For non-involutive distributions, a tangency set of positive measure is possible exactly when its fractional Sobolev regularity is low enough relative to the surface gradient regularity.
Tangency sets of non-involutive distributions and unrectifiability in Carnot-Carath\'{e}odory spaces
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abstract
In this paper, we establish refined versions of the Frobenius Theorem for non-involutive distributions and use these refinements to prove an unrectifiability result for Carnot-Carath\'{e}odory spaces. We also introduce a new class of metric spaces that extends the framework of Carnot-Carath\'{e}odory geometry and show that, within this class, Carnot-Carath\'{e}odory spaces are, in some sense, extremal. Our results provide new insights into the relationship between integrability, non-involutivity, and rectifiability in both classical and sub-Riemannian settings.
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Frobenius theorem and fine structure of tangency sets to non-involutive distributions
For non-involutive distributions, a tangency set of positive measure is possible exactly when its fractional Sobolev regularity is low enough relative to the surface gradient regularity.