Einstein Four-Manifolds with Sectional Curvature Bounded from Above

Given an Einstein structure g ¯ with positive scalar curvature on a four-dimensional Riemannian manifold, that is R ¯ i c = λ g ¯ for some positive constant λ , 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 ¯ i c = 1 . In this paper, we firstly show that if the sectional curvature of g ¯ satisfies K ¯ ≤ 3 2 ≈ 0.866025 , then g ¯ 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 + s K ¯ ij ≥ K s for every orthonormal basis { e i } with K ¯ ik ≥ K ¯ ij , where s is some nonnegative constant. More precisely, we show that such Einstein manifolds must be isometric to either S 4 , or R P 4 , or C P 2 (with standard metrics respectively). As a corollary, we obtain a rigidity result of Einstein four-manifolds with R ¯ i c = 1 and the sectional curvature satisfying the upper bound K ¯ ≤ M 2 ≈ 0.750912 .