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

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 ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) 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, Ricciflat, Ricci-parallel and Einstein metrics is also given.

Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups

“…Şukiloviç [71] gave a classification of left invariant metrics on 4-dimensional solvable Lie groups in terms of curvatures.…”

On the moduli spaces of left invariant metrics on cotangent bundle of Heisenberg group

“…The classification in the case of nilpotent Lie groups in small dimensions was extensively studied in both the Riemannian [29] and the pseudo-Riemannian setting [5,23,41]. Recent results include the classification of pseudo-Riemannian metrics for 4-dimensional solvable Lie groups [42] and in positive definite case, the moduli space for 6-dimensional nilpotent Lie groups admitting complex structure with the first Betti number equal to 4 has been determined [36]. In arbitrary dimension, one must mention the Lorentz classification of left invariant metrics on Heisenberg group H 2n+1 [43] and classification of Ricci solitons on nilmanifolds [30].…”

Classification of Left Invariant Riemannian metrics on Complex hyperbolic space

“…The first method is systematically outlined in [12], and the second one in [9], where Tamaru with coauthors called it Milnor-type theorems in reference to the grounding work of Milnor [19], who used it for the first time to classify all left invariant Riemannian metrics on three-dimensional unimodular Lie groups. Although Milnor's method relies on existence of the cross product in dimension three, lots of results have been later obtained for Riemannian and Lorentzian cases in dimensions three and four (see for example [3,4,5,6,7,16,21]), and recently, for dimension four with neutral signature as well [22,23].…”