Scalar curvature of definable CAT-spaces

We study the scalar curvature measure for sets belonging to o-minimal structures (e.g. semialgebraic or subanalytic sets) from the viewpoint of metric differential geometry. Theorem: Let S be a compact connected definable pseudo-manifold with curvature bounded from above, then the singular part of the scalar curvature measure is non-positive. The topological restrictions cannot be removed, as is shown in examples.