Nonpositively curved implies conjugate-free
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.
This observation is crucial to the proof of the Cartan-Hadamard theorem.