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High-dimensional helicities and rigidity of linked foliations. (English) Zbl 1045.57017
The paper opens with a very brief description of work of M. H. Freedman and Z.-X. He [Ann. Math. (2) 134, No. 1, 189–229 (1991; Zbl 0746.57011)] on the generalized Hopf invariant and rigidity of knotted magnetic tubes. The work by Freedman and He builds on earlier ideas of V. I. Arnol’d [Sel. Math. Sov. 5, 327–345 (1986; Zbl 0623.57016)], who termed the generalized Hopf invariant helicities in this context, and even earlier work by S. P. Novikov [Russ. Math. Surv. 39, No. 5, 113–124 (1984; Zbl 0619.58002)]. The present author gives an ergodic interpretation of Hopf-Novikov helicities as conjectured by Arnol’d. Furthermore, the topological bounds obtained by Freedman and He for energies of invariant forms of linked foliations are extended to higher dimensions.

57R30 Foliations in differential topology; geometric theory
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
57Q25 Comparison of PL-structures: classification, Hauptvermutung
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