Abstract
PurposeIn this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics.Design/methodology/approachIn the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature.FindingsIn the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.Originality/valueIn this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of left-invarainat Randers metrics will be studied.
Highlights
Research into left-invariant Riemannian metrics on Lie groups is an active subject of research and this topic is mentioned among many author’s works so far
We develop the results of [4] for a special case of five-dimensional Heisenberg group by investigating the geometry of left-invariant Randers metrics on the Heisenberg group H2nþ1, of dimension 2n þ 1 [5]
We show the Heisenberg group H2nþ1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types
Summary
Research into left-invariant Riemannian metrics on Lie groups is an active subject of research and this topic is mentioned among many author’s works so far. Similar to the Riemannian case, the notion of left-invariant Randers metrics on a Lie group G is defined, and the geometry of such spaces is part of many author’s interest topic. The Heisenberg groups play a crucial role in theoretical physics, and they are well understood from the viewpoint of sub-Riemannian geometry These groups arise in the description of one-dimensional quantum mechanical systems. We develop the results of [4] for a special case of five-dimensional Heisenberg group by investigating the geometry of left-invariant Randers metrics on the Heisenberg group H2nþ, of dimension 2n þ 1 [5]. The full terms of this licence may be seen at http:// creativecommons.org/licences/by/4.0/legalcode
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