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

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
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