On some new properties of the spherical curvature of stereographically projected analytic curves (Q1811936)

Let \(\Pi\) denote the stereographic projection of the complex plane onto the unit sphere. Let \(r,\theta\), \(0<r<1\), \(-\pi<\theta \leq\pi\) be given numbers and let \(f\) be a function of the class \(S\). Assume that \(\Gamma_r= \Pi(f(|z|=r))\), \({\mathfrak L}_\theta= \Pi(f(\arg z=\theta))\), \(|z|<1\). The author derives various formulas for the spherical curvature and the spherical torsion of \(\Gamma_r\), \({\mathfrak L}_\theta\), gave a relation between them. He then investigates properties of the curvature of the curves \(\Gamma_r\), \({\mathfrak L}_\theta\), both as functions of one or two variables. He also studies topological properties of level sets he encountered in the process.