On the Riemannian geometry of tangent Lie groups
type: Item
instance of: scholarly article
title: On the Riemannian geometry of tangent Lie groups
zbMATH Open document ID: 1401.53011
author: F. Asgari
author: Hamid Reza Salimi Moghaddam
publication date: 2018-09-21T00:00:00Z
review text: Let \(G\) denote an \(m\)-dimensional Lie group. It is well known that if \(G\) is a Lie group, then its tangent bundle \(TG\) is also a Lie group. Moreover, the complete and vertical lifts of any left-invariant vector field of \(G\) are left-invariant vector fields on the Lie group \(TG\). In the present paper, the authors consider a Lie group \(G\) equipped with a left-invariant Riemannian metric \(g\). Using complete and vertical lifts of left-invariant vector fields, they induce a left-invariant Riemannian metric \(\widetilde{g}\) on the tangent Lie group \(TG\). The Levi-Civita connection and sectional curvature of \((TG,\overline{g})\) are given in terms of the Levi-Civita connection and the sectional curvature of \((G,g)\). Then, the formulas of Levi-Civita connection, sectional curvature and Ricci tensor of \((TG,\overline{g})\) are given in terms of structure constants of the Lie algebra of \(G\). These formulas are useful in constructing tangent Lie groups with some curvature properties. Finally, some examples of tangent Lie groups of strictly negative and non-negative Ricci curvatures are given.
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 6940995
zbMATH Keywords: left-invariant Riemannian metric
zbMATH Keywords: tangent Lie group
zbMATH Keywords: complete and vertical lifts
zbMATH Keywords: sectional curvature
zbMATH Keywords: Ricci curvatures
Wikidata QID: Q57705988
reviewed by: Nabil L. Youssef
MaRDI profile type: Publication
full work available at URL: s12215-017-0304-z
OpenAlex ID: W2607045135
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DOI: 10.1007/S12215-017-0304-Z
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 6940995