Solvability of the cohomological equation for regular vector fields on the plane

Details Solvability of the cohomological equation for regular vector fields on the plane Solvability of the cohomological equation for regular vector fields on the plane

2010-07-18

arxiv.org/abs/1007.3016v1

Mathematics

Differential Geometry

21 pages, 2 figures

Scientific paper

We consider planar vector field without zeroes X and study the image of the associated Lie derivative operator LX acting on the space of smooth functions. We show that the cokernel of LX is infinite-dimensional as soon as X is not topologically conjugate to a constant vector field and that, if the topology of the integral trajectories of X is ``simple enough'' (e.g. if X is polynomial) then X is transversal to a Hamiltonian foliation. We use this fact to find a large explicit subalgebra of the image of LX and to build an embedding of R^2 into R^4 which rectifies X. Finally we use this embedding to characterize the functions in the image of LX.

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