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Diffeomorphism groups of compact manifolds. (English. Russian original) Zbl 1147.58012
J. Math. Sci., New York 146, No. 6, 6213-6312 (2007); translation from Sovrem. Mat. Prilozh. 37, 3-100 (2006).
This is a survey on diffeomorphism groups of manifolds viewed as infinite-dimensional Lie-Frechét groups. The intensive studies on diffeomorphism groups began after V. I. Arnold [Ann. Inst. Fourier 16, No. 1, 319–361 (1966; Zbl 0148.45301)] had shown that motions of the ideal incompressible fluid are geodesics on the group of volume-element-preserving diffeomorphisms. Many researchers (Omori, Ebin, Marsden, Hamilton, McDuff, Lukatskii, Shnirelman, Milnor, to mention a few) contributed to these studies, where the curvature and metric properties of the diffeomorphism groups, and application of the diffeomorphism groups to ideal fluid hydrodynamics were considered.
This work is divided into 13 parts.
Let $$M$$ be a smooth compact orientable manifold without boundary of dimension $$n$$. The set $$C^ 1{\mathcal D}$$ consisting of $$C^ 1$$-diffeomorphisms of $$M$$ is open in $$C^ 1(M,M)$$ and is a topological group. After introductory information on differential operators and Lie derivatives, the author presents the inverse limit Hilbert (ILH) Lie group of diffeomorphisms on ILH-Lie and Hamilton Fréchet manifolds. In Part 3, various subgroups of the diffeomorphism group are studied: subgroups of strict ILH-Lie groups, groups of volume-element-preserving diffeomorphisms, symplectic diffeomorphism groups, contact transformation groups, and groups of diffeomorphisms leaving a vector field fixed. Next, exponential mappings and weak Riemannian structures on the diffeomorphism groups are defined.
Parts 6 and 7 deal with volume-element-preserving diffeomorphism groups and the ideal barotropic fluids.
The internal energy, Hamiltonian property of the hydromechanics equations, first integrals of motion of the ideal barotropic fluid, hydromechanical interpretations of geodesics, and Euler-Poincaré reduction are discussed. Curvatures of various groups of diffeomorphisms preserving the Riemannian volume element are calculated in Part 8. Symplectic and exact contact diffeomorphism groups and their properties are studied in Parts 10 and 11. Diffeomorphisms preserving certain tensor fields, harmonic diffeomorphisms, and diffeomorphism without periodic points are presented in Part 12. The last part deals with metric and topological properties of diffeomorphism groups: the diameter of the diffeomorphism group, symplectic transformation groups, algebraic properties, homotopy type, and homologies and cohomologies of diffeomorphism groups.
The list of references contains 274 entries.

##### MSC:
 58D17 Manifolds of metrics (especially Riemannian) 53D05 Symplectic manifolds, general 76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena 58-02 Research exposition (monographs, survey articles) pertaining to global analysis
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