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

Abstract

We discover new information about the spherical curvature of stereographically projected analytic curves. To do so, we first state formulas for the spherical curvature and spherical torsion of the curves on S2 which result after stereographically projecting the image curves of analytic, univalent functions belonging to the class 𝒮. We then derive results concerning the location of the critical points of the spherical curvature, considered both as a function of one and two variables. Further analysis leads to a maximum principle for the spherical curvature functions.