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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
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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.
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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.
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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)
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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.
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length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.
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Recall that any Jacobi field can be written as the derivative of a geodesic variation (see the article on
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are, roughly, points that can almost be joined by a 1-parameter family of
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the antipodal point; beyond that, there are shorter paths.
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386:, then the end point of the variation, namely
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620:Comparison Theorems in Riemannian Geometry
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606:The large scale structure of space-time
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572:Operation Argus § USS Albemarle
47:adding citations to reliable sources
452:. Then, at a conjugate point, the
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532:{\displaystyle \mathbb {R} ^{n}}
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567:Aurora § Conjugate auroras
34:needs additional citations for
415:{\displaystyle \gamma _{s}(1)}
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345:{\displaystyle \gamma _{s}(t)}
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608:. Cambridge university press.
382:generates the Jacobi field
237:if there exists a non-zero
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426:only up to first order in
172:pseudo-Riemannian manifold
16:In differential geometry
458:Raychaudhuri's equation
297:{\displaystyle \gamma }
254:{\displaystyle \gamma }
229:{\displaystyle \gamma }
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639:Riemannian geometry
544:sectional curvature
454:expansion parameter
450:geodesic congruence
446:Lorentzian manifold
375:{\displaystyle s=0}
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41:Please help
36:verification
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578:References
158:Definition
99:March 2019
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557:Cut locus
395:γ
325:γ
292:γ
249:γ
224:γ
182:γ
135:geodesics
633:Category
551:See also
465:Examples
196:geodesic
162:Suppose
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143:meridian
507:On the
472:sphere
470:On the
448:with a
312:end at
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