Folding maps on spacelike and timelike surfaces and duality

Abstract

We study the reflectional symmetry of a generically embedded $2$-dimensional surface $M$ in the hyperbolic or de Sitter $3$-dimensional spaces. This symmetry is picked up by the singularities of folding maps that are defined and studied here. We also define the evolute and symmetry set of $M$ and prove duality results that relate them to the bifurcation sets of the family of folding maps.