A new variational characterization of $n$-dimensional space forms

A Riemannian manifold $(M^n,g)$ is associated with a Schouten $(0,2)$-tensor $C_g$ which is a naturally defined Codazzi tensor in case $(M^n,g)$ is a locally conformally flat Riemannian manifold. In this paper, we study the Riemannian functional $\mathcal{F}_k[g]=\int_M\sigma_k(C_g)dvol_g$ defined on $\mathcal{M}_1=\{g\in\mathcal{M}\vert Vol(g)=1\}$, where $\mathcal{M}$ is the space of smooth Riemannian metrics on a compact smooth manifold $M$ and $\{\sigma_k(C_g), 1\leq k\leq n\}$ is the elementary symmetric functions of the eigenvalues of $C_g$ with respect to $g$. We prove that if $n\geq 5$ and a conformally flat metric $g$ is a critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$, then $g$ must have constant sectional curvature. This is a generalization of Gursky and Viaclovsky's very recent theorem that the critical point of $\mathcal{F}_2\vert _{\mathcal{M}_1}$ with $\mathcal{F}_2[g]\geq0$ characterized the three-dimensional space forms.