Scalar Curvature Question [?3]: ○What are geometries ofindividual manifoldswith Sc > σ

Definitions and mathematical statement

For a smooth Riemannian manifold $(M^n,g)$, scalar curvature is the trace $\operatorname{Sc}_g=\operatorname{tr}_g\operatorname{Ric}_g$. For a prescribed scalar threshold $\sigma$, the condition in the source is the strict pointwise inequality \[ \operatorname{Sc}_g(x)>\sigma\qquad(x\in M). \] For a constant threshold one can denote the metrics satisfying it by $\mathcal R_{>\sigma}(M)$. The source's question, with spacing normalized, is: what are the geometries of individual manifolds with this curvature inequality?

Describe the geometric possibilities and restrictions for individual Riemannian manifolds satisfying this bound. The item does not state a comparison theorem, geometric invariant to estimate, manifold category, dimension range, completeness or compactness assumption, or a proposed classification. It is an umbrella program and is not silently replaced by a question about the topology of the space $\mathcal R_{>\sigma}(M)$.

The supplied summary points to comparison and rigidity results for scalar-curvature bounds, but supplies no full hypotheses. In particular, a spherical lower bound alone is not here asserted to force a round metric, and the summary's references to tori/aspherical manifolds do not establish an unrestricted theorem for every such manifold. These are related areas of partial progress, while the exact source condition remains $\operatorname{Sc}>\sigma$, not $\operatorname{Sc}\geq\sigma$.

Short English statement

Describe what geometry is possible for an individual manifold with scalar curvature strictly greater than a prescribed threshold. The source poses a broad research program without a specific comparison or classification assertion.

Sources

• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?3], occurrence 3, PDF page 1, 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 author-hosted PDF as incomplete.
• Related references named in the supplied summary: Llarull; Gromov--Lawson; Brendle--Marques--Neves; and Gromov, Four Lectures on Scalar Curvature (2023). No precise theorem citations or full hypotheses are given for these comparisons.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700003$, source code catalogue entry, Scalar Curvature Question [?3]. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.