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. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. Geodesics are locally length-minimizing, but, for example, on a sphere, any geodesic from the north-pole fail to be length-minimizing if it passes through the south-pole.

Definition

Suppose p and q are points on a Riemannian manifold, and $\gamma$ is a geodesic that connects p and q. Then p and q are conjugate points along $\gamma$ if there exists a non-zero Jacobi field along $\gamma$ 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 $\gamma$ , one can construct a family of geodesics that start at p and almost end at q. In particular, if $\gamma _{s}(t)$ is the family of geodesics whose derivative in s at $s=0$ generates the Jacobi field J, then the end point of the variation, namely $\gamma _{s}(1)$ , 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 $S^{2}$ , 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.