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In [[differential geometry]], '''conjugate points''' are, roughly, points that can almost be joined by a 1-parameter family of [[geodesic]]s. For example, on a [[Spherical geometry|sphere]], the north-pole and south-pole are connected by any [[Meridian (geography)|meridian]].

==Definition==
Suppose ''p'' and ''q'' are points on a [[Riemannian manifold]], and <math>\gamma</math> is a [[geodesic]] that connects ''p'' and ''q''. Then ''p'' and ''q'' are '''conjugate points along <math>\gamma</math>''' if there exists a non-zero [[Jacobi field]] along <math>\gamma</math> 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 field]]s). Therefore, if ''p'' and ''q'' are conjugate along <math>\gamma</math>, one can construct a family of geodesics which start at ''p'' and ''almost'' end at ''q''. In particular,
if <math>\gamma_s(t)</math> is the family of geodesics whose derivative in ''s'' at <math>s=0</math> generates the Jacobi field ''J'', then the end point
of the variation, namely <math>\gamma_s(1)</math>, 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 <math>S^2</math>, [[antipodal point]]s are conjugate.
* On <math>\mathbb{R}^n</math>, there are no conjugate points.
* On Riemannian manifolds with non-positive [[sectional curvature]], there are no conjugate points.

==See also==
* [[Cut locus (Riemannian manifold)|cut locus]]
* [[Jacobi field]]

{{DEFAULTSORT:Conjugate Points}}
[[Category:Riemannian geometry]]

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