Positive Ricci Observer Homotopy Beyond Spheres

Definitions and mathematical statement

Let $M$ be a smooth manifold of dimension $d$, and let $x\in M$ be a chosen observer point. Define \[ \mathcal R_{\operatorname{Ric}>0}(M) =\{g\text{ a smooth Riemannian metric on }M: \operatorname{Ric}_g(v,v)>0\text{ for every }0\ne v\in TM\}. \] Equip this space with the $C^\infty$ topology. Using the usual observer convention, set \[ \operatorname{Diff}_{x,1}(M) =\{\phi\in\operatorname{Diff}(M): \phi(x)=x,\ {\,\mathrm{d}}\phi_x=\operatorname{id}_{T_xM}\}, \qquad \mathcal M^{\mathrm{obs}}_{\operatorname{Ric}>0}(M) =\mathcal R_{\operatorname{Ric}>0}(M)/\operatorname{Diff}_{x,1}(M), \] where the action is by pullback and the orbit space has the quotient topology. Fixing the derivative is equivalent to fixing a chosen frame at the observer point. These conventions make the term observer moduli space explicit; the supplied record does not itself specify its construction.

Source question. Can the nontrivial rational homotopy occurring in the stable range for $\mathcal M^{\mathrm{obs}}_{\operatorname{Ric}>0}(S^d)$ be extended to other manifolds?

For degrees $k\ge2$, a precise individual nonvanishing assertion of the relevant type would be \[ \pi_k\!\left(\mathcal M^{\mathrm{obs}}_{\operatorname{Ric}>0}(M),[g_0]\right) \otimes_{{\mathbb{Z}}}{\mathbb{Q}}\ne0, \qquad g_0\in\mathcal R_{\operatorname{Ric}>0}(M). \] The question asks for extensions of the sphere results of this kind. However, the supplied record does not specify the dimensions, homotopy degrees, stable-range inequalities, hypotheses on the other manifolds, or quantifiers over those manifolds and base metrics. Thus the displayed expression is a template for the requested nonvanishing, not a fully specified conjecture. The record also does not say whether degree-one information is intended; the displayed tensor-product formulation is restricted to $k\ge2$. A precise extension must state these missing qualifications and ensure that the positive-Ricci metric space is nonempty.

Short English statement

The observer moduli space identifies positive-Ricci-curvature metrics under diffeomorphisms fixing a point and its tangent frame. The question asks whether the stable-range rational homotopy nonvanishing supplied for spheres extends to other manifolds. The intended manifolds, dimensions, degrees, stable range, and base components are unspecified, so the source supports a research question rather than a uniquely quantified conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003362$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. Title: Positive Ricci Observer Homotopy Beyond Spheres. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied source citation: Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics (2018), Oberwolfach Reports, pp. 133--166, DOI 10.4171/owr/2017/3. https://doi.org/10.4171/owr/2017/3. The supplied literature-source label is "Oberwolfach report on observer moduli." The citation's 2018 designation differs from the record's proposed year and 2017 year tag; this discrepancy is unresolved.
• Supplied literature source: "Observer moduli of Ricci-positive metrics." https://doi.org/10.2140/gt.2019.23.3003.
• Supplied literature source: "Positive curvature moduli beyond spheres." https://doi.org/10.1007/s00209-026-03947-3.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that positive-Ricci observer-moduli nontriviality is proved on high-dimensional spheres, that related results concern other curvature conditions and manifolds, and that the requested positive-Ricci extension to other manifolds remains unsettled. This is supplied dated curation derived from the OWR material, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.