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

TL;DR

This paper describes a specific parametric curve lying entirely within the CR-umbilical locus of a family of ellipsoids in complex two-space, revealing geometric properties of these loci.

Contribution

It introduces an explicit parametric description of the CR-umbilical locus on ellipsoids in ext{C}^2, expanding understanding of their geometric structure.

Findings

The curve is contained in the CR-umbilical locus of the ellipsoid.

The parametric form depends on parameters a and b, with (a,b) ≠ (1,1).

The locus is explicitly characterized by trigonometric functions.

Abstract

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