Fernand Pelletier

Almost Lie structures on an anchored Banach bundle

Abstract Under appropriate assumptions, we generalize the concept of linear almost Poisson structures, almost Lie algebroids, almost differentials in the framework of Banach anchored bundles and the relation between these objects. We then obtain an adapted formalism for mechanical systems which is illustrated by the evolutionary problem of the “Hilbert snake” as exposed in Pelletier and Saffidine (2011) [9] .

Abnormality of trajectory in sub-Riemannian structure

In the sub-Riemannian framework, we give geometric necessary and sufficient conditions for the existence of abnormal extremals of the Maximum Principle. We give relations between abnormality, C1-rigidity and length minimizing. In particular, in the case of three dimensional manifolds we show that, if there exist abnormal extremals, generically, they are locally length minimizing and in the case of four dimensional manifolds we exhibit abnormal extremals which are not C1-rigid and which can be minimizing or non minimizing according to different metrics.

SECTIONS QUADRATIQUES FREDHOLMIENNES

Abstract A Fredholm quadratic section on a Hilbert manifold M is a smooth section of the Banach bundle of symmetric bilinear forms whose associated operators are Fredholm; such a section defines a singular pseudo-Riemannian structure on M. These structures appear naturally after induction on submanifolds of finit codimension in a pseudo-Riemannian Hilbert manifold (W, Q). The germs of singular pseudo-Riemannian Hilbert structures are classified

On Darboux Theorem for symplectic forms on direct limits of symplectic Banach manifolds

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show in general, without very strong conditions, the answer is negative. In particular we give an example of an ascending weak symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In a second part, we illustrate this discussion in the context of an ascending sequences of Sobolev manifolds of loops in symplectic finite dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux theorem is true around any point.

Localisation des courbes anormales et couples de tenseurs de Poisson en petite dimension

Resume Nous construisons des stratifications en dimension 4 et 5 permettant de localiser les courbes anormales pour des distributions generiques de codimension 2 en dimension 6 et 7 . Ces stratifications permettent aussi, en dimension 4 et 5 , de decrire les singularites des feuilletages de Weinstein associes a un couple de tenseurs de Poisson compatibles intervenant dans le contexte des varietes bihamiltoniennes.