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Advances in Geometry

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The index of symmetry of three-dimensional Lie groups with a left-invariant metric

Silvio Reggiani
  • Corresponding author
  • CONICET and Universidad Nacional de Rosario, Dpto. de Matemática, ECEN-FCEIA, Av. Pellegrini 250, 2000 Rosario, Argentina
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Published Online: 2018-01-24 | DOI: https://doi.org/10.1515/advgeom-2017-0061


We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3-dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the so-called foliation of symmetry, and we explain in what cases the group fibers over a 2-dimensional space of constant curvature.

Keywords: Index of symmetry; unimodular Lie group; distribution of symmetry; naturally reductive space

MSC 2010: 53C30; 53C35


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About the article

Received: 2016-07-11

Published Online: 2018-01-24

Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0061.

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