On geometrically realizable Möbius triangulations

Let MM be a map on a surface F2F2. A geometric realization   of MM is an embedding of F2F2 into a Euclidian 3-space R3R3 with no self-intersection such that each face of MM is a flat polygon. In this paper, we characterize geometrically realizable triangulations on the Möbius band.

► In 1983, Brehm showed a Möbius triangulation with no geometric realization.
► However, he did not characterize Möbius triangulations with geometric realizations.
► In this paper, we characterize Möbius triangulations with geometric realizations.