The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid. (English. Russian original) Zbl 0725.58005

Shnirel’man, A. I.

Math. USSR, Sb. 56, No. 1, 79-105 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 1, 82-109 (1985).

Let D be the space of volume-preserving diffeomorphisms of the 3- dimensional cube \(K^ 3\) with the Riemannian metric \(<V,W>=\int_{K}(V(x),W(x))dx\) (V,W vector fields tangent to D). The induced distance in D is studied. The main theorem of the paper shows that the Riemannian distance is bounded by the \(L_ 2\)-distance. This is used to produce examples of \(\xi\in D\) with no shortest path in D joining \(\xi\) with the identity. Another application shows that the completion of D is the space of all measure-preserving maps \(K^ 3\to K^ 3\). The proof of the main theorem is based on approximation of D by transformations of finite configuration spaces.

MSC:

58D05	Groups of diffeomorphisms and homeomorphisms as manifolds
58D30	Applications of manifolds of mappings to the sciences
37N99	Applications of dynamical systems
76A02	Foundations of fluid mechanics

Keywords:

space of volume-preserving diffeomorphisms; Riemannian distance; \(L_ 2\)-distance; space of all measure-preserving maps