nLab
hyperkähler manifold

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Contents

Definition

For nn \in \mathbb{N} a natural number, a 4n4n-dimensional Riemannian manifold is a hyperkähler manifold if its holonomy group is (a subgroup of) the quaternionic unitary group Sp(n). Regarded as a subgroup of the central product group Sp(n).Sp(1) this means that hyper-Kähler manifolds are special cases of quaternion-Kähler manifolds, though the latter are often taken to be only those Riemannian manifolds with full Sp(n).Sp(1)-holonomy.

Equivalently, a hyperkähler manifold is a Riemannian manifold (M,g)(M,g) with three complex structures I,J,KI, J, K which are Kähler with respect to the metric gg and satisfy the quaternionic identities

I 2=J 2=K 2=IJK=1.I^2=J^2=K^2=I J K=-1.

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(k)\,2k\,2k\,\,Kähler forms ω 2\omega_2\,
\,Calabi-Yau manifold\,\,SU(k)\,2k\,2k\,
\,\mathbb{H}\,\,quaternionic Kähler manifold\,\,Sp(n).Sp(1)\,4k\,4k\,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(k)\,4k\,4k\,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 special \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)

References

Last revised on April 15, 2019 at 04:15:18. See the history of this page for a list of all contributions to it.