Four-dimensional Einstein manifolds with sectional curvature bounded from above

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=\lambda g$ for some positive constant $\lambda$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A basic problem is to classify four-dimensional Einstein manifolds with positive or nonnegative curvature and $Ric=1$. In this paper, we firstly show that if the sectional curvature satisfies $K\le M_1= \frac{\sqrt{3}}2\approx 0.866025$, then the sectional curvature will be nonnegative. Next, we prove a family of rigidity theorems of Einstein four-manifolds with nonnegative sectional curvature, and satisfies $K_{ik}+sK_{ij}\ge K_s = \frac{1 + \sqrt{2}}3 - \frac{\sqrt{4+2\sqrt{2}}}4 + \frac{2-\sqrt{2}}6 s$ for every orthonormal basis $\{e_i\}$ with $K_{ik}\ge K_{ij}$, where $s$ is any nonnegative constant. Indeed, we will show that these Einstein manifolds must be isometric either $S^4$, $RP^4$ or $CP^2$ with standard metrics. As a corollary, we give a rigidity result of Einstein four-manifolds with $Ric=1$, and the sectional curvature satisfies $K \le M_2 = \frac {2-\sqrt{2}}6 + \frac{\sqrt{4+2\sqrt{2}}}4 \approx 0.750912$.