Homotopy Types of Curvature-Metric Moduli on Surfaces

Definitions and mathematical statement

Let $M$ be either the sphere $S^2$ or the plane ${\mathbb{R}}^2$. For a smooth Riemannian metric $g$ on $M$, let $K_g(x)$ denote its Gaussian curvature at $x\in M$. Thus nonnegative curvature means $K_g(x)\geq 0$ for every $x\in M$, and positive curvature means $K_g(x)>0$ for every $x\in M$.

The supplied question does not specify the topology on the space of metrics, the precise equivalence relation defining the moduli space, or whether metrics on ${\mathbb{R}}^2$ must be complete. Its background discusses complete metrics, making completeness a relevant but unresolved qualification. Completeness is automatic for smooth Riemannian metrics on $S^2$.

To describe the requested objects while displaying these missing choices, let $\mathcal C(M)$ be a specified class of smooth Riemannian metrics on $M$, such as all such metrics or all complete such metrics. Give $\mathcal C(M)$ a specified topology $\tau$, and let $G(M)$ be a specified subgroup of the diffeomorphism group preserving $\mathcal C(M)$ and acting continuously by pullback. Define \[ \begin{aligned} \mathcal R_{\geq 0}^{\mathcal C}(M) &=\{g\in\mathcal C(M):K_g(x)\geq 0\text{ for every }x\in M\},\\ \mathcal R_{>0}^{\mathcal C}(M) &=\{g\in\mathcal C(M):K_g(x)>0\text{ for every }x\in M\}. \end{aligned} \] Equip these spaces with the topology induced by $\tau$, and define their moduli spaces by \[ \mathcal M_{\diamond}^{\mathcal C,\tau,G}(M) =\mathcal R_{\diamond}^{\mathcal C}(M)/G(M), \qquad \diamond\in\{\geq 0,>0\}, \] with the quotient topology, where $g\sim h$ means $h=\phi^*g$ for some $\phi\in G(M)$. In particular, choosing the full diffeomorphism group gives equivalence by isometry through arbitrary diffeomorphisms.

Question (with unresolved conventions). Determine the homotopy types of the three spaces \[ \mathcal M_{\geq 0}^{\mathcal C,\tau,G}(S^2),\qquad \mathcal M_{>0}^{\mathcal C,\tau,G}({\mathbb{R}}^2),\qquad \mathcal M_{\geq 0}^{\mathcal C,\tau,G}({\mathbb{R}}^2) \] for the metric class, topology, and equivalence relation intended by the source. The parameters above expose missing conventions; the supplied record does not ask for a classification over all their possible choices. In particular, it does not establish that the intended topology is the compact-open $C^\infty$ topology or that the intended quotient uses the full diffeomorphism group. A unique precise formulation requires these conventions to be confirmed.

Short English statement

Determine the homotopy types of three moduli spaces: metrics of nonnegative Gaussian curvature on $S^2$, and metrics of positive or nonnegative Gaussian curvature on ${\mathbb{R}}^2$. The background concerns complete metrics, but completeness on the plane, the topology on metrics, and the quotient convention need confirmation. The supplied dated curation labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30003373$, source problem code catalogue entry, title Homotopy Types of Curvature-Metric Moduli on Surfaces, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified.
• 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 record separately gives proposed year 2017; the relationship between that year and the citation's 2018 designation is unresolved.
• Banakh and Belegradek, Spaces of nonnegatively curved surfaces. Supplied literature link: https://www.jstage.jst.go.jp/article/jmath/70/2/70_733/_article.
• Belegradek and Hu, Complete nonnegatively curved planes. Supplied literature link: https://arxiv.org/abs/1307.3167.
• Supplied literature assessment dated 2026-08-21: partially solved. The curation reports substantial analysis of spaces of complete nonnegative metrics on $S^2$ and ${\mathbb{R}}^2$, including homeomorphism and connectedness results, while stating that the three requested moduli homotopy types are not all determined. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.