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 . (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.

在本文中，我们研究了 Q 曲率的刚性。具体来说，我们考虑一个封闭的、有向的n维 ( \(n\ge 6\) ) 黎曼流形 ( M ,  g ) 并在 条件下证明以下结果。 (1) 如果 ( M ,  g ) 局部共形平坦且具有非负里奇曲率，则 ( M ,  g ) 等距于商\(\mathbb {R}^n\) , \(\mathbb {S}^ n\)或\(\mathbb {R}\times \mathbb {S}^{n-1}\)。 (2) 如果 ( M ,  g ) 的\(\delta ^2 W=0\)具有非负截面曲率，则 ( M ,  g ) 与爱因斯坦流形乘积的商等距。此外，我们还研究了一些涉及单连通空间形式超曲面 Q 曲率的刚性定理。我们还展示了在固定共形类中具有恒定标量曲率和恒定 Q 曲率的度量的唯一性。