Tangent groupoid and tangent cones in sub-Riemannian geometry

TL;DR

This paper generalizes Connes's tangent groupoid to sub-Riemannian manifolds, enabling the calculation of tangent cones in the Gromov-Hausdorff sense, extending Bella"iche's earlier results.

Contribution

It introduces a new completion of the tangent groupoid tailored for sub-Riemannian geometry, facilitating tangent cone computations.

Findings

Constructed a generalized tangent groupoid for sub-Riemannian manifolds.

Calculated all tangent cones of the sub-Riemannian metric in Gromov-Hausdorff sense.

Extended Bella"iche's results to a broader class of manifolds.

Abstract

Let $X_1,\cdots,X_m$ be vector fields satisfying H\"ormander's Lie bracket generating condition on a smooth manifold $M$. We generalise Connes's tangent groupoid, by constructing a completion of the space $M\times M\times \mathbb{R}_+^\times$ using the sub-Riemannian metric. We use our space to calculate all the tangent cones of the sub-Riemannian metric in the sense of the Gromov-Hausdorff distance. This generalises a result of Bella\"iche.