Leafwise hyperbolicity of proper foliations (Q1812687)

Let \(M\) be a compact three-dimensional orientable manifold equipped with a codimension-one transversely orientable foliation \(F\). Assume that all the leaves are proper and each component of the boundary of \(M\) is a leaf. The authors prove the existence of a leafwise hyperbolic (constant curvature \(-1\)) Riemannian metric on \(M\) if and only if no leaf of \(F\) is a torus or a sphere.