Conjugate points
In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian.
Definition
Suppose p and q are points on a Riemannian manifold, and 
is a geodesic that connects p and q. Then p and q are conjugate points along if there exists a non-zero Jacobi field along that vanishes at p and q.
Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on Jacobi fields). Therefore, if p and q are conjugate along , one can construct a family of geodesics that start at p and almost end at q. In particular,
if 
is the family of geodesics whose derivative in s at 
generates the Jacobi field J, then the end point
of the variation, namely 
, is the point q only up to first order in s. Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them.
Examples
- On the sphere
, antipodal points are conjugate. - On
, there are no conjugate points. - On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.