Lie groups with a bi-invariant distance

Abstract

We show that a Lie group $G$ admitting a bi-invariant distance must be the product $G=H\times K$ of an abelian group $H$ and a compact group $K$ with discrete center. Moreover, the distance in $G$ must come from the infima of lengths of paths for a unique infinitesimal metric (a Finsler norm) defined in the Lie algebra of $G$. From this we derive the distance minimizing paths which are left or right translations of one-parameter groups (though these are not the unique minizing paths if the norm is not smooth or strictly convex). Then we introduce a notion of sectional curvature $sec(\pi)$ for a bi-invariant distance, following Milnor's ideas, and we show that this curvature is bounded and non-negative, and it is null when the $2$-plane $\pi$ is an abelian Lie subalgebra of $Lie(G)$. We show that when the distance is strictly convex, our sectional curvature vanishes if and only if the $2$-plane is abelian. We give finer characterizations for the case of vanishing curvature, for the case of non-strictly convex norms