Euclidean symmetry of closed surfaces immersed in 3-space

Abstract

Given a finite group G of orientation-preserving isometries of euclidean 3-space E3 and a closed surface S, an immersion f:S→E3 is in G-general position if f(S) is invariant under G, points of S have disk neighborhoods whose images are in general position, and no singular points of f(S) lie on an axis of rotation of G. For such an immersion, there is an induced action of G on S whose Riemann–Hurwitz equation satisfies certain natural restrictions. We classify which restricted Riemann–Hurwitz equations are realized by a G-general position immersion of S. This generalizes work by various authors on euclidean symmetry of closed surfaces embedded in E3. The analysis involves a detailed study of immersions of the quotient surface S/G in the orbifold E3/G.