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Abnormal Curve

An abnormal curve of a smooth distribution is a horizontal curve at which the differential of the fixed-start endpoint map is not surjective (Wang and Zelenko 2026).

The sub-Riemannian Sard conjecture asks whether the endpoints of all abnormal curves from a fixed point have measure zero. Restricting to abnormal curves that minimize length gives a different assertion.

See also

Endpoint Map, Horizontal Curve, Sub-Riemannian Sard Conjecture

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References

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. "Abnormal Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbnormalCurve.html

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