# What is the relation between universal covering space and curvature of base manifolds?

Does anyone know what is the relation between universal covering space and curvature of base manifolds?

Edit: For example If universal covering of a complete $3$-manifold $(M,g)$ isometric to a Riemann product $N^2\times \Bbb R$ where $N^2$ is a complete $2$-manifold with non-negative sectional curvature then what we can say about $(M,g)$?

Thanks

• This is a rather broad and vague question. There is no single answer. Can you make your question more specific? – Lee Mosher Oct 4 '17 at 13:22
• In the 3-manifold case, we know what the possible geometries are (on the universal cover). See homepages.warwick.ac.uk/~masgar/Teach/2012_MA4J2/geometry.pdf – Steve D Oct 4 '17 at 17:13