Invariant Positive-Scalar-Curvature Metrics

Definitions and mathematical statement

For the differential-geometric interpretation of the supplied question, let $M$ be a closed smooth manifold, meaning a compact smooth manifold without boundary, and let $G$ be a compact connected Lie group acting smoothly and effectively on $M$. Effectiveness means that if $a\cdot x=x$ for every $x\in M$, then $a=e_G$. Smoothness of the manifold and action is the regularity interpretation used here; the supplied statement does not expressly specify it.

For $a\in G$, write $\phi_a(x)=a\cdot x$. A Riemannian metric $g$ is $G$-invariant if $\phi_a^*g=g$ for every $a\in G$. It has positive scalar curvature if $\operatorname{Scal}_g(x)>0$ for every $x\in M$.

Supplied existence question. For every such group, manifold, and action, does there exist a smooth Riemannian metric $g$ on $M$ satisfying \[ \phi_a^*g=g\quad\text{for all }a\in G, \qquad \operatorname{Scal}_g(x)>0\quad\text{for all }x\in M? \] No dimension restriction, connectedness assumption on $M$, nontriviality assumption on $G$, or prior existence of a positive-scalar-curvature metric is stated in the supplied formulation. These hypotheses have not been added.

Unresolved scope qualification. The supplied dated curation labels the problem partially solved and reports that the unrestricted existence question has no uniform affirmative answer. Thus the displayed question preserves the recorded formulation, but the supplied material does not identify a precise remaining conjecture or the additional hypotheses defining an unresolved case. The short original wording, "Does M admit an invariant metric of positive scalar curvature?", does not independently supply that missing context.

Short English statement

Given an effective action of a compact connected Lie group on a closed manifold, the recorded question asks whether there is a Riemannian metric preserved by the action whose scalar curvature is positive everywhere. A smooth interpretation is used explicitly. The supplied curation reports partial results and obstructions, but does not specify the remaining unresolved cases.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002639$, title Invariant Positive-Scalar-Curvature Metrics, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record lists proposed year $2014$; no proposer is supplied.
• Supplied source citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2015), Oberwolfach Reports, pp. 1991--2046, DOI 10.4171/owr/2014/36. Original OWR report: https://doi.org/10.4171/owr/2014/36. The background describes these reports as documenting workshops and their open-problem discussions.
• Supplied literature reference, labeled "circle actions and scalar curvature": https://doi.org/10.1090/tran/6666. No authors or fuller bibliographic details are supplied for this reference.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified here. It labels the problem partially_solved and reports invariant positive-scalar-curvature metrics for nonabelian actions and several circle-action classes, together with explicit circle-action obstructions and examples without invariant metrics showing that the unrestricted existence question has no uniform affirmative answer. The background identifies this assessment as literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and says that it was checked against the official Oberwolfach report and that the clean statement restores source hypotheses and notation. This is a supplied verification claim, not a claim of source inspection or mathematical verification in this draft.