The Sphere Theorem

The sphere theorem is a landmark result in comparison geometry, showing that sufficiently positively curved manifolds must be topologically spheres. Its proof combines Toponogov comparison with delicate geometric arguments.

Classical Sphere Theorem

Theorem9.6Classical sphere theorem

Let $(M^n, g)$ be a complete, simply connected Riemannian manifold with sectional curvature satisfying

\frac{1}{4} < K \leq 1.

Then $M$ is homeomorphic to the sphere $S^n$ .

The pinching constant $1/4$ is optimal: the complex projective spaces $\mathbb{C}P^n$ have $1/4 \leq K \leq 1$ but are not homeomorphic to spheres for $n \geq 2$ .

Proof Sketch

Proof

Step 1: Diameter bound. By Bonnet-Myers, the diameter satisfies $\mathrm{diam}(M) \leq \pi$ (since $K \geq 1/4$ gives $\mathrm{Ric} \geq (n-1)/4$ , but actually $K > 1/4$ gives $\mathrm{diam} < 2\pi$ by the Klingenberg injectivity radius estimate).

Step 2: Injectivity radius bound. Klingenberg showed that for even-dimensional manifolds with $1/4 < K \leq 1$ , the injectivity radius satisfies $\mathrm{inj}(M) \geq \pi$ . For odd-dimensional manifolds, the same bound holds under the stronger pinching $1/4 < K$ .

Step 3: Two-ball decomposition. Choose $p, q \in M$ with $d(p,q) = \mathrm{diam}(M)$ . The geodesic balls $B_{\mathrm{inj}}(p)$ and $B_{\mathrm{inj}}(q)$ are diffeomorphic to open disks. Using Toponogov comparison with the $1/4$ -pinching, one shows that every point of $M$ lies in $B_\pi(p) \cup B_\pi(q)$ .

Step 4: Topological argument. The balls $B_\pi(p)$ and $B_\pi(q)$ are topological disks (since $\exp_p$ and $\exp_q$ are diffeomorphisms on balls of radius $\pi$ by the injectivity radius bound). Their union covers $M$ , and the intersection is connected (by a Toponogov argument). By a topological lemma, a manifold covered by two open disks with connected intersection is homeomorphic to $S^n$ . $\blacksquare$

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The Differentiable Sphere Theorem

Theorem9.7Differentiable sphere theorem (Brendle-Schoen 2009)

If $(M^n, g)$ is a complete, simply connected Riemannian manifold with pointwise $1/4$ -pinched sectional curvature (i.e., at each point, the ratio of the maximum and minimum sectional curvature is less than 4), then $M$ is diffeomorphic to $S^n$ .

RemarkRicci flow proof

Brendle and Schoen proved the differentiable sphere theorem using Hamilton's Ricci flow. They showed that the $1/4$ -pinching condition is preserved under Ricci flow, and the flow converges to a constant curvature metric. Since the sphere is the only simply connected manifold admitting a round metric, $M$ must be diffeomorphic to $S^n$ . This resolved a longstanding conjecture.

ExampleExotic spheres

In dimension 7, there exist exotic smooth structures on $S^7$ (Milnor). The differentiable sphere theorem implies that these exotic spheres cannot carry any $1/4$ -pinched metric, showing a deep connection between curvature and smooth structure.

RemarkOptimal pinching

The constant $1/4$ cannot be improved to $\leq 1/4$ because $\mathbb{C}P^n$ satisfies $1/4 \leq K \leq 1$ but is not a sphere. For manifolds that are not simply connected, the situation is more complex and involves the fundamental group.