Rigidity of manifolds with equal first n+1 widths

Determine whether the sphere is the only closed n-dimensional manifold, for n≥3, that admits a Riemannian metric whose first n+1 widths are all equal.

Background

The paper observes that, for dimensions at least three, the sphere is the only currently known closed manifold admitting a metric with equal first n+1 widths. It raises the unresolved question of whether any other closed manifold can have this spectral-width degeneracy, a problem motivated by the relationship between width equalities and Zoll-type geometric rigidity.

References

We note that, at the time of writing this paper, $Sn$ is the only known closed manifold that admits a metric with equal first $n+1$ widths for $n\geq 3$. Optimistically, one may suspect that it is the only one (see also for what is known about $RP3$). Whether this is true or not, we hope that our results will help future work on this problem.

— Topological and spectral rigidity of hypersurface Zoll manifolds  (2609.20689 - Martins, 17 Sep 2026) in Section 1, Introduction, immediately before Organization of the paper