Lorentz geometry of 4-dimensional nilpotent Lie groups

Bokan, Neda; Šukilović, Tijana; Vukmirovic, Srdjan

doi:10.1007/s10711-014-9980-4

Cited by 16 publications

(16 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Calvaruso and Zaeim [9] have classified Lorentz left invariant metrics on the Lie groups that are Einstein or Ricci-parallel. Also, the classification in thecase of nilpotent Lie groups was extensively studied in both the Riemannian [16] and the pseudo-Riemannian setting [8,20].…”

Section: Preliminariesmentioning

confidence: 99%

“…In other notations, these two algebras are often referred to as h 3 + R and n 4 . The classification of inner products on these two algebras has been made in [16] for the Riemannian case and later completed for the pseudo-Riemannian case in [8] (Lorentz case) and [20] (neutral signature case). The results are summarized in the following theorem, whose proof is omitted.…”

Section: Classification Of Left Invariant Metricsmentioning

confidence: 99%

“…The non-isometric inner products on g 1 4,2 are represented by (2.5), where is one of the matrices: 8) or given by one of the matrices:…”

Section: Classification Of Left Invariant Metricsmentioning

confidence: 99%

See 2 more Smart Citations

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Šukilović

2020

Theor appl mech (Belgr)

View full text Add to dashboard Cite

In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.

show abstract

Section: Preliminariesmentioning

confidence: 99%

Section: Classification Of Left Invariant Metricsmentioning

confidence: 99%

See 1 more Smart Citation

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Šukilović

2020

Theor appl mech (Belgr)

View full text Add to dashboard Cite

show abstract

“…In the Riemannian case, P. Topalov [10] considered the conditions under which the left invariant metric on a Lie group admits a non-trivial geodesically equivalent metric. In our paper [1] we classified left invariant metrics of Lorentz signature on 4-dimensional nilpotent Lie groups and noticed that if two of these metrics are geodesically equivalent they have to be affinely equivalent. The same is true for much wider class of metrics, namely for any two G-invariant metrics on homogenous space G/H.…”

Section: Introductionmentioning

confidence: 99%

Geodesically equivalent metrics on homogenous spaces

Bokan¹,

Šukilović²,

Vukmirovic³

2018

Czech.Math.J.

View full text Add to dashboard Cite

Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that existence of non-proportional, geodesically equivalent, G-invariant metrics on homogenous space G/H implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metric, of any signature, on sphere S 3 are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, non-proportional, left-invariant metrics.

show abstract

“…In Section , we use the automorphism group of

frakturh

to determine the equivalence classes of left invariant Lorentzian metrics on H , finding seven distinct families of left invariant Lorentzian metrics in Theorem . For recent work on classifying the left invariant Lorentzian metrics on nilpotent Lie groups see .…”

Section: Introductionmentioning

confidence: 99%

On Lorentzian Ricci solitons on nilpotent Lie groups

Wears

2016

Math. Nachr.

View full text Add to dashboard Cite

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.

show abstract

Lorentz geometry of 4-dimensional nilpotent Lie groups

Cited by 16 publications

References 16 publications

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Classification of left invariant metrics on 4-dimensional solvable Lie groups

Geodesically equivalent metrics on homogenous spaces

On Lorentzian Ricci solitons on nilpotent Lie groups

Contact Info

Product

Resources

About