Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CLASSIFICATION OF HOMOGENEOUS STRUCTURES ON 4-DIMENSIONAL NILPOTENT LIE GROUPS

  • Wafaa Batat (Departement de la Formation Preparatoire en Sciences et Technologies Ecole Nationale Polytechnique d'Oran-Maurice Audin) ;
  • Rabea Taleb (Departement de la Formation Preparatoire en Sciences et Technologies Ecole Nationale Polytechnique d'Oran-Maurice Audin)
  • Received : 2023.11.03
  • Accepted : 2024.08.14
  • Published : 2025.01.01
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Abstract

We determine, for all left-invariant Lorentzian metrics, the set of homogeneous structures on the four-dimensional 3-step nilpotent Lie group G4. Combined with the results of [17], this provides a complete classification of homogeneous structures on four-dimensional nilpotent Lie groups. As an application, we explore the distinct characteristics of each structure and demonstrate the existence of homogeneous structures that are not canonical. We then identify scenarios in which the metrics exhibit natural reductiveness, proving that a naturally reductive homogeneous structure can exist for left-invariant Lorentzian metrics admitting a parallel null vector on G4. This highlights a significant distinction between Riemannian and pseudo-Riemannian geometries, as Gordon's result [13] does not apply in the Lorentzian context, where the Lie group is not restricted to being 2-step nilpotent.

Keywords

Acknowledgement

This work was conducted as part of the requirements for the second author's doctoral degree. Financial support was provided by PRFU under Grant Agreement C00L03ES310120200001.

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