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Equiregular Distribution

An equiregular distribution is a smooth distribution for which the dimension of each successive span of iterated Lie brackets is constant over the manifold. For a bracket-generating distribution Delta, write Delta^j(q) for the span of brackets of length at most j at q. Equiregularity means that the growth vector (dimDelta^1(q),...,dimDelta^s(q)) is independent of q (Wang and Zelenko 2026).

The sub-Riemannian Sard conjecture concerns endpoints of abnormal curves for such distributions.

See also

Bracket-Generating Distribution, Smooth Distribution, Sub-Riemannian Sard Conjecture

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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. "Equiregular Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EquiregularDistribution.html

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