Comptes Rendus
no. 2
Analytic geometry/Differential geometry
Parametric CR-umbilical locus of ellipsoids in C2
[Détermination paramétrique du lieu CR-ombilic d'ellipsoïdes dans C2]

Présenté par : the Editorial Board

Wei-Guo Foo1 ; Joël Merker1 ; The-Anh Ta1
1 Departement of Mathematics, Orsay University, Paris, France
Comptes Rendus. Mathématique, Volume 356 (2018) no. 2, pp. 214-221.

Pour tous nombres réels a⩾1, b⩾1 avec (a,b)≠(1,1), la courbe paramétrée par θ∈R à valeurs dans C2≅R4

γ:θ⟼(x(θ)+−1y(θ),u(θ)+−1v(θ))
ayant pour composantes :
x(θ):=a−1a(ab−1)cos⁡θ,y(θ):=b(a−1)ab−1sin⁡θ,u(θ):=b−1b(ab−1)sin⁡θ,v(θ):=−a(b−1)ab−1cos⁡θ,
est d'image contenue dans le lieu CR-ombilic :
γ(R)⊂UmbCR(Ea,b)⊂Ea,b
de l'ellipsoïde Ea,b⊂C2 d'équation ax2+y2+bu2+v2=1, où le lieu CR-ombilic d'une hypersurface Levi non dégénérée M3⊂C2 est l'ensemble des points en lesquels la courbure de Cartan de M s'annule.

For every real numbers a⩾1, b⩾1 with (a,b)≠(1,1), the curve parametrized by θ∈R valued in C2≅R4

γ:θ⟼(x(θ)+−1y(θ),u(θ)+−1v(θ))
with components:
x(θ):=a−1a(ab−1)cos⁡θ,y(θ):=b(a−1)ab−1sin⁡θ,u(θ):=b−1b(ab−1)sin⁡θ,v(θ):=−a(b−1)ab−1cos⁡θ,
has image contained in the CR-umbilical locus:
γ(R)⊂UmbCR(Ea,b)⊂Ea,b
of the ellipsoid Ea,b⊂C2 of equation ax2+y2+bu2+v2=1, where the CR-umbilical locus of a Levi nondegenerate hypersurface M3⊂C2 is the set of points at which the Cartan curvature of M vanishes.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.11.019
Wei-Guo Foo 1 ; Joël Merker 1 ; The-Anh Ta 1

1 Departement of Mathematics, Orsay University, Paris, France 
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     title = {Parametric {CR-umbilical} locus of ellipsoids in $ {\mathbb{C}}^{2}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {214--221},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2017.11.019},
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Wei-Guo Foo; Joël Merker; The-Anh Ta. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.11.019/

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[9] J. Merker; M. Sabzevari The Cartan equivalence problem for Levi-non-degenerate real hypersurfaces M3⊂C2, Izv. Ross. Akad. Nauk, Ser. Mat., Volume 78 (2014) no. 6, pp. 103-140 (in Russian); translation in: Izv. Math., 78, 6, 2014, pp. 1158-1194

[10] P. Nurowski; G.A.J. Sparling 3-dimensional Cauchy–Riemann structures and 2nd order ordinary differential equations, Class. Quantum Gravity, Volume 20 (2003), pp. 4995-5016

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