Metrics and tangent cones of uniformly regular Carnot—Carathéodory spaces

Abstract

Given a uniformly regular Carnot—Caratheodory space, we prove equivalence of the quasimetrics generated by various bases of vector fields which agree with filtration of the space. We prove a theorem on a nilpotent tangent cone for a uniformly regular Carnot—Caratheodory space furnished with quasimetrics. As a consequence, we obtain a theorem on isomorphism of nilpotent tangent cones defined at a common distinguished point.