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On the geodesic flow of a foliation of a compact manifold of negative constant curvature. (English) Zbl 0703.58047
Proc. Winter Sch. Geom. Phys., Srní/Czech. 1988, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 21, 349-354 (1989).
[For the entire collection see Zbl 0672.00006.]
Let M be a compact Riemannian manifold of constant negative curvature K, equipped with a complete foliation F. Let SF denote the unitary tangent bundle of F and $$h_{\mu}(\phi)$$ the measure entropy of the flow $$\phi$$ on SF with respect to any $$\phi$$-invariant Borel measure $$\mu$$ on SF. The main result of the paper asserts that then $(1)\quad \mu (\{v\in SF| \quad rank(F,v)\leq (n+p-2)\})=1$ and, if $$\mu$$ is smooth then $(2)\quad h_{\mu}(\phi)\geq \sqrt{-K/2}rank(F,\mu).$
Reviewer: D.Repovš
##### MSC:
 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry 57R30 Foliations in differential topology; geometric theory 53C20 Global Riemannian geometry, including pinching