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{{Short description|In differential geometry}}
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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><ref>{{cite book |last1=Hawking |first1=Stephen |last2=Ellis |first2=George |title=The large scale structure of space-time |date=1973 |publisher=Cambridge university press}}</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 not 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>
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Up to the first conjugate point, a geodesic between two points is unique. Beyond this, there can be multiple geodesics connecting two points.
Suppose we have a [[Lorentzian manifold]] with a [[congruence (general relativity)|geodesic congruence]]. Then, at a conjugate point, the [[expansion parameter]] θ in [[Raychaudhuri's equation]] becomes negative infinite in a finite amount of proper time, indicating that the geodesics are focusing to a point. This is because the cross-sectional area of the congruence becomes zero, and hence the rate of change of this area (which is what θ represents) diverges negatively.
==Examples==
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{{DEFAULTSORT:Conjugate Points}}
[[Category:Riemannian geometry]]
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