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

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

2008-03-07

arxiv.org/abs/0803.1060v2

Mathematics

Differential Geometry

39 pages, 11 figures. To appear in Comm. Math. Helv

Scientific paper

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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