Analytic geometry/Differential geometry
Parametric CR-umbilical locus of ellipsoids in C2
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 214-221.

For every real numbers a1, b1 with (a,b)(1,1), the curve parametrized by θR valued in C2R4

γ:θ(x(θ)+1y(θ),u(θ)+1v(θ))
with components:
x(θ):=a1a(ab1)cosθ,y(θ):=b(a1)ab1sinθ,u(θ):=b1b(ab1)sinθ,v(θ):=a(b1)ab1cosθ,
has image contained in the CR-umbilical locus:
γ(R)UmbCR(Ea,b)Ea,b
of the ellipsoid Ea,bC2 of equation ax2+y2+bu2+v2=1, where the CR-umbilical locus of a Levi nondegenerate hypersurface M3C2 is the set of points at which the Cartan curvature of M vanishes.

Pour tous nombres réels a1, b1 avec (a,b)(1,1), la courbe paramétrée par θR à valeurs dans C2R4

γ:θ(x(θ)+1y(θ),u(θ)+1v(θ))
ayant pour composantes :
x(θ):=a1a(ab1)cosθ,y(θ):=b(a1)ab1sinθ,u(θ):=b1b(ab1)sinθ,v(θ):=a(b1)ab1cosθ,
est d'image contenue dans le lieu CR-ombilic :
γ(R)UmbCR(Ea,b)Ea,b
de l'ellipsoïde Ea,bC2 d'équation ax2+y2+bu2+v2=1, où le lieu CR-ombilic d'une hypersurface Levi non dégénérée M3C2 est l'ensemble des points en lesquels la courbure de Cartan de M s'annule.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.11.019
Foo, Wei-Guo 1; Merker, Joël 1; Ta, The-Anh 1

1 Departement of Mathematics, Orsay University, Paris, France
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Foo, Wei-Guo; Merker, Joël; Ta, The-Anh. Parametric CR-umbilical locus of ellipsoids in $ {\mathbb{C}}^{2}$. Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 214-221. doi : 10.1016/j.crma.2017.11.019. http://www.numdam.org/articles/10.1016/j.crma.2017.11.019/

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