the sphere theorem
How round does a space have to be before it is forced to be a sphere? If a manifold's curvature is strictly positive and never varies by more than a factor of four from point to point and plane to plane, the answer is: round enough — it must be a sphere, topologically. The sphere theorem is the landmark result that turns a purely local, numerical pinching condition on curvature into a global topological identification, one of the cleanest 'curvature determines shape' statements in all of geometry.
Precisely (the classical topological version): let M be a complete, simply connected Riemannian manifold whose sectional curvature is strictly 1/4-pinched, meaning 1/4 < K <= 1 at every point and for every tangent 2-plane (after scaling the maximum to 1). Then M is homeomorphic to the sphere S^n. The pinching is sharp: at exactly 1/4 the conclusion can fail, because the complex projective space CP^m with its Fubini-Study metric has curvature pinched between 1/4 and 1 inclusive and is not a sphere. The proof (Berger, Klingenberg, building on Rauch) combines Toponogov triangle comparison with a lower bound on the injectivity radius to build an explicit homeomorphism by gluing two exponential-image balls. The much harder differentiable sphere theorem (Brendle-Schoen, 2007, via Ricci flow) strengthens 'homeomorphic' to 'diffeomorphic' under the same pinching, even allowing pointwise rather than global pinching.
The sphere theorem is the crown of comparison geometry: it shows that a quantitative curvature bound, with no topological input beyond simple connectivity, recovers the entire global shape. Honest caveats that matter. First, 1/4 is strict in the topological version — CP^m is the borderline example showing you cannot include the value 1/4 and still force a sphere. Second, the original theorem gave only a homeomorphism, and the gap to diffeomorphism mattered enormously because exotic spheres exist (a homotopy sphere need not be diffeomorphic to the standard one); closing that gap took fifty years and Ricci flow. Third, the simply connected hypothesis is essential — a space form like a spherical space form S^n/Gamma is pinched but not simply connected and not a sphere.
Any compact, simply connected manifold whose curvature lives strictly between 1/4 and 1 — for instance, a slightly squashed but not-too-eccentric round sphere — is homeomorphic to S^n; Brendle-Schoen upgrade this to diffeomorphic. The sharpness shows in CP^2 with the Fubini-Study metric: its sectional curvatures fill exactly the closed interval [1/4, 1], so it is 1/4-pinched only in the non-strict sense, and indeed CP^2 is NOT a sphere (it has nontrivial H^2 and the wrong Euler characteristic). This single borderline example is why the inequality K > 1/4 must be strict.
Strict 1/4 < K <= 1 forces a (homotopy) sphere; CP^m sits exactly at the 1/4 boundary and is not a sphere.
Homeomorphic is not diffeomorphic. The classical theorem gave only a homeomorphism — crucial because exotic spheres exist — and upgrading to diffeomorphic required Brendle-Schoen's Ricci-flow proof. Never state the differentiable version as if it were elementary.