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.

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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.
DOI : 10.5817/AM2021-2-101
Classification : 53C30, 53C60
Keywords: Heisenberg groups; oscillator groups; left-invariant Douglas $(\alpha, \beta )$-metrics 
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     title = {On the {Finsler} geometry of the {Heisenberg} group $H_{2n+1}$ and its extension},
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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. http://geodesic.mathdoc.fr/articles/10.5817/AM2021-2-101/

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