GTA Seminar | Speaker: Salah Chaib (CMAT, Univ. Minho)
Title: Geodesic Completeness of Left-Invariant Lorentzian Metrics on the Pseudo-Homothetic Lie Group
Abstract: In this talk, we address the geodesic completeness problem for the 3-dimensional pseudo-homothetic group, a non-unimodular Lie group structured as a semidirect product of $\R^2$ by $\R$ under a non-semisimple action. We reveal the existence of a surprising family of left-invariant Lorentzian metrics that are geodesically complete with bounded velocity, despite lacking a quadratic definite first integral for the associated Euler-Arnold vector field. In addition, we identify a unique complete metric with unbounded geodesics. Our study also highlights that this group's set of complete metrics is neither open nor closed, presenting a counterexample to the closeness of complete metrics in 3-dimensional Lie groups. This is a joint work with Ana Cristina Ferreira and Abdelghani Zeghib.