On completeness of transversely projective foliations (Q1196976)

A codimension-one foliation \(\mathcal F\) of a manifold \(M\) is transversely projective if \(M\) admits an \(\mathcal F\)-distinguished atlas for which the transverse coordinate transformations are of the form (*) \(x\mapsto \frac{ax+b}{cx+d}\). \(\mathcal F\) is reducible if all the transformations (*) belong to some proper closed Lie subgroup of PSL\((2,\mathbb{R})\). The main result says that, for a fixed compact 3-manifold \(M\) and the space of all transversely projective irreducible foliations of \(M\) endowed with the fine topology, the subset of complete foliations is open. Some homology type criteria for completeness of \(\mathcal F\) are also obtained.