Scalar Curvature Question [?9]: Identify the most general classes of geometric objects having properties analogous to those of $C^2$ Riemannia

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a $C^2$ Riemannian metric $g$ on a manifold, $\operatorname{Sc}_g$ denotes the scalar curvature, the trace of its Ricci tensor. A lower bound $\operatorname{Sc}_g\geq\sigma$, for a real number $\sigma$, holds pointwise. The source asks about generalized geometric objects that retain analogous scalar-curvature features, including possible singular spaces obtained through limits or completions. It does not fix a particular singular category, convergence notion, or list of axioms in advance.

Statement recorded in the supplied source.

Identify the most general classes of geometric objects displaying properties analogous to those of $C^2$ Riemannian manifolds with positive scalar curvature and, more generally, with scalar curvature bounded below by $\sigma$. [S2]

Short English statement

Find the broadest geometric setting that retains the essential features of manifolds with positive or lower-bounded scalar curvature.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6700009 (AMR-066-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Misha Gromov, Four Lectures on Scalar Curvature (2021 version). Discussion preceding Section 3.20 and Question 1, p. 204. https://arxiv.org/pdf/1908.10612v6.