Lie groups with a bi-invariant distance

TL;DR

This paper characterizes Lie groups that admit bi-invariant distances in the general Ad-invariant Finsler setting, proving that such groups split as a product G = H × K with H abelian and K compact with discrete center, and that the distance arises from a unique Ad-invariant Finsler norm on the Lie algebra. It develops a Milnor-style curvature framework for bi-invariant distances by introducing the form S and a corresponding sectional curvature sec(π), showing S ≥ 0 and sec(π) ∈ [0,1], with flatness precisely tied to abelian 2-planes under suitable smoothness or convexity assumptions. The work also analyzes the Ad-invariant norm geometry via norming functionals, complexifications, and the root structure of the compact factor, linking algebraic structure to distance-minimizing geodesics and curvature behavior. Collectively, the results extend Milnor's Riemannian insights to a broad Finsler setting and provide concrete criteria for flatness and curvature in Lie groups with bi-invariant distances.

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 $sec(π)$ 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 $π$ 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

Theorems & Definitions (77)

• Theorem : A
• Theorem : B
• Theorem : C
• Definition 2.1: Bi-invariant intrinsic distances
• Remark 2.2: Left-invariant metrics
• Definition 2.3: Rectifiable paths and length
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Lemma 2.7