451 research outputs found

    On Representation of the Reeb Graph as a Sub-Complex of Manifold

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    The Reeb graph R(f)\mathcal{R}(f) is one of the fundamental invariants of a smooth function f ⁣:MRf\colon M\to \mathbb{R} with isolated critical points. It is defined as the quotient space M/ ⁣M/_{\!\sim} of the closed manifold MM by a relation that depends on ff. Here we construct a 11-dimensional complex Γ(f)\Gamma(f) embedded into MM which is homotopy equivalent to R(f)\mathcal{R}(f). As a consequence we show that for every function ff on a manifold with finite fundamental group, the Reeb graph of ff is a tree. If π1(M)\pi_1(M) is an abelian group, or more general, a discrete amenable group, then R(f)\mathcal{R}(f) contains at most one loop. Finally we prove that the number of loops in the Reeb graph of every function on a surface MgM_g is estimated from above by gg, the genus of MgM_g.Comment: 18 page

    Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics

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    We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V.Arnold's problem on describing all invariants of generic isovorticed fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties.Comment: 38 pages, 11 figures; to appear in Annales de l'Institut Fourie
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