Privileged coordinates and tangent groupoid for Carnot manifolds

This paper is part of a series of papers on the construction of a full hypoelliptic pseudodifferential calculus on Carnot manifolds. The first goal of the paper is to produce a refinement of privileged coordinates for Carnot manifolds for which the extrinsic tangent group agree with the intrinsic tangent group. We call these coordinates Carnot coordinates. These coordinates have various applications and we present a number of them in this paper. This lead us to the second goal of this paper, which is to use these coordinates to establish an approximation result for maps between Carnot manifolds that are compatible with the Carnot manifold structures. Here the map is approximated in a suitable way by its tangent map, which is defined as a Lie group map between the corresponding tangent groups. This result is new and is the complete analogue in the setup of Carnot manifolds of the tangent linear approximation of smooth maps between manifolds. The third goal of this paper is to make use of the Carnot coordinates and their properties to construct an analogue for Carnot manifolds of the tangent groupoid of Connes. We then obtain a differentiable groupoid which encodes the smooth deformation of the pair groupoid $M\times M$ to the tangent Lie group bundle of the Carnot manifold. The existence of such a groupoid was conjectured by Bella\"iche. This shows that the tangent group of Carnot manifold at a given point is tangent in a true differential-geometric fashion.