Geodesics and Isometry Group of Defrise Universe as a Lie Group with Left-invariant Lorentz Metric

The authors investigate a model of the Defrise Universe DU as a Lie group $$G_{4}$$ with a left-invariant Lorentz metric and find by two methods its geodesics (they are not closed and may be incomplete), all Killing vector fields, structures of the Lie algebra of $$G_{4},$$ the isometry Lie group $$G_{6}$$ of the Defrise Universe DU and its subgroups. We also describe all possible types of induced sub-Lorentz structures on DU up to isomorphisms of the Lie group $$G_{4}$$ .