Conformal arc-length via osculating circles

The set of osculating circles of a given curve in S 3 forms a curve in the set of oriented circles in S 3. We show that its “ 1-dimensional measure” 2 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. Key words and phrases. Conformal arc-length, osculating circles, pseudo-Riemannian manifolds 1991 Mathematics Subject Classification. 53A30 1