Space of subspheres and conformal invariants of curves

TL;DR

This paper introduces a conformally invariant framework for analyzing space curves using osculating circles and spheres, providing new tools for characterizing curves and canal surfaces in Minkowski space.

Contribution

It develops a conformally defined moving frame based on osculating circles and spheres, enabling the derivation of conformal invariants and normal forms of space curves.

Findings

Conformal arc-length, curvature, and torsion determine space curves up to Möbius transformations.

A conformally invariant moving frame is constructed using osculating circles and spheres.

Characterization of canal surfaces via curves in the space of circles.

Abstract

A space curve is determined by conformal arc-length, conformal curvature, and conformal torsion, up to M\"obius transformations. We use the spaces of osculating circles and spheres to give a conformally defined moving frame of a curve in the Minkowski space, which can naturally produce the conformal invariants and the normal form of the curve. We also give characterization of canal surfaces in terms of curves in the set of circles.