nLab
Berger's theorem

Contents

Context

Riemannian geometry

Symplectic geometry

Contents

Statement

Berger’s theorem says that if a manifold XX is

then the possible special holonomy groups are the following

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\,

References

  • M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955)

Last revised on March 19, 2019 at 06:54:39. See the history of this page for a list of all contributions to it.