We prove that an n-dimensional $(n\ge 4)$ compact Bach-flat manifold with positive scalar curvature and positive constant ${\sigma }_2$ is an Einstein manifold, provided that its Weyl curvature tensor satisfies a suitable pinching condition. Moreover, we prove that under some pinching conditions, the manifold is isometric to a quotient of the round ${\mathbb{S}}^n$.

中文翻译：

---

具有正常数 σ2 的紧凑巴赫平流形的刚性定理

我们证明了一个n维$(n\ge 4)$ 具有正标量曲率和正常数的紧凑巴赫平流形 ${\sigma }_2$是爱因斯坦流形，只要其外尔曲率张量满足合适的收缩条件。此外，我们证明了在某些夹点条件下，流形与圆的商等距${\mathbb{秒}}^n$.