Metric geometry of nonregular weighted Carnot-Carath\'eodory spaces

TL;DR

This paper explores the local and metric geometry of weighted Carnot-Carathéodory spaces, generalizing sub-Riemannian manifolds, and introduces new methods to analyze their tangent cones and local algebraic structures.

Contribution

It provides new insights into the local algebraic structure and tangent cones of weighted Carnot-Carathéodory spaces, extending classical results to nonregular cases.

Findings

Describes local algebraic structure using a quasimetric

Compares local geometries of the space and its tangent cone

Provides new proofs for classical theorems in sub-Riemannian geometry

Abstract

We investigate local and metric geometry of weighted Carnot-Carath\'eodory spaces which are a wide generalization of sub-Riemannian manifolds and arise in nonlinear control theory, subelliptic equations etc. For such spaces the intrinsic Carnot-Carath\'eodory metric might not exist, and some other new effects take place. We describe the local algebraic structure of such a space, endowed with a certain quasimetric (first introduced by A. Nagel, E.M. Stein and S. Wainger), and compare local geometries of the initial C-C space and its tangent cone at some fixed (possibly nonregular) point. The main results of the present paper are new even for the case of sub-Riemannian manifolds. Moreover, they yield new proofs of such classical results as the Local approximation theorem and the Tangent cone theorem, proved for H\"ormander vector fields by M. Gromov, A.Bellaiche, J.Mitchell etc.