Solid angles and Seifert hypersurfaces

Abstract

Given a smooth closed oriented manifold $M$ of dimension $n$ embedded in $\mathbb{R}^{n+2}$ we study properties of the `solid angle' function $\Phi\colon\mathbb{R}^{n+2}\setminus M\to S^1$. It turns out that a non-critical level set of $\Phi$ is an explicit Seifert hypersurface for $M$.