In this paper, we attempt to introduce the singularity theory into multiplicative Euclidean space and investigate the singularities of multiplicative spherical Darboux image, multiplicative rectifying Gaussian surface, and multiplicative rectifying developable surface. By constructing three families of multiplicative functions and applying unfolding theory in multiplicative Euclidean space, we establish the relationships between singularities of these objects and the multiplicative geometric invariants. In addition, we find that the multiplicative geometric invariants of multiplicative space curves are closely related to the order of contact with multiplicative helices. Finally, to enhance visual comprehension, an example is used to demonstrate our theoretical results.

The motivation of this paper is that we wish to add some new information on Legendrian dualities for continuous families of pseudo-spheres in Minkowski space. We construct all contact diffeomorphic mappings between those contact manifolds and display them in a table which contains all information about Legendrian dualities, by the way, we extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space.