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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