Integrability and Adapted Complex Structures to Smooth Vector Fields on the Plane

8-th International Workshop on Complex structures and vector fields

On the Integrability of quasihomogeneous and Related Planar Vector Fields

In this work we consider planar quasihomogeneous vector fields and we show, among other qualitative properties, how to calculate all the inverse integrating factors of such C systems. Additionally, we obtain a necessary condition in order to have analytic inverse integrating factors and first integrals for planar positively semi-quasihomogeneous vector fields which is related with the existence...

Deformations of Vector Fields and Hamiltonian Vector Fields on the Plane

For the Lie algebras L\(H(2)) and L\(W(2)), we study their infinitesimal deformations and the corresponding global ones. We show that, as in the case of L\{W(\)), each integrable infinitesimal deformation of L\(H(2)) and L1(W/(2)) can be represented by a 2-cocycle that defines a global deformation by means of a trivial extension. We also illustrate that all deformations of L\{H{2)) arise as res...

Adapted complex structures and the geodesic flow

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real analytic Riemannian manifold. Motivated by the “complexifier” approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the “imaginary...

The Maslov Cocycle, Smooth Structures and Real-analytic Complete Integrability

This paper proves two main results. First, it is shown that if Σ is a smooth manifold homeomorphic to the standard n-torus Tn = Rn/Zn and H is a real-analytically completely integrable convex hamiltonian on T Σ, then Σ is diffeomorphic to Tn. Second, it is proven that for some topological 7-manifolds, the cotangent bundle of each smooth structure admits a realanalytically completely integrable ...