the Bonnet-Myers theorem
/ boh-NAY MY-erz /
If a complete surface curves at least as much as a sphere everywhere, it cannot run off to infinity — it has to close up and be small, just like the sphere it imitates. The Bonnet-Myers theorem is the high-dimensional, Ricci-curvature version of that intuition. A uniform positive lower bound on curvature forces the whole manifold to have a bounded diameter, hence to be compact, and even constrains its topology by making its fundamental group finite. Positive curvature, applied everywhere, literally squeezes the space shut.
Precisely: let M be a complete connected Riemannian n-manifold whose Ricci curvature satisfies Ric >= (n-1) k g for some constant k > 0 (equivalently, Ric is bounded below by that of the sphere of radius 1/sqrt(k)). Then the diameter of M is at most pi / sqrt(k), M is compact, and its fundamental group pi_1(M) is finite. The mechanism is a second-variation argument: along any geodesic longer than pi/sqrt(k), the index form admits a negative direction built from a sine-shaped variation field, so the geodesic cannot be minimizing — but in a complete manifold every two points are joined by a minimizing geodesic (Hopf-Rinow), so no two points can be that far apart. The finiteness of pi_1 follows by applying the same diameter bound upstairs on the universal cover, which inherits the Ricci bound and so is also compact, forcing the deck group to be finite.
Bonnet-Myers is the prototypical 'curvature controls topology' theorem and the positive-curvature counterpart to Cartan-Hadamard. Three honest caveats. First, the hypothesis is on Ricci curvature, which is weaker than sectional curvature — this is the strength of the theorem (Ricci is only an average), and it cannot be relaxed to scalar curvature, which is weaker still and does not bound the diameter. Second, the diameter bound pi/sqrt(k) is sharp, attained exactly by the round sphere of radius 1/sqrt(k); manifolds achieving it are rigid (Cheng's maximal-diameter theorem). Third, Ric strictly positive but not bounded below by a positive constant is not enough — the paraboloid has positive curvature everywhere yet is non-compact with infinite diameter, because its curvature decays to zero.
The round sphere S^n of radius r has constant sectional curvature 1/r^2, hence Ric = (n-1)/r^2 g, so k = 1/r^2 and the theorem predicts diameter <= pi/sqrt(k) = pi r. This is exactly the true diameter (antipodal distance pi r) — the bound is achieved. Now consider real projective space RP^n = S^n / {±1}: it satisfies the same Ricci bound, so it too is compact with finite (indeed order-2) fundamental group pi_1 = Z/2 — a concrete instance of the topological conclusion. By contrast, the flat plane (Ric = 0) and hyperbolic plane (Ric < 0) violate the hypothesis and are non-compact with infinite diameter, as the theorem allows.
Round sphere of radius r: Ricci bound k = 1/r^2 gives the sharp diameter pi r; RP^n shows the finite-pi_1 conclusion.
The bound is on Ricci, not scalar, curvature. Positive scalar curvature alone does NOT imply compactness — there are complete non-compact manifolds of positive scalar curvature. Never weaken the hypothesis to scalar.