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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
==Definition==
Suppose ''p'' and ''q'' are points on a [[pseudo-Riemannian manifold]], and <math>\gamma</math> is a [[geodesic]] that connects ''p'' and ''q''. Then ''p'' and ''q'' are '''conjugate points along <math>\gamma</math>''' if there exists a non-zero [[Jacobi field]] along <math>\gamma</math> that vanishes at ''p'' and ''q''.
Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on [[Jacobi field]]s). Therefore, if ''p'' and ''q'' are conjugate along <math>\gamma</math>, one can construct a family of geodesics that start at ''p'' and ''almost'' end at ''q''. In particular,
if <math>\gamma_s(t)</math> is the family of geodesics whose derivative in ''s'' at <math>s=0</math> generates the Jacobi field ''J'', then the end point
of the variation, namely <math>\gamma_s(1)</math>, is the point ''q'' only up to first order in ''s''. Therefore, if two points are conjugate, it is not necessary that there exist two distinct geodesics joining them.
For Riemannian geometries, beyond a conjugate point, the geodesic is no longer locally the shortest path between points, as there are nearby paths that are shorter. This is analogous to the Earth's surface, where the geodesic between two points along a great circle is the shortest route only up to the antipodal point; beyond that, there are shorter paths.
Beyond a conjugate point, a geodesic in Lorentzian geometry may not be maximizing proper time (for timelike geodesics), and the geodesic may enter a region where it is no longer unique or well-defined. For null geodesics, points beyond the conjugate point are now timelike separated.
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==
* On the [[Sphere#Dimensionality|sphere <math>S^2</math>]], [[antipodal point]]s are conjugate.
* On the [[real coordinate space]] <math>\mathbb{R}^n</math>, there are no conjugate points.
* On Riemannian manifolds with non-positive [[sectional curvature]], there are no conjugate points.
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* [[Cut locus (Riemannian manifold)|Cut locus]]
* [[Jacobi field]]
* {{slink|Aurora#Conjugate auroras}}
* {{slink|Operation Argus#USS Albemarle}}
==References==
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