## Scalar curvature of defineable CAT-spaces.(English)Zbl 1028.53031

For a compact connected set belonging to some $$o$$-minimal structure (e.g. a semialgebraic or subanalytic set) the author defined the notion of scalar curvature measure which shares many of the properties of the standard scalar curvature of Riemannian manifolds. Carrying on previous work [A. Bernig, Adv. Geom. 2, 29-55 (2002; Zbl 1027.53041)] the author relates the scalar curvature measure to curvature bounds in the sense of metric differential geometry and proves here the following result. For a compact connected definable pseudo-manifold $$S$$ with curvature bounded from above, the singular part of the scalar curvature measure is non-positive. More precisely, if the dimension of $$S$$ equals $$m$$ and the curvature bound is $$\kappa$$, then $$\text{scal}(S,-)\leq\kappa m(m-1)\text{vol}(S,-)$$.

### MSC:

 53C20 Global Riemannian geometry, including pinching 14P10 Semialgebraic sets and related spaces 57N80 Stratifications in topological manifolds 58A35 Stratified sets 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)

Zbl 1027.53041
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