---
tag: 7GRSF20H15
kind: statement
role: theorem
name: Sphere theorem
msc: [53C20, 53C21]
uses: [7CBT4SM83Y, 1X42JAA349, T62369RFH6, 0QT00PB5G2, 0KZZ70ZHXP]
about: [1X42JAA349]
sources:
  - url: https://en.wikipedia.org/wiki/Sphere_theorem
same_as:
  wikidata: Q2309682
  wikipedia: Sphere theorem
---

## Statement

Let $(M, g)$ be a connected, simply connected, complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \ge 2$.
Suppose that every [[sectional curvature|1X42JAA349]] $K$ of $M$ satisfies
$$\frac{1}{4} < K \le 1.$$
Then:

1. $M$ is [[homeomorphic|0QT00PB5G2]] to the [[sphere|T62369RFH6]] $S^n$ (Berger and Klingenberg).
2. $M$ is [[diffeomorphic|0KZZ70ZHXP]] to $S^n$ (Brendle and Schoen).

## Remarks

By Myers's theorem, $M$ is compact.
The idea of the topological proof: Klingenberg's estimate shows that the injectivity radius of $M$ is at least $\pi$.
With Toponogov's comparison theorem one then shows that $M$ is the union of two embedded open balls centred at two points at maximal distance.
A closed manifold that is the union of two embedded open discs is homeomorphic to a sphere.
The differentiable version uses the Ricci flow: Brendle and Schoen showed that the flow, normalized, takes the metric to one of constant curvature.

The constant $1/4$ is sharp: take the complex projective space $\C P^m$ with $m \ge 2$ and the Fubini–Study metric scaled so that $1/4 \le K \le 1$.
It is simply connected and compact but not homeomorphic to a sphere.
Simple connectedness is needed: real projective space $\R P^n$ with the round metric has $K = 1$ and is not a sphere.

## Explanations

### L1 human

Suppose a simply connected space bends positively.
If the bending in its most curved direction is less than four times that in its least curved direction, then the space has the shape of a sphere.
Not only roughly but also smoothly: it can be deformed into a round sphere.

### L2 human

The topological sphere theorem was proved by Berger and Klingenberg around 1960, after a version with a weaker constant by Rauch.
The smooth version, proved by Brendle and Schoen in 2009, answered the question whether exotic spheres can carry quarter-pinched metrics.
Their method also gives the result under a pointwise pinching condition.