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quaternion-Kähler manifold

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Contents

Definition

A Riemannian manifold (X,g)(X,g) of dimension 4n4n for n2n \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 nnth quaternionic unitary group, and in particular Sp(1)SU(2)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 /2\mathbb{Z}/2).

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

Quaternion-Kähler manifolds are necessarily Einstein manifolds (see below). In particular their scalar curvature RR is constant, and hence a real number RR \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\,2n\,\,Kähler forms ω 2\omega_2\,
\,Calabi-Yau manifold\,\,SU(n)\,2n\,2n\,
\,\mathbb{H}\,\,quaternionic Kähler manifold\,\,Sp(n).Sp(1)\,4n\,4n\,ω 4=ω 1ω 1+ω 2ω 2+ω 3ω 3\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,
\,hyper-Kähler manifold\,\,Sp(n)\,4n\,4n\,ω=aω 2 (1)+bω 2 (2)+cω 2 (3)\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\, (a 2+b 2+c 2=1a^2 + b^2 + c^2 = 1)
𝕆\,\mathbb{O}\,\,Spin(7) manifold\,\,Spin(7)\,\,8\,\,Cayley form\,
\,G2 manifold\,\,G2\,7\,7\,\,associative 3-form\,

As \mathbb{H}-Riemannian manifolds

\;normed division algebra\;𝔸\;\mathbb{A}\;\;Riemannian 𝔸\mathbb{A}-manifolds\;\;special Riemannian 𝔸\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\;𝕆\;\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 MM has holonomy group contained in the direct product group Sp(n)×\timesSp(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 quat\nabla_\text{quat} of \nabla however is not unique, since quat+𝒮\nabla_\text{quat} + \mathcal{S} is another Sp(n)Sp(1)-preserving connection, where 𝒮\mathcal{S} is a (1, 2)-tensor such that for every pMp \in M, 𝒮(p)\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 XX be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for

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

then the Euler class χ\chi, the second Pontryagin class p 2p_2 and the cup product-square (p 1) 2(p_1)^2 of the first Pontryagin class of the frame bundle/tangent bundle are related by

(1)8χ=4p 2(p 1) 2. 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)Spin(7)-structure, see there

See also at C-field tadpole cancellation.

Reduction to hyper-Kähler structure

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

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

Positive quaternion-Kähler manifolds

Definition

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

  1. it is a geodesically complete

  2. its scalar curvature, which is a constant by (X,g)(X,g) being an Einstein manifold, is a positive number, R(g)>0R(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)dim(X) there is a finite number of isometry classes of positive quaternion-Kähler manifolds (Def. ).

(LeBrun-Salamon 94, Theorem 0.1)

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


Examples

The archetypical example is

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

See around Prop. above.


References

Original articles:

Exposition

Textbook references include:

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

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

See also

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.

  • 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.

  • 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)

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:

Last revised on July 21, 2019 at 11:47:39. See the history of this page for a list of all contributions to it.