On the Finsler geometry of the Heisenberg group $H_{2n+1}$ and its extension
- 1 January 2021
- journal article
- research article
- Published by Masaryk University Press in Archivum Mathematicum
- Vol. 57 (2), 101-111
- https://doi.org/10.5817/am2021-2-101
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.
Keywords
This publication has 14 references indexed in Scilit:
- Classification of left-invariant metrics on the Heisenberg groupJournal of Geometry and Physics, 2015
- Some remarks on the oscillator groupDifferential Geometry and its Applications, 2014
- Left invariant Randers metrics on the 3-dimensional Heisenberg groupPublicationes Mathematicae Debrecen, 2014
- On Flag Curvature of Homogeneous Randers SpacesCanadian Journal of Mathematics, 2013
- The S-curvature of homogeneous Randers spacesDifferential Geometry and its Applications, 2009
- Invariant Randers metrics on homogeneous Riemannian manifoldsJournal of Physics A: General Physics, 2004
- Homogeneous Lorentzian structures on the oscillator groupsArchiv der Mathematik, 1999
- Métriques de lorentz sur les groupes de lie unimodulaires, de dimension troisJournal of Geometry and Physics, 1992
- Curvatures of left invariant metrics on lie groupsAdvances in Mathematics, 1976
- Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaitsAnnales de l'institut Fourier, 1966