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A group of diffeomorphisms which leave a vector field fixed. (English. Russian original) Zbl 0552.58007

Sib. Math. J. 25, 313-317 (1984); translation from Sib. Mat. Zh. 25, No. 2(144), 180-185 (1984).
H. Omori [Proc. Symp. Pure Math. 15, 167-183 (1970; Zbl 0214.488)] showed that the group \({\mathcal D}\) of all diffeomorphisms of a compact manifold is a strong ILH group (where ILH \(=\) inverse limit of Hilbert spaces). In subsequent papers he has shown that such groups have many of the nice properties of finite dimensional Lie groups, that they are closely related to their Lie algebras, and so on. Thus it is important to understand which subgroups of \({\mathcal D}\) are strong ILH subgroups. Known examples are \({\mathcal D}_{\mu}\) and \({\mathcal D}_{\omega}\), the subgroups of \({\mathcal D}\) consisting of all diffeomorphisms which preserve a volume form \(\mu\), resp. a symplectic form \(\omega\). The author considers the group \({\mathcal D}_ X\) of all diffeomorphisms of M which preserve a given vector field X. His main results are: (1) \({\mathcal D}_ X\cap {\mathcal D}_{\mu}\) is an ILH subgroup of \({\mathcal D}_{\mu}\) if X is \(\mu\)- divergence free and without zeros, or if X is \(\mu\)-divergence free and M is a 3-dimensional manifold with \(H^ 1(M,{\mathbb{R}})=0\); (ii) \({\mathcal D}_ X\cap {\mathcal D}_{\omega}\) is an ILH subgroup of \({\mathcal D}_{\omega}\) if X is Hamiltonian and \(H^ 1(M;{\mathbb{R}})=0\).
Reviewer: D.McDuff

MSC:

58C25 Differentiable maps on manifolds

Citations:

Zbl 0214.488
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References:

[1] H. Omori, ?On the group of diffeomorphisms on a compact manifold,? Proc. Symp. Pure Math.,15, Am. Math. Soc. (1970). · Zbl 0214.48805
[2] D. J. Eben and J. E. Marsden, ?Groups of diffeomorphisms and the motion of an incompressible fluid,? Ann. Math.,92, No. 1, 102-163 (1970). · Zbl 0211.57401 · doi:10.2307/1970699
[3] H. Omori, ?On smooth extension theorems,? J. Math. Soc. Jpn.,24, No. 3, 405-432 (1972). · Zbl 0235.58003 · doi:10.2969/jmsj/02430405
[4] N. K. Smolentsev, ?Bivariant metric on the group of diffeomorphisms of a three-dimensional manifold,? Sib. Mat. Zh.,24, No. 1, 152-159 (1983).
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