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.


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
Full Text: DOI