×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI