Geodesic
On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension
Archivum mathematicum, Tome 57 (2021) no. 2, pp. 101-111 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2021-2-101
Classification : 53C30, 53C60
Keywords: Heisenberg groups; oscillator groups; left-invariant Douglas $(\alpha, \beta )$-metrics 
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Nasehi, Mehri. On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension. Archivum mathematicum, Tome 57 (2021) no. 2, pp. 101-111. doi: 10.5817/AM2021-2-101

[1] Aradi, B.: Left invariant Finsler manifolds are generalized Berwald. Eur. J. Pure Appl. Math. 8 (2015), 118–125. | MR

[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. | DOI | MR

[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

[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

[5] Biggs, R., Remsing, C.C.: Some remarks on the oscillator group. Differential Geom. Appl. 35 (2014), 199–209. | DOI | MR

[6] Chern, S.S., Shen, Z.: Riemann-Finsler geometry. World Scientific, Singapore, 2005. | MR

[7] Deng, S.: The S-curvature of homogeneous Randers spaces. Differential Geom. Appl. 27 (2010), 75–84. | DOI | MR

[8] Deng, S.: Homogeneous Finsler spaces. Springer, New York, 2012. | MR

[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

[10] Deng, S., Hou, Z.: Invariant Randers metrics on homogeneous Riemannian manifolds. J. Phys. A Math. Gen. 37 (2004), 4353–4360. | DOI | MR | Zbl

[11] Deng, S., Hu, Z.: On flag curvature of homogeneous Randers spaces. Canad. J. Math. 65 (2013), 66–81. | DOI | MR

[12] Fasihi-Ramandi, Gh., Azami, S.: Geometry of left invariant Randers metric on the Heisenberg group. submitted.

[13] Gadea, P.M., Oubina, J.A.: Homogeneous Lorentzian structures on the oscillator groups. Arch. Math. (Basel) 73 (1999), 311–320. | DOI | MR

[14] Kowalski, O., Vanhecke, L.: Riemannian manifolds with homogeneous geodesics. Boll. Un. Mat. Ital. B (7) 5 (1) (1991), 189–246. | MR | Zbl

[15] Latifi, D.: Bi-invariant Randers metrics on Lie groups. Publ. Math. Debrecen 76 (1–2) (2010), 219–226. | MR

[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. | DOI | MR

[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

[18] Liu, H., Deng, S.: Homogeneous $(\alpha ,\beta )$-metrics of Douglas type. Forum Math. (2014), 1–17. | MR

[19] Milnor, J.: Curvatures of left-invariant metrics on Lie groups. Adv. Math. 21 (3) (1976), 293–329. | DOI | MR

[20] Nasehi, M.: On 5-dimensional 2-step homogeneous Randers nilmanifolds of Douglas type. Bull. Iranian Math. Soc. 43 (2017), 695–706. | MR

[21] Nasehi, M.: On the Geometry of Higher Dimensional Heisenberg Groups. Mediterr. J. Math. 29 (2019), 1–17. | MR

[22] Nasehi, M., Aghasi, M.: On the geometry of Douglas Heisenberg group. 48th Annual Irannian Mathematics Conference, 2017, pp. 1720–1723.

[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

[24] Rahmani, S.: Metriques de Lorentz sur les groupes de Lie unimodulaires de dimension 3. J. Geom. Phys. 9 (1992), 295–302. | DOI | MR

[25] Vukmirovic, S.: Classification of left-invariant metrics on the Heisenberg group. J. Geom. Phys. 94 (2015), 72–80. | DOI | MR

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