On the geodesic flow of a foliation of a compact manifold of negative constant curvature. (English) Zbl 0703.58047

Walczak, Paweł G.

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

Keywords:

geodesic flow; probability measure; compact Riemannian manifold of constant negative curvature; foliation

Citations:

Zbl 0672.00006