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

TL;DR

This paper introduces bi-Lipschitz Riemannian Clairaut metrics on Lie groups, providing conditions under which all left-invariant semi-Riemannian metrics are complete, and applies these to various classes of Lie groups.

Contribution

It defines Clairaut metrics that guarantee completeness of all left-invariant semi-Riemannian metrics on Lie groups and identifies classes of groups satisfying the growth conditions.

Findings

All Clairaut metrics are complete for compact and 2-step nilpotent groups.

Semidirect products with certain properties also satisfy the completeness condition.

The affine group of the real line illustrates the techniques beyond linear growth cases.

Abstract

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 completeness of $g$. When the adjoint representation of $G$ satisfies 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 $K \ltimes_\rho \mathbb{R}^n$ , where $K$ is the direct product of a compact and an abelian Lie group and $\rho(K)$ 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 the absence of linear growth and suggest new questions.