Helicity is the only integral invariant of volume-preserving transformations

TL;DR

This paper proves that helicity is the unique regular integral invariant under volume-preserving transformations on compact 3-manifolds, establishing its fundamental role in the study of divergence-free vector fields.

Contribution

It demonstrates that any regular integral invariant of volume-preserving transformations must be a function of helicity, showing its uniqueness among such invariants.

Findings

Helicity is the only regular integral invariant under volume-preserving diffeomorphisms.

Any invariant functional with a well-behaved integral kernel depends solely on helicity.

The result applies to divergence-free vector fields on compact 3-manifolds.

Abstract

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional $\mathcal I$ defined on exact divergence-free vector fields of class $C^1$ on a compact 3-manifold that is associated with a well-behaved integral kernel, we prove that $\mathcal I$ is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.