On Ahlfors' imaginary Schwarzian

Abstract

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.