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

TL;DR

This paper proves that certain 2-convex non-orientable surfaces in four-dimensional space can be mapped onto a torus with degree one, and shows that the projective plane and Klein bottle cannot be embedded in this manner.

Contribution

It establishes conditions under which 2-convex non-orientable surfaces in four-dimensional space admit degree-one mappings to a torus, and rules out embeddings of specific surfaces.

Findings

Surfaces with vertex incident to at most five edges admit a degree-one torus mapping.

The projective plane and Klein bottle cannot be 2-convexly embedded in E^4.

Provides conditions for embeddings of non-orientable surfaces in four-dimensional space.

Abstract

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 $\mod 2$ 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 $E^4$.