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Tangency sets of non-involutive distributions and unrectifiability in Carnot-Carath\'{e}odory spaces

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arxiv 2503.01373 v1 pith:LPJCWZ5M submitted 2025-03-03 math.DG math.FAmath.MG

Giovanni Alberti , Annalisa Massaccesi , Andrea Merlo This is my paper · ORCID
classification math.DGmath.FAmath.MG
keywords carnot-carathodoryspacesclassdistributionsnon-involutiveunrectifiabilityclassical
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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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  1. Frobenius theorem and fine structure of tangency sets to non-involutive distributions

    math.DG 2025-06 conditional novelty 7.0 of 10

    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.

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