Helicity is the only integral invariant of volume-preserving transformations

2016/02/15 by Alberto Enciso, Daniel Peralta-Salas, Francisco Torres de Lizaur
Mathematics · Physics and Astronomy · #math.DS #math-ph #math.DG #math.MP

paper · pdf · doi:10.1073/pnas.1516213113

12 pages, 1 figure

arxiv created 2016/02/15 · arxiv updated 2016/02/16

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 C¹ 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.

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