In [[differential geometry]], '''conjugate points''' are, roughly, points that can be connectedjoined withby [[geodesic]]s in more than one way. For example, on a [[Spherical geometry|sphere]], the north-pole and south-pole are connected by any [[Meridian (geography)|meridian]].
==Definition==
==Formal definition==
Suppose ''p'' and ''q'' are points on a [[Riemannian manifold]], and ''c''<math>\gamma</math> is a [[geodesic]] that connects ''p'' and ''q''. Then ''p'' and ''q'' are '''conjugate points along <math>\gamma</math>''' if there isexists a non-zero [[Jacobi field]] onalong ''c'' that is orthogonal to ''c'' and<math>\gamma</math> that vanishes at ''p'' and ''q''.
Let us recallRecall 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 thatwhich connectstart conjugateat points''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.
