Scalar Curvature Question [?11]: What could be a, possibly non-geometric, extension of the concept ofSc ≥0, where one would be able perform sym

Definitions and mathematical statement

For a smooth Riemannian metric $g$, write $\operatorname{Sc}_g=\operatorname{tr}_g\operatorname{Ric}_g$; nonnegative scalar curvature means $\operatorname{Sc}_g(x)\geq0$ at every point. The orthogonal group $O(n-1)$ consists of real $(n-1)\times(n-1)$ matrices $A$ with $A^TA=I$.

Find an extension of the concept $\operatorname{Sc}\geq0$, possibly to objects that are not themselves Riemannian geometries, in which a symmetrization procedure can reduce general neighborhoods $V$ to $O(n-1)$-symmetric ones. A complete proposal should identify the enlarged object class, explain how its nonnegativity notion relates to ordinary smooth scalar curvature, define the symmetry action and symmetrization, and state the preservation/comparison principle that makes the intended local problem reducible to the symmetric case.

The source does not identify the neighborhoods' underlying local problem, their boundary or normalization data, the action of $O(n-1)$, or which quantities symmetrization must preserve. It also does not prescribe an analytic substitute for scalar curvature. Consequently it requests a framework and reduction method, not a specified inequality for a named symmetrization operator. The possibility of a non-geometric extension is retained rather than requiring every intermediate object to be a smooth metric.

Short English statement

Extend nonnegative scalar curvature, possibly beyond geometric objects, so that symmetrization reduces general neighborhoods to ones with $O(n-1)$ symmetry. The source does not specify the local problem or preservation rules.

Sources

• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?11], occurrence 11, PDF page 11, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; accessed 2026-07-29, PDF-text-normalized extraction. The provenance describes the public author PDF as incomplete.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700011$, source code catalogue entry, Scalar Curvature Question [?11]. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.