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.
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$\,$ |
$\;$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)
(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.
quaternion-Kähler manifolds are Einstein manifolds (e.g. Cortés 05, slide 22)
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
(Čadek-Vanžura 98, Theorem 8.1 with Remark 8.2)
See also at C-field tadpole cancellation.
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)
A quaternion-Kähler manifold $(X,g)$ is called positive if
it is a geodesically complete
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)
A connected positive quaternion-Kähler manifold (Def. ) is necessarily compact.
(Salamon 82, p. 158 (16 of 29))
A connected positive quaternion-Kähler manifold (Def. ) is necessarily simply connected.
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)
(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)
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.
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
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.