| Title:
|
On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension (English) |
| Author:
|
Nasehi, Mehri |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
57 |
| Issue:
|
2 |
| Year:
|
2021 |
| Pages:
|
101-111 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We first classify left invariant Douglas $(\alpha , \beta )$-metrics on the Heisenberg group $H_{2n+1}$ of dimension $2n + 1$ and its extension i.e., oscillator group. Then we explicitly give the flag curvature formulas and geodesic vectors for these spaces, when equipped with these metrics. We also explicitly obtain $S$-curvature formulas of left invariant Randers metrics of Douglas type on these spaces and obtain a comparison on geometry of these spaces, when equipped with left invariant Douglas $(\alpha , \beta )$-metrics. More exactly, we show that although the results concerning bi-invariant Douglas $(\alpha ,\beta )$-metrics on these spaces are similar, several results concerning left invariant Douglas $(\alpha ,\beta )$-metrics on these spaces are different. For example we prove that the existence of left-invariant Matsumoto, Kropina and Randers metrics of Berwald type on oscillator groups can not extend to Heisenberg groups. Also we prove that oscillator groups have always vanishing $S$-curvature, whereas this can not occur on Heisenberg groups. Moreover, we prove that there exist new geodesic vectors on oscillator groups which can not extend to the Heisenberg groups. (English) |
| Keyword:
|
Heisenberg groups |
| Keyword:
|
oscillator groups |
| Keyword:
|
left-invariant Douglas $(\alpha ,\beta )$-metrics |
| MSC:
|
53C30 |
| MSC:
|
53C60 |
| idZBL:
|
Zbl 07361068 |
| idMR:
|
MR4306171 |
| DOI:
|
10.5817/AM2021-2-101 |
| . |
| Date available:
|
2021-05-11T14:25:06Z |
| Last updated:
|
2021-11-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148893 |
| . |
| Reference:
|
[1] Aradi, B.: Left invariant Finsler manifolds are generalized Berwald.Eur. J. Pure Appl. Math. 8 (2015), 118–125. MR 3313971 |
| Reference:
|
[2] Arnold, V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications a l’hydrodynamique des fluides parfaits.Ann. Inst. Fourier (Grenoble) 16 (1) (1966), 319–361. MR 0202082, 10.5802/aif.233 |
| Reference:
|
[3] Bacso, S., Cheng, X., Shen, Z.: Curvature properties of $(\alpha ,\beta )$-metrics.Finsler Geometry, Sapporo 2005, Adv. Stud. Pure Math., vol. 48, 2007, pp. 73–110. MR 2389252 |
| Reference:
|
[4] Berndt, J., Tricceri, F., Vanhecke, L.: Generalized Heisenberg Groups and Damek Ricci Harmonic Spaces.Lecture Notes in Math., vol. 1598, Springer, Heidelberg, 1995. MR 1340192 |
| Reference:
|
[5] Biggs, R., Remsing, C.C.: Some remarks on the oscillator group.Differential Geom. Appl. 35 (2014), 199–209. MR 3254303, 10.1016/j.difgeo.2014.03.003 |
| Reference:
|
[6] Chern, S.S., Shen, Z.: Riemann-Finsler geometry.World Scientific, Singapore, 2005. MR 2169595 |
| Reference:
|
[7] Deng, S.: The S-curvature of homogeneous Randers spaces.Differential Geom. Appl. 27 (2010), 75–84. MR 2488989, 10.1016/j.difgeo.2008.06.007 |
| Reference:
|
[8] Deng, S.: Homogeneous Finsler spaces.Springer, New York, 2012. MR 2962626 |
| Reference:
|
[9] Deng, S., Hosseini, M., Liu, H., Salimi Moghaddam, H.R.: On the left invariant $(\alpha ,\beta )$-metrics on some Lie groups.to appear in Houston Journal of Mathematics. MR 4102870 |
| Reference:
|
[10] Deng, S., Hou, Z.: Invariant Randers metrics on homogeneous Riemannian manifolds.J. Phys. A Math. Gen. 37 (2004), 4353–4360. Zbl 1049.83005, MR 2063598, 10.1088/0305-4470/37/15/004 |
| Reference:
|
[11] Deng, S., Hu, Z.: On flag curvature of homogeneous Randers spaces.Canad. J. Math. 65 (2013), 66–81. MR 3004458, 10.4153/CJM-2012-004-6 |
| Reference:
|
[12] Fasihi-Ramandi, Gh., Azami, S.: Geometry of left invariant Randers metric on the Heisenberg group.submitted. |
| Reference:
|
[13] Gadea, P.M., Oubina, J.A.: Homogeneous Lorentzian structures on the oscillator groups.Arch. Math. (Basel) 73 (1999), 311–320. MR 1710084, 10.1007/s000130050403 |
| Reference:
|
[14] Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics.Boll. Un. Mat. Ital. B (7) 5 (1) (1991), 189–246. Zbl 0731.53046, MR 1110676 |
| Reference:
|
[15] Latifi, D.: Bi-invariant Randers metrics on Lie groups.Publ. Math. Debrecen 76 (1–2) (2010), 219–226. MR 2598183 |
| Reference:
|
[16] Lengyelné Tóth, A., Kovács, Z.: Left invariant Randers metrics on the 3-dimensional Heisenberg group.Publ. Math. Debrecen 85 (1–2) (2014), 161–179. MR 3231513, 10.5486/PMD.2014.5894 |
| Reference:
|
[17] Lengyelné Tóth, A., Kovács, Z.: Curvatures of left invariant Randers metric on the five-dimensional Heisenberg group.Balkan J. Geom. Appl. 22 (1) (2017), 33–40. MR 3678008 |
| Reference:
|
[18] Liu, H., Deng, S.: Homogeneous $(\alpha ,\beta )$-metrics of Douglas type.Forum Math. (2014), 1–17. MR 3393392 |
| Reference:
|
[19] Milnor, J.: Curvatures of left-invariant metrics on Lie groups.Adv. Math. 21 (3) (1976), 293–329. MR 0425012, 10.1016/S0001-8708(76)80002-3 |
| Reference:
|
[20] Nasehi, M.: On 5-dimensional 2-step homogeneous Randers nilmanifolds of Douglas type.Bull. Iranian Math. Soc. 43 (2017), 695–706. MR 3670890 |
| Reference:
|
[21] Nasehi, M.: On the Geometry of Higher Dimensional Heisenberg Groups.Mediterr. J. Math. 29 (2019), 1–17. MR 3911142 |
| Reference:
|
[22] Nasehi, M., Aghasi, M.: On the geometry of Douglas Heisenberg group.48th Annual Irannian Mathematics Conference, 2017, pp. 1720–1723. |
| Reference:
|
[23] Parhizkar, M., Salimi Moghaddam, H.R.: Geodesic vector fields of invariant $(\alpha ,\beta )$-metrics on homogeneous spaces.Int. Electron. J. Geom. 6 (2) (2013), 39–44. MR 3125830 |
| Reference:
|
[24] Rahmani, S.: Metriques de Lorentz sur les groupes de Lie unimodulaires de dimension 3.J. Geom. Phys. 9 (1992), 295–302. MR 1171140, 10.1016/0393-0440(92)90033-W |
| Reference:
|
[25] Vukmirovic, S.: Classification of left-invariant metrics on the Heisenberg group.J. Geom. Phys. 94 (2015), 72–80. MR 3350270, 10.1016/j.geomphys.2015.01.005 |
| . |
