Conformal arc-length as $\frac{1}{2}$-dimensional length of the set of osculating circles

Abstract

The set of osculating circles of a given curve in ${\mathbf{S}}^3$ forms a lightlike curve in the set of oriented circles in ${\mathbf{S}}^3$. We show that its “$\frac{1}{2}$-dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.