Scalar Curvature Question [?43]: Would it be more prudent to replace the conditionSc(g) > 0 byRicci> 0

Definitions and mathematical statement

For a Riemannian metric $g$ on $M$, scalar curvature positivity means $\operatorname{Sc}_g(x)>0$ at every point. Positive Ricci curvature means \[ \operatorname{Ric}_g(v,v)>0\qquad\text{for every nonzero tangent vector }v. \] Since $\operatorname{Sc}_g=\operatorname{tr}_g\operatorname{Ric}_g$, positive Ricci curvature implies positive scalar curvature. It is a stronger hypothesis, not a change of notation.

The source asks whether it would be "more prudent" to replace an earlier condition $\operatorname{Sc}(g)>0$ by $\operatorname{Ric}_g>0$. The earlier proposed theorem or construction is absent, so no antecedent or conclusion is specified for that replacement. This is a methodological question about strengthening assumptions, rather than a conjecture that the two curvature conditions are equivalent or that every positive-scalar-curvature manifold admits a positive-Ricci metric.

The summary mentions Bonnet--Myers motivations, but pointwise positive Ricci curvature is not by itself a stated uniform positive lower bound on a noncompact manifold. Any compactness or diameter conclusion needs its own completeness and uniform-lower-bound hypotheses. No such conclusion is added to this fragment.

Short English statement

Would strengthening a preceding scalar-curvature hypothesis to positive Ricci curvature make the intended programme more appropriate? The preceding statement and a criterion for prudence are not supplied.

Sources

• Local UnsolvedMath registry, permanent ID 6700041, exact source code catalogue entry; proposed by Misha Gromov, 2017. Registry status partially solved, classification PARTIAL-PROGRESS. The source printed tag is [?43], occurrence 43.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?43], PDF page 48, 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 research summary calls this partial/interpretive, mentions Myers and Bonnet and the stronger nature of positive Ricci curvature, and says prudence is a design choice rather than a theorem. No precise preceding target or reference is supplied.
• Provenance flags current status and statement-reuse terms NEEDS_REVIEW; the source marks the passage as a numbered open item.