Distinguishing curvature classes defined by successive intermediate-Ricci bounds

Determine whether, for k ≥ 2, the class of closed manifolds admitting a Riemannian metric with Ric_k ≥ K and diameter at most D differs from the class admitting a Riemannian metric with Ric_{k+1} ≥ K and diameter at most D.

Background

The curvature conditions satisfy Ric_k ≥ K ⇒ Ric_{k+1} ≥ K, so the associated classes of manifolds are nested. The paper notes that, although a distinction is known between sectional curvature and Ricci curvature in a related setting, it is not known whether successive intermediate-Ricci conditions yield genuinely different classes for k ≥ 2.

This problem concerns the structural sharpness of the intermediate-Ricci hierarchy and is not resolved by the local Betti-number vanishing theorem established in the paper.

References

Nevertheless, when $k\geq2$, it remains unclear whether the class of closed manifolds admitting a Riemannian metric with $Ric_k\geq K$ and $\mathrm{diam}\leq D$ differs from the class of closed manifolds admitting a Riemannian metric with $Ric_{k+1}\geq K$ and $\mathrm{diam}\leq D$ (see also for such a distinction when $k=1$).

— Local Topology of Riemannian Manifolds with Lower Intermediate Ricci Curvature Bounds  (2609.10268 - Reiser et al., 9 Sep 2026) in Section 1, immediately after Theorem A