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:
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
The classification in Berger's theorem is due to
For more see
Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monogrophs (200)
Luis J. Boya, Special Holonomy Manifolds in Physics Monografías de la Real Academia de Ciencias de Zaragoza. 29: 37–47, (2006). (pdf)