Lorentzian flat lie groups admitting a timelike left-invariant killing vector field

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Lorentzian flat lie groups admitting a timelike left-invariant killing vector field

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Title: Lorentzian flat lie groups admitting a timelike left-invariant killing vector field
Author: Lebzioui, Hicham
Abstract: We call a connected Lie group endowed with a left-invariant Lorentzian at metric Lorentzian at Lie group. In this Note, we determine all Lorentzian at Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. Moreover, we show that a Lorentzian at Lie group (G,μ) admits a timelike left-invariant Killing vector field if and only if G admits a left-invariant Riemannian metric which has the same Levi-Civita connection of μ. Finally, we give an useful characterization of left-invariant pseudo-Riemannian at metrics on Lie groups G satisfying the property: for any couple of left invariant vector fields X and Y their Lie bracket [X, Y ] is a linear combination of X and Y.
URI: http://hdl.handle.net/10662/10150
Date: 2014


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