Tangent Maps and Tangent Groupoid for Carnot Manifolds

TL;DR

This paper develops a differential calculus for Carnot manifolds, introducing the Carnot differential and tangent groupoid, which generalize known derivatives and structures to a broader class of geometric spaces.

Contribution

It introduces the Carnot differential for maps between Carnot manifolds and constructs a tangent groupoid, extending the Pansu derivative and Connes' tangent groupoid to this setting.

Findings

Carnot differential generalizes Pansu derivative.

Constructed a tangent groupoid for Carnot manifolds.

Proved Carnot maps are osculated by their Carnot differential.

Abstract

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold $(M,H)$ we get a smooth groupoid that encodes the smooth deformation of the pair $M\times M$ to the tangent group bundle $GM$. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bella\"iche.