Conformal arc-length as \(\frac 1 2 \)-dimensional length of the set of osculating circles (Q2269702)

Following \textit{A. Fialkov} [Trans. Am. Math. Soc. 51, 435--501 (1942; Zbl 0063.01358)], a curve in the conformal \(3\)-sphere is uniquely (up to Möbius transformation) determined by its conformal arc-length, curvature and torsion. The authors relate the conformal arc-length of a curve to the \(L^{1\over2}\) arc-length of its curve of osculating circles: in the case of a planar curve (Theorem 5.1), its curve of osculating circles is a light-like curve in de Sitter space; in the case of a space curve (Theorem 7.3), its curve of osculating circles can be thought of as a light-like curve in the Grassmannian of \(3\)-dimensional Minkowski subspaces of Minkowski \(5\)-space. The authors provide a careful and comprehensive exposition, including a limit construction for the \(L^{1\over2}\) arc-length of a light-like curve and a characterization of curves of osculating circles (Theorem 7.11).