Holonomy Classification

Every smooth manifold admits a Riemannian metric, so consider Unsorted/Riemannian manifold.
Here \(H\leq {\operatorname{SO}}(n)\) is the holonomy group :
Berger’s classification for smooth Riemannian geometry, one of 7 possibilities. \begin{align*} \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} \end{align*}

Types in bold: amenable to Algebraic Geometry. \(G2\) shows up in Physics!