For a space equipped with a -connection on a bundle (for some Lie group ) and for any point, the parallel transport of assigns to each curve in starting and ending at an element : the holonomy of along that curve.
The holonomy group of at is the subgroup of on these elements.
then the possible special holonomy groups are the following
|G-structure||special holonomy||dimension||preserved differential form|
|Kähler manifold||U(k)||Kähler forms|
|quaternionic Kähler manifold||Sp(k)Sp(1)|
|Spin(7) manifold||Spin(7)||8||Cayley form|
|G2 manifold||G2||associative 3-form|
A manifold having special holonomy means that there is a corresponding reduction of structure groups.
Moreover, the space of such -structures is the coset , where is the group of elements suchthat conjugating with them lands in .
This appears as (Joyce prop. 3.1.8)
|normed division algebra||Riemannian -manifolds||Special Riemannian -manifolds|
|real numbers||Riemannian manifold||oriented Riemannian manifold|
|complex numbers||Kähler manifold||Calabi-Yau manifold|
|quaternions||quaternion-Kähler manifold||hyperkähler manifold|
The classification in Berger's theorem is due to
For more see
Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monographs (2000)
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