Conjugate points: Difference between revisions

Revision as of 17:24, 10 September 2007

Revision as of 17:27, 10 September 2007

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

geodesics 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]].

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

==Formal definition==

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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.

==Examples==

* On the sphere<math>S^n</math>, any two points are conjugate.

* On <math>\mathbb{R}^n</math>, there are no conjugate points.