2025/11/12 by Larotonda, Gabriel, Rey, Iván
#22E65 (Primary) 58B20 #53C22 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2511.09699
We show that a Lie group G admitting a bi-invariant distance must be the product G=H× 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