Calculus of Variations and Geometric Measure Theory
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D. Barilari - M. Kohli

On sub-Riemannian geodesic curvature in dimension three

created by barilari on 22 Oct 2019
modified on 12 May 2021


Accepted Paper

Inserted: 22 oct 2019
Last Updated: 12 may 2021

Journal: Advances in Calculus of Variations
Year: 2021
Doi: 10.1515/acv-2020-0021

ArXiv: 1910.13132 PDF


We introduce a notion of geodesic curvature for a smooth horizontal curve in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riemannian distance between two points on a unit speed horizontal curve.

The sub-Riemannian distance is not smooth on the diagonal, hence the result contains the existence of such an asymptotics. In this sense, we show a higher-order differentiability property of the sub-Riemannian distance along smooth horizontal curves. This generalizes the previously known results on the Heisenberg group.

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