The asymptotic Hopf invariant and its applications

Abstract

The classical Hopf invariant distinguishes among the homotopy classes of continuous mappings from the three-sphere to the two-sphere and is equal to the linking number of the two curves that are the preimages of any two regular points of the two-sphere.

The asymptotic Hopf invariant is an invariant of a divergence-free vector field on a three-dimensional manifold with given volume element. It is invariant under the group of volume-preserving diffeomorphisms, and describes the ″helicity″ of the field, i.e., the mean asymptotic rotation of the phase curves around each other. The asymptotic Hopf invariant coincides with the classical Hopf invariant for the unitary vector field that is tangent to the Hopf bundle. In the general case the asymptotic Hopf invariant can have any real value (whereas the classical Hopf invariant is always an integer).

The asymptotic Hopf invariant can also be considered as a quadratic form on the Lie algebra of the volume-preserving diffeomorphisms of the three-dimensional manifold that is invariant under the adjoint action of the group on the algebra.

In this paper we present the definition and simplest properties of the asymptotic Hopf invariant, as well as some of its applications to an unusual variational problem that arises in magnetohydrodynamics which was called to the author’s attention by Ya. B. Zel’dovich. In connection with this problem there arise a whole series of unsolved mathematical problems, some of which appear to be difficult. The main object of this paper is to discuss these unsolved problems; all the theorems in the paper are obvious.

Attention was first called to the problems considered here by Voltjer [7] in connection with magnetohydrodynamics. Applications to ordinary hydrodynamics were given by Moffatt [4], [5] and Kraichnam [3].