nLab
special holonomy

Idea

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 ${\mathrm{hol}}_{\nabla }(\gamma )\in G$: 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 .

Properties

Classification

Berger's theorem says that if a manifold $X$ is

• simply connected

• 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
Calabi-Yau manifold	SU(k)	$2k$
hyper-Kähler manifold	Sp(k)	$4k$
G2 manifold	G2	$7$	associative 3-form
Spin(7) manifold	Spin(7)	8	Cayley form

Relation to $G$-reductions

A manifold having special holonmy means that there is a corresponding reduction of structure groups.

Theorem

Let $(X,g)$ be a connected Riemannian manifold of dimension $n$ with holonomy group $\mathrm{Hol}(g)\subset O(n)$.

For $G\subset O(n)$ some other subgroup, $(X,g)$ admits a torsion-free G-structure precisely if $\mathrm{Hol}(g)$ 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 $\mathrm{Hol}(g)$ with them lands in $G$.

This appears as (Joyce prop. 3.1.8)

Related concepts

• connection on a bundle

  • connection on a 2-bundle, connection on an infinity-bundle

• parallel transport, higher parallel transport

• holonomy

  • holonomy group

  • special holonomy, reduction of structure groups, G-structure, exceptional geometry

References

The classification in Berger's theorem is due to

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

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)