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 $a\,x^2+y^2+b\,u^2+y^2 = 1$.
Categories: math.CV