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On Flat Pseudo-Euclidean Nilpotent Lie Algebras
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Abstract
A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie algebra is degenerate and all these Lie algebras are obtained by using a sequence of double extension from an abelian Lie algebra. In particular, we determine all flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature . The paper contains also some examples in low dimension.
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We prove that the Ricci operator of any nonunimoular solvable metric Lie algebra having a two-step nilpotent derived Lie algebra of dimension 6 has at least two negative eigenvalues.
A Lorentzian flat Lie group is a Lie group G with a flat left invariant
metric with signature . The Lie algebra
of G endowed with is called flat Lorentzian Lie
algebra. It is known that the metric of a flat Lorentzian Lie group is
geodesically complete if and only if its Lie algebra is unimodular. In this
paper, we determine all nonunimodular flat Lorentzian Lie algebras leading to
the determination of all incomplete Lorentzian flat Lie groups.
The main purpose of this paper is to prove that there are no closed timelike geodesics in a (compact or noncompact) flat Lorentz 2-step nilmanifold N / Γ , N/\Gamma , where N N is a simply connected 2-step nilpotent Lie group with a flat left-invariant Lorentz metric, and Γ \Gamma a discrete subgroup of N N acting on N N by left translations. For this purpose, we shall first show that if N N is a 2-step nilpotent Lie group endowed with a flat left-invariant Lorentz metric g , g, then the restriction of g g to the center Z Z of N N is degenerate. We shall then determine all 2-step nilpotent Lie groups that can admit a flat left-invariant Lorentz metric. We show that they are trivial central extensions of the three-dimensional Heisenberg Lie group H 3 H_{3} . If ( N , g ) \left ( N,g\right ) is one such group, we prove that no timelike geodesic in ( N , g ) \left ( N,g\right ) can be translated by an element of N . N. By the way, we rediscover that the Heisenberg Lie group H 2 k + 1 H_{2k+1} admits a flat left-invariant Lorentz metric if and only if k = 1. k=1.
The purpose of this paper is to investigate Ricci-flatness of left-invariant Lorentzian metrics on 2-step nilpotent Lie groups. We first show that if 〈,〉 is a Ricci-flat left-invariant Lorentzian metric on a 2-step nilpotent Lie group N, then the restriction of 〈,〉 to the center of the Lie algebra of N is degenerate. We then characterize the 2-step nilpotent Lie groups which can be endowed with a Ricci-flat left-invariant Lorentzian metric, and we deduce from this that a Heisenberg Lie group H2n+1 can be endowed with Ricci-flat left-invariant Lorentzian metric if and only if n = 1. We also show that the free 2-step nilpotent Lie group on m generators Nm,2 admits a Ricci-flat left-invariant Lorentzian metric if and only if m = 2 or m = 3, and we determine all Ricci-flat left-invariant Lorentzian metrics on the free 2-step nilpotent Lie group on 3 generators N3,2.
This article outlines what is known to the author about the Riemannian geometry of a Lie group which has been provided with a Riemannian metric invariant under left translation.
We call the Lie algebra of a Lie group with a left invariant
pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new
proof of a classical result of Milnor on Riemannian flat Lie algebras. We
reduce the study of Lorentzian flat Lie algebras to those with trivial center
or those with degenerate center. We show that the double extension process can
be used to construct all Lorentzian flat Lie algebras with degenerate center
generalizing a result of Aubert-Medina on Lorentzian flat nilpotent Lie
algebras. Finally, we give the list of Lorentzian flat Lie algebras with
degenerate center up to dimension 6.
The determination of affine Lie groups (i.e., which carry a left-invariant affine structure) is an open problem. In this work we begin the study of Lie groups with a left-invariant, flat pseudo-Riemannian metric (flat pseudo-Riemannian groups). We show that in such groups the left-invariant affine structure defined by the Levi-Civita connection is geodesically complete if and only if the group is unimodular. We also show that the cotangent manifold of an affine Lie group is endowed with an affine Lie group structure and a left-invariant, flat hyperbolic metric. We describe a double extension process which allows us to construct all nilpotent, flat Lorentzian groups. We give examples and prove that the only Heisenberg group which carries a left invariant, flat pseudo-Riemannian metric is the three dimensional one.
Ricci flat left-invariant pseudo-Riemannian metrics on 2-step nilpotent Lie groups
- M Boucetta
Boucetta, M. (2009). Ricci flat left-invariant pseudo-Riemannian metrics on 2-step nilpotent Lie groups, arXiv:
0910.2563v1[math. D.G].
Annales de la faculté des sciences de Toulouse, 5éme série
- P Revoy
Revoy, P. (1980). Algèbres de Lie métabéliennes, Annales de la faculté des sciences de Toulouse, 5éme série, tome
2, No. 2, 93-100.