3-manifolds with(out) metrics of nonpositive curvature. (English) Zbl 0840.53031
In the context of Thurston’s geometrisation program we address the question which compact aspherical 3-manifolds admit Riemannian metrics of nonpositive curvature. We prove that a Haken manifold with possibly empty boundary of zero Euler characteristic admits metrics of nonpositive curvature if the boundary is non-empty or if at least one atoroidal component occurs in its canonical topological decomposition. Our arguments are based on Thurston’s hyperbolisation theorem. We give examples of closed graph-manifolds with linear gluing graph and arbitrarily many Seifert components which do not admit metrics of nonpositive curvature.
Reviewer: B.Leeb (Bonn)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 57M50 | General geometric structures on low-dimensional manifolds |
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