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arXiv · 0803.1060

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

Rémi Langevin·Jun O'Hara
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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.

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Rémi Langevin, Jun O'Hara. 2008-03-07. Conformal arc-length as $\frac12$ dimensional length of the set of osculating circles. https://arxiv.org/abs/0803.1060

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