We prove that the total positive Gauss-Kronecker curvature of any closed hypersurface embedded in a complete simply connected manifold of nonpositive curvature M n , n ≥ 2, is bounded below by the volume of the unit sphere in Euclidean space R n . This yields the optimal isoperimetric inequality for bounded regions of finite perimeter in M , via Kleiner's variational approach, and thus settles the Cartan-Hadamard conjecture. The proof employs a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for smooth approximation of the signed distance function. Immediate applications include sharp extensions of the