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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?

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  • $\begingroup$ The paper you cite completely solves the problem except for the case of closed graph manifolds. The latter case seems to be solved in m.mathnet.ru/links/8f77e6d1ef31437a2b3a52664f36aead/aa556.pdf $\endgroup$ – ThiKu Dec 17 '18 at 22:17
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    $\begingroup$ But if you are only interested in circle bundles, then actually Section 2.3 of Leeb‘s paper already does the job. $\endgroup$ – ThiKu Dec 17 '18 at 22:18
  • $\begingroup$ 1. Only virtually trivial circle bundles admit NPC metrics. For 2 and 3: Sometimes yes, sometimes not, but there are no general theorems (in higher dimensions). $\endgroup$ – Misha Dec 27 '18 at 3:04

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