Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups

Guediri, Mohammed; Bin-Asfour, Mona

doi:10.5817/am2014-3-171

Cited by 8 publications

(6 citation statements)

References 15 publications

(16 reference statements)

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“…In contrast to the Riemannian case, there exist Ricci-flat pseudo-metrics on non-Abelian nilpotent Lie algebras (see [3,4,7,14,15,16,18,22]). Some of these examples are not flat, indicating another difference with the Riemannian case.…”

Section: Introductionmentioning

confidence: 99%

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

Fernández¹,

Freibert²,

Sánchez³

2020

Preprint

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We answer in the affirmative the question posed by Conti and Rossi [7,8] on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudometric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric g of signature (3, 4) on a nilpotent Lie group of dimension 7, such that g is Einstein and not Ricci-flat. We show that the pseudo-metric g cannot be induced by any left-invariant closed G * 2 -structure on the Lie group. Moreover, some results on closed and harmonic G * 2 -structures on an arbitrary 7-manifold M are given. In particular, we prove that the underlying pseudo-Riemannian metric of a closed and harmonic G * 2 -structure on M is not necessarily Einstein, but if it is Einstein then it is Ricci-flat.

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Section: Introductionmentioning

confidence: 99%

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

Fernández¹,

Freibert²,

Sánchez³

2020

Preprint

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“…It would be a natural problem to consider whether the above three correspondences hold for any Lie group or not. In fact, some papers study the relations between the curvature properties and the signatures of the restrictions ( [3,9]).…”

Section: Table 2 the Number Of Equivalence Classesmentioning

confidence: 99%

A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups

Kondo

2021

Preprint

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It is known that a connected and simply-connected Lie group admits only one left-invariant Riemannian metric up to scaling and isometry if and only if it is isomorphic to the Euclidean space, the Lie group of the real hyperbolic space, or the direct product of the three dimensional Heisenberg group and the Euclidean space of dimension n − 3. In this paper, we give a classification of left-invariant pseudo-Riemannian metrics of an arbitrary signature for the third Lie groups with n ≥ 4 up to scaling and automorphisms. This completes the classifications of left-invariant pseudo-Riemannian metrics for the above three Lie groups up to scaling and automorphisms.

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“…In particular, we do not obtain Ricci-flat metrics of Lorentzian signature. In fact, it was proved in [16] that Ricci-flat Lorentzian metrics on 2-step nilpotent Lie algebras have degenerate center; diagonal metrics on a nice Lie algebra never have this property. We note that these metrics are not ad-invariant, i.e.…”

Section: Ricci-flat Metricsmentioning

confidence: 99%

“…a compact quotient (see [24]); thus, the solutions that we obtain typically determine compact Einstein manifolds. Examples of Ricci-flat nilpotent Lie algebras appear in the literature in particular contexts: four-dimensional ( [30]), bi-invariant ( [9,14,19]), nearly parakähler ( [5]), G * 2 -holonomy ( [13]), or 2-step ( [16]). The first example of an Einstein metric with nonzero scalar curvature on a nilpotent Lie algebra was constructed by the authors in [7].…”

mentioning

confidence: 99%

“…= {125, 346}.In particular, we do not obtain Ricci-flat metrics of Lorentzian signature. In fact, it was proved in[16] that Ricci-flat Lorentzian metrics on 2-step nilpotent Lie algebras have degenerate center; diagonal metrics on a nice Lie algebra never have this property.We note that these metrics are not ad-invariant, i.e. they do not satisfy[x, y], z + y, [x, z] = 0, x, y, z ∈ g.In fact, a diagonal metric on a 2-step nice Lie algebra cannot be ad-invariant, as one can see by taking x and y to be elements of the nice basis with z = [x, y] = 0.…”

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confidence: 99%

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Ricci-flat and Einstein pseudoriemannian nilmanifolds

Conti

Rossi

2019

Complex Manifolds

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This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group. Classifications of special classes of Ricci-flat metrics on nilpotent Lie groups of dimension ≤ 8 are obtained. Some related open questions are presented. MSC 2010 : 53C25; 53C50, 53C30, 22E25.

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Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups

Cited by 8 publications

References 15 publications

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

A non Ricci-flat Einstein pseudo-Riemannian metric on a 7-dimensional nilmanifold

A classification of left-invariant pseudo-Riemannian metrics on some nilpotent Lie groups

Ricci-flat and Einstein pseudoriemannian nilmanifolds

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