Conformal arc-length as -dimensional length of the set of osculating circles
Rémi Langevin
Université de Bourgogne, Dijon, FranceJun O'Hara
Chiba University, Japan
![Conformal arc-length as $\frac 1 2$-dimensional length of the set of osculating circles cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserials%2Fcover-cmh.png&w=3840&q=90)
Abstract
The set of osculating circles of a given curve in forms a lightlike curve in the set of oriented circles in . We show that its “-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 -dimensional length of the set of osculating circles. Comment. Math. Helv. 85 (2010), no. 2, pp. 273–312
DOI 10.4171/CMH/196