the Cartan-Hadamard theorem
/ kar-TAHN ah-dah-MAR /
Negative curvature does the opposite of a sphere: instead of pulling geodesics back together, it pushes them apart, so rays from a point spread forever and never refocus. The Cartan-Hadamard theorem turns that picture into a complete topological verdict. A complete, simply connected manifold with curvature that never turns positive is, no matter its dimension, just a smoothly bent copy of ordinary Euclidean space — topologically as simple as a space can be. There are no holes, no handles, no surprises; it is contractible.
Precisely: let M be a complete, simply connected Riemannian n-manifold with sectional curvature K <= 0 everywhere. Then for every point p the exponential map exp_p : T_p M -> M is a diffeomorphism. Consequently M is diffeomorphic to R^n, hence contractible, and any two points are joined by a unique minimizing geodesic. The proof has two clean steps. First, nonpositive curvature means there are no conjugate points: in the Jacobi equation J'' + R(J, gamma') gamma' = 0 the curvature term has the 'wrong' sign, so a Jacobi field vanishing at 0 has |J| convex and growing (like sinh, not sin), never returning to zero — therefore exp_p is a local diffeomorphism everywhere (a local immersion that is nonsingular). Second, completeness plus simple connectedness upgrades this local diffeomorphism to a global one (a covering map onto a simply connected target is a diffeomorphism).
Cartan-Hadamard is the negative-curvature bookend to Bonnet-Myers and the gateway to the geometry of nonpositive curvature. It says, for example, that hyperbolic space H^n and any complete nonpositively curved Lie group are all diffeomorphic to R^n. Crucial honest caveats. First, simple connectedness is essential — a compact hyperbolic surface of genus 2 has K = -1 and is complete, but it is not simply connected and certainly not R^2; the theorem applies to its universal cover, which IS the hyperbolic plane. Second, the conclusion is a diffeomorphism, not an isometry: the manifold is bent, with genuine negative curvature, even though it is topologically Euclidean. Third, the smooth statement requires the manifold be smooth; the metric-space generalization to CAT(0) spaces lives in metric geometry.
Hyperbolic space H^n (constant K = -1) is complete and simply connected, so Cartan-Hadamard says exp_p is a global diffeomorphism — indeed in the disk or upper-half-space model H^n is visibly diffeomorphic to R^n, with unique geodesics between any two points. Contrast a genus-2 surface Sigma with a hyperbolic metric: K = -1 and complete, but pi_1(Sigma) is a nontrivial surface group, so the theorem does NOT make Sigma into R^2. Instead it makes the universal cover of Sigma into H^2, on which the surface group acts by isometries — the standard way hyperbolic surfaces are built.
H^n is diffeomorphic to R^n; a compact hyperbolic surface is not, but its universal cover is.
Diffeomorphic to R^n does NOT mean flat or isometric to R^n — the curvature is genuinely negative. And drop simple connectedness and the conclusion collapses: only the universal cover is R^n.