Feuilletages transversalement projectifs sur les variétés de Seifert

Barbot, Thierry

Annales de l'Institut Fourier, Tome 53 (2003) no. 5, pp. 1551-1613.

Soit $M$ une variété de Seifert de groupe fondamental non virtuellement résoluble. Soit $\Phi$ un feuilletage de dimension $1$ sur $M$, muni d’une structure projective réelle transverse. On suppose que $\Phi$ satisfait la propriété de relèvement des chemins, i.e., que l’espace des feuilles du relèvement de $\Phi$ dans le revêtement universel de $M$ est séparé au sens de Hausdorff. On montre qu’à revêtements finis près, $\Phi$ est soit une fibration projective, soit un feuilletage géodésique convexe, soit un feuilletage horocyclique projectif.

Let $M$ be a Seifert manifold with non-solvable fundamental group. Let $\Phi$ be a one- dimensional foliation on $M$, equipped with a transverse real projective structure. We assume moreover that $\Phi$ satisfies the Homotopy Lifting Property, i.e., that the leaf space of the lifting of $\Phi$ in the universal covering of $M$ satisfies the Hausdorff separation property. Then, up to finite coverings, $\Phi$ belongs to one of the following three families of transversely projective foliations: the family of projective fibrations, the family of convex geodesic foliations, or the family of projective horocyclic foliations.

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