A hyper-Kähler manifold can be defined as a Riemannian manifold of dimension 
with three covariantly
constant orthogonal automorphisms 
, 
, 
of the tangent bundle which
satisfy the quaternionic identities
 |
(1)
|
where 
denotes the negative of the identity automorphism 
on the tangent bundle. The term hyper-Kähler is sometimes
written without a hyphen (as hyperKähler) or without capitalization (as
hyperkähler).
This definition is equivalent to several others commonly encountered in the literature; indeed, a manifold 
is said to be hyper-Kähler if and only if:
1. 
is a holomorphically symplectic Kähler manifold with holonomy
in 
.
2. 
is a holomorphically symplectic Kähler manifold which is Ricci-flat
(i.e., which has zero scalar curvature.
The first of the above equivalences is referring to Berger's classification of the holonomy groups of Riemannian manifolds and implies that parallel transport preserves
,
,
and 
.
Both this criterion and the criterion listed in the second of the above equivalences
is used to differentiate hyperkähler manifolds from the similarly-named quaternionic-Kähler
manifolds which have nonzero Ricci curvature and, in general, fail to be Kähler.
Hyper-kähler manifolds are necessarily Calabi-Yau manifolds and are Einstein manifolds with constant 0.
Generally, the automorphisms 
are assumed to be integrable.
The presence of these three complex structures
induces three Kähler 2-forms 
, 
, on 
, namely
 |
(2)
|
 |
(3)
|
and
 |
(4)
|
for all 
where, here, 
is the Kähler/Riemannian metric on 
. As the two equivalent definitions above indicate, hyperkähler
manifolds are holomorphically symplectic, i.e., they have three holomorphic
symplectic 2-forms induced by each of 
, 
, and 
. For example, the 2-form 
of the form
 |
(5)
|
is holomorphic and symplectic on 
(where 
denotes the standard imaginary
unit). Calabi proved a partial converse which says that a compact holomorphically
symplectic Kähler manifold admits a unique hyper-Kähler metric with respect
to any of its Kähler forms.
All even-dimensional complex vector spaces and tori are hyper-Kähler. Further examples
include the quaternions 
, the cotangent bundle 
of 
-dimensional
complex projective space, K3
surfaces, Hilbert schemes of points on compact
hyper-kähler 4-manifolds, and generalized
Kummer varieties, as well as various moduli spaces,
spaces of solutions to Nahm's equations, and the
Nakajima quiver varieties.
