Invariant positive Ricci curvature for the 21 spin cohomology S2 × S5 types with only non-spin orbit spaces

Determine which of the 21 oriented diffeomorphism types of smooth, closed, simply-connected spin 7-manifolds with the cohomology ring of S2 × S5 that admit free S1-actions exclusively with non-spin orbit spaces also admit S1-invariant Riemannian metrics of positive Ricci curvature.

Background

The paper proves that 441 oriented diffeomorphism types of spin cohomology S2 × S5 admit infinitely many non-equivalent free S1-actions, and for each such action an invariant metric of positive Ricci curvature. These include all cases whose free S1-actions have spin orbit spaces.

However, 21 oriented diffeomorphism types admit free S1-actions only with non-spin orbit spaces. For these remaining cases, the existence of invariant positive Ricci curvature metrics is not settled.

References

The remaining 21 oriented diffeomorphism types that admit a free circle action by Theorem A but are not covered by Theorem B therefore only have free circle actions with non-spin orbit space. It remains open whether these manifolds admit invariant Riemannian metrics of positive Ricci curvature, cf. Remark 4.3 below.

— 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), after Theorem B; see also Remark 4.3