Ricci-positive invariant metric versus quotient Ricci-positivity for free circle actions

Establish whether, for every closed smooth manifold M admitting a free smooth S1-action with quotient N = M/S1, the existence of a Riemannian metric of positive Ricci curvature on N is equivalent to the existence of an S1-invariant Riemannian metric of positive Ricci curvature on M. In particular, determine if the full "if and only if" analogue of Bérard-Bergery’s positive scalar curvature result holds for positive Ricci curvature beyond the currently known implication when M has finite fundamental group.

Background

Bérard-Bergery proved that a closed manifold with a free S1-action admits an invariant metric of positive scalar curvature if and only if the orbit space admits a metric of positive scalar curvature. The paper asks whether an analogous equivalence holds for positive Ricci curvature.

A partial result due to Gilkey–Park–Tuschmann shows one direction: if M has finite fundamental group and the quotient N admits a metric of positive Ricci curvature, then M admits an S1-invariant metric of positive Ricci curvature. The missing direction(s) of the equivalence remain unknown.

References

It is open whether an analogous equivalence holds for positive Ricci curvature, although one half of it has been established by Gilkey-Park-Tuschmann [15], namely if the manifold has finite fundamental group and the quotient space admits a Riemannian metric of positive Ricci curvature, then there exists an invariant Riemannian metric of positive Ricci curvature.

— Free circle actions and positive Ricci curvature on manifolds with the cohomology ring of $S^2\times S^5$  (2603.29838 - Reiser, 31 Mar 2026) in Section 1 (Introduction and Main Results)