×

zbMATH — the first resource for mathematics

On the asymptotic linking number. (English) Zbl 1015.57018
Let \(M\) be a smooth closed oriented \(3\)-dimensional manifold such that its real cohomology is the same as that of the \(3\)-sphere and let \(\mu\) be a volume form on \(M\). For smooth vector fields \(X\) and \(Y\) on \(M\) which are divergence-free with respect to \(\mu\), the author proves a theorem formulated by V. I. Arnol’d in [Sel. Math. Sov. 5, 327-345 (1986; Zbl 0623.57016)]: The average linking number of \(X\), \(Y\) is equal to the Hopf invariant of \(X\), \(Y\).
As remarked by the author, his proof is analogous to the proof due to Arnol’d, but the asymptotic linking number \(\lambda_{X,Y}(x,y)\) (the average linking number is defined to be the integral of \(\lambda_{X,Y}(x,y)\) over \(M\times M\)) is no longer defined as a pointwise limit, but as a limit existing in the \(L^1\)-sense.

MSC:
57R25 Vector fields, frame fields in differential topology
37C10 Dynamics induced by flows and semiflows
57R30 Foliations in differential topology; geometric theory
76W05 Magnetohydrodynamics and electrohydrodynamics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] V. I. Arnol\(^{\prime}\)d, The asymptotic Hopf invariant and its applications, Selecta Math. Soviet. 5 (1986), no. 4, 327 – 345. Selected translations. · Zbl 0623.57016
[2] Vladimir I. Arnold and Boris A. Khesin, Topological methods in hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. · Zbl 0902.76001
[3] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. · Zbl 0496.55001
[4] Georges de Rham, Differentiable manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 266, Springer-Verlag, Berlin, 1984. Forms, currents, harmonic forms; Translated from the French by F. R. Smith; With an introduction by S. S. Chern. · Zbl 0534.58003
[5] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. · Zbl 0084.10402
[6] Michael H. Freedman and Zheng-Xu He, Divergence-free fields: energy and asymptotic crossing number, Ann. of Math. (2) 134 (1991), no. 1, 189 – 229. · Zbl 0746.57011 · doi:10.2307/2944336 · doi.org
[7] Hans-Otto Georgii, Gibbs measures and phase transitions, De Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. · Zbl 0657.60122
[8] Ulrich Krengel, Ergodic theorems, De Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. · Zbl 0575.28009
[9] H. K. Moffat, The degree of knottedness of tangled vortex lines, J. Fluid. Mech. 35 (1969), 117-129. · Zbl 0159.57903
[10] A. A. Tempel\(^{\prime}\)man, Ergodic theorems for general dynamical systems, Dokl. Akad. Nauk SSSR 176 (1967), 790 – 793 (Russian).
[11] L. Woltjer, A theorem on force-free magnetic fields, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 489 – 491. · Zbl 0081.21703
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.