$\mathbb{C}^2$ 中椭球面的参数化 CR-脐点轨迹
复变函数
2017-07-24 v1
摘要
对于任意满足 ( a , b ) ≠ ( 1 , 1 ) (a,b) \neq (1,1) ( a , b ) = ( 1 , 1 ) a ⩾ 1 a \geqslant 1 a ⩾ 1 b ⩾ 1 b \geqslant 1 b ⩾ 1 θ ∈ R \theta \in \mathbb{R} θ ∈ R C 2 ≅ R 4 \mathbb{C}^2 \cong \mathbb{R}^4 C 2 ≅ R 4 γ : θ ⟼ ( x ( θ ) + − 1 y ( θ ) , u ( θ ) + − 1 v ( θ ) ) \gamma\, \colon \ \ \ \theta \,\,\,\longmapsto\,\,\, \big( x(\theta)+{\scriptstyle{\sqrt{-1}}}\,y(\theta),\,\, u(\theta)+{\scriptstyle{\sqrt{-1}}}\,v(\theta) \big) γ : θ ⟼ ( x ( θ ) + − 1 y ( θ ) , u ( θ ) + − 1 v ( θ ) ) x ( θ ) : = a − 1 a ( a b − 1 ) cos θ , y ( θ ) : = b ( a − 1 ) a b − 1 sin θ , ( ˘ θ ) : = b − 1 b ( a b − 1 ) sin θ , v ( θ ) : = − a ( b − 1 ) a b − 1 cos θ , 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, x ( θ ) := a ( ab − 1 ) a − 1 cos θ , y ( θ ) := ab − 1 b ( a − 1 ) sin θ , ( ˘ θ ) := b ( ab − 1 ) b − 1 sin θ , v ( θ ) := − ab − 1 a ( b − 1 ) cos θ , a x 2 + y 2 + b u 2 + y 2 = 1 a\,x^2+y^2+b\,u^2+y^2 = 1 a x 2 + y 2 + b u 2 + y 2 = 1 E a , b ⊂ C 2 {\sf E}_{a,b} \subset \mathbb{C}^2 E a , b ⊂ C 2 γ ( R ) ⊂ U m b C R ( E a , b ) ⊂ E a , b \gamma(\mathbb{R}) \,\subset\, {\sf UmbCR} \big({\sf E}_{a,b}\big) \,\subset\, {\sf E}_{a,b} γ ( R ) ⊂ UmbCR ( E a , b ) ⊂ E a , b
引用
@article{arxiv.1707.06787,
title = {Parametric CR-umbilical Locus of Ellipsoids in $\mathbb{C}^2$},
author = {Wei-Guo Foo and Joel Merker and The-Anh Ta},
journal= {arXiv preprint arXiv:1707.06787},
year = {2017}
}