Scalar curvature of singular spaces (Q2726230)

The geometric objects studied in this work are special singular sets, so called \textit{definable sets}. On the one hand they are more general than Riemannian manifolds since they can contain complicated singularities and on the other hand they are more special than pure metric spaces. Examples for such singular sets are semialgebraic or subanalytic sets. The author introduces a scalar measure of curvature on these singular sets and shows that this curvature can be described by mere geometric quantities. Theorems concerning the density and the geometry of geodesics are proved. The connection between the sectional curvatures and the scalar curvature is investigated. It is also shown that the scalar curvature is a quantity which only depends on the inner geometry of the singular set to which it belongs. Finally the author discusses how a conform transformation of the metric influences the scalar curvature of the singular set.