On 2-convex non-orientable surfaces in four-dimensional Euclidean space

We prove that a 2-convex closed surface $S\subset E^4$ in the four-dimensional Euclidean space $E^4$, which is either $C^2$-smooth or polyhedral, provided that each vertex is incident to at most five edges, admits a mapping of degree one to a two-dimensional torus, where the degree is assumed to be $\mathrm{m}\mathrm{o}\mathrm{d}2$ if $S$ is nonorientable. As a corollary, we show that the projective plane and the Klein bottle do not admit such a 2-convex embedding in $E^4$.