A locally CAT(0)-metric metric on length space means that every point in it has a geodesically convex neighborhood such that every triangle in it is slimmer than the comparison triangle in the Euclidean plane. For example, the Riemannian metric with nonpositive sectional curvature is a locally CAT(0)-metric metric.
Let $M$ be a closed manifold and $M\times S^1$ admits locally CAT(0)-metrics, if $M$ also admits locally CAT(0)-metric?
Let $M$ be a closed manifold such that $M\times S^1$ admits a locally CAT(0) metric. Does $M$ also admit a locally CAT(0) metric?
