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

The index of symmetry of three-dimensional Lie groups with a left-invariant metric AbstractWe 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Geometry de Gruyter

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

Reggiani, Silvio
Advances in Geometry , Volume 18 (4): 10 – Oct 25, 2018
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Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1615-7168
eISSN
1615-7168
DOI
10.1515/advgeom-2017-0061
Publisher site
See Article on Publisher Site

Abstract

AbstractWe 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.

Journal

Advances in Geometry – de Gruyter

Published: Oct 25, 2018

References

  • The index of symmetry of a flag manifold
  • Classification of left-invariant metrics on the Heisenberg group
  • Curvatures of left invariant metrics on Lie groups
  • The index of symmetry of compact naturally reductive spaces
  • Left invariant metrics and curvatures on simply connected three-dimensional Lie groups
  • Compact homogeneous Riemannian manifolds with low coindex of symmetry
  • Curvature invariants, Killing vector fields, connections and cohomogeneity
  • Degenerations of Lie algebras and geometry of Lie groups
  • The index of symmetry of a flag manifold
  • The space of left-invariant metrics on a Lie group up to isometry and scaling
  • The isometry groups of simply connected 3-dimensional unimodular Lie groups
  • The space of left-invariant metrics on a Lie group up to isometry and scaling
  • Classification of left-invariant metrics on the Heisenberg group
  • Compact homogeneous Riemannian manifolds with low coindex of symmetry
  • Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold
  • Left invariant metrics and curvatures on simply connected three-dimensional Lie groups
  • Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold
  • Degenerations of Lie algebras and geometry of Lie groups
  • Curvatures of left invariant metrics on Lie groups
  • Curvature invariants, Killing vector fields, connections and cohomogeneity
  • The index of symmetry of compact naturally reductive spaces
  • The isometry groups of simply connected 3-dimensional unimodular Lie groups