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The asymptotic Hopf invariant and its applications. (English. Russian original) Zbl 0623.57016
Sel. Math. Sov. 5, 327-345 (1986); translation from Mater. Vses. Shk. Differ. Uravn. Beskonechnym Chislom Nezavisimykh Din. Syst. Beskonechnym Chislom Stepeneĭ Svobodny, Dilizhane, May-June 1973, Akad. Arm. SSR, Erevan, 229-256 (1974).
The author presents the definition and simplest properties of the asymptotic Hopf invariant which coincides with the classical Hopf invariant for the unitary vector field. The asymptotic Hopf invariant can also be considered as a quadratic form on the Lie algebra of volume- preserving diffeomorphisms of a three-dimensional manifold that is invariant under the adjoint action of the group on the algebra. The main object of this paper is to discuss the unsolved problems connected with the asymptotic Hopf invariant.

MSC:
57R25 Vector fields, frame fields in differential topology
37C10 Dynamics induced by flows and semiflows