A sphere theorem for pinching positive scalar curvature manifolds

Abstract

Let ( M n , g ) , n ⩾ 3 , be a smooth closed Riemannian manifold with positive scalar curvature R g . There exists a positive constant C = C ( M , g ) , which is a geometric invariant, such that R g ⩽ n ( n − 1 ) C . In this paper we prove that R g = n ( n − 1 ) C if and only if ( M n , g ) is isometric to the Euclidean sphere S n ( C ) with constant sectional curvature C. Also, there exists a Riemannian metric g on M n such that the scalar curvature satisfies the pinched condition, n 2 ( n − 2 ) n − 1 C < R g ⩽ n ( n − 1 ) C , if and only if M n is diffeomorphic to the standard sphere S n .