Rigidity of Einstein manifolds with positive scalar curvature

We prove that if $M^n(nge 4)$ is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition $R_0>sigma _{n}K_{max }$ , where $sigma _nin (frac{1}{4},1)$ is an explicit positive constant depending only on $n$ , then $M$ must be isometric to a spherical space form. Moreover, we prove that if an $n(ge {!!4})$ -dimensional compact Einstein manifold satisfies $K_{min }ge eta _n R_0,$ where $eta _nin (frac{1}{4},1)$ is an explicit positive constant, then $M$ is locally symmetric. It should be emphasized that the pinching constant $eta _n$ is optimal when $n$ is even. We then obtain some rigidity theorems for Einstein manifolds under $(n-2)$ -th Ricci curvature and normalized scalar curvature pinching conditions. Finally we extend the theorems above to Einstein submanifolds in a Riemannian manifold, and prove that if $M$ is an $n(ge {!!4})$ -dimensional compact Einstein submanifold in the simply connected space form $F^{N}(c)$ with constant curvature $cge 0$ , and the normalized scalar curvature $R_0$ of $M$ satisfies $R_0>frac{A_n}{A_n+4n-8}(c+H^2),$ where $A_n=n^3-5n^2+8n$ , and $H$ is the mean curvature of $M$ , then $M$ is isometric to a standard $n$ -sphere.