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Cône normal et régularités de Kuo-Verdier. (Normal cone and Kuo-Verdier regularities). (French) Zbl 1014.58004
Let \(Z\) be a stratified closed set of \(\mathbb{R}^n\) whose strata are the \(C^k\) differentiable varieties, \(k \geq 2\). If \(Y\) is an stratum of \(Z\), the normal cone of \(Z\) along \(Y\) consists on the restriction to \(Y\) of the closure of the set \[ \{ (x, \mu(x\pi(x))) \mid x \in Z - Y \}, \] where \(\pi\) is the local projection onto \(Y\) and \(\nu(x) = x / \|x\|\).
In the paper the authors introduce a new type of Kuo-Verdier regularities, called \(r^e\), and it is devoted to prove that for an \((a + r^e)\) regular \(C^2\)-stratification, (\((a)\) being the (a) condition of Whitney), the fiber of the normal cone along a stratum \(Y\) is equal to the tangent cone of the fiber of a retraction onto \(Y\). This result generalizes a previous one by the authors which holds for \((\omega + \delta)\)-differentiable stratifications, \(\omega\) (respectively, \(\delta\)) being the Kuo-Verdier (respectively, Bekka-Trotman) conditions.

58A35 Stratified sets
32S15 Equisingularity (topological and analytic)
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