Cartan–Hadamard conjecture

The Cartan–Hadamard conjecture is a statement in differential geometry regarding the behavior of geodesics on Riemannian manifolds. Specifically, it deals with the topology of simply connected, complete Riemannian manifolds with non-positive sectional curvature. The conjecture asserts that if a Riemannian manifold \( M \) is simply connected and complete, and if its sectional curvature is non-positive throughout, then the manifold is contractible.