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1. Conformal arc-length as ½-dimensional length of the set of osculating circles

Author

Jun O'Hara and Rémi Langevin

Subjects
General Mathematics, Mathematical analysis, Conformal map, Mathematics::Differential Geometry, Invariant (mathematics), Arc length, Mathematics, Osculating circle
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 “ 1 -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.

Published
2010

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