# Conformal arc-length as $21 $-dimensional length of the set of osculating circles

### Rémi Langevin

Université de Bourgogne, Dijon, France### Jun O'Hara

Chiba University, Japan

## Abstract

The set of osculating circles of a given curve in $S_{3}$ forms a lightlike curve in the set of oriented circles in $S_{3}$. We show that its “$21 $-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.

## Cite this article

Rémi Langevin, Jun O'Hara, Conformal arc-length as $21 $-dimensional length of the set of osculating circles. Comment. Math. Helv. 85 (2010), no. 2, pp. 273–312

DOI 10.4171/CMH/196