On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

In this article we study the validity of the Whitney C 1 extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a s...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2018-04, Vol.57 (2), p.1-34, Article 59
Main Authors: Sacchelli, Ludovic, Sigalotti, Mario
Format: Article
Language:English
Subjects:
Analysis
Approximation
Calculus of Variations and Optimal Control
Optimization
Control
Differential Geometry
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Metric Geometry
Optimization and Control
Riemann manifold
Systems Theory
Theorems
Theoretical
Topological manifolds
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Summary:In this article we study the validity of the Whitney C 1 extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the C 1 extension property. We conclude by showing that the C 1 extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-018-1336-8