Sphere theorem

In Riemannian geometry, 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 ${\displaystyle M}$ is a complete, simply-connected, n-dimensional Riemannian manifold with sectional curvature taking values in the interval ${\displaystyle (1,4]}$ then ${\displaystyle M}$ 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 ${\displaystyle (1,4]}$.) Another way of stating the result is that if ${\displaystyle M}$ is not homeomorphic to the sphere, then it is impossible to put a metric on ${\displaystyle M}$ with quarter-pinched curvature.

Note that the conclusion is false if the sectional curvatures are allowed to take values in the closed interval ${\displaystyle [1,4]}$. The standard counterexample is complex projective space with the Fubini–Study metric; sectional curvatures of this metric take on values between ${\displaystyle 1}$ and ${\displaystyle 4}$, with endpoints included. Other counterexamples may be found among the rank one symmetric spaces.