Every tight immersion in three-space of the projective plane with one handle is asymmetric

Abstract

Kuiper's original analysis of tight surfaces showed that every surface has a tight immersion in three-space except for the Klein bottle and the projective plane, which have none, and the projective plane with one handle, for which he was unable to determine whether a tight immersion was possible. The latter obtained a unique position among surfaces when it was shown that no smooth tight immersion of it can be formed, while a polyhedral one does exist. Continuing in its role as an unusual example, this surface has another unexpected property, demonstrated here: Any tight immersion is necessarily asymmetric, while every other surface can be immersed tightly and symmetrically in space.