Differential geometry Connections and curvature Theorem

Berger's classification of holonomy groups

Builds on Holonomy Riemannian manifold Levi-Civita connection Symmetric space

Statement

Let $(M, g)$ be a simply connected Riemannian manifold of dimension $n$ with Levi-Civita connection $\nabla$ , and let $p \in M$ . Assume that the holonomy group $\mathrm{Hol}_p(\nabla)$ acts irreducibly on $T_pM$ , and that $(M, g)$ is not locally symmetric, that is, $\nabla R \ne 0$ for its curvature tensor $R$ . Then $\mathrm{Hol}_p(\nabla)$ , as a subgroup of $\mathrm{O}(T_pM) \cong \mathrm{O}(n)$ , is conjugate to one of the following:

$\mathrm{SO}(n)$ ;
$\mathrm{U}(m)$ , with $n = 2m \ge 4$ ;
$\mathrm{SU}(m)$ , with $n = 2m \ge 4$ ;
$\mathrm{Sp}(k) \cdot \mathrm{Sp}(1)$ , with $n = 4k \ge 8$ ;
$\mathrm{Sp}(k)$ , with $n = 4k \ge 8$ ;
$G_2$ , with $n = 7$ ;
$\mathrm{Spin}(7)$ , with $n = 8$ .

Remarks

Berger's proof combines the Ambrose–Singer theorem with the Bianchi identity: the holonomy algebra must be spanned by curvature-type tensors. Berger checked which irreducible subgroups of $\mathrm{SO}(n)$ admit enough of them while allowing $\nabla R \ne 0$ . Berger's original list also contained $\mathrm{Spin}(9)$ in dimension $16$ ; later work showed that such a metric is locally symmetric, so this case drops out. All seven groups do occur; the existence of metrics with holonomy $G_2$ and $\mathrm{Spin}(7)$ was shown much later, first locally by Bryant and then on compact manifolds by Joyce. The hypotheses are needed. By the de Rham decomposition theorem, a reducible holonomy splits $M$ locally as a product. A locally symmetric space can have the holonomy of its isotropy group, which is not in this list in general. The geometric meaning: $\mathrm{U}(m)$ gives Kähler manifolds, $\mathrm{SU}(m)$ Ricci-flat Kähler (Calabi–Yau) manifolds, $\mathrm{Sp}(k)$ hyperkähler manifolds, and $\mathrm{Sp}(k)\cdot\mathrm{Sp}(1)$ quaternion-Kähler manifolds.

Going deeper

For a student

The theorem organises much of modern Riemannian geometry: each group on the list defines a class of manifolds with special holonomy and parallel tensors. For example holonomy in $\mathrm{U}(m)$ means there is a parallel complex structure, which is the Kähler condition. The groups $\mathrm{SU}(m)$ , $\mathrm{Sp}(k)$ , $G_2$ and $\mathrm{Spin}(7)$ force Ricci-flatness, and these are the geometries used in string theory compactifications. Together with the de Rham decomposition and Cartan's classification of symmetric spaces, it describes the holonomy of every simply connected complete Riemannian manifold.

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Holonomy concept
Levi-Civita connection concept
Riemannian manifold concept
Symmetric space concept
Berger's classification of holonomy groups statement

Tag	`20GTNPF5V6`
URI	`m2://20GTNPF5V6`
Kind	statement, theorem
Writing	W0 stub
MSC	53C29
Status	This copy of the site has no build data. Run `m2 status 20GTNPF5V6`.
About	Holonomy
Uses	Holonomy, Riemannian manifold, Levi-Civita connection, Symmetric space
Changed	2 November 2026