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

2013/12/08 by Stephan Stadler, Stadler, Stephan
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Metric Geometry (math.MG) #math.DG #math.MG

paper · pdf · doi:10.48550/arxiv.1312.2198

13 pages, 2 figures

openalex publication_date 2013/12/08 · arxiv created 2015/08/11 · arxiv updated 2015/08/12 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

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.

Citations

• A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry

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