Existence of smooth invariant measures for geodesic flows of foliations of Riemannian manifolds

We construct a nontrivial smooth finite measure invariant under the geodesic flow of a foliation 9 of a compact Riemannian manifold M assuming that the transverse mean curvature of Y is distributed "nicely" along some leaf geodesics. In [W], we proved that if a foliation Y of a Riemannian manifold M is transversely minimal (i.e., if H' = 0, where H' is the trace of the second fundamental form a1I of the orthogonal complement NF of . ), then the volume form Q on the unit tangent bundle SY of Y equipped with the socalled Sasaki metric defines a smooth measure A invariant under the geodesic flow (gt) of Y. This is an analogue of the standard result of Riemannian geometry (and classical mechanics) saying that the Liouville measure is invariant under the geodesic flow of a Riemannian manifold [K]. In this note, we show the existence of a smooth (gt)-invariant measure under a weaker assumption of a "nice" distribution of the transverse mean curvature H' of 5 along the leaf geodesics. The existence of smooth invariant measures for dynamical systems is of some importance because of the following: Pesin's theory [P] allows one to estimate the topological entropy of a smooth dynamical system from below if such a measure exists (compare Remark 3 below). Theorem. If M is compact and the smooth functions h, : S5 -* R, t > 0, given by hl(v) = exp (H' (7rgv), gv) ds, are uniformly bounded from below by a constant c > 0 on an open subset U c S5-, then theflow (gt) admits a smooth invariant measure. Received by the editors November 1, 1991. 1991 Mathematics Subject Classification. Primary 53C12, 58F1 1, 28D10, 28D20.