In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian manifold whose Riemannian holonomy group

Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...

 is a subgroup of Sp(n)·Sp(1).

Another, more explicit, definition, uses a 3-dimensional subbundle H of End(TM) of endomorphisms of the tangent bundle to a Riemannian M. For M to be quaternion-Kähler, H should be preserved by the Levi-Civita connection and pointwise isomorphic to the imaginary quaternions, in such a way that unit imaginary quaternions in H act on TM preserving the metric.

Notice that this definition includes hyperkähler manifold

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...

s. However, these are often excluded from the definition of a quaternion-Kähler manifold by imposing the condition that the scalar curvature

Scalar curvature

In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...

 is nonzero, or that the holonomy group is equal to Sp(n)·Sp(1).

Ricci curvature

Quaternion-Kähler manifolds appear in Berger's list of Riemannian holonomies

Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections,...

 as the only manifolds of special holonomy with
non-zero Ricci curvature. In fact, these manifolds are Einstein

Einstein manifold

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor is proportional to the metric...

.

If an Einstein constant of a quaternion-Kähler manifold is zero, it is hyperkähler

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k) In differential geometry, a hyperkähler...

. This case is often excluded from the definition. That is, quaternion-Kähler is defined as one with holonomy reduced to Sp(n)·Sp(1) and with non-zero Ricci curvature (which is constant).

Quaternion-Kähler manifolds divide naturally into those with positive and negative Ricci curvature.

Examples

There are no known examples of compact quaternion-Kähler manifolds which are not locally symmetric or hyperkähler. Symmetric quaternion-Kähler manifolds are also known as Wolf spaces. For any simple Lie group G, there is a unique Wolf space G/K obtained as a quotient of G by a subgroup


Here, SU(2) is the subgroup associated with the highest root of G, and K₀ is its centralizer in G. The Wolf spaces with positive Ricci curvature are compact and simply connected.

If G is Sp(n+1), the corresponding Wolf space is the quaternionic projective space

Quaternionic projective space

In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions H. Quaternionic projective space of dimension n is usually denoted byand is a closed manifold of dimension 4n...

It can be identified with a space of quaternionic lines in Hⁿ⁺¹.

It is conjectured that all quaternion-Kähler manifolds with positive Ricci curvature are symmetric.

Twistor spaces

Questions about quaternion-Kähler manifolds of positive Ricci curvature can be translated into the language of algebraic geometry using the
methods of twistor theory (this approach is due to Penrose and Salamon). Let M be a quaternionic-Kähler manifold, and H the corresponding subbundle of End(TM), pointwise isomorphic to the imaginary quaternions. Consider the corresponding S²-bundle
S of all h in H satisfying h² = -1. The points of S are identified with the complex structures on its base. Using this, it is can be shown that the total space Z of S is equipped with an almost complex structure

Almost complex manifold

In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. The existence of this structure is a necessary, but not sufficient, condition for a manifold to be a complex manifold. That is, every complex manifold is an almost...

.

Salamon proved that this almost complex structure is integrable, hence Z is a complex manifold. When the Ricci curvature of M is positive, Z is a projective Fano manifold

Fano variety

In algebraic geometry, a Fano variety, introduced by , is a non-singular complete variety whose anticanonical bundle is ample.Fano varieties are quite rare, compared to other families, like Calabi–Yau manifolds and general type surfaces....

, equipped with a holomorphic contact structure.

The converse is also true: a projective Fano manifold which admits a holomorphic contact structure is always a twistor space, hence quaternion-Kähler geometry with positive Ricci curvature is essentially equivalent to the geometry of holomorphic contact Fano manifolds.