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

2025/03/03 by Giovanni Alberti, Alberti, Giovanni, Annalisa Massaccesi +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #35R03 #53C17 #58A25 #58A30 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2503.01373

openalex publication_date 2025/03/03 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

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.

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• Frobenius theorem and fine structure of tangency sets to non-involutive distributions

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