Rigidity of locally homogeneous metrics of negative curvature on the leaves of a foliation (Q911062)

Let \({\mathcal F}\) and \({\mathcal F}'\) be \(C^ 1\)-equivalent foliations of a compact manifold M. Assume that the foliations are equipped with smooth Riemannian metrics along the leaves which vary continuously in the transverse direction. Theorem. If the leaves of \({\mathcal F}\) and \({\mathcal F}'\) are locally isometric to: 1) a quaternionic hyperbolic space of dimension \(\leq 8\), or 2) a Cayley hyperbolic plane, or 3) a real or complex hyperbolic space of dimension \(\leq 3\), then \({\mathcal F}\) and \({\mathcal F}'\) are metrically equivalent provided, in the case (3), that \({\mathcal F}\) admits a holonomy invariant transverse measure.