---
aliases: ["Berger's classification"]

---

- Every smooth manifold admits a [Riemannian metric](Riemannian%20metric), so consider [Unsorted/Riemannian manifold](Unsorted/Riemannian%20manifold.md).

- Here $H\leq \SO(n)$ is the [holonomy group](holonomy%20group.md) :

- [Berger's classification](Berger's%20classification) for smooth [Riemannian geometry](Riemannian%20geometry.md), one of 7 possibilities.
$$
\begin{array}{|c|c|c|c|c|}
\hline n=\operatorname{dim} M & H & \text { Parallel tensors } & \text { Name } & \text { Curvature } \\
\hline n & \mathrm{SO}(n) & g, \mu & \text {orientable} & \\
\hline 2 m(m \geq 2) & \mathrm{U}(m) & g, \omega & \textbf{Kähler} & \\
\hline 2 m(m \geq 2) & \mathrm{SU}(m) & g, \omega, \Omega & \textbf{Calabi-Yau} & \text {Ricci-flat} \\
\hline 4 m(m \geq 2) & \mathrm{Sp}(m) & g, \omega_{1}, \omega_{2}, \omega_{3}, J_{1}, J_{2}, J_{3} & \textbf{hyper-Kähler} & \text {Ricci-flat} \\
\hline 4 m(m \geq 2) & (\mathrm{Sp}(m) \times \mathrm{Sp}(1)) / \mathbb{Z}_{2} & g, \Upsilon & \text {quaternionic-Kähler} & \text {Einstein} \\
\hline 7 & \mathrm{G}_{2} & g, \varphi, \psi & \mathrm{G}_{2} & \text {Ricci-flat} \\
\hline 8 & \operatorname{Spin}(7) & g, \Phi & \operatorname{Spin}(7) & \text {Ricci-flat} \\
\hline
\end{array}
$$


> Types in bold: amenable to Algebraic Geometry.
> $G2$ shows up in Physics!

- Ricci-flat, i.e. [Ricci curvature](Ricci%20curvature.md) tensor vanishes