Lie groups with all left-invariant semi-riemannian metrics complete - Ferreira, Ana Cristina (2024)
For each left-invariant semi-Riemannian metric g on a Lie group G, we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the com- pleteness of g. When the adjoint representation of Gsatisfies an at most linear growth bound, then all the Clairaut metrics are complete for any g. We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products K⋉ρ Rn , where K is the direct product of a compact and an abelian Lie group and ρ(K) is pre-compact; they include all the known examples of Lie groups with all left-invariant metrics complete. The affine group of the real line is considered to illustrate how our techniques work even in the absence of linear growth and suggest new questions.