Tangent cones and C¹ regularity of definable sets

2017/03/15 by Krzysztof Kurdyka, Kurdyka, Krzysztof, Olivier Le Gal +3 · 2 citations
Mathematics · #Advanced Topology and Set Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.1703.05421

openalex publication_date 2017/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X⊂ \mathbb Rⁿ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) X is a C¹ manifold, (ii) the tangent cone and the paratangent cone of X coincide at every point in X, (iii) for every x ∈ X, the tangent cone of X at the point x is a k-dimensional linear subspace of \mathbb Rⁿ (k does not depend on x) varies continuously in x, and the density θ(X, x) < 3/2.

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