Kähler hyperbolicity and the Euler number of a compact manifold with geodesic flow of Anosov type

Abstract

Let M be a 2m-dimensional compact Riemannian manifold with Anosov geodesic flow. We prove that every closed bounded k form, k≥2, on the universal covering of M is d(bounded). Further, if M is homotopy equivalent to a compact Kahler manifold, then its Euler number χ(M) satisfies (−1)mχ(M)>0.