On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension

Mehri Nasehi

Address: Faculty of Basic Sciences, University of Shahreza, P.O. Box 86149-56841, Shahreza, Iran

E-mail: nasehi.mehri@gmail.com

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.

AMSclassification: primary 53C30; secondary 53C60.

Keywords: Heisenberg groups, oscillator groups, left-invariant Douglas (\alpha ,\beta )-metrics.

DOI: 10.5817/AM2021-2-101