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Regularity of the sub-Riemannian distance and cut locus. (English) Zbl 1239.53043

Isidori, Alberto (ed.) et al., Nonlinear control in the year 2000. Vol. 1. Papers from the 2nd workshop of the nonlinear control network, Paris, France, June 5–9, 2000. London: Springer (ISBN 1-85233-363-4). Lect. Notes Control Inf. Sci. 258, 521-533 (2001).
This article will not be reviewed individually. See the review of the whole volume (2001; Zbl 0968.93008).
Summary: Sub-Riemannian distances are obtained by minimizing the length of curves whose velocity is constrained to be tangent to a given sub-bundle of the tangent bundle. We study the regularity properties of the function \(x\mapsto d(x_0,x)\) for a given sub-Riemannian distance \(d\) on a neighborhood of a point \(x_0\) of a manifold \(M\). We already know that this function is not \(C^1\) on any neighborhood of \(x_0\) (see A. A. Agrachev, Rend. Semin. Mat., Torino 56, No. 4, 1–12 (1998; Zbl 1039.53038)] and, even if the data are analytic, the distance may fail to be subanalytic [A. A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, ESAIM, Control Optim. Calc. Var. 2, 377–448 (1997; Zbl 0902.53033)]. In this paper we make the link between the singular support of \(x\mapsto d(x_0,x)\), the cut locus and the set of points reached from \(x_0\) by singular minimizers.
For the entire collection see [Zbl 0968.93008].

MSC:

53C17 Sub-Riemannian geometry
93B20 Minimal systems representations
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