Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. If ''M'' is a complete, simply-connected, ''n''-dimensional Riemannian manifold with

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

taking values in the interval

(1,4]

then ''M'' is homeomorphic to the ''n''-sphere. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in

(1,4]

.) Another way of stating the result is that if ''M'' is not homeomorphic to the sphere, then it is impossible to put a metric on ''M'' with quarter-pinched curvature. Note that the conclusion is false if the sectional curvatures are allowed to take values in the ''closed'' interval

,4 /math>.  The standard counterexample is

complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...

with the Fubini–Study metric; sectional curvatures of this metric take on values between 1 and 4, with endpoints included. Other counterexamples may be found among the rank one symmetric spaces.

Differentiable sphere theorem

The original proof of the sphere theorem did not conclude that ''M'' was necessarily

diffeomorphic In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two man ...

to the ''n''-sphere. This complication is because spheres in higher dimensions admit

smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. Definition A smooth structure on a manifold M is ...

s that are not diffeomorphic. (For more information, see the article on exotic spheres.) However, in 2007 Simon Brendle and

Richard Schoen Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984. Career Born in Celina, Ohio, and a ...

utilized

Ricci flow In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...

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.

History of the sphere theorem

Heinz Hopf Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry. Early life and education Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Elizabeth ( ...

conjectured that a simply connected manifold with pinched sectional curvature is a sphere. In 1951, Harry Rauch showed that a simply connected manifold with curvature in /4,1is homeomorphic to a sphere. In 1960, Marcel Berger and Wilhelm Klingenberg proved the topological version of the sphere theorem with the optimal pinching constant.

References

* * *{{cite journal , last1=Brendle , first1=Simon , last2=Schoen , first2=Richard , title=Curvature, Sphere Theorems, and the Ricci Flow , year=2011 , journal=

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...

, volume=48 , pages=1–32 , doi=10.1090/s0273-0979-2010-01312-4 , issue=1 , mr=2738904, arxiv=1001.2278 Riemannian geometry Theorems in topology Theorems in Riemannian geometry