Rigidity of surfaces with no conjugate points
E. Hopf proved that any complete Riemannian metric with no conjugate points on the torus \(T^ 2\) is flat. Our paper extends Hopf's argument to obtain sufficient conditions for metrics with no conjugate points on a cylinder or the plane to be flat. We show that a cylinder with no conjugate points, curvature bounded from below whose ends do not open out, must be flat. In the case of the plane, we show that a metric with no conjugate points and curvature bounded from below that satisfies a version of Euclid's parallel axiom must be flat.
scientific article; zbMATH DE number 4192560