A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Fréchet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.2010 Mathematics Subject Classification. 58D10 (primary); 37K65; 53C30; 53D20; 58D05. Key words and phrases. nonlinear flag manifolds; nonlinear Grassmannians; groups of diffeomorphisms; spaces of embeddings; Fréchet manifold; moment map; coadjoint orbits. ID-PCE-2016-0778 of the Romanian Ministry of Research and Innovation CNCS-UEFISCDI, within PNCDI III. Both authors would like to express their gratitude to Martin Bauer and Peter Michor for helpful bibliographical comments.

International audienceThis article is a rigorous study of the dual pair structure of the ideal fluid (Phys D 7:305-323, 1983) and the dual pair structure for the n-dimensional Camassa-Holm (EPDiff) equation (The breadth of symplectic and poisson geometry: Festshrift in honor of Alan Weinstein, 2004), including the proofs of the necessary transitivity results. In the case of the ideal fluid, we show that a careful definition of the momentum maps leads naturally to central extensions of diffeomorphism groups such as the group of quantomorphisms and the Ismagilov central extension. © 2011 Springer Science+Business Media B.V

International audienceWe formulate Euler-Poincaré equations on the Lie group Aut(P) of automorphisms of a principal bundle P. The corresponding flows are referred to as EP A ut flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle P, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing ?-like momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group Aut vol(P) of volume-preserving bundle automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group. © American Institute of Mathematical Sciences

Abstract. We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant L 2 or H 1 metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.

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