Scalar Curvature Question [?41]: For instance, ifX =SO(n) withn≥5, then no known method can rule out metricsg ≥g on X with Sc(g) > Sc(g)

Definitions and mathematical statement

Let $n\geq5$ and $X=\operatorname{SO}(n)=\{A\in M_n({\mathbb{R}}):A^{\mathsf T}A=I,\ \det A=1\}$. Let $\bar g$ be a bi-invariant Riemannian metric on $X$, meaning invariant under both left and right translations. The bar distinguishes this reference metric from a competing smooth metric $g$; the extraction prints both as $g$, which would otherwise make the strict scalar curvature comparison impossible.

With the usual quadratic-form order on metrics, the question is whether there exists $g$ such that \[ g_x(v,v)\geq\bar g_x(v,v)\quad(x\in X,\ v\in T_xX),\qquad \operatorname{Scal}_g(x)>\operatorname{Scal}_{\bar g}(x)\quad(x\in X). \] Equivalently, can one rule out this simultaneous metric enlargement and everywhere strict scalar curvature improvement? The source says only that no known method rules it out, in its 2017 discussion; that is neither a proof of existence nor a verified statement of present knowledge. The competing metric is not assumed left invariant or bi-invariant.

The bi-invariant reference comes from the supplied summary and the adjacent local record ID 6700037. That context discusses area extremality, whereas the present extracted inequality is $g\geq\bar g$. The displayed formulation uses the usual tensor inequality; whether an area comparison was intended instead needs source checking. The reference metric's normalization is not supplied.

Short English statement

For $\operatorname{SO}(n)$, $n\geq5$, can a metric dominate a bi-invariant reference metric while having strictly larger scalar curvature everywhere? The source describes a gap in available methods rather than asserting that such a metric exists.

Sources

• Local UnsolvedMath registry, permanent ID 6700038, source code catalogue entry; Misha Gromov, 2017; update 2026-08-06. Adjacent local ID 6700037, source code AMR-066-0037, supplies the bi-invariant-metric and area-extremality context.
• Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?41], occurrence 40, PDF page 48: https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Registry access date 2026-07-29; normalized PDF extraction from an incomplete public author-hosted PDF; current status and reuse terms need review.
• The supplied summary reports the case as open and connects the difficulty to trivial tangent bundles and Dirac-index methods. It supplies no additional exact reference or verified present-day resolution.