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 poin ...

, 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 Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...

, ''n''-dimensional

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

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 In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

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

; 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 Inverse function, invertible Function (mathematics), function that maps one differentiable manifold to another such that both the function and its inverse function ...

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 sphere In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of al ...

s.) However, in 2007

Simon Brendle Simon Brendle (born June 1981) is a German mathematician working in differential geometry and nonlinear partial differential equations. He received his Dr. rer. nat. from Tübingen University under the supervision of Gerhard Huisken (2001). He ...

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 1 ...

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 analo ...

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 Harry Ernest Rauch (November 9, 1925 – June 18, 1979) was an American mathematician, who worked on complex analysis and differential geometry. He was born in Trenton, New Jersey, and died in White Plains, New York. Rauch earned his PhD i ...

showed that a simply connected manifold with curvature in /4,1is homeomorphic to a sphere. In 1960,

Marcel Berger Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Las ...

and

Wilhelm Klingenberg Wilhelm Paul Albert Klingenberg (28 January 1924 – 14 October 2010) was a German mathematician who worked on differential geometry and in particular on closed geodesics. Life Klingenberg was born in 1924 as the son of a Protestant minister. I ...

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