Tangent groupoid and tangent cones in sub-Riemannian geometry

Omar Mohsen

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arXiv · 2212.01652

Tangent groupoid and tangent cones in sub-Riemannian geometry

Omar Mohsen

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Abstract

Let $X_1,\cdots,X_m$ be vector fields satisfying Hörmander'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 Bellaïche.

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Omar Mohsen. 2024-09-02. Tangent groupoid and tangent cones in sub-Riemannian geometry. https://arxiv.org/abs/2212.01652

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