In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem is a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature is homeomorphic to the n-sphere with quarter-pinched curvature.

Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed. The standard counterexample is complex projective space with the Fubini–Study metric, with endpoints included. Other counterexamples may be found among the rank one symmetric spaces.

Differentiable sphere theorem

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 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.

History of the sphere theorem

Heinz Hopf conjectured that a simply connected manifold with pinched sectional curvature is a sphere. In 1951, Harry Rauch is homeomorphic to a sphere. In 1960, both Marcel Berger and Wilhelm Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant. Berger discusses the history of the theorem in his book A Panoramic View of Riemannian Geometry, originally published in 2003.

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