Complete classification of homogeneous structures on Lorentzian direct extensions of the Heisenberg group

Abstract

The Heisenberg Lie group is one of the most famous and important Lie groups among the family of three dimensional Lie groups. The direct extension of this group to the fourth dimension was taken into consideration in the study of the nilpotent Lie algebras from the fourth dimension, and as a result, the classification of these extensions up to isometric classes was previously presented in some research. Homogeneous structures provide us with a tensor approach to investigate the homogeneity of space. Perhaps the most important feature of homogeneous structures can be summarized in this statement that in Riemannian geometry, the existence of homogeneous structures is equivalent to being reductive locally homogeneous of space. In this paper, based on the existing classification of the direct Lorentzian extension of the Heisenberg group with dimension four, which are isometrically classified in the form of five families, we study the family of homogeneous structures on this space and classify them completely. In non-flat cases, we determine the homogeneous structures separately in each class.

Keywords

• direct extension
• heisenberg group
• homogeneous structure