  By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean
space must be a cylinder over a plane curve. However, if we admit some
singularities, there are many non-trivial examples. Flat fronts are flat
hypersurfaces with admissible singularities. Murata--Umehara gave a
representation formula for complete flat fronts with non-empty singular set in
Euclidean $3$-space, and proved the four vertex type theorem. In this paper, we
prove that, unlike the case of $n=2$, there do not exist any complete flat
fronts with non-empty singular set in Euclidean $(n+1)$-space $(n\geq 3)$.
