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Horizontal Curve

A horizontal curve for a smooth distribution Delta on a manifold M is an absolutely continuous curve gamma satisfying gamma^.(t) in Delta(gamma(t)) for almost every t. Thus its velocity is constrained to the prescribed tangent directions (Wang and Zelenko 2026).

For a fixed initial point, the endpoint map assigns to each such finite-energy curve its final point. A horizontal curve at which the differential of this map is not onto is an abnormal curve.

See also

Abnormal Curve, Endpoint Map, Smooth Distribution

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References

Agrachev, A.; Barilari, D.; and Boscain, U. A Comprehensive Introduction to Sub-Riemannian Geometry: From the Hamiltonian Viewpoint. Cambridge, England: Cambridge University Press, 2019. https://doi.org/10.1017/9781108677325.Wang, B. and Zelenko, I. "An Equiregular Counterexample to the Sard Conjecture for Sub-Riemannian Endpoint Maps." 16 Oct 2026. https://arxiv.org/abs/2610.24080.

Cite this as:

Weisstein, Eric W. "Horizontal Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HorizontalCurve.html

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