Classification of left invariant metrics on $4$-dimensional solvable Lie groups

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.