High-dimensional helicities and rigidity of linked foliations.
The paper opens with a very brief description of work of \textit{M. H. Freedman} and \textit{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 \textit{V. I. Arnol'd} [Sel. Math. Sov. 5, 327--345 (1986; Zbl 0623.57016)], who termed the generalized Hopf invariant \textit{helicities} in this context, and even earlier work by \textit{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.
scientific article; zbMATH DE number 1925770