Example of locally symmetric spaces

Question

A locally symmetric manifold is a manifold with parallel curvature tensor $\nabla R=0$.

Can you give an example except spheres, projective spaces and hyperbolic spaces?

Answer 1

Let $G$ be any compact Lie group with a bi-invariant metric $\mathtt{g}$. For each $g\in G$, define $\sigma_g\colon G \to G$ by $\sigma_g(x) = gx^{-1}g$. Note that each $\sigma_g$ is an isometry for $\mathtt{g}$. Moreover, we have that $$\sigma_g(g) = g \quad \mbox{and} \quad {\rm d}(\sigma_g)_g(v) = -v.$$With this in place, note that $\nabla R$ is a tensor of rank $5$, so that $[\sigma_g^*(\nabla R)]_g = - (\nabla R)_g$, as $(-1)^5=-1$. On the other hand, $[\sigma_g^*(\nabla R)]_g = (\nabla R)_g$ because $\sigma_g$ is an isometry. Thus $(\nabla R)_g = 0$ but, as $g$ was arbitrary, $\nabla R = 0$.