Higher-dimensional linking integrals

Abstract

We derive an integral formula for the linking number of two submanifolds of the $n$-sphere $S^n$, of the product $S^n \times \mathbb {R}^m$, and of other manifolds which appear as “nice” hypersurfaces in Euclidean space. The formulas are geometrically meaningful in that they are invariant under the action of the special orthogonal group on the ambient space.