Parametric CR-umbilical Locus of Ellipsoids in ${\mathbb{C}}^2$

Abstract

For every real numbers $a⩾1$, $b⩾1$ with $(a,b)\ne (1,1)$, the curve parametrized by $\theta \in \mathbb{R}$ valued in ${\mathbb{C}}^2\cong {\mathbb{R}}^4$ \[ γ\, \colon \ \ \ θ\,\,\,\longmapsto\,\,\, \big( x(θ)+{\scriptstyle{\sqrt{-1}}}\,y(θ),\,\, u(θ)+{\scriptstyle{\sqrt{-1}}}\,v(θ) \big) \] with components: \[ x(θ) \,:=\, {\textstyle{\sqrt{\frac{a-1}{a\,(ab-1)}}}}\, \cos\,θ, \ \ \ \ \ y(θ) \,:=\, {\textstyle{\sqrt{\frac{b\,(a-1)}{ab-1}}}}\, \sin\,θ, \ \ \ \ \ u(θ) \,:=\, {\textstyle{\sqrt{\frac{b-1}{b\,(ab-1)}}}}\, \sin\,θ, \ \ \ \ \ v(θ) \,:=\, -\, {\textstyle{\sqrt{\frac{a\,(b-1)}{ab-1}}}}\, \cos\,θ, \] has image contained in the CR-umbilical locus: \[ γ(\mathbb{R}) \,\subset\, {\sf UmbCR} \big({\sf E}_{a,b}\big) \,\subset\, {\sf E}_{a,b} \] of the ellipsoid ${\mathsf{E}}_{a,b}\subset {\mathbb{C}}^2$ of equation $a{x}^2+y^2+b{u}^2+y^2=1$.