Conformal arc-length as $\frac12$ dimensional length of the set of osculating circles

TL;DR

This paper establishes that the conformal arc-length of a curve in the 3-sphere is proportional to the half-dimensional measure of its osculating circles set, revealing a new geometric invariant in conformal geometry.

Contribution

It introduces a novel interpretation of conformal arc-length as a measure of the osculating circles set in a pseudo-Riemannian framework, linking local conformal invariants to geometric measures.

Findings

Conformal arc-length is proportional to the half-dimensional measure of osculating circles.

The set of osculating circles forms a curve in the space of oriented circles with a pseudo-Riemannian structure.

The measure is conformally invariant and relates to classical conformal invariants.

Abstract

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "${\frac12}$-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.