Orientation-reversing symmetry of closed surfaces immersed in euclidean 3-space
Undine Leopold, Thomas W. Tucker · Contemporary mathematics - American Mathematical Society · 2021
Given a finite group G G of isometries of euclidean 3-space E 3 \mathbb {E}^3 and a closed surface S S , an immersion f : S → E 3 f: S\rightarrow \mathbb {E}^3 is in G G -general position if f ( S ) f(S) is invariant under G G , points of S S have disk neighborhoods mapped homeomorphically onto their images and these images are in general position, and all double curves of f f are in general position with respect to axes of rotations and reflection planes. For such an immersion, there is an induced action of the orientation-preserving subgroup G + G^+ on S S whose Riemann–Hurwitz equation satisfies certain natural restrictions. We classify which restricted Riemann–Hurwitz equations for G + G^+ are realized by a G G -general position immersion of S S , extending results of the authors for the orientation-preserving case G = G + G=G^+ . The analysis involves a detailed study of immersions of the quotient surface