Jean-Paul Dufour

Nondegeneracy of the Lie algebra aff(n)

Abstract We show that aff (n) , the Lie algebra of affine transformations of R n , is formally and analytically nondegenerate in the sense of A. Weinstein. This means that every analytic (resp., formal) Poisson structure vanishing at a point with a linear part corresponding to aff (n) is locally analytically (resp., formally) linearizable. To cite this article: J.-P. Dufour, N.T. Zung, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 1043–1046.

Formes normales de structures de poisson ayant un 1 jet nul en un point

Abstract Singularities of Poisson structures where 1-jet vanishes appear in a stable manner because they are generically not destroyed by small perturbations of the Poisson structure. In this paper we study these singularities. We give first a normal form for Poisson structures with zero 1-jet but with a “generic” 2-jet at a point. We give also a “quadratisation” result in class C∞.

Rigidity of Webs and Families of Hypersurfaces

Publisher Summary This chapter discusses rigidity of webs and families of hypersurfaces. The chapter discusses local smooth classification of (p+l)-webs of codimension n of R qn , with p > q, and shows that it is the same as the topological one. A (p+l)-web of codimension n of R qn is a (p+l)-tuple of p to p transversal foliations of codimension n of R qn . So it is locally equivalent to consider a diagram of p+l p to p transversal submersions f l , f 2 ,…, f p+l . The study is confined to a neighborhood of the origin, in the case p > q, assuming f i (0) = 0 for all i, 1 ≤ I ≤ p+l. The results are applied in the chapter to the classification of families of hypersurfaces of R n , to settle that the local topological stability of these families is not generic. It is proved that the existence of a smooth equivalence between two webs implies that the diffeomorphism k is linear, at least on some neighborhood of the origin.

Hyperbolic Actions of Rp on Poisson Manifolds

Unless otherwise explicitly stated all manifolds and mappings are C ∞ Recall that a Poisson manifold ([W]) is a manifold V with a Lie algebra structure (f,g) ↦ {f,g} on C ∞(V) (the set of C ∞ mappings f: V → R) such that

On the local structure of Dirac manifolds

\{ f,gh\} = \{ f,g\} h + g\{ f,h\}