Helicity is the only integral invariant of volume-preserving transformations.

Enciso, Alberto; Peralta-Salas, Daniel; de Lizaur, Francisco Torres · Proc Natl Acad Sci U S A · 2016

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Abstract

We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional 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 I is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.