Topology of Complete Noncompact Metric Moduli Spaces

Definitions and mathematical statement

Let $V$ be a noncompact smooth manifold. Define \[ \mathcal R_{\geq 0}^{\mathrm{comp}}(V) =\{g: g\text{ is a complete smooth Riemannian metric on }V, \ \sec_g(\sigma)\geq 0\text{ for every }x\in V \text{ and every two-dimensional subspace }\sigma\subset T_xV\}. \] Here $\sec_g(\sigma)$ denotes sectional curvature. Let $\operatorname{Diff}(V)$ be the group of smooth diffeomorphisms of $V$, acting on metrics by pullback. Define the moduli set \[ \mathcal M_{\geq 0}^{\mathrm{comp}}(V) =\mathcal R_{\geq 0}^{\mathrm{comp}}(V)/{\sim}, \qquad g\sim h\ \Longleftrightarrow\ h=\phi^*g\text{ for some }\phi\in\operatorname{Diff}(V), \] and let $q$ denote the projection to this quotient.

Source question. What topology is appropriate on $\mathcal R_{\geq 0}^{\mathrm{comp}}(V)$, and what topology is appropriate on $\mathcal M_{\geq 0}^{\mathrm{comp}}(V)$? If the moduli space is assigned the quotient topology induced by a topology $\tau$ on the metric space, that topology is defined by \[ U\subseteq\mathcal M_{\geq 0}^{\mathrm{comp}}(V)\text{ is open} \quad\Longleftrightarrow\quad q^{-1}(U)\in\tau. \] This describes the quotient-topology construction; the supplied record does not explicitly prescribe all required properties of the sought topologies.

Unresolved qualifications. The supplied background mentions metrizability, lifting, continuity, and convergence requirements, but does not state those requirements. In particular, it does not identify the maps or families to be lifted, the maps whose continuity is required, or the intended notion of convergence. Thus "appropriate" does not specify a mathematical predicate, and the record does not determine a precise existence or uniqueness assertion. The dimension, connectedness, and boundary conventions for $V$ are also unspecified; any boundary convention affecting completeness requires clarification.

Short English statement

For a noncompact smooth manifold, seek suitable topologies on the space of complete smooth Riemannian metrics with nonnegative sectional curvature and on the space of their equivalence classes under diffeomorphisms. The source leaves the suitability criteria unspecified, so this remains a formulation question requiring clarification.

Sources

• Local UnsolvedMath registry, permanent ID 30003372, source problem code catalogue entry, title Topology of Complete Noncompact Metric Moduli Spaces, in the supplied Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 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 background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 3/2017. https://ems.press/content/serial-article-files/46667.
• Supplied literature source: On the topology of moduli spaces. https://doi.org/10.1007/s00208-021-02327-y.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that no generally accepted topology satisfying the mentioned metrizability, lifting, continuity, and convergence requirements for noncompact moduli has been established. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied record requests independent verification of time-sensitive status and cited claims.