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RESEARCH PRODUCT

The transcendence required for computing the sphere and wave front in the Martinet sub-Riemannian geometry

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Consider a \it{sub-Riemannian geometry} $(U,D,g)$ where $U$ is a neighborhood of $O$ in $\mathbb{R}^3$, $D$ is a \it{Martinet type distribution} identified to $Ker \,\omega$, $\omega =dz-\f{y^2}{2}dx$, $q=(x,y,z)$ and $g$ is a \it{metric on $D$} which can be taken in the normal form : \mbox{$a(q)dx^2+c(q)dy^2$}, \mbox{$a=1+yF(q)$}, \mbox{$c=1+G(q)$}, \mbox{$G_{|x=y=0}=0$}. In a previous article we analyzed the \it{flat case} : \mbox{$a=c=1$} ; we showed that the set of geodesics is integrable using \it{elliptic integrals} of the \it{first and second kind} ; moreover we described the sphere and the wave front near the abnormal direction using the \it{\mbox{exp-log} category}. The objective of this article is to analyze the transcendence we need to compute the sphere and the wave front of small radius in the abnormal direction and globally when we consider the gradated normal form of order $0$ : \mbox{$a=(1+\alpha y)^2$}, \mbox{$c=(1+\beta x + \gamma y)^2$}, where $\alpha, \beta, \gamma$ are real parameters.