If the total space of circle bundle over higher genus surface admit Riemannian metric of non-positive sectional curvature? I wish to use the result about the question and find Leeb's work 3-manifolds with(out) metrics of nonpositive curvature relate it. But I can not understand the paper since I am a novice in 3-dimension manifold, so I ask for help .
Leeb asked in 1995: which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive sectional curvature? Do we know more results about the question though the progress of those years in 3-manifold?
Add: Thanks to Thiku's answer, it is clearly for aspherical 3-manifold. It may too ambitious to ask which compact aspherical smooth n-manifolds (n>3) admit Riemannian metrics of nonpositive sectional curvature?, since Davis and Januszkiewicz proved some exotic aspherical closed manifolds in the paper Hyperbolization of polyhedra. For example, for each n≥4 there exists an aspherical closed n-dimensional manifold such that its universal covering is not homeomorphic to n-dimension Euclid space. So I reduce it to the following questions:
1) If the total space of circle bundle over closed manifold with Riemannian metric of non-positive sectional curvature also admits Riemannian metric of non-positive sectional curvature?
2) If the total space of fiber bundle over circle, which the fiber are closed manifold with Riemannian metric of non-positive sectional curvature, also admits Riemannian metric of non-positive sectional curvature?
3) In general, given the fiber bundle with fiber and base space are closed manifolds with Riemannian metric of non-positive sectional curvature, if its total space also admits Riemannian metric of non-positive sectional curvature?