Submitted Paper

Inserted: 4 jun 2025

Year: 2025

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.

Keywords: Rectifiability, Carnot-Carathéodory spaces, non-involutive distributions, Frobenius theorem