Core Metrics and Positive Ricci Curvature on Simply Connected Manifolds

Definitions and mathematical statement

A manifold is understood here to be a finite-dimensional smooth manifold. A manifold is closed if it is compact and has no boundary, and simply connected if it is path connected and has trivial fundamental group. For such a manifold $M$, let \[ \mathcal R_{\operatorname{Ric}>0}(M) = \left\{g:\ g\text{ is a smooth Riemannian metric on }M,\quad \operatorname{Ric}_{g,p}(v,v)>0\text{ for every }p\in M\text{ and }0\ne v\in T_pM\right\}. \] Let $\operatorname{Core}(M)$ stand for the assertion that $M$ admits a core metric in the sense intended by the supplied source. The record does not define a core metric, including its geometric requirements or any boundary and normal conventions. Thus $\operatorname{Core}(M)$ is an unresolved source-dependent predicate here, and the first question is not yet a self-contained geometric formulation.

The supplied problem asks whether both of the following assertions hold:

1. For every closed simply connected smooth manifold $M$, \[ \mathcal R_{\operatorname{Ric}>0}(M)\ne\varnothing \quad\Longrightarrow\quad \operatorname{Core}(M). \]
2. For every closed simply connected smooth manifold $M$ with $\dim M=6$, \[ \mathcal R_{\operatorname{Ric}>0}(M)\ne\varnothing. \]

The first assertion concerns existence of a core metric; it does not require the originally assumed positive-Ricci-curvature metric itself to be a core metric. No lower dimension bound is supplied for the first assertion. The smooth interpretation is the usual setting for Riemannian metrics, but the record does not explicitly specify a manifold category or address dependence on smooth structure.

Short English statement

Does every closed simply connected manifold admitting a positive-Ricci-curvature metric admit a core metric, and does every closed simply connected 6-manifold admit a positive-Ricci-curvature metric? These are two universal existence questions. The meaning of "core metric" must be recovered from the cited source before the first question is fully specified. The supplied record labels both assertions open; this status is not independently verified.

Sources

• Local registry: data/unsolvedmath/problems.json, supplied record with permanent UnsolvedMath ID $30004723$, source problem code catalogue entry, entitled "Core Metrics and Positive Ricci Curvature on Simply Connected Manifolds." Collection: Oberwolfach Reports Open Problems. The record marks the statement as exact and reports byte-for-byte retention; this draft does not independently verify that designation.
• Supplied report citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2022), Oberwolfach Reports, pp. 1579--1649, DOI 10.4171/owr/2021/30. https://doi.org/10.4171/owr/2021/30. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions. It gives proposed year 2021 and supplies no named proposer.
• Supplied literature reference: "Reiser, Journal of Topology 2024." https://doi.org/10.1112/topo.70007.
• Supplied literature reference: Twisted suspensions and positive Ricci curvature. https://arxiv.org/abs/2401.06587.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that many new families with positive Ricci curvature and core metrics are known, while the universal core-metric assertion and the assertion for every simply connected six-manifold remain open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.