A quaternion Kähler manifold is a Riemannian manifold of dimension 
, 
, with nonzero scalar
curvature, whose full holonomy group is, up
to conjugacy, a subgroup
of
 |
The nonzero scalar curvature condition is an explicit convention excluding the locally hyper-Kähler case. These manifolds are sometimes called quaternionic Kähler and are sometimes written hyphenated as quaternion-Kähler, quaternionic-Kähler, etc.
Despite their name, quaternion-Kähler manifolds need not be Kähler due to the fact that
all Kähler manifolds have holonomy
groups which are subgroups of 
, whereas 
. Depending on the literature, such
manifolds are sometimes assumed to be connected
and/or orientable. In the above definition,
the case for 
is usually excluded due to the fact that 
which, under Berger's classification of holonomy, implies merely that the manifold
is Riemannian. The above classification can
be extended to the case where 
by requiring that the manifold
be an Einstein manifold with self-dual Weyl tensor and nonzero scalar curvature.
Salamon (2001) omits the nonzero scalar curvature condition and defines quaternion-Kähler manifolds
solely by the holonomy group containment, with
the additional Einstein manifold and self-dual curvature requirements in dimension
4. Under this broader convention, hyper-Kähler
manifolds are included. In dimensions 
with 
, zero scalar
curvature implies that the metric is locally hyper-Kähler.
Its full holonomy group may still contain elements
outside 
,
so excluding full holonomy group containment in
alone does not exclude zero scalar
curvature.
Berger (1955) showed that for 
, quaternionic-Kähler manifolds
are necessarily Einstein manifolds.
Because the definition of quaternion-Kähler manifolds excludes the possibility of having zero scalar curvature, it is natural to investigate the cases of quaternion-Kähler manifolds with positive and negative scalar curvatures (referred to as positive quaternion-Kähler and negative quaternion-Kähler manifolds, respectively) separately. The work of LeBrun (1991) shows a number of significant differences in these two cases, and while many advances have been made towards the understanding of positive quaternion-Kähler manifolds, little seems to be known regarding their negative scalar curvature counterparts.
Amann (2009b) reported that the only known complete positive quaternion-Kähler examples were the symmetric Wolf spaces. Moreover, it has been conjectured by LeBrun among others (Amann 2009a) that all positive quaternion-Kähler manifolds that are connected and complete are symmetric. In particular, the conjecture has been proved for dimensions 4 and 8 (Amann 2009a). Compact positive quaternion-Kähler symmetric spaces are known as Wolf spaces.
