Sphere Theorem - Differentiable Sphere Theorem

Differentiable Sphere Theorem

The original proof of the sphere theorem did not conclude that M was necessarily diffeomorphic to the n-sphere. This complication is because spheres in higher dimensions admit smooth structures that are not diffeomorphic. (For more information, see the article on exotic spheres.) 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. Moreover, the proof of Brendle and Schoen only uses the weaker assumption of pointwise rather than global pinching. This result is known as the Differentiable Sphere Theorem.