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Leafwise hyperbolicity of proper foliations. (English) Zbl 0768.57013
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.

MSC:
57R30 Foliations in differential topology; geometric theory
57N10 Topology of general \(3\)-manifolds (MSC2010)
30F99 Riemann surfaces
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