Geodesic
Classification of left invariant metrics on 4-dimensional solvable Lie groups
Theoretical and applied mechanics, Tome 47 (2020) no. 2, p. 181

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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.
DOI : 10.2298/TAM200826014S
Classification : 22E25, 53B30
Keywords: solvable Lie groups, left invariant metrics, metric algebra, Ricci-parallel metrics, Einstein spaces 
Tijana Šukilović. Classification of left invariant metrics on 4-dimensional solvable Lie groups. Theoretical and applied mechanics, Tome 47 (2020) no. 2, p. 181 . doi: 10.2298/TAM200826014S
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     title = {Classification of left invariant metrics on 4-dimensional solvable {Lie} groups},
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     year = {2020},
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     doi = {10.2298/TAM200826014S},
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