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Helicity is the only integral invariant of volume-preserving transformations. (English) Zbl 1359.58006

Summary: We prove that any regular integral invariant of volume-preserving transformations is equivalent to the helicity. Specifically, given a functional \(\mathscr J\) 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 \(\mathscr J\) is invariant under arbitrary volume-preserving diffeomorphisms if and only if it is a function of the helicity.

MSC:

58C20 Differentiation theory (Gateaux, Fréchet, etc.) on manifolds
53A45 Differential geometric aspects in vector and tensor analysis
58C35 Integration on manifolds; measures on manifolds
76B47 Vortex flows for incompressible inviscid fluids
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