Conjugate points: Difference between revisions

Revision as of 00:31, 7 October 2005

Revision as of 17:15, 10 September 2007

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In differential geometry, conjugate points are points that can be connected with

#REDIRECT[[Glossary of Riemannian and metric geometry]]

geodesics in more than one way. For example, on a sphere, the north-pole and south-pole

are connected by [[Meridian (geography)|meridians]].

==Formal definition==

Suppose ''p'' and ''q'' are points on a [[Riemannian manifold]], and ''c'' is a

[[geodesic]] that connects ''p'' and ''q''. Then ''p'' and ''q''

are '''conjugate points''' if there is a non-zero [[Jacobi field]] on ''c'' that vanishes

on ''p'' and ''q''.

Let us recall that any Jacobi field can be written as the derivative of a geodesic

variation. Therefore one can construct a family of geodesics that connect conjugate points.