| In this thesis, we focus on the rigidity of certain non-smooth metric structures on manifolds. A well-known theorem of Berestovskii states that a finite dimensional geodesic metric space with transitive isometry group is isometric to a homogeneous space G/H equipped with a Finsler-Carnot-Caratheodory metric; here G is a connected Lie group and H is a closed subgroup. Any such metric is defined by a bracket generating sub-bundle of the tangent bundle and a norm on the sub-bundle.;We consider the problem of describing biLipschitz homogeneous geodesic manifolds, i.e., path metric spaces which are homeomorphic to manifolds and have a transitive group of biLipschitz homeomorphisms. In the discussion it will be crucial the fact that the group of biLipschitz maps, unlikely the isometry group, is not locally compact.;We present a first result that shows that a biLipschitz homogeneous geodesic manifold has to be biLipschitz equivalent to one of the above isometrically homogeneous spaces under the assumption that there is a locally compact group acting transitively by biLipschitz maps.;We afterwards restrict to the two-dimensional case without any extra assumption. We exhibit the fact that such biLipschitz homogeneous geodesic surfaces are locally doubling metric spaces. Moreover, there exists a special doubling measure that behaves like the Haar measure for locally compact groups. By the fact that such properties hold, one can start applying the general theory of Analysis on Metric Spaces to further study such objects. |
