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Communications in Mathematics

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Isometries of Riemannian and sub-Riemannian structures on three-dimensional Lie groups

Rory Biggs
Published Online: 2018-01-11 | DOI: https://doi.org/10.1515/cm-2017-0010

Abstract

We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.

MSC 2010: 53C20; 53C17; 22E30

Keywords: Riemannian structures; sub-Riemannian structures; three-dimensional Lie groups

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About the article

Received: 2016-10-29

Accepted: 2017-08-17

Published Online: 2018-01-11

Published in Print: 2017-12-20


Citation Information: Communications in Mathematics, Volume 25, Issue 2, Pages 99–135, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2017-0010.

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© 2018. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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