Lorentz Ricci Solitons on 3-dimensional Lie groups

Abstract

The three-dimensional Heisenberg group H 3 has three left-invariant Lorentzian metrics g 1 , g 2 , and g 3 as in Rahmani (J. Geom. Phys. 9(3), 295-302 (1992)). They are not isometric to each other. In this paper, we characterize the left-invariant Lorentzian metric g 1 as a Lorentz Ricci Soliton. This Ricci Soliton g 1 is a shrinking non-gradient Ricci Soliton. We also prove that the group E(2) of rigid motions of Euclidean 2-space and the group E(1, 1) of rigid motions of Minkowski 2-space have Lorentz Ricci Solitons.

On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups

“…Therefore, this pseudo-orthonormal basis satisfies the same bracket relations as in Theorem 6.2 (1) with α = 1, and hence the metric coincides with g 1 in the above notation. It has been known that g 1 is not Einstein, but algebraic Ricci soliton ( [16], see also [15]). (2) The case of λ = 1.…”

Ricci solitons on Lorentzian manifolds with large isometry groups

“…Indecomposable Lorentzian symmetric spaces are either irreducible or the so-called CahenWallach symmetric spaces which are given as follows [4], [5]. Take M = R n+2 and define a metric tensor by (12) g cw (u, v, x 1 , . .…”

“…Locally conformally flat Cahen-Wallach symmetric spaces are precisely the ε-spaces introduced in Section 1. In this case the metric is given by (12) with κ 1 = · · · = κ n = ε.…”

“…Riemannian manifolds admitting a group of isometries of dimension at least 1 2 n(n − 1) + 1 are either of constant curvature or products of an (n − 1)dimensional space of constant curvature with a line or a circle. Lorentzian metrics allow other solutions that are related to the existence of null submanifolds on M that are left invariant by Lie group actions (see, for instance, [2,12]). If the group of isometries has dimension at least 1 2 n(n − 1) + 2, then the sectional curvature is constant [14].…”

A New Class of Almost Ricci Solitons and Their Physical Interpretation

“…Basic details and a collection of research papers on this area of research are available in [2, 3], respectively. There are very limited papers on Ricci solitons for semi-Riemannian (in particular, Lorentzian) manifolds (e.g., see [4, 5]). On the other hand, the study on MCF is primarily focused on the fixed points of a submanifold of minimal volume embedded in some fixed space.…”