Scalar Curvature Question [?58]: there is no apparent non-trivial bound on the width ofX = Σn−1× [−1, 1]even we assume that thesectional curvat

Definitions and mathematical statement

Let $X=\Sigma^{n-1}\times[-1,1]$ be the smooth product band, endowed with a Riemannian metric $g$. The product notation describes the topology, not necessarily a product metric. The stated curvature hypothesis is \[ \operatorname{sec}_g(\Pi)=1 \] for every tangent two-plane $\Pi$; it is not a scalar-curvature-only assumption. A literal Riemannian product metric would have mixed sectional curvatures zero and would not satisfy this condition.

The source observes that there is "no apparent non-trivial bound" on the width even under this constant-sectional-curvature hypothesis. It does not define width or state a numerical conjecture. If ordinary band width is intended, write \[ \operatorname{width}_g(X) =\inf\{d_g(p,q):p\in\Sigma\times\{-1\},\ q\in\Sigma\times\{1\}\}. \] A precise investigation under that convention would ask for the supremum of this quantity over admissible curvature-one metrics and whether a bound depends only on dimension or also on $\Sigma$ and boundary data. The source does not fix those quantifiers, compactness/completeness assumptions, or even whether this is the intended notion of width. Its absence of an apparent bound is not asserted here as a proved unboundedness theorem.

The supplied research summary speculates about scalar-curvature limitations, but that discussion does not replace the actual sectional-curvature-equals-one hypothesis.

Short English statement

Investigate width bounds for a product-topology band with sectional curvature exactly one. The extract states that no bound is apparent but does not define width or give the manifold and metric hypotheses needed for a precise extremal question.

Sources

• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?58], occurrence 57, PDF page 59, 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 $6700054$, source code catalogue entry, Scalar Curvature Question [?58]. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.