LeBrun–Salamon conjecture on quaternionic-Kähler manifolds

Prove that every complete quaternionic-Kähler manifold, namely any 4n-dimensional Riemannian manifold whose Levi-Civita holonomy is contained in Sp(n)Sp(1), with positive scalar curvature is symmetric.

Background

Quaternionic-Kähler manifolds arise in Berger’s list as those with holonomy Sp(n)Sp(1). The notes summarize known examples and properties, noting that all compact examples presently known are locally symmetric (Wolf spaces). The LeBrun–Salamon conjecture posits a global characterization of such manifolds under the assumption of positive scalar curvature, strengthening the connection between holonomy and symmetric space structure.

Establishing this conjecture would settle the existence question for compact quaternionic-Kähler manifolds of positive scalar curvature beyond the known symmetric cases and clarify the landscape of special holonomy geometries with Sp(n)Sp(1).

References

A conjecture of LeBrun-Salamon asserts that all complete quaternionic-Kähler manifolds of positive scalar curvature are symmetric.

— Topics in representation theory and Riemannian geometry  (2504.13689 - Russo, 18 Apr 2025) in Example 5.13, Section 5.4

On the other hand, the LeBrun--Salamon conjecture, which states that every positive quaternionic Kähler manifold is symmetric, has been proved in real dimensions 8, 12, and 16, see.

— Inhomogeneous Einstein metrics on complex projective spaces  (2608.16880 - Cao-Labora et al., 17 Aug 2026) in Remark 2.14, Section 2.3.3 (remark labeled “nonNK”)

Indeed, the LeBrun--Salamon conjecture asserts:

Every connected complete positive quaternionic K"ahler manifold is, up to rescaling, a Wolf space.

— The Lebrun--Salamon conjecture is true in dimension up to 56  (2610.06336 - Serrano et al., 5 Oct 2026) in Section 1.1, “The LeBrun–Salamon conjecture”