Given an Einstein structure g¯\documentclass[12pt]{minimal}
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\begin{document}$${\bar{g}}$$\end{document} with positive scalar curvature on a four-dimensional Riemannian manifold, that is R¯ic=λg¯\documentclass[12pt]{minimal}
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\begin{document}$${\bar{R}}ic=\lambda {\bar{g}}$$\end{document} for some positive constant λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, a basic problem is to classify such Einstein 4-manifolds with positive or nonnegative sectional curvature. For convenience, the Ricci curvature is always normalized to be R¯ic=1\documentclass[12pt]{minimal}
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\begin{document}$${\bar{R}}ic=1$$\end{document}. In this paper, we firstly show that if the sectional curvature of g¯\documentclass[12pt]{minimal}
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\begin{document}$${\bar{g}}$$\end{document} satisfies K¯≤32≈0.866025\documentclass[12pt]{minimal}
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\begin{document}$${\bar{K}}\le \frac{\sqrt{3}}{2}\approx 0.866025$$\end{document}, then g¯\documentclass[12pt]{minimal}
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\begin{document}$${\bar{g}}$$\end{document} must have nonnegative sectional curvature. Next, we prove a rigidity theorem of Einstein four-manifolds with nonnegative sectional curvature satisfying the additional condition that K¯ik+sK¯ij≥Ks\documentclass[12pt]{minimal}
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\begin{document}$${\bar{K}}_{ik}+s{\bar{K}}_{ij}\ge K_s$$\end{document} for every orthonormal basis {ei}\documentclass[12pt]{minimal}
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\begin{document}$$\{e_i\}$$\end{document} with K¯ik≥K¯ij\documentclass[12pt]{minimal}
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\begin{document}$${\bar{K}}_{ik}\ge {\bar{K}}_{ij}$$\end{document}, where s is some nonnegative constant. More precisely, we show that such Einstein manifolds must be isometric to either S4\documentclass[12pt]{minimal}
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\begin{document}$$S^4$$\end{document}, or RP4\documentclass[12pt]{minimal}
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\begin{document}$$RP^4$$\end{document}, or CP2\documentclass[12pt]{minimal}
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\begin{document}$$CP^2$$\end{document} (with standard metrics respectively). As a corollary, we obtain a rigidity result of Einstein four-manifolds with R¯ic=1\documentclass[12pt]{minimal}
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\begin{document}$${\bar{R}}ic=1$$\end{document} and the sectional curvature satisfying the upper bound K¯≤M2≈0.750912\documentclass[12pt]{minimal}
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\begin{document}$${\bar{K}} \le M_2 \approx 0.750912$$\end{document}.
