nLab
Berger's theorem
Contents
Context
Riemannian geometry
Symplectic geometry
Contents
Statement
Berger’s theorem says that if a manifold X X 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 \, ℂ \,\mathbb{C}\, \, Kähler manifold \, \, U(n) \, 2 n \,2n\, \, Kähler forms ω 2 \omega_2\,
\, Calabi-Yau manifold \, \, SU(n) \, 2 n \,2n\,
ℍ \,\mathbb{H}\, \, quaternionic Kähler manifold \, \, Sp(n).Sp(1) \, 4 n \,4n\, ω 4 = ω 1 ∧ ω 1 + ω 2 ∧ ω 2 + ω 3 ∧ ω 3 \,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,
\, hyper-Kähler manifold \, \, Sp(n) \, 4 n \,4n\, ω = a ω 2 ( 1 ) + b ω 2 ( 2 ) + c ω 2 ( 3 ) \,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\, (a 2 + b 2 + c 2 = 1 a^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
Original article:
Marcel Berger , Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes , Bull. Soc. Math. France 83 (1955) (doi:10.24033/bsmf.1464 )
Carlos Olmos, A Geometric Proof of the Berger Holonomy Theorem , Annals of Mathematics Second Series, Vol. 161, No. 1 (Jan., 2005), pp. 579-588 (10 pages) (jstor:3597350 )
See also
Last revised on July 13, 2020 at 10:29:55.
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