nLab special holonomy table

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\,

Last revised on May 12, 2019 at 13:42:27. See the history of this page for a list of all contributions to it.