Enciso, Alberto; Peralta-Salas, Daniel; Torres de Lizaur, Francisco Helicity is the only integral invariant of volume-preserving transformations. (English) Zbl 1359.58006 Proc. Natl. Acad. Sci. USA 113, No. 8, 2035-2040 (2016). 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. Cited in 1 ReviewCited in 20 Documents 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 Keywords:integral invariant; volume-preserving transformations; helicity PDFBibTeX XMLCite \textit{A. Enciso} et al., Proc. Natl. Acad. Sci. USA 113, No. 8, 2035--2040 (2016; Zbl 1359.58006) Full Text: DOI arXiv References: [1] DOI: 10.1017/S0022112069000991 · Zbl 0159.57903 · doi:10.1017/S0022112069000991 [2] DOI: 10.1073/pnas.1400277111 · doi:10.1073/pnas.1400277111 [3] Arnold VI Khesin B (1999) Topological Methods in Hydrodynamics (Springer, New York) · Zbl 0743.76019 [4] Kudryavtseva EA Helicity is the only invariant of incompressible flows whose derivative is continuous in C 1 -topology. ArXiv:1511.03746v2 [5] DOI: 10.1134/S0001434614050332 · Zbl 1370.37034 · doi:10.1134/S0001434614050332 [6] DOI: 10.1016/j.aim.2014.09.009 · Zbl 1303.37035 · doi:10.1016/j.aim.2014.09.009 [7] DOI: 10.1134/S0081543812060028 · Zbl 1304.78003 · doi:10.1134/S0081543812060028 [8] Arnold, The asymptotic Hopf invariant and its applications, Selecta Math Soviet 5 pp 327– (1986) [9] Baader, Asymptotic concordance invariants for ergodic vector fields, Comment Math Helv 86 (1) pp 1– (2011) · Zbl 1209.57007 · doi:10.4171/CMH/215 [10] Baader, Asymptotic Vassiliev invariants for vector fields, Bull Soc Math Fr 140 pp 569– (2012) · Zbl 1278.57017 · doi:10.24033/bsmf.2637 [11] DOI: 10.1215/S0012-7094-01-10613-3 · Zbl 1010.37010 · doi:10.1215/S0012-7094-01-10613-3 [12] Khesin, Geometry of higher helicities, Moscow Math J 3 pp 989– (2003) · Zbl 1156.55300 [13] DOI: 10.1017/etds.2014.83 · Zbl 1375.37056 · doi:10.1017/etds.2014.83 [14] DOI: 10.1088/0305-4470/23/13/017 · Zbl 0711.57008 · doi:10.1088/0305-4470/23/13/017 [15] DOI: 10.1007/s00220-009-0896-z · Zbl 1206.57007 · doi:10.1007/s00220-009-0896-z [16] DOI: 10.1063/1.533299 · Zbl 0971.57019 · doi:10.1063/1.533299 [17] DOI: 10.1016/0167-2789(84)90273-2 · Zbl 0586.76025 · doi:10.1016/0167-2789(84)90273-2 [18] DOI: 10.1016/j.crma.2008.07.012 · Zbl 1157.37014 · doi:10.1016/j.crma.2008.07.012 [19] DOI: 10.1007/BF02570870 · Zbl 0676.47012 · doi:10.1007/BF02570870 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.