakh kelerovykh mnogoobrazii”, Sibirskii matematicheskii zhurnal, 55:4 (2014), 719–723 | DOI | MR | Zbl

[4] M. B. Banaru, “O pochti kontaktnykh metricheskikh giperpoverkhnostyakh s malymi tipovymi chislami v $W_4$-mnogoobraziyakh”, Vestnik Moskovskogo universiteta. Ser. 1. Matematika, mekhanika, 2018, no. 1, 67–70 | DOI | MR | Zbl

[5] N. K. Smolentsev, “Levoinvariantnye para-sasakievy struktury na gruppakh Li”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 62 (2019), 27–37 | DOI | MR | Zbl

[6] N. K. Smolentsev, I. Yu. Shagabudinova, “O parasasakievykh strukturakh na pyatimernykh algebrakh Li”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 69 (2021), 37–52 | DOI | MR | Zbl

[7] V. I. Panzhensky, T. R. Klimova, “Contact metric connection on the Heisenberg group”, Russian Mathematics, 2018, 45–52 | DOI

[8] V. I. Panzhenskii, A. O. Rastrepina, “Kontaktnaya i pochti kontaktnaya struktury na veschestvennom rasshirenii ploskosti Lobachevskogo”, Uchenye zapiski Kazanskogo universiteta. Ser. Fiziko-matematicheskie nauki, 163:3-4 (2021), 291–303 | DOI

[9] V.I. Panzhenskii, A. O. Rastrepina, “Levoinvariantnaya parasasakieva struktura na gruppe Geizenberga”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika, 75 (2022), 38–51 | DOI | Zbl

[10] A. Diatta, “Left inuariant contact structures on Lie groups”, Diff. Geom. and its Appl., 26:5 (2008), 544–552 | DOI | Zbl

[11] G. Calvaruso, “Three-dimensional homogeneous almost contact metric structures”, Journal of Geometry and Physics, 69 (2013), 60–73 | DOI | MR | Zbl

[12] G. Calvaruso, A. Perrone, “Left-invariant hypercontact structures on three-dimensional Lie groups”, Periodica Mathematica Hungarica, 69 (2014), 97–108 | DOI | MR | Zbl

[13] G. Calvaruso, V. Martin-Molina, “Paracontact metric structures on the unit tangent sphere bundle”, Annadi di Matematica Pura ed Applicata (1923-), 194 (2015), 1359–1380 | DOI | MR | Zbl

[14] G. Calvaruso, A. Perrone, “Five-dimensional paracontact Lie algebras”, Differential Geometry and Its Applications, 45 (2016), 115–129 | DOI | MR | Zbl

[15] A. M. Vershik, L. D. Fadeev, “Lagranzheva mekhanika v invariantnom izlozhenii”, Problemy teoreticheskoi fiziki, Izd-vo LGU, L., 1975, 129–141

[16] A. M. Vershik, V. Ya. Gershkovich, “Negolonomnye dinamicheskie sistemy. Geometriya raspredelenii i variatsionnye zadachi”, Itogi nauki i tekhniki. Ser. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 16, VINITI, M., 1987, 5–85

[17] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Springer-Verlag, Berlin–Heidelberg–New York, 1976, 148 pp. | DOI | Zbl

[18] V. F. Kirichenko, Differentsialno-geometricheskie struktury na mnogoobraziyakh, Pechatnyi dom, Odessa, 2013, 458 pp.

[19] D. Gromol, V. Klingenberg, V. Meier, Rimanova geometriya v tselom, per. s nem. Yu.D. Burago, pod red. i s dop. V.A. Toponogova, Mir, M., 1971, 343 pp.