451 research outputs found
On Representation of the Reeb Graph as a Sub-Complex of Manifold
The Reeb graph is one of the fundamental invariants of a
smooth function with isolated critical points. It is
defined as the quotient space of the closed manifold by a
relation that depends on . Here we construct a -dimensional complex
embedded into which is homotopy equivalent to .
As a consequence we show that for every function on a manifold with finite
fundamental group, the Reeb graph of is a tree. If is an abelian
group, or more general, a discrete amenable group, then
contains at most one loop. Finally we prove that the number of loops in the
Reeb graph of every function on a surface is estimated from above by ,
the genus of .Comment: 18 page
Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics
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
- …