On the Finsler geometry of the Heisenberg group H 2 n + 1 and its extension

Mehri Nasehi

Archivum Mathematicum (2021)

  • Volume: 057, Issue: 2, page 101-111
  • ISSN: 0044-8753

Abstract

top
We first classify left invariant Douglas ( α , β ) -metrics on the Heisenberg group H 2 n + 1 of dimension 2 n + 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 ( α , β ) -metrics. More exactly, we show that although the results concerning bi-invariant Douglas ( α , β ) -metrics on these spaces are similar, several results concerning left invariant Douglas ( α , β ) -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.

How to cite

top

Nasehi, Mehri. "On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension." Archivum Mathematicum 057.2 (2021): 101-111. <http://eudml.org/doc/298086>.

@article{Nasehi2021,
abstract = {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.},
author = {Nasehi, Mehri},
journal = {Archivum Mathematicum},
keywords = {Heisenberg groups; oscillator groups; left-invariant Douglas $(\alpha ,\beta )$-metrics},
language = {eng},
number = {2},
pages = {101-111},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the Finsler geometry of the Heisenberg group $H_\{2n+1\}$ and its extension},
url = {http://eudml.org/doc/298086},
volume = {057},
year = {2021},
}

TY - JOUR
AU - Nasehi, Mehri
TI - On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension
JO - Archivum Mathematicum
PY - 2021
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 057
IS - 2
SP - 101
EP - 111
AB - 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.
LA - eng
KW - Heisenberg groups; oscillator groups; left-invariant Douglas $(\alpha ,\beta )$-metrics
UR - http://eudml.org/doc/298086
ER -

References

top
  1. Aradi, B., Left invariant Finsler manifolds are generalized Berwald, Eur. J. Pure Appl. Math. 8 (2015), 118–125. (2015) MR3313971
  2. Arnold, V.I., 10.5802/aif.233, Ann. Inst. Fourier (Grenoble) 16 (1) (1966), 319–361. (1966) MR0202082DOI10.5802/aif.233
  3. Bacso, S., Cheng, X., Shen, Z., Curvature properties of ( α , β ) -metrics, Finsler Geometry, Sapporo 2005, Adv. Stud. Pure Math., vol. 48, 2007, pp. 73–110. (2007) MR2389252
  4. Berndt, J., Tricceri, F., Vanhecke, L., Generalized Heisenberg Groups and Damek Ricci Harmonic Spaces, Lecture Notes in Math., vol. 1598, Springer, Heidelberg, 1995. (1995) MR1340192
  5. Biggs, R., Remsing, C.C., 10.1016/j.difgeo.2014.03.003, Differential Geom. Appl. 35 (2014), 199–209. (2014) MR3254303DOI10.1016/j.difgeo.2014.03.003
  6. Chern, S.S., Shen, Z., Riemann-Finsler geometry, World Scientific, Singapore, 2005. (2005) MR2169595
  7. Deng, S., 10.1016/j.difgeo.2008.06.007, Differential Geom. Appl. 27 (2010), 75–84. (2010) MR2488989DOI10.1016/j.difgeo.2008.06.007
  8. Deng, S., Homogeneous Finsler spaces, Springer, New York, 2012. (2012) MR2962626
  9. Deng, S., Hosseini, M., Liu, H., Salimi Moghaddam, H.R., On the left invariant ( α , β ) -metrics on some Lie groups, to appear in Houston Journal of Mathematics. MR4102870
  10. Deng, S., Hou, Z., 10.1088/0305-4470/37/15/004, J. Phys. A Math. Gen. 37 (2004), 4353–4360. (2004) Zbl1049.83005MR2063598DOI10.1088/0305-4470/37/15/004
  11. Deng, S., Hu, Z., 10.4153/CJM-2012-004-6, Canad. J. Math. 65 (2013), 66–81. (2013) MR3004458DOI10.4153/CJM-2012-004-6
  12. Fasihi-Ramandi, Gh., Azami, S., Geometry of left invariant Randers metric on the Heisenberg group, submitted. 
  13. Gadea, P.M., Oubina, J.A., 10.1007/s000130050403, Arch. Math. (Basel) 73 (1999), 311–320. (1999) MR1710084DOI10.1007/s000130050403
  14. Kowalski, O., Vanhecke, L., Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. B (7) 5 (1) (1991), 189–246. (1991) Zbl0731.53046MR1110676
  15. Latifi, D., Bi-invariant Randers metrics on Lie groups, Publ. Math. Debrecen 76 (1–2) (2010), 219–226. (2010) MR2598183
  16. Lengyelné Tóth, A., Kovács, Z., 10.5486/PMD.2014.5894, Publ. Math. Debrecen 85 (1–2) (2014), 161–179. (2014) MR3231513DOI10.5486/PMD.2014.5894
  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. (2017) MR3678008
  18. Liu, H., Deng, S., Homogeneous ( α , β ) -metrics of Douglas type, Forum Math. (2014), 1–17. (2014) MR3393392
  19. Milnor, J., 10.1016/S0001-8708(76)80002-3, Adv. Math. 21 (3) (1976), 293–329. (1976) MR0425012DOI10.1016/S0001-8708(76)80002-3
  20. Nasehi, M., On 5-dimensional 2-step homogeneous Randers nilmanifolds of Douglas type, Bull. Iranian Math. Soc. 43 (2017), 695–706. (2017) MR3670890
  21. Nasehi, M., On the Geometry of Higher Dimensional Heisenberg Groups, Mediterr. J. Math. 29 (2019), 1–17. (2019) MR3911142
  22. Nasehi, M., Aghasi, M., On the geometry of Douglas Heisenberg group, 48th Annual Irannian Mathematics Conference, 2017, pp. 1720–1723. (2017) 
  23. Parhizkar, M., Salimi Moghaddam, H.R., Geodesic vector fields of invariant ( α , β ) -metrics on homogeneous spaces, Int. Electron. J. Geom. 6 (2) (2013), 39–44. (2013) MR3125830
  24. Rahmani, S., 10.1016/0393-0440(92)90033-W, J. Geom. Phys. 9 (1992), 295–302. (1992) MR1171140DOI10.1016/0393-0440(92)90033-W
  25. Vukmirovic, S., 10.1016/j.geomphys.2015.01.005, J. Geom. Phys. 94 (2015), 72–80. (2015) MR3350270DOI10.1016/j.geomphys.2015.01.005

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.