Conjugate-free Riemannian manifold
The study of conjugate-free Riemannian manifolds (or Riemannian manifolds without conjugate points) originated with a theorem by Hopf that the only Riemannian metric on the torus without conjugate points is the flat one. Green generalized this to say that any Riemannian metric without conjugate points, must have everywhere nonpositive sectional curvature.
A Riemannian manifold is said to be conjugate-free or without conjugate points if it does not contain any pair of conjugate points. In other words, there is no pair of points for which there is a smoothly varying family of geodesics joining them.
Relation with other properties
- Nonpositively curved manifold if we assume compactness