Conjugate points: Difference between revisions

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Revision as of 04:23, 14 January 2022

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==Definition==

Suppose ''p'' and ''q'' are points on a [[Riemannian manifold]], and <math>\gamma</math> is a [[geodesic]] that connects ''p''ath 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>XDXD 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 that start at ''p'' and ''almost'' end at ''q''. In particular,

X x d d xjoining them.

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==