Creating an interesting Riemannian manifold: let \(G\in\mathsf{Lie}{\mathsf{Grp}}^{\mathrm{ss}}\) with \(K\leq G\) its maximal compact subgroup. The Riemannian structure comes from an invariant metric on \(G\), and so \(G\to \mathop{\mathrm{Isom}}(G/K)\) via left translation and \(G\curvearrowright G/K\). Take \(\Gamma \leq G\) and form the double quotient \(\dcoset{\Gamma}{G}{K}\). These are some of the most celebrated manifolds in mathematics.
Take \(G={\operatorname{SL}}_2({\mathbf{R}})\) to get uniformization: every \(\Sigma_g\) for \(g\geq 2\) can be described this way.
A riemannian manifold\(X\) of even real dimension \(\dim_{\mathbf{R}}(X) = 2n\) with Holonomy\({\mathrm{holon}}(X) \subseteq {\operatorname{SU}}_n \subset {\operatorname{O}}_{2n}({\mathbf{R}})\).