Conjugate Point

A conjugate point to a point on a geodesic occurs when a nonzero Jacobi field along the geodesic vanishes at both points. Equivalently, the differential of the exponential map based at the first point is singular at the corresponding tangent vector. Antipodal points on a round sphere are conjugate along a great circle. Nonpositive sectional curvature implies absence of conjugate points. The converse for a particular metric, and the existence of some nonpositively curved metric on the same manifold, are different questions.