# 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.[1]

## 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.

## See also

## References

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