For example, on a sphere, the north-pole and south-pole are connected by any meridian.
Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing.
For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) globally length minimizing.
, one can construct a family of geodesics that start at p and almost end at q.
generates the Jacobi field J, then the end point of the variation, namely
For Riemannian geometries, beyond a conjugate point, the geodesic is no longer locally the shortest path between points, as there are nearby paths that are shorter.
This is analogous to the Earth's surface, where the geodesic between two points along a great circle is the shortest route only up to the antipodal point; beyond that, there are shorter paths.
Suppose we have a Lorentzian manifold with a geodesic congruence.
Then, at a conjugate point, the expansion parameter θ in Raychaudhuri's equation becomes negative infinite in a finite amount of proper time, indicating that the geodesics are focusing to a point.