The tangent groupoid of a Carnot manifold

Abstract: This talk will deal with 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 Bellaiche. This is joint work with Woocheol Choi.

analysis of PDEsdifferential geometrymetric geometryoptimization and controlspectral theory

Audience: researchers in the topic

( video )

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Sub-Riemannian Seminars

Series comments: The "Sub-Riemannian seminars" are the union of the "Séminaire de géométrie et analyse sous-riemannienne" (held in Paris since 2011) and the "International Sub-Riemannian Seminars", which were born in spring 2020 as a reaction to the COVID-19 pandemic.

The new format will gather every 3 weeks on average, alternating between these types of sessions:

- physical session in Paris (Laboratoire Jacques-Louis Lions), also transmitted online on Zoom.

- fully online session on Zoom.

- special session hosted physically somewhere else, and transmitted online.

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