A quaternion Kähler manifold is a Riemannian manifold of dimension 
, 
, whose holonomy is,
up to conjugacy, a subgroup of
 |
but is not a subgroup of 
. 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 both an Einstein
manifold and self-dual.
Some authors exclude this last criterion, thereby classifying manifolds as quaternion-Kähler provided that they are Riemannian and have a holonomy group
which is a subgroup of 
. Under this less-restrictive definition, Hyper-Kähler
manifolds-manifolds with holonomy group a subgroup of 
-would be considered quaternion-Kähler, though it
is not uncommon for literature to distinguish between manifolds which are quaternion-Kähler
and hypkerkähler. In place of the last criterion, some authors instead impose
the condition that the manifold have nonzero scalar
curvature, whereby manifolds which are hypkerkähler (and hence are Ricci-flat)
are again precluded.
Berger 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 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.
There are no known examples of compact quaternion Kähler manifolds which are neither locally symmetric nor hyper-Kähler. Moreover, it has been conjectured by LeBrun among others that all positive quaternion-Kähler manifolds are symmetric with proved confirmation for dimensions 4 and 8. Quaternion-Kähler manifolds which are locally symmetric are known as Wolf spaces.
