Pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy (Q1622887)

The authors prove a sort of integrability theorem for high-dimensional almost quaternionic homogeneous pseudo-Hermitian manifolds. Recall that one can generalize the concept of a Kähler manifold to the quaternionic setting in two different ways. Referring to Berger's famous list of holonomy groups (hence restricting to the irreducible Riemannian case but the ``Lorentzian'', i.e., the index \(4\) situation is similar), the first generalization is the concept of a \textit{hyper-Kähler manifold} having holonomy group \(\mathrm{Sp}(n)\subset\mathrm{U}(2n)\) while the second one is that of a \textit{quaternionic Kähler manifold} having holonomy group \(\mathrm{Sp}(n)\cdot\mathrm{Sp}(1)\subset \mathrm{U}(2n)\). From the point of view of (for example) mathematical physics, these structures are important because both carry solutions to the vacuum Einstein's equation hence play a role in plain gravity theory (i.e., classical general relativity); however in fact with their extra structures they fit better into various more advanced supergravity theories. The authors establish the following result. Let \((M,g)\) be a connected almost quaternionic pseudo-Hermitian manifold of index \(4\) and dimension \(\mathrm{dim}_{{\mathbb R}}M=4n+4\geqq 16\). Assume that \((M,g)\) is homogeneous and irreducible in the sense that there exists a connected subgroup \(G\) of the almost-quaternionic isometry group of \((M,g)\) which acts transitively on \(M\) such that for some (hence all) \(p\in M\) the real tangent space \(T_pM\) has the structure of an irreducible \({\mathbb H}\)-module with respect to the corresponding isotropy subgroup \(G_p\subset G\). Then \((M,g)\) is locally isometric to a quaternionic Kähler symmetric space (Theorem 1.1). The theorem does not work in dimension \(12\) as an explicit counterexample shows (Section 4.2).