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

Ahmed Elshafei, Ana Cristina Ferreira, Miguel Sánchez, Abdelghani Zeghib
Mathematics, Differential Geometry, Differential Geometry (math.DG)
2023-08-30 16:00:00
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.
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