Conjugate points: Difference between revisions

Revision as of 16:45, 18 January 2020

Revision as of 16:49, 18 January 2020

Line 1:

{{Refimprove|date=March 2019}}

In [[differential geometry]], '''conjugate points''' are, roughly, points that can almost be joined by a 1-parameter family of [[geodesic]]s. For example, on a [[Spherical geometry|sphere]], the north-pole and south-pole are connected by any [[Meridian (geography)|meridian]]. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. Geodesics are locally length-minimizing, but, for example, on a sphere, any geodesic frompassing through the north-pole failscan be extended to bereach globallythe lengthsouth-minimizingpole, becauseand theyhence allare passnot through(uniquely) theglobally south-polelength minimizing. <ref>Cheeger, Ebin. ''Comparison Theorems in Riemannian Geometry''. North-Holland Publishing Company, 1975, pp. 17-18.</ref>

==Definition==