On Ahlfors' imaginary Schwarzian

Kirjoittajat

  • Martin Chuaqui Pontificia Universidad Católica de Chile, Facultad de Matemáticas

Avainsanat:

Ahlfors' Schwarzian for curves, imaginary Schwarzian, osculating sphere, Möbius transformation, overdetermined problem

Abstrakti

We study geometric aspects of the imaginary Schwarzian \(S_2f\) for curves in 3-space, as introduced by Ahlfors in [1]. We show that \(S_2f\) points in the direction from the center of the osculating sphere to the point of contact with the curve. We also establish an important law of transformation of \(S_2f\) under Möbius transformations. We finally study questions of existence and uniqueness up to Möbius transformations of curves with given real and imaginary Schwarzians. We show that curves with the same generic imaginary Schwarzian are equal provided they agree to second order at one point, while prescribing in addition the real Schwarzian becomes an overdetermined problem.
Osasto
Articles

Julkaistu

2021-06-21

Viittaaminen

Chuaqui, M. (2021). On Ahlfors’ imaginary Schwarzian. Annales Fennici Mathematici, 46(1), 345–353. Noudettu osoitteesta https://afm.journal.fi/article/view/109587