Higher-dimensional linking integrals

TL;DR

This paper develops integral formulas for linking numbers of submanifolds in spheres, products, and certain hypersurfaces, emphasizing their geometric invariance under orthogonal transformations.

Contribution

It introduces new integral formulas for linking numbers that are invariant under orthogonal group actions in various manifolds.

Findings

Derived integral formulas for linking numbers in S^n and product spaces

Formulas are invariant under special orthogonal group actions

Applicable to hypersurfaces in Euclidean space

Abstract

We derive an integral formula for the linking number of two submanifolds of the n-sphere S^n, of the product S^n x 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.