On Non-Orientable Closed Surfaces in Euclidean Spaces

Abstract

Let us begin with a simple result.Proposition. Let X be a non-orientable closed surface differentiably imbedded into the euclidean 4-space R4. Then there is a line in R4 which intersects X at more than two points.Proof. Let c be any point of R4 and let r be the smallest number such that X is contained in the closed 4-spheroid W of centre c and radius r. Clearly the boundary 3-sphere S of W intersects X.