Before revision:  In this paper, we show that the case of four mutually non-congruent curved foldings with the same crease and crease pattern occurs if and only if  \Gamma and C do not admit any symmetries.
After revision:  In this paper, we show that the case of four mutually non-congruent curved foldings with the same crease and crease pattern occurs if and only if both \Gamma and C do not admit any symmetries.