Double point-homogeneous spherical curves

Abstract

A curve is, in this paper, the image of the circle $S^1$ under an immersion $f$ into $S^2$, $\mathbb{R}^2$ or the real projective plane $P_2(\mathbb{R})$, such that every multiple point of $f$ is an ordinary double point. Such a curve $C$ is double point-homogeneous or DP-homogeneous when the group of diffeomorphisms (of $S^2, \mathbb{R}^2$ or $P_2(\mathbb{R})$) preserving $C$ has a transitive action on the set of its double points. The orbits of DP-homogeneous curves in $S^2$ are totally determined; using combinatorial methods, we prove that they fall into five countably infinite families ; the description of every family is illustrated by drawings of some representatives with a small number of double points. As a corollary, we obtain a similar classification of the DP-homogeneous curves in $\mathbb{R}^2$. We also propose a conjecture about the classification of DP-homogeneous curves in $P_2(\mathbb{R})$.