Equi-Gaussian curvature folding

Abstract

In this paper we introduce a new type of folding called equi-Gaussian curvature folding of connected Riemannian 2-manifolds. We prove that the composition and the cartesian product of such foldings is again an equi-Gaussian curvature folding. In case of equi-Gaussian curvature foldings, f: M → P _n , of an orientable surface M onto a polygon P _n we prove that

$$ f \in \mathcal{F}_{EG} (S^2 ) \Leftrightarrow n = 3 $$

((i))

$$ f \in \mathcal{F}_{EG} (T^2 ) \Rightarrow n = 4 $$

((ii))

$$ f \in \mathcal{F}_{EG} (\# 2T^2 ) \Rightarrow n = 5,6 $$

((iii))

and we generalize (iii) for #nT ².