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

Abstract : 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.
Document type :
Conference papers

https://hal.archives-ouvertes.fr/hal-00086419
Contributor : Emmanuel Trélat <>
Submitted on : Wednesday, July 19, 2006 - 11:27:00 AM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM
Long-term archiving on : Friday, May 13, 2011 - 11:34:44 PM

### Identifiers

• HAL Id : hal-00086419, version 1

### Citation

Bernard Bonnard, Geneviève Launay, Emmanuel Trélat. The transcendence required for computing the sphere and wave front in the Martinet sub-Riemannian geometry. 1999, pp.82--117. ⟨hal-00086419⟩

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