Transversely projective foliations on Seifert manifolds.
In this paper, a projective structure means a \((PGL(3,\mathbb{R}),\mathbb{R} P^2)\) structure. In the Memoir [Affine flows on 3-manifolds, Mem. Am. Math. Soc. 771 (2003; Zbl 1026.57022)], \textit{S. Matsumoto} introduced a property which he called the ``homotopy lifting property'', which says that the space of leaves of the lift of the foliation to the universal cover of the ambient manifold is Hausdorff separated. In the paper under review, the author studies transversely projective foliations on three-manifolds, and the main result is the following Theorem: Let \(M\) be a Seifert fibered three-manifold whose base is an orbifold of nonpositive Euler characteristic and let \(\Phi\) be a transversely projective foliation on \(M\) satisfying the homotopy lifting property. Then, up to passing to a finite cover, \(\Phi\) is conjugate to a foliation which is of one of the following types: -- a projective fibration, that is, a foliation whose leaves are the fibres of a Seifert fibration over an orbifold equipped with a real projective structure; -- a ``convex geodesic foliation'', which is typically the foliation of a geodesic flow of a surface equipped with a metric of constant negative curvature; -- a ``horocyclic projective foliation'' associated to the geodesic foliation above. In each case, the conjugacy respects the projective structures.
scientific article; zbMATH DE number 2014686
scientific article