Solid angles and Seifert hypersurfaces

Given a smooth closed oriented manifold M of dimension n embedded in $${\mathbb {R}}^{n+2}$$, we study properties of the ‘solid angle’ function $$\varPhi :{\mathbb {R}}^{n+2}{{\setminus }} M\rightarrow S^1$$. It turns out that a non-critical level set of $$\varPhi$$ is an explicit Seifert hypersurface for M. This gives an explicit analytic construction of a Seifert surface in higher dimensions.