Helicity is the only integral invariant of volume-preserving transformations

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.

arXiv comment12 pages, 1 figure

References in corpus (3)

• Helicity is the only invariant of incompressible flows whose derivative is continuous in $C^1$-topology
• The third order helicity of magnetic fields via link maps
• On volume-preserving vector fields and finite type invariants of knots

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