Scalar Curvature Question [?25]: (iii) TheSn- andS2-product inequalities seems to hold fornon-trivial sphere fibrations, but I have not checked

Definitions and mathematical statement

The source's tentative assertion is: the $S^n$-product and $S^2$-product inequalities appear to extend to nontrivial sphere fibrations, but the author has not checked this carefully.

A common smooth candidate setting is a sphere bundle $\pi:E\to B$ with fiber $S^n$, locally trivial but not globally isomorphic over $B$ to $B\times S^n$. The source uses the broader word fibration and does not specify whether this is the intended category. The separate $S^2$ case must be retained, even though it is formally a sphere dimension, because the referenced product inequalities may differ.

Neither product inequality is reproduced in the record. Its quantities, constants, dimension ranges, metric structure, curvature hypotheses, and target conclusion are absent. Consequently the proposal can only be expressed as extending the two preceding product statements to the corresponding nontrivial-fibration setting while retaining their as-yet-unspecified hypotheses. No numerical scalar-curvature inequality, uniform estimate over all fibrations, or positive-scalar-curvature existence theorem is inferred. Existence of a positive-scalar-curvature metric on a bundle would in any case be a different question from a sharp inequality for prescribed geometric data.

Short English statement

Check whether the source's $S^n$- and $S^2$-product inequalities remain valid for nontrivial sphere fibrations. The inequalities and their hypotheses are missing, and the author explicitly describes the extension as unchecked.

Sources

• Local UnsolvedMath registry, permanent ID 6700026, exact source code catalogue entry; proposed by Misha Gromov, 2017. Registry status open, classification OPEN-TRIAGE. The source's printed tag is [?25], occurrence 26; the occurrence and permanent ID are not the printed tag.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), item (iii) at printed tag [?25], PDF page 36, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Supplied normalized pdf-text extraction accessed July 29, 2026; incomplete author-hosted PDF. No PDF was accessed here.
• The supplied summary mentions related sphere-bundle scalar-curvature themes and Gromov--Lawson surgery, without exact hypotheses or citations, and says it located no dedicated proof of the particular extension. Those themes do not identify or prove the missing product inequalities.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW. The source's explicit "I have not checked this carefully" qualification is preserved.