Let $(M, g)$ be a closed smooth Riemannian manifolds with Anosov geodesic flow, does it implies that $M$ is an aspherical manifold?
I am thinking it is not known yet?
W. Klingenberg in [Riemannian Manifolds With Geodesic Flow of Anosov Type, Annals of Mathematics, Vol. 99, No. 1, 1974, pp. 1-13] proved among other things that a closed Riemannian manifold with Anosov geodesic flow does not have conjugate points, and hence the exponential map at any point is a covering map. In particular, the manifold is aspherical.