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Exponential mapping, spheres and waves front in SR-geometry : The Martinet case. (Méthodes géométriques et analytiques pour étudier l’application exponentielle, la sphère et le front d’onde en géométrie sous-riemannienne dans le cas Martinet.) (French) Zbl 0929.53016

Summary: Consider a sub-Riemannian geometry \((U,D,g)\) where \(U\) is a neighborhood of 0 in \(\mathbb{R}^3\), \(D\) is a Martinet type distribution identified with \(\ker\omega\), \(\omega\) being the 1-form \(\omega= dz- \frac{y^2}{2} dx\), \(q= (x,y,z)\) and \(g\) is a metric on \(D\) which can be taken in the normal form \(g= a(q) dx^2+ c(q) dy^2\), \(a= 1+yF(q)\), \(c= 1+G(q)\), \(G_{| x=y=0}=0\).
In a previous article [ibid. 2, 377-448 (1997; Zbl 0902.53033)] we analyzed the flat case \(a=c=1\); we described the conjugate and cut loci, the sphere and the wave front.
The objective of this article is to provide a geometric and computational framework to analyze the general case. This is achieved by analyzing three one-parameter deformations of the flat case which clarify the role of the three parameters \(\alpha, \beta, \gamma\) in the graded normal form of order 0 where \(a= (1+ \alpha y)^2\), \(c= (1+ \beta x+\gamma y)^2\).
More generally, this analysis provides an explanation of the role of abnormal minimizers in SR-geometry.

MSC:

53C17 Sub-Riemannian geometry
49J15 Existence theories for optimal control problems involving ordinary differential equations
53C22 Geodesics in global differential geometry

Citations:

Zbl 0902.53033
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References:

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