On some rigidity theorems of Q-curvature

In this paper, we investigate the rigidity of Q-curvature. Specifically, we consider a closed, oriented $n$-dimensional ($n\ge 6$) Riemannian manifold $(M,g)$ and prove the following results under the condition ${\int }_M\mathrm{\nabla }R\cdot \mathrm{\nabla }\mathrm{Q}\mathrm{d}{V}_g\le 0$. (1) If $(M,g)$ is locally conformally flat with nonnegative Ricci curvature, then $(M,g)$ is isometric to a quotient of ${\mathbb{R}}^n$, ${\mathbb{S}}^n$, or $\mathbb{R}\times {\mathbb{S}}^{n-1}$. (2) If $(M,g)$ has ${\delta }^{2}W=0$ with nonnegative sectional curvature, then $(M,g)$ is isometric to a quotient of the product of Einstein manifolds. Additionally, we investigate some rigidity theorems involving Q-curvature about hypersurfaces in simply-connected space forms. We also show the uniqueness of metrics with constant scalar curvature and constant Q-curvature in a fixed conformal class.