# Nonpositively curved implies conjugate-free

## Statement

### Verbal statement

If the sectional curvature of a Riemannian manifold is everywhere nonpositive, then the Riemannian manifold is conjugate-free, viz it does not contain a pair of conjugate points.

## Proof

Let be the Riemannian manifold, a geodesic in and a Jacobi field about that vanishes at the endpoints. We must show that is everywhere zero.

Consider the Jacobi equation:

Taking the inner product of both these with , the second term becomes the sectional curvature of the tangent plane spanned by and , which is negative. Hence we conclude that:

This tells us that is nondecreasing, and hence, that is nondecreasing. Thus cannot be zero at both endpoints unless it is identically zero.

## Application

This observation is crucial to the proof of the Cartan-Hadamard theorem.