On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groups

We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is the trivial bundle. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known by the work of Brooks and Goldman on the Seifert volume. As a consequence, we verify the following strong version of a problem of Hopf for the above class of manifolds: Every self-map of non-zero degree of a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group is either homotopic to a homeomorphism or homotopic to a non-trivial covering and the bundle is trivial.