Privileged Coordinates and Nilpotent Approximation for Carnot Manifolds, II. Carnot Coordinates

TL;DR

This paper introduces Carnot coordinates for Carnot manifolds, providing a systematic way to approximate the manifold locally by its tangent group, enhancing the understanding of their geometric structure.

Contribution

It defines Carnot coordinates, a special class of privileged coordinates, and constructs $ ext{ε}$-Carnot coordinates that depend smoothly on the base point and approximate the tangent group law.

Findings

Carnot coordinates include Darboux and Goodman-Rothschild-Stein coordinates.

$ ext{ε}$-Carnot coordinates provide an effective local approximation of Carnot manifolds.

Coordinates depend smoothly on the base point and are osculated by the tangent group's law.

Abstract

This paper is a sequel of arxiv:1709.09045 and deals with privileged coordinates and nilpotent approximation of Carnot manifolds. By a Carnot manifold it is meant a manifold equipped with a filtration by subbundles of the tangent bundle which is compatible with the Lie bracket of vector fields. In this paper, we single out a special class of privileged coordinates in which the nilpotent approximation at a given point of a Carnot manifold is given by its tangent group. We call these coordinates Carnot coordinates. Examples of Carnot coordinates include Darboux coordinates on contact manifolds and the canonical coordinates of the first kind of Goodman and Rothschild-Stein. By converting the privileged coordinate of Bella\"iche into Carnot coordinates we obtain an effective construction of Carnot coordinates, which we call $\varepsilon$-Carnot coordinates. They form the building block of all systems of Carnot coordinates. On a graded nilpotent Lie group they are given by the group law of the group. For general Carnot manifolds, they depend smoothly on the base point. Moreover, in Carnot coordinates at a given point, they are osculated in a very precise manner by the group law of the tangent group at the point.