Geodesic manifolds with a transitive subset of smooth biLipschitz maps

Abstract

This paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let $X = G/H$ be a homogeneous manifold of a Lie group $G$ and let $d$ be a geodesic distance on $X$ inducing the same topology. Suppose that there exists a subgroup $G\_S$ of $G$ that acts transitively on $X$ such that each element $g \in G\_S$ induces a locally biLipschitz homeomorphism of the metric space $(X,d)$. Then the metric is locally biLipschitz equivalent to a sub-Riemannian metric. Any such metric is defined by a bracket generating $G\_S$-invariant sub-bundle of the tangent bundle. The result is a consequence of a more general fact that requires a transitive family of uniformly biLipschitz diffeomorphisms with a control on their differentials. It will be relevant that the group acting transitively on the space is a Lie group and so it is locally compact, since, in general, the whole group of biLipschitz maps, unlikely the isometry group, is not locally compact. Our method also gives an elementary proof of the following fact. Given a Lipschitz sub-bundle of the tangent bundle of a Finsler manifold, both the class of piecewise differentiable curves tangent to the sub-bundle and the class of Lipschitz curves almost everywhere tangent to the sub-bundle give rise to the same Finsler–Carnot–Carathéodory metric, under the condition that the topologies induced by these distances coincide with the manifold topology.