Generalized and Standard Almost-Nonnegative Curvature

Definitions and mathematical statement

Let $M$ be a closed smooth manifold. In the standard sectional-curvature convention, almost nonnegative curvature means that for every $\varepsilon>0$ there is a Riemannian metric $g_\varepsilon$ on $M$ such that \[ \sec(M,g_\varepsilon)\ge-\varepsilon, \qquad\operatorname{diam}(M,g_\varepsilon)\le1. \] This is a scale-normalized convention; the report's precise normalization should be confirmed. It does not require a single metric with nonnegative sectional curvature.

The source distinguishes this standard, "strict $k=0$" notion from a generalized notion whose defining collapsing sequence is allowed on \[ M\times{\mathbb{R}}^k\qquad\text{for some fixed integer }k\ge0. \] The question is whether every closed smooth $M$ satisfying that generalized definition also satisfies the standard definition on $M$ itself. In quantifier form, the existence of an allowed generalized collapsing sequence for some $k$ should imply the existence of metrics on $M$ obeying the displayed curvature and diameter bounds for every $\varepsilon>0$.

The excerpt gives only the product-manifold feature of the generalized definition. It does not give the curvature domains, completeness, basepoints, limit space, convergence notion, or fiber conditions for that sequence. Since $M\times{\mathbb{R}}^k$ is noncompact for $k>0$, the compact-manifold diameter condition above cannot simply be imposed on the entire product to define the generalized property. Nor is the generalized sequence stated to consist of product metrics. Those details must be recovered from the cited definition rather than supplied by guesswork.

Here "strict" labels the unstabilized $k=0$ case; it does not mean strictly positive sectional curvature. The desired implication removes the Euclidean stabilization at the level of the existence property, not necessarily by restricting each particular generalized metric to a slice.

Short English statement

Does allowing an auxiliary Euclidean factor in the generalized collapsing definition enlarge the class of closed manifolds with almost nonnegative curvature? The target is the usual unstabilized metric property on $M$, while the full generalized definition is missing.

Sources

• Local UnsolvedMath registry, permanent ID 30000597, code catalogue entry, "Generalized and Standard Almost-Nonnegative Curvature." Proposed year 2006; proposer unspecified.
• Geometrie (2007), Oberwolfach Reports, pp. 2699--2756, DOI 10.4171/owr/2006/45; https://doi.org/10.4171/owr/2006/45. The displayed report year 2007 differs from the DOI and proposed year 2006.
• Supplied reference "Kapovitch--Petrunin--Tuschmann": https://www.math.toronto.edu/vtk/nilpotency.pdf.
• Statement status corrected_verified; the supplied note says parameter, product, and range notation was enclosed in balanced delimiters, retaining the question and explanatory definition.
• Literature triage checked 2026-08-21: open. It says the generalized/strict distinction is not known to disappear in general and the foundational literature explicitly records it as unknown. Precise definitions and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.