Conjugate points: Difference between revisions

Revision as of 13:32, 20 April 2014

Revision as of 20:43, 12 September 2016

Line 3:

==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]] that is not identically zero along <math>\gamma</math> thatbut 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 that start at ''p'' and ''almost'' end at ''q''. In particular,