Möbius parametrizations of curves in ${\mathbb{R}}^n$

Abstract.

We use Ahlfors’ definition of Schwarzian derivative for curves in euclidean spaces to present new results about Möbius or projective parametrizations. The class of such parametrizations is invariant under compositions with Möbius transformations, and the resulting curves are simple. The analysis is based on the oscillatory behavior of the associated linear equation $u^{\prime\prime} + \frac{1}{4}k^{2}u = 0$, where k = k(s) is the curvature as a function of arclength.