Tangency sets of non-involutive distributions and unrectifiability in Carnot-Carathéodory spaces

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-Carathéodory spaces. We also introduce a new class of metric spaces that extends the framework of Carnot-Carathéodory geometry and show that, within this class, Carnot-Carathéodory 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.