Non subanalyticity of sub-Riemannian Martinet spheres

Emmanuel Trélat (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

Consider the sub-Riemannian Martinet structure $(M,\Delta,g)$ where $M=\R^3$, $\Delta={\rm{Ker }}(dz-{{y^2}\over{2}}dx)$ and $g$ is the general gradated metric of order $0$~: $g=(1+\alpha y)^2dx^2+(1+\beta x+\gamma y)^2dy^2$. We prove that if $\alpha\neq 0$ then the sub-Riemannian spheres $S(0,r)$ with small radii are not subanalytic.

Domaines: Optimisation et contrôle [math.OC]

Emmanuel Trélat. Non subanalyticity of sub-Riemannian Martinet spheres. Comptes rendus de l'Académie des sciences. Série I, Mathématique, 2001, 332, pp.527--532. ⟨hal-00086298⟩