Comparing Positive and Nonnegative Ricci Metric Spaces

Definitions and mathematical statement

For a fixed smooth manifold $M$, let $\mathcal R(M)$ denote the set of smooth Riemannian metrics on $M$. Define \[ \begin{aligned} R^{>0}(M)&=\{g\in\mathcal R(M):\operatorname{Ric}_g(v,v)>0\text{ for every }x\in M\text{ and }0\ne v\in T_xM\},\\ R^{\geq0}(M)&=\{g\in\mathcal R(M):\operatorname{Ric}_g(v,v)\geq0\text{ for every }x\in M\text{ and }v\in T_xM\}. \end{aligned} \] Here $\operatorname{Ric}_g$ is the Ricci curvature tensor of $g$. These definitions concern Ricci curvature, as specified in the supplied statement.

The source does not specify the class of manifolds, dimension, compactness, boundary conditions, or topology on the metric spaces. To interpret the following questions, fix a topology on $\mathcal R(M)$ and give both subsets the induced topology. No particular topology or class of manifolds is asserted to be the intended one.

The supplied problem asks how the homotopy types of these spaces compare. In particular, for the inclusion \[ i:R^{>0}(M)\hookrightarrow R^{\geq0}(M), \] is $i$ a homotopy equivalence? Explicitly, does there exist a continuous map $r:R^{\geq0}(M)\to R^{>0}(M)$ such that \[ r\circ i\simeq\operatorname{id}_{R^{>0}(M)},\qquad i\circ r\simeq\operatorname{id}_{R^{\geq0}(M)}? \] The source also asks whether $i$ is a "surjection on homotopy." One possible precise interpretation is that, for every $g\in R^{>0}(M)$ and every integer $k\geq1$, the induced map \[ i_*:\pi_k(R^{>0}(M),g)\longrightarrow\pi_k(R^{\geq0}(M),i(g)) \] is surjective. Whether this is required for every basepoint and degree, and whether it also includes surjectivity of \[ \pi_0(R^{>0}(M))\longrightarrow\pi_0(R^{\geq0}(M)), \] where $\pi_0$ denotes path components, is unresolved in the supplied record. Nonemptiness assumptions are also unspecified. Thus these are conditional formulations of the supplied questions; the record does not determine a fully quantified conjecture over manifolds.

Short English statement

The problem asks how the space of metrics with positive Ricci curvature compares with the space of metrics with nonnegative Ricci curvature, especially whether their inclusion is a homotopy equivalence or induces surjections on homotopy. The intended manifolds, topology, nonemptiness assumptions, and meaning of homotopy surjectivity require clarification. The supplied literature assessment discusses scalar curvature and a higher-index map, whose connection to this Ricci-curvature question is unspecified.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30003358$, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded 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 literature-source label for this same link is "Oberwolfach report on invertible Dirac metrics." The relationship between that label and the Ricci-curvature statement has not been independently verified.
• Supplied literature source: "Higher rho invariants and positive scalar curvature moduli." https://doi.org/10.1016/j.aim.2016.11.030.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that higher-index and rho-invariant theories give substantial information about positive-scalar-curvature spaces, while the proposed homotopy equivalence and general surjectivity of the higher-index map are not established. This is supplied dated curation, not an independently verified assessment. Its scalar-curvature and higher-index assertions do not specify how they apply to the Ricci-curvature inclusion above. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified, with a qualification that only notation, typography, and TeX delimiters were cleaned up. This supplied label does not constitute independent mathematical or source verification of this draft.