In this paper, we first prove a topological obstruction for a four-dimensional manifold carrying an Einstein metric. More precisely, assume (M, g) is a closed Einstein four-manifold with $$Ric=\rho g$$. Denote by K the sectional curvature of M. If $$K\ge \delta \ge \frac{2\rho -\sqrt{5}\left|\rho \right|}{6}$$ (or $$K\le \delta \le \frac{2\rho +\sqrt{5}}{6}$$) for some constant $$\delta $$, then the Euler characteristic $$\chi $$ and the signature $$\tau $$ of M satisfy $$\begin{aligned}\chi \ge \left( \dfrac{3}{8\left( 1-3\delta /\rho \right) ^2} +\dfrac{3}{2}\right) \left|\tau \right|.\end{aligned}$$Our second result is a rigidity theorem for closed oriented Einstein four-manifolds with positive sectional curvature. Assume $$\lambda _1$$ is the first eigenvalue of the Laplacian of an oriented closed Einstein four-manifold (M, g) with $$Ric = g$$. We show that M must be isometric to a round 4-sphere or $$\mathbb {CP}^2$$ with the (normalized) Fubini-Study metric if the sectional curvature bounded above by $$1-\frac{4}{9\lambda _1+12}$$ (or bounded below by $$\frac{2}{9\lambda _1+12}$$).

中文翻译：

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关于具有正截面曲率的爱因斯坦四流形刚度的注记

在本文中，我们首先证明了带有爱因斯坦度量的四维流形的拓扑障碍。更准确地说，假设 (M, g) 是一个封闭的爱因斯坦四流形，其中 $$Ric=\rho g$$。用 K 表示 M 的截面曲率。如果 $$K\ge \delta \ge \frac{2\rho -\sqrt{5}\left|\rho \right|}{6}$$（或 $$K \le \delta \le \frac{2\rho +\sqrt{5}}{6}$$) 对于一些常数 $$\delta $$，然后是欧拉特征 $$\chi $$ 和签名 $$ \tau $$ of M 满足 $$\begin{aligned}\chi \ge \left( \dfrac{3}{8\left( 1-3\delta /\rho \right) ^2} +\dfrac{3 }{2}\right) \left|\tau \right|.\end{aligned}$$我们的第二个结果是具有正截面曲率的封闭定向爱因斯坦四流形的刚性定理。假设 $$\lambda _1$$ 是具有 $$Ric = g$$ 的有向封闭爱因斯坦四流形 (M, g) 的拉普拉斯算子的第一个特征值。