Abstract
We prove that if \(M^n(n\ge 4)\) is a compact Einstein manifold whose normalized scalar curvature and sectional curvature satisfy pinching condition \(R_0>\sigma _{n}K_{\max }\), where \(\sigma _n\in (\frac{1}{4},1)\) 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 \(\eta _n\in (\frac{1}{4},1)\) 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 \(c\ge 0\), 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.
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The authors would like to thank the referee for his valuable suggestions.
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Research supported by the NSFC, Grant No. 11071211, 10771187; the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.
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Xu, Hw., Gu, Jr. Rigidity of Einstein manifolds with positive scalar curvature. Math. Ann. 358, 169–193 (2014). https://doi.org/10.1007/s00208-013-0957-7
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DOI: https://doi.org/10.1007/s00208-013-0957-7