On Lorentzian Ricci solitons on nilpotent Lie groups

Abstract

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors boldE1,boldE2,boldE3,boldE4 and E5 and non‐trivial bracket relations E1,E5=boldE3 and E2,E5=boldE4, investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.

Lorentz Ricci solitons of 4-dimensional non-Abelian nilpotent Lie groups

“…We can obtain the coefficient functions P i s of the vector field (9), by solving the above system of PDE's and additionally it concludes α = − 3a 2 a 1 . Putting a 1 = 1, a 2 = λ in metric g + λ we have α = −3λ and and then it is a shrinking Lorentz Ricci soliton while in g − λ and g µ substituting a 1 = 1, a 2 = −λ and a 1 = −1, a 2 = µ respectively, we have α = 3λ and α = 3µ respectively that shows they are expanding.…”

“…In, [9], Wears carried out a complete investigation of the left invariant Lorentzian metrics on a five-dimensional, connected, simply-connected, two-step nilpotent Lie group and its left invariant Ricci soliton and algebraic Ricci soliton metrics. One of his student in her dissertation, [8], tried to presented classification of Lorentzian scalar products of some four dimensional Lie algebras as Ricci solitons.…”

ًRicci bi-conformal vector fields on Lorentzian five dimensional two-step nilpotent Lie groups

“…which is a natural generalization of Einstein metric. Wears in [16] studied Lorentzian Ricci solitons on simply-connected five-dimensional two-step nilpotent Lie groups which are also connected. For more details, see [1][2][3][4][5][6][13][14][15].…”

“…Motivated by [9,16], we study the Ricci bi-conformal vector fields on simply-connected five-dimensional two-step nilpotent Lie groups (G, g) with Lorentzian left invariant metric g which are also connected. The paper is organized as follows.…”

Indefinite nilsolitons and Einstein solvmanifolds

“…The notions of algebraic Ricci soliton and nilsoliton carry over naturally to the pseudo-Riemannian setting, simply by imposing (1) on an indefinite metric. It is known (see [26]) that pseudo-Riemannian algebraic Ricci solitons are Ricci solitons; however, a left-invariant indefinite metric on a Lie group can be a Ricci soliton without satisfying (1), see [3,4,28]. The variational nature of nilsoliton metrics as critical points of the scalar curvature for an appropriately restricted class of metrics also carries over to the indefinite case (see [32]).…”