Kähler hyperbolicity and the Euler number of a compact manifold with geodesic flow of Anosov type (Q1566333)

\textit{M. Gromov} [J. Differ. Geom. 33, 263--292 (1991; Zbl 0719.53042)] proved that if \(M\) is a \(2m\)-dimensional compact Riemannian manifold of negative sectional curvature and \(M\) is homotopy equivalent to a compact Kähler manifold then its Euler number \(\chi (M)\) satisfies \((-1)^m\chi (M) >0.\) The author mainly proved that if a compact Riemannian manifold \(M\) whose geodesic flow is of Anosov type is homotopy equivalent to a compact Kähler manifold then \((-1)^m\chi (M) >0.\) He proves that if \(M\) is a simply connected complete Riemannian manifold without conjugate points with Anosov geodesic flow and sectional curvature \(\geq -c^2, \quad c > 0\) then every closed bounded k-form on \(M\), \(k\geq 2\), is the differential of a bounded \((k-1)\)-form. For compact \(M\) with Anosov geodesic flow every closed \(k\)-form (\(k\geq 2\)) on the universal covering of \(M\) is the differential of a bounded form.