Refinements of the Frobenius Theorem reveal new insights into integrability and rectifiability in Carnot-Carathéodory spaces, suggesting connections with non-involutive distributions.
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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Alberti et al. (2025) studied this question.
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