# Compact quaternionic Kahler manifolds of negative curvature: examples

There is a well known problem of LeBrun-Salamon: are there any non-symmetric compact quaternionic-Kahler manifolds of positive scalar (and Ricci) curvature? It is hard and still unsolved: Quaternionic-Kahler metrics whose universal covers have only discrete isometry groups?

The symmetric compact quaternionic-Kahler manifolds ("Wolf spaces") are understood and classified. However, for each Wolf space there is the dual symmetric space, say, $G/H$, which is quaternionic-Kahler of negative scalar (and Ricci) curvature.

For any lattice $\Gamma\subset G$, the double quotient $\Gamma\backslash G/H$ is a locally symmetric quaternionionic-Kahler orbifold of negative curvature and finite volume. However, it can have cusp points, and then it is non-compact.

Are there any compact locally symmetric quaternionionic-Kahler orbifolds? Manifolds? I could not find a reference to either existence or non-existence results.

• A comment for now. Here's a construction that doesn't work. Choose a quadratic form q over Q (the rationals) of signature (4,4) over the reals. But require q to be anisotropic at some nonempty set of finite places. Take $\Gamma$ an arithmetic subgroup of $Spin(q)$. Then $\Gamma \backslash Spin(4,4) / K$ might do it. But sadly quadratic forms in 8 variables over $Q_p$ are never anisotropic, so this fails. Similar failure for $G_2$ instead of $Spin(4,n)$. Sad! – Marty Jul 23 '17 at 22:12

## 1 Answer

Any (Riemannian) symmetric space admits a cocompact lattice. This is due to A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2, 1963, pp.111-122. The quaternionic hyperbolic space is symmetric and quaternionic-Kahler.