Conjugate points: Difference between revisions

Revision as of 16:52, 14 September 2007

Revision as of 04:54, 16 September 2007

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In [[differential geometry]], '''conjugate points''' are points that can be connected with geodesics[[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]].

==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 is orthogonal to ''c'' and that vanishes at ''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.

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

==Examples==