Toral manifolds and positive scalar curvature — Question 2

Definitions and mathematical statement

Let $M$ be a connected closed smooth manifold. The source asks whether \[ \left[\exists\text{ Riemannian }g\text{ on }M:\ \operatorname{Sc}_g(x)>0 \text{ for every }x\in M\right] \quad\Longrightarrow\quad \left[M\text{ is }p\text{-atoral for every odd }p\right]. \] No spin, orientability, fundamental-group, or dimension restriction is stated.

The record does not define $p$-atorality. A common mod-$p$ torality convention for an oriented closed $n$-manifold tests products of degree-one cohomology classes: torality would mean existence of $a_1,\ldots,a_n\in H^1(M;{\mathbb{F}}_p)$ with \[ \langle a_1\smile\cdots\smile a_n,[M]_{{\mathbb{F}}_p}\rangle\ne0, \] and atorality would mean all such evaluations vanish. This is an indicated possible interpretation only; the source may involve covers or a different homological definition. For odd primes the displayed fundamental class requires orientability or a specified local-coefficient convention, which is missing. The phrase odd $p$ also does not explicitly say prime rather than another allowed modulus.

The supplied summary's language about essential tori or degree maps does not give a definition. Maps from a torus, maps to a torus, and mod-$p$ degree-one cup-product tests are different formulations. None is substituted for the source's undefined predicate without the original convention.

Short English statement

Does existence of positive scalar curvature force $p$-atorality for every odd parameter $p$? The meaning of atorality, prime/modulus range, and orientation conventions must be recovered from the source.

Sources

• Local UnsolvedMath registry, permanent ID 6800011, exact source code catalogue entry, "Toral manifolds and positive scalar curvature---Question 2"; proposed by Bernhard Hanke, Augsburg, 2018. Registry status open, classification OPEN-TRIAGE.
• Morgan and Pansu, A List of Open Problems in Differential Geometry (2018), question environment 11, https://www.imo.universite-paris-saclay.fr/~pansu/problems_MTDG.tex. Supplied source-TeX extraction accessed July 29, 2026. The environment number and title question number are distinct.
• The supplied summary mentions Hanke--Schick, index theory and Schoen--Yau-type arguments as related structural progress, without precise publications, definitions or hypotheses. It reports no definitive solution located. Its informal essential-torus/degree-map description is not treated as the definition of $p$-atorality.
• Provenance flags current status and redistribution terms NEEDS_REVIEW; source TeX is publicly available on a coauthor's publication page.