The timelike cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups

Abstract

We investigate the timelike cut locus and the locus of conjugate points in Lorentz 2-step nilpotent Lie groups. For these groups with a timelike center, we give some criteria for the existence of conjugate points along timelike geodesics. We show for instance that a nonsingular timelike geodesic which is translated by an element of the group has a conjugate point. For those that we will call of GH-type, and for those which are globally hyperbolic with a timelike center and a one-dimensional derived subgroup, we show that if in addition the derived subgroup is spacelike, then nonsingular timelike geodesics maximize distance up to and including the first conjugate point. Other related results and corollaries are also obtained.