Sur les structures transversalement affines des feuilletages de codimension un

Abstract

The notion of a transversely affine structure on a codimension one foliation on paracompact Hausdorff manifolds is defined, and translated in terms of pfaffian forms when the foliation is transversely orientable. In the later case on can construct interesting examples (foliations almost wthout holonomy on S n ×S 1 (n≥3) where the universal coverings of leaves are not the same, foliations on bundles over S 1 with fiber T n (n≥2) having dense leaves with holonomy, foliations on closed manifolds which are not fibered over S 1 ) and study some geometrical and topological properties of the foliated manifold (invariant of Godbillon-Vey, holonomy of leaves, holonomy and invariant of transversely affine structures, modification of transversely affine structures along closed transversals, (classic) classification of affine structures on R and S 1 , classification of transversely affine structures in dimension greater than one...).