Conjugate points: Difference between revisions

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{{Short description|In differential geometry}}

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]].

{{More citations needed|date=March 2019}}

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>

==Definition==

==Formal definition==

Suppose ''p'' and ''q'' are points on a [[pseudo-Riemannian manifold]], and ''c''<math>\gamma</math> is a [[geodesic]] that connects ''p'' and ''q''. Then ''p'' and ''q'' are '''conjugate points along <math>\gamma</math>''' if there isexists a non-zero [[Jacobi field]] onalong ''c'' that is orthogonal to ''c'' and<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,

Let us recall that any Jacobi field can be written as the

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

derivative of a geodesic variation. Therefore one can construct a family of geodesics that connect conjugate points.

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]] pointss 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.

==See also==

* [[Cut locus (Riemannian manifold)|Cut locus]]

* [[Jacobi field]]

* {{slink|Aurora#Conjugate auroras}}

* {{slink|Operation Argus#USS Albemarle}}

==References==

{{Reflist}}

{{DEFAULTSORT:Conjugate Points}}

[[Category:Riemannian geometry]]