For complete Riemannian manifolds M the condition to have no conjugate points is equivalent to the following strong minimality property of all of its geodesics: If c: R → M is a geodesic and if s < t ∈ R,then c|[s,t] is the (up to parametrization) unique shortest curve connecting c(s) and c(t) in the homotopy class of c|[s,t].Non-positivity of sectional curvature implies this condition.There is a long list of rigidity results showing that under appropriate conditions a complete Riemannian manifold without conjugate points is of a very special type.The first and prototypical result in this list is by E.Hopf (1948) and states that on the 2-torus only the flat metrics have no conjugate points.