Lie groups with all left-invariant semi-Riemannian metrics complete

Abstract

For each left-invariant semi-Riemannian metric g g on a Lie group G G , we introduce the class of bi-Lipschitz Riemannian Clairaut metrics, whose completeness implies the completeness of g g . When the adjoint representation of G G satisfies an at most linear growth bound, then all the Clairaut metrics are complete for any g g . We prove that this bound is satisfied by compact and 2-step nilpotent groups, as well as by semidirect products K ⋉ ρ R n K \ltimes _\rho \mathbb {R}^n , where K K is the direct product of a compact and an abelian Lie group and ρ ( K ) \rho (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.