Sphere theorem

If M is a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval

Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval

However, in 2007 Simon Brendle and Richard Schoen utilized Ricci flow to prove that with the above hypotheses, M is necessarily diffeomorphic to the n-sphere with its standard smooth structure.

[2] In 1960, both Marcel Berger and Wilhelm Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant.

[3][4] Berger discusses the history of the theorem in his book A Panoramic View of Riemannian Geometry, originally published in 2003.