Infinite Components of Positive Ricci Metric Moduli

Definitions and mathematical statement

Let $n\in{\mathbb{Z}}$ with $n\geq 2$, and let $M$ be a smooth closed spin manifold of dimension $4n-1$, where closed means compact without boundary and spin means that the oriented tangent bundle admits a spin structure. Define \[ \mathcal R_{\operatorname{Ric}>0}(M)=\{g\in\Gamma^\infty(\operatorname{Sym}^2T^*M): g\text{ is a Riemannian metric and }\operatorname{Ric}_g(v,v)>0\text{ for every nonzero }v\in TM\}. \] Assume $\mathcal R_{\operatorname{Ric}>0}(M)\neq\varnothing$, as explicitly required by the supplied corrected statement.

For a precise conventional reading, equip this space with the $C^\infty$ topology and define \[ \mathcal M_{\operatorname{Ric}>0}(M)=\mathcal R_{\operatorname{Ric}>0}(M)/\operatorname{Diff}(M), \] where the full diffeomorphism group acts by pullback and the quotient carries the quotient topology. The question is whether, for every such $n$ and $M$, the space $\mathcal M_{\operatorname{Ric}>0}(M)$ has infinitely many connected components.

Unresolved formulation qualifications. The supplied record does not specify the topology or the group used in forming the moduli space. The displayed definition uses the customary smooth topology and full diffeomorphism group; its agreement with the intended source formulation requires review. Quotients by orientation-preserving or spin-structure-preserving diffeomorphisms are possible alternatives that the record does not resolve. Connected components are meant literally, without replacement by path components. The record also does not explicitly require $M$ to be connected; no such hypothesis has been added.

Short English statement

Does every closed spin manifold of dimension $4n-1\geq 7$ admitting a metric of positive Ricci curvature have infinitely many connected components in its moduli space of such metrics? The precise quotient and topology require confirmation. The supplied dated curation labels the problem partially solved and reports results for particular families, while leaving the universal assertion unestablished; this assessment is not independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003352$, title Infinite Components of Positive Ricci Metric Moduli, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied primary 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 background describes Oberwolfach Reports as documenting workshops and their open-problem discussions. The supplied title year 2018 and the recorded proposed year 2017 require bibliographic reconciliation.
• Supplied literature source, labelled "Oberwolfach report on curvature moduli": https://doi.org/10.4171/owr/2017/3.
• Supplied literature source, labelled "Dessai--Klaus--Tuschmann nonconnected moduli": https://arxiv.org/abs/1601.04877.
• Supplied literature source, labelled "Sphere bundles with positive Ricci moduli": https://doi.org/10.1007/s10711-022-00713-7.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem partially solved and reports infinite-component results for substantial families, including generalized Witten-type manifolds and related sphere-bundle examples, while stating that the universal assertion is not established. The supplied background describes this as literature triage derived from OWR curation and asks that time-sensitive status and cited claims be verified independently.
• The registry labels the statement corrected_verified and describes its verification as repairs to notation, typography, and sentence structure without changing mathematical content. This is supplied editorial metadata, not an independent mathematical verification here.