Manifolds with 1/4-pinched Curvature Are Space Forms Simon Brendle and Richard Schoen

One of the basic problems of Riemannian geometry is the classification of manifolds of positive sectional curvature. The known examples include the spherical space forms which carry constant curvature metrics and the rank 1 symmetric spaces whose canonical metrics have sectional curvatures at each point varying between 1 and 4. In 1951 H.E. Rauch [18] introduced the notion of curvature pinching for Riemannian manifolds and posed the question of whether a simply connected compact manifold Mn whose sectional curvatures all lie in the interval (1, 4] is necessarily diffeomorphic to the sphere Sn. (In this paper pinching will always mean strict pinching unless otherwise specified.) This question is known as the Differentiable Sphere Theorem, and the purpose of this paper is to prove this and a more general result which we describe. We will say that a manifold has (pointwise) 1/4-pinched curvature if all sectional curvatures are positive and for any point p ∈ M the ratio of the maximum to the minimum sectional curvature at that point is less than 4; more precisely, for any pair of two planes P1 and P2 contained in the tangent space TpM we have 0 < K(P1) < 4K(P2). Our main result is the following.