Parametric CR-umbilical locus of ellipsoids in $ {\mathbb{C}}^{2}$

Foo, Wei-Guo; Merker, Joël; Ta, The-Anh

doi:10.1016/j.crma.2017.11.019

Analytic geometry/Differential geometry

Parametric CR-umbilical locus of ellipsoids in

C^{2}

[Détermination paramétrique du lieu CR-ombilic d'ellipsoïdes dans

C^{2}

]

Présenté par : the Editorial Board

Foo, Wei-Guo ¹ ; Merker, Joël ¹ ; Ta, The-Anh ¹

¹ Departement of Mathematics, Orsay University, Paris, France

Comptes Rendus. Mathématique, Tome 356 (2018) no. 2, pp. 214-221.

Résumé
Abstract

Pour tous nombres réels $a ⩾ 1$ , $b ⩾ 1$ avec $(a, b) \neq (1, 1)$ , la courbe paramétrée par $θ \in R$ à valeurs dans $C^{2} ≅ R^{4}$

γ : θ ⟼ (x (θ) + \sqrt{- 1} y (θ), u (θ) + \sqrt{- 1} v (θ))

ayant pour composantes :

\begin{matrix} x (θ) : = \sqrt{\frac{a - 1}{a (a b - 1)}} \cos θ, y (θ) : = \sqrt{\frac{b (a - 1)}{a b - 1}} \sin θ, \\ u (θ) : = \sqrt{\frac{b - 1}{b (a b - 1)}} \sin θ, v (θ) : = - \sqrt{\frac{a (b - 1)}{a b - 1}} \cos θ, \end{matrix}

est d'image contenue dans le lieu CR-ombilic :

γ (R) \subset UmbCR (E_{a, b}) \subset E_{a, b}

de l'ellipsoïde

E_{a, b} \subset C^{2}

d'équation

a x^{2} + y^{2} + b u^{2} + v^{2} = 1

, où le lieu CR-ombilic d'une hypersurface Levi non dégénérée

M^{3} \subset C^{2}

est l'ensemble des points en lesquels la courbure de Cartan de M s'annule.

For every real numbers $a ⩾ 1$ , $b ⩾ 1$ with $(a, b) \neq (1, 1)$ , the curve parametrized by $θ \in R$ valued in $C^{2} ≅ R^{4}$

γ : θ ⟼ (x (θ) + \sqrt{- 1} y (θ), u (θ) + \sqrt{- 1} v (θ))

with components:

\begin{matrix} x (θ) : = \sqrt{\frac{a - 1}{a (a b - 1)}} \cos θ, y (θ) : = \sqrt{\frac{b (a - 1)}{a b - 1}} \sin θ, \\ u (θ) : = \sqrt{\frac{b - 1}{b (a b - 1)}} \sin θ, v (θ) : = - \sqrt{\frac{a (b - 1)}{a b - 1}} \cos θ, \end{matrix}

has image contained in the CR-umbilical locus:

γ (R) \subset UmbCR (E_{a, b}) \subset E_{a, b}

of the ellipsoid

E_{a, b} \subset C^{2}

of equation

a x^{2} + y^{2} + b u^{2} + v^{2} = 1

, where the CR-umbilical locus of a Levi nondegenerate hypersurface

M^{3} \subset C^{2}

is the set of points at which the Cartan curvature of M vanishes.

Reçu le : 2017-07-12
Accepté le : 2017-11-13
Publié le : 2018-01-12

DOI : 10.1016/j.crma.2017.11.019

Affiliations des auteurs :

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

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