The time-like cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups (Q1882759)

A Lie group \(G\) is called Lorentz 2-step nilpotent if its Lie algebra \(L\) satisfies \([[L,L],L]= 0\) and admits a left-invariant Lorentzian scalar product. The author investigates geodesics in just this type of Lorentzian manifolds. He derives criteria for the existence of conjugate points on time-like geodesics in the case where \(G\) has a time-like center. He shows for instance that a nonsingular time-like geodesic which is translated by a group element has a conjugate point. For a generalized Heisenberg group and for a globally hyperbolic group with one-dimensional spacelike derived group and time-like center the nonsingular time-like geodesics maximize the distance up to the first conjugate point. In deriving these results the paper reviews or develops many important facts of Lorentzian geometry.