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

TL;DR

This work sharpens the connection between integrability and rectifiability by refining Frobenius-type behavior for $h$-non-involutive distributions and deriving unrectifiability results in Carnot–Carathéodory spaces. It develops a cohesive framework that couples Euclidean tangency analysis with CC geometry, establishing $(h-1)$-rectifiability for $C^{2}$ tangency sets and $h$-pure unrectifiability for $C^{1,1}$ tangency; it then transfers these ideas to CC spaces under the Hörmander condition, proving intrinsic unrectifiability. A novel concept of $ta$-squeezed metrics is introduced to interpolate between rectifiable and unrectifiable regimes, with CC spaces shown to be extremal among these metrics and to arise as Gromov–Hausdorff limits of squeezed-geometry spaces. The results illuminate how non-involutivity governs rectifiability in classical and sub-Riemannian settings and advance geometric measure theory in CC geometries through precise metric-geometry constructions and limit arguments.

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.

Theorems & Definitions (34)

• Theorem 1.1.3
• Proposition 1.1.4
• Lemma 2.3.2
• Proposition 2.3.3
• Proposition 2.3.5
• Proposition 2.4.3
• Proposition 2.4.4
• Lemma 2.4.7
• Theorem 2.6.3: [Chow-Rashevskii, see Gromov1996Carnot-CaratheodoryWithin]
• Theorem 2.6.5: Gromov1996Carnot-CaratheodoryWithin