nLab quaternion-Kähler manifold

Definition

A Riemannian manifold $(X,g)$ of dimension $4n$ for $n \geq 2$ is called a quaternion-Kähler manifold if its holonomy group is a subgroup of Sp(n).Sp(1) (where Sp(n) is the $n$th quaternionic unitary group, and in particular $Sp(1) \simeq SU(2) \simeq$ Spin(3), and the central product is the quotient group of the direct product group by the diagonal center $\mathbb{Z}/2$).

If the holonomy group is in fact a subgroup of just the $Sp(n)$-factor, one speaks of a hyperkähler manifold.

Quaternion-Kähler manifolds are necessarily Einstein manifolds (see below). In particular their scalar curvature $R$ is constant, and hence a real number $R \in \mathbb{R}$. If the scalar curvature is positive, then one speaks of a positive quaternion-Kähler manifold.

Properties

As part of the Berger classification

classification of special holonomy manifolds by Berger's theorem:

	$\,$G-structure$\,$	$\,$special holonomy$\,$	$\,$dimension$\,$	$\,$preserved differential form$\,$
$\,\mathbb{C}\,$	$\,$Kähler manifold$\,$	$\,$U(n)$\,$	$\,2n\,$	$\,$Kähler forms $\omega_2\,$
	$\,$Calabi-Yau manifold$\,$	$\,$SU(n)$\,$	$\,2n\,$
$\,\mathbb{H}\,$	$\,$quaternionic Kähler manifold$\,$	$\,$Sp(n).Sp(1)$\,$	$\,4n\,$	$\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,$
	$\,$hyper-Kähler manifold$\,$	$\,$Sp(n)$\,$	$\,4n\,$	$\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,$ ($a^2 + b^2 + c^2 = 1$)
$\,\mathbb{O}\,$	$\,$Spin(7) manifold$\,$	$\,$Spin(7)$\,$	$\,$8$\,$	$\,$Cayley form$\,$
	$\,$G₂ manifold$\,$	$\,$G₂$\,$	$\,7\,$	$\,$associative 3-form$\,$

As $\mathbb{H}$-Riemannian manifolds

$\;$normed division algebra$\;$	$\;\mathbb{A}\;$	$\;$Riemannian $\mathbb{A}$-manifolds$\;$	$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$	$\;\mathbb{R}\;$	$\;$Riemannian manifold$\;$	$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$	$\;\mathbb{C}\;$	$\;$Kähler manifold$\;$	$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$	$\;\mathbb{H}\;$	$\;$quaternion-Kähler manifold$\;$	$\;$hyperkähler manifold$\;$
$\;$octonions$\;$	$\;\mathbb{O}\;$	$\;$Spin(7)-manifold$\;$	$\;$G₂-manifold$\;$

(Leung 02)

As quaternionic manifolds

Example

(quaternion-Kähler manifolds are quaternionic manifolds)

By definition, a quaternion-Kähler manifold $M$ has holonomy group contained in the direct product group Sp(n)$\times$Sp(1), admitting an extension of the Levi-Civita connection $\nabla$ on the holonomy bundle as torsion-free. Thus a quaternion-Kähler manifold is automatically a quaternionic manifold.

Such extension $\nabla_\text{quat}$ of $\nabla$ however is not unique, since $\nabla_\text{quat} + \mathcal{S}$ is another Sp(n)Sp(1)-preserving connection, where $\mathcal{S}$ is a (1, 2)-tensor such that for every $p \in M$, $\mathcal{S}(p)$ takes values in the first prolongation of the Lie algebra for the G-structure.

As Einstein manifolds

quaternion-Kähler manifolds are Einstein manifolds (e.g. Cortés 05, slide 22)

Characteristic classes

Proposition

Let $X$ be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for

$\;\;G =$ Sp(2).Sp(1) $\hookrightarrow$ SO(8)

then the Euler class $\chi$, the second Pontryagin class $p_2$ and the cup product-square $(p_1)^2$ of the first Pontryagin class of the frame bundle/tangent bundle are related by

(1)$$8 \chi \;=\; 4 p_2 - (p_1)^2 \,.$$

(Čadek-Vanžura 98, Theorem 8.1 with Remark 8.2)

Remark

The same conclusion (1) also holds for $Spin(7)$-structure, see there

See also at C-field tadpole cancellation.

Reduction to hyper-Kähler structure

A quaternion-Kähler manifold $(X,g)$ is a hyper-Kähler manifold, hence has $Sp(n) \hookrightarrow Sp(n)\cdot Sp(1)$-structure, precisely if its scalar curvature, which is a constant by $(X,g)$ being an Einstein manifold, vanishes: $R(g) = 0$.

(e.g. Amann 09, below Def. 1.5)

Positive quaternion-Kähler manifolds

Definition

A quaternion-Kähler manifold $(X,g)$ is called positive if

1. it is a geodesically complete

2. its scalar curvature, which is a constant by $(X,g)$ being an Einstein manifold, is a positive number, $R(g) \gt 0$.

(Salamon 82, Section 6, see e.g. Amann 09, Def. 1.5)

Proposition

A connected positive quaternion-Kähler manifold (Def. ) is necessarily compact.

(Salamon 82, p. 158 (16 of 29))

Proposition

A connected positive quaternion-Kähler manifold (Def. ) is necessarily simply connected.

(Salamon 82, Theorem 6.6)

Proposition

For each dimension $dim(X)$ there is a finite number of isometry classes of positive quaternion-Kähler manifolds (Def. ).

(LeBrun-Salamon 94, Theorem 0.1)

Proposition

(Wolf spaces are positive quaternion-Kähler manifolds)

Every Wolf space is a positive quaternion-Kähler manifold.

In fact the Wolf spaces are the only known examples of positive quaternion-Kähler manifold (which is not hyper-Kähler ?!), as of today (e.g. Salamon 82, Section 5).

This leads to the conjecture that in every dimension, the Wolf spaces are the only positive quaternion-Kähler manifolds.

The conjecture has been proven for the following dimensions

• $d = 4$ (Hitchin)

• $d = 8$ (Poon-Salamon 91, LeBrun-Salamon 94)

Examples

The archetypical example is

• quaternionic projective plane$\,\mathbb{H}P^2$

This is the first of the list of examples of spaces that are both quaternion-Kähler manifolds as well as symmetric spaces, called Wolf spaces.

See around Prop. above.

Related concepts

• quaternionic manifold, Kähler manifold, hyper-Kähler manifold

References

Original articles:

• Simon Salamon, Quaternionic Kähler manifolds, Invent Math (1982) 67: 143. (doi:10.1007/BF01393378)

• Y. S. Poon, Simon Salamon, Quaternionic Kähler 8-manifolds with positive scalar curvature, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 (euclid:1214446322)

• Claude LeBrun, Simon Salamon, Strong rigidity of positive quaternion Kähler manifolds, Inventiones Mathematicae 118, 1994, 109–132 (dml:144231, doi:10.1007/BF01231528)

Exposition:

• Vicente Cortés, Quaternionic Kähler manifolds, 2005 (pdf)

Textbook accounts:

• Arthur Besse, Einstein Manifolds, Springer-Verlag 1987.

• Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.

See also

• Wikipedia, Quaternion-Kähler manifold

In terms of G-structure:

• Edmond Bonan, Sur les $G$-structures de type quaternionien, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Volume 9 (1967) no. 4, p. 389-463 (numdam:CTGDC_1967__9_4_389_0)

• Simon Salamon, Differential Geometry of Quaternionic Manifolds, Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55 (numdam:ASENS_1986_4_19_1_31_0)

• Dmitri V. Alekseevsky, Stefano Marchiafava, Quaternionic-like structures on a manifold: Note I. 1-integrability and integrability conditions, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 43-52 (dml:244082)

• Dmitri V. Alekseevsky, Stefano Marchiafava, Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 (dml:244299)

• Dmitry V. Alekseevsky, Stefano Marchiafava, Quaternionic structures on a manifold and subordinated structures, Annali di Matematica pura ed applicata 171, 205–273 (1996) (doi:10.1007/BF01759388)

• Joseph Thurman, Quaternionic Geometry and Special Holonomy, 2018 (pdf)

Articles discussing quaternion-Kähler holonomy, connection, and relation to other hypercomplex structures:

• Andrei Moroianu, Uwe Semmelmann, Killing Forms on Quaternion-Kähler Manifolds, Annals of Global Analysis and Geometry, November 2005, Volume 28, Issue 4, pp 319–335 (arXiv:math/0403242, doi:10.1007/s10455-005-1147-y)

• Pedersen, Poon, and Swann. “Hypercomplex structures associated to quaternionic manifolds”, Differential Geometry and its Applications (1998) 273-293 North-Holland.

• Misha Verbitsky, Hyperkähler manifolds with torsion, supersymmetry and Hodge theory, Asian J. Math, V. 6 No. 4, pp. 679-712, Dec. 2002.

See also

• Claude LeBrun, On complete quaternionic-Kähler manifolds, Duke Math. J. Volume 63, Number 3 (1991), 723-743 (euclid:1077296077)

• Simon G. Chiossi, Óscar Maciá, SO(3)-Structures on 8-manifolds, Ann. Glob. Anal. Geom. 43 (1) (2013), 1–18 (arXiv:1105.1746)

On positive quaternion-Kähler manifolds

• Amann, Positive Quaternion Kähler Manifolds, 2009 (pdf)

• Amann, Partial Classification Results for Positive Quaternion Kaehler Manifolds (arXiv:0911.4587)

Discussion of characteristic classes:

• Martin Čadek, Jiří Vanžura, Almost quaternionic structures on eight-manifolds, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 (euclid:1200787905)

On quaternion-Kähler orbifolds:

• K. Galicki, H. Blaine Lawson, Quaternionic Reduction and Quaternionic Orbifolds, Mathematische Annalen (1988) Volume: 282, Issue: 1, page 1-22 (dml:164446)