Cyclic Lorentzian Lie groups

Abstract

We consider Lie groups equipped with a left-invariant cyclic Lorentzian metric. As in the Riemannian case, in terms of homogeneous structures, such metrics can be considered as different as possible from bi-invariant metrics. We show that several results concerning cyclic Riemannian metrics do not extend to their Lorentzian analogues, and obtain a full classification of three-and four-dimensional cyclic Lorentzian metrics.2010 Mathematics Subject Classification. 53C30, 53C50, 22E25, 22E46.

Conformally Einstein Lorentzian Lie groups: extensions of the Euclidean and Poincaré groups

“…Since dim g ′ 3 = 2 in the cases under consideration g 3 = e(2) and g 3 = e(1, 1), one has that dim g ′ is either two or three. The case when dim g ′ = 3 is inconsistent with dim g ′ 3 = 2 due to the Jacobi identity (see [10]). Moreover, it was shown in [18] (see also [10]) that if dim g ′ = 2, since the metric is of Lorentzian signature, there is a non-null vector y ∈ g ′ acting as a derivation on g so that g = h ⋊ span{y} and the restriction of the metric to h is non-degenerate.…”

“…The case when dim g ′ = 3 is inconsistent with dim g ′ 3 = 2 due to the Jacobi identity (see [10]). Moreover, it was shown in [18] (see also [10]) that if dim g ′ = 2, since the metric is of Lorentzian signature, there is a non-null vector y ∈ g ′ acting as a derivation on g so that g = h ⋊ span{y} and the restriction of the metric to h is non-degenerate. Therefore, this situation is covered by the previous analysis.…”

“…Hence we consider all these possibilities separately. Aiming to be as self-contained as possible, we describe all left-invariant Lorentzian metrics on semi-direct extensions G 3 ⋊ R following our previous work [18] (see also [10]).…”

Ricci solitons on four-dimensional Lorentzian Lie groups

“…The Lorentzian situation is more subtle due to the fact that the restriction of the metric to the threedimensional subalgebras su(2), sl(2, R), e(2), e(1, 1), h or r 3 may be a positive definite, Lorentzian or degenerate inner product. We follow [8] and consider separately the three possibilities above.…”

On the geometry of some solvable extensions of the Heisenberg group

“…Homogeneous structures belonging to S 1 ⊕ S 2 are as different as possible from the naturally reductive ones. The study of this kind of structures was recently undertaken in [10] and [8]. S eiej e k = 0 which implies that they are of types S 1 ⊕ S 2 and S 2 .…”