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This article is cited in 2 scientific papers (total in 2 papers)
Classification of left invariant metrics on $4$-dimensional solvable Lie groups
Tijana Šukilović Faculty of Mathematics, University of Belgrade, Belgrade, Serbia
Abstract:
In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group $G$, the inner product $\langle \cdot ,\cdot \rangle$ on $\mathfrak{g}=\operatorname{Lie}G$ extends uniquely to a left invariant metric $g$ on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs $(\mathfrak{g},\langle\cdot ,\cdot\rangle)$ known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the $4$-dimensional solvable case isometric means isomorphic.
Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricci-flat, Ricci-parallel and Einstein metrics is also given.
Keywords:
solvable Lie groups, left invariant metrics, metric algebra, Ricci-parallel metrics, Einstein spaces.
Received: 26.08.2020
Citation:
Tijana Šukilović, “Classification of left invariant metrics on $4$-dimensional solvable Lie groups”, Theor. Appl. Mech., 47:2 (2020), 181–204
Linking options:
https://www.mathnet.ru/eng/tam85 https://www.mathnet.ru/eng/tam/v47/i2/p181
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