Conjugate points

In differential geometry, conjugate points are points that can be connected with geodesics in more than one way. For example, on a sphere, the north-pole and south-pole are connected by any meridian.

Formal definition

Suppose p and q are points on a Riemannian manifold, and c is a geodesic that connects p and q. Then p and q are conjugate points if there is a non-zero Jacobi field on c that vanishes on p and q.

Let us recall that any Jacobi field can be written as the derivative of a geodesic variation. Therefore one can construct a family of geodesics that connect conjugate points.

Examples

• On the sphere$S^{n}$, any two points are conjugate.
• On $\mathbb {R} ^{n}$, there are no conjugate points.