Geodesic
On Ahlfors' imaginary Schwarzian
Martin Chuaqui1
1 Pontificia Universidad Católica de Chile, Facultad de Matemáticas
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 345-353.

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

Martin Chuaqui 1

1 Pontificia Universidad Católica de Chile, Facultad de Matemáticas 
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Martin Chuaqui. On Ahlfors' imaginary Schwarzian. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 345-353. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a18/