# Conjugate-free Riemannian manifold

*This article defines a property that makes sense for a Riemannian metric over a differential manifold*

alt text="BEWARE"*This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.*

## Contents

## History

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.

## Definition

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

### Stronger properties

- Nonpositively curved manifold:
*For full proof, refer: Nonpositively curved implies conjugate-free*

### Weaker properties

- Nonpositively curved manifold if we assume compactness