For $X$ a space equipped with a $G$-connection on a bundle $\nabla$ (for some Lie group $G$) and for $x \in X$ any point, the parallel transport of $\nabla$ assigns to each curve $\Gamma : S^1 \to X$ in $X$ starting and ending at $x$ an element $hol_\nabla(\gamma) \in G$: the holonomy of $\nabla$ along that curve.
The holonomy group of $\nabla$ at $x$ is the subgroup of $G$ on these elements.
If $\nabla$ is the Levi-Civita connection on a Riemannian manifold and the holonomy group is a proper subgroup $H$ of the special orthogonal group, one says that $(X,g)$ is a manifold of special holonomy .
Berger's theorem says that if a manifold $X$ is
neither locally a product nor a symmetric space
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 |
---|---|---|---|
Kähler manifold | U(k) | $2k$ | Kähler forms $\omega_2$ |
Calabi-Yau manifold | SU(k) | $2k$ | |
hyper-Kähler manifold | Sp(k) | $4k$ | $\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2$ ($a^2 + b^2 + c^2 = 1$) |
quaternionic Kähler manifold | $4k$ | $\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3$ | |
G2 manifold | G2 | $7$ | associative 3-form |
Spin(7) manifold | Spin(7) | 8 | Cayley form |
A manifold having special holonmy means that there is a corresponding reduction of structure groups.
Let $(X,g)$ be a connected Riemannian manifold of dimension $n$ with holonomy group $Hol(g) \subset O(n)$.
For $G \subset O(n)$ some other subgroup, $(X,g)$ admits a torsion-free G-structure precisely if $Hol(g)$ is conjugate to a subgroup of $G$.
Moreover, the space of such $G$-structures is the coset $G/L$, where $L$ is the group of elements suchthat conjugating $Hol(g)$ with them lands in $G$.
This appears as (Joyce prop. 3.1.8)
special holonomy, reduction of structure groups, G-structure, exceptional geometry, Walker coordinates
The classification in Berger's theorem is due to
For more see
Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)
Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monographs (200o)
Luis J. Boya, Special Holonomy Manifolds in Physics Monografías de la Real Academia de Ciencias de Zaragoza. 29: 37–47, (2006). (pdf)
Discussion of the relation to Killing spinors includes