The Cartan–Hadamard theorem in conventional Riemannian geometry asserts that the universal covering space of a connected complete Riemannian manifold of non-positive sectional curvature is diffeomorphic to Rn. In fact, for complete manifolds on non-positive curvature the exponential map based at any point of the manifold is a covering map. The theorem holds also for Hilbert manifolds in the sense that the exponential map of a non-positively curved geodesically complete connected manifold is a covering map. Completeness here is understood in the sense that the exponential map is defined on the whole tangent space of a point.
Metric geometry
In metric geometry, the Cartan–Hadamard theorem is the statement that the universal cover of a connected non-positively curved complete metric spaceX is a Hadamard space. In particular, if X is simply connected then it is a geodesic space in the sense that any two points are connected by a unique minimizing geodesic, and hence contractible. A metric spaceX is said to be non-positively curved if every point p has a neighborhood U in which any two points are joined by a geodesic, and for any point z in U and constant speed geodesic γ in U, one has This inequality may be usefully thought of in terms of a geodesic triangle Δ = zγγ. The left-hand side is the square distance from the vertex z to the midpoint of the opposite side. The right-hand side represents the square distance from the vertex to the midpoint of the opposite side in a Euclidean triangle having the same side lengths as Δ. This condition, called the CAT condition is an abstract form of Toponogov's triangle comparison theorem.
The assumption of non-positive curvature can be weakened, although with a correspondingly weaker conclusion. Call a metric space X convex if, for any two constant speed minimizing geodesics a and b, the function is a convex function of t. A metric space is then locally convex if every point has a neighborhood that is convex in this sense. The Cartan–Hadamard theorem for locally convex spaces states:
If X is a locally convex complete connected metric space, then the universal cover of X is a convex geodesic space with respect to the induced length metricd.
In particular, the universal covering of such a space is contractible. The convexity of the distance function along a pair of geodesics is a well-known consequence of non-positive curvature of a metric space, but it is not equivalent.
Significance
The Cartan–Hadamard theorem provides an example of a local-to-global correspondence in Riemannian and metric geometry: namely, a local condition and a global condition together imply a strong global property ; or in the Riemannian case, diffeomorphism with Rn. The metric form of the theorem demonstrates that a non-positively curved polyhedral cell complex is aspherical. This fact is of crucial importance for modern geometric group theory.
