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.Cheeger, Ebin. ''Comparison Theorems in Riemannian Geometry''. North-Holland Publishing Company, 1975, pp. 17-18.

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 $\mathbb^n$, there are no conjugate points.
* On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.

See also

* Cut locus
* Jacobi field

References

{{DEFAULTSORT:Conjugate Points
Riemannian geometry