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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.

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
Full Text: DOI arXiv
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