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?