Conjugate points

In differential geometry, conjugate points or focal 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. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are locally length-minimizing, but not globally. 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. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.

Definition

Suppose p and q are points on a Riemannian manifold, and $\gamma$ is a geodesic that connects path>XDXD that vanishes at p and q.

X x d d xjoining them.

Examples

• On the sphere $S^{2}$, antipodal points are conjugate.
• On $\mathbb {R} ^{n}$, there are no conjugate points.
• On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.

See also

• Cut locus
• Jacobi field