An obstruction to the smoothability of singular nonpositively curved metrics on 4-manifolds by patterns of incompressible tori

Abstract

We give new examples of closed smooth 4-manifolds which support singular metrics of nonpositive curvature, but no smooth ones, thereby answering affirmatively a question of Gromov. The obstruction comes from patterns of incompressible 2-tori sufficiently complicated to force branching of geodesics for nonpositively curved metrics.

13 pages, 2 figures

References in corpus (2)

• A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry
• 4-dimensional locally CAT(0)-manifolds with no Riemannian smoothings