Localization of abnormal curves and pairs of Poisson tensors in low dimension (Q1591582)

The paper starts with a detailed analysis of the construction of stratifications in spaces of jets of pairs of bivectors and 2-forms on 4 and 5 dimensional manifolds, and pairs of 1-forms on 6 and 7 dimensional manifolds. This analysis has two objectives, which are developed as applications in the next two sections. Section 3 is about the localization of abnormal curves for generic codimension 2 distributions on 6 and 7 dimensional manifolds. It appears that in dimension 6, generically, there exists an open neigbourhood in which there are no abnormal curves apart from the constant ones, whereas a completely different situation occurs in dimension 7. A subsection is devoted to the particular case of homogeneous spaces. The second application concerns a geometrical study, on 4 and 5 dimensional manifolds, of the singularities of the recursion operator associated to a pair of compatible Poisson tensors, or the singularities of the associated Weinstein foliation. Necessary and sufficient conditions are obtained for a vector field to be bi-Hamiltonian or bi-dynamical.