Conformal arc-length as ½-dimensional length of the set of osculating circles

Abstract

The set of osculating circles of a given curve in S3 forms a lightlike curve in the set of oriented circles in S3. We show that its “½-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.