Some Unstable Critical Metrics for $L^{\frac{n}{2}}$-norm of the Curvature Tensor

We consider the Riemannian functional defined on the space of Riemannian metrics with unit volume on a closed smooth manifold $M$ given by ${\mathcal{R}}_{\frac{n}{2}}(g):={\int }_M\mathrm{\mid }R(g){\mathrm{\mid }}^{\frac{n}{2}}d{v}_g$ where $R(g)$, $d{v}_g$ denote the Riemannian curvature and volume form corresponding to $g$. We show that there are locally symmetric spaces which are unstable critical points for this functional.