# Non-positive sectional curvature in 3-dimensional manifold

The answer of the following question may be well-known in the field of Geometric Topology, so I ask for help in here.

Does the total space of circle bundle over a closed hyperbolic surface admit a Riemannian metric with non-positive sectional curvature?

In particular, the circle bundle which comes from the tangent bundle of the hyperbolic surface is directly related to my thesis about positive scalar curvature.

Since if it does not carry metrics with non-positive sectional curvature, it may not be an enlargeable manifold, which is an obstruction of admitting positive scalar curvature metrics.

From Theorem 5.3 in https://homepages.warwick.ac.uk/~masgar/Teach/2012_MA4J2/geometry.pdf a circle bundle over a hyperbolic surface has a geometry locally modeled on either $$H^2\times R$$ or $$\widetilde{SL(2,R)}$$.
The product metric on $$H^2\times R$$ is nonpositively curved, while (by Igor‘s comment below) closed $$\widetilde{SL(2,R)}$$-manifolds don‘t admit metrics of nonpositive curvature. (However $$\widetilde{SL(2,R)}$$-manifolds with nonempty incompressible boundary admit nonpositively curved metrics by https://arxiv.org/pdf/dg-ga/9410002.pdf)
• In CAT(0) groups centralizers virtually split. Hence closed manifolds modelled on $\widetilde{SL}(2,\mathbf R)$ don't admit metrics of nonpositive curvature. – Igor Belegradek Oct 19 '19 at 14:09