A New Approach to Investigation of Carnot-Caratheodory Geometry

A new approach for investigating a local geometry of Carnot- Carathéodory spaces under minimal smoothness of basis vector fields is discussed. A connected Riemannian manifold of a topological dimension is considered and this manifold is called the Carnot-Carathéodory space if in the tangent bundle there exists a filtration such that, for each point, there exists a neighborhood with a collection of smooth vector fields. It is also proved that the vector fields are locally continuous. A local Carnot group is defined by a push-forward of a structure into the neighborhood of a unity by the mapping in a such a way that an isomorphism of local groups is defined. The quasi-isometric coefficients of the mapping are found to be independent.