In Riemannian geometry

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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, 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

Riemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

 with sectional curvature

Sectional curvature

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K depends on a two-dimensional plane σp in the tangent space at p...

 taking values in the interval 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 .) 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 . The standard counterexample is complex projective space

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 complex projective space label the complex lines...

 with the Fubini-Study metric

Fubini-Study metric

In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study....

; 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

Riemannian symmetric space

In differential geometry, representation theory and harmonic analysis, a symmetric space is a smooth manifold whose group of symmetries contains an inversion symmetry about every point. There are two ways to formulate the inversion symmetry, via Riemannian geometry or via Lie theory...

.

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 structure

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....

s that are not diffeomorphic. (For more information, see the article on exotic spheres

Exotic sphere

In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...

.) However, in 2007 Simon Brendle and Richard Schoen

Richard Schoen

Richard Melvin Schoen is an American mathematician. Born in Fort Recovery, Ohio, he received his PhD in 1977 from Stanford University where he is currently the Anne T. and Robert M. Bass Professor of Humanities and Sciences...

 of Stanford University utilized Ricci flow

Ricci flow

In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

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

Hopf conjectured that a simply connected manifold with pinched sectional curvature is a sphere. In 1951, Harry Rauch showed that a simply connected manifolds with curvature in [3/4,1] is homeomorphic to a sphere. In 1960, Berger and Klingenberg proved the topological version of the sphere with the optimal pinching constant.