Conjugate points: Difference between revisions

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In [[differential geometry]], '''conjugate points''' or '''focal points'''<ref>Bishop, Richard L. and Crittenden, Richard J. ''Geometry of Manifolds''. AMS Chelsea Publishing, 2001, pp.224-225.</ref> 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. All geodesics are ''locally'' length-minimizing, but, fornot globally. For example, on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) ''globally'' length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.<ref>Cheeger, Ebin. ''Comparison Theorems in Riemannian Geometry''. North-Holland Publishing Company, 1975, pp. 17-18.</ref>

==Definition==