Extremals of the Induced Sub-Lorentz Structure on the Gödel Universe

We study the Gödel universe as the Lie group with the inducedleft-invariant sub-Lorentz structure defined by some proper subspace of the Liealgebra that generates it. We find the expressions for timelike and isotropic extremalsin elementary functions by methods of geometric optimal control.Also, we prove that these extremals are not closed but as a rule not complete, having properopen subintervals of the real line as maximal connected domains.