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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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