Legendrian dual surfaces in hyperbolic 3-space
We consider surfaces in hyperbolic $3$-space and their duals. We study flat dual surfaces in hyperbolic $3$-space by using extended Legendrian dualities between pseudo-hyperspheres in Lorentz–Minkowski 4-space. We define the flatness of a surface in hyperbolic $3$-space by the degeneracy of its dual, which is similar to the case of the Gauss map of a surface in Euclidean $3$-space. Such surfaces are a kind of ruled surfaces. Moreover, we investigate the singularities of these surfaces and the dualities of the singularities.