Privileged Coordinates and Nilpotent Approximation of Carnot Manifolds, I. General Results

TL;DR

This paper systematically studies privileged coordinates and nilpotent approximations in Carnot manifolds, providing descriptions of coordinate systems and algebraic characterizations of associated nilpotent groups, laying groundwork for future research.

Contribution

It offers a comprehensive description of privileged coordinates and characterizes nilpotent groups in Carnot manifolds, advancing understanding of their geometric and algebraic structures.

Findings

All systems of privileged coordinates at a point are described.

An algebraic characterization of nilpotent groups as approximations is provided.

Transformations between privileged coordinate systems are explicitly characterized.

Abstract

In this paper we attempt to give a systematic account on privileged coordinates and the nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold with a distinguished filtration of subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. This paper lies down the background for its sequel by clarifying a few points on privileged coordinates and the nilpotent approximation of Carnot manifolds. In particular, we give a description of all the systems of privileged coordinates at a given point. We also give an algebraic characterization of all nilpotent groups that appear as the nilpotent approximation at a given point. In fact, given a nilpotent group $G$ satisfying this algebraic characterization, we exhibit all the changes of variables that transform a given system of privileged coordinates into another system of privileged coordinates in which the nilpotent approximation is given by $G$.