Conjugate points

conjugateConjugate point
In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics.wikipedia
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Jacobi field

Jacobi equation
Then p and q are conjugate points along \gamma if there exists a non-zero Jacobi field along \gamma that vanishes at p and q.

Cut locus (Riemannian manifold)

cut locusRiemannian cut
A standard result is that either (1) there is more than one minimizing geodesic joining p to q, or (2) p and q are conjugate along some geodesic

Differential geometry

differentialdifferential geometerdifferential geometry and topology
In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics.

Geodesic

geodesicsgeodesic flowgeodesic equation
In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. Suppose p and q are points on a Riemannian manifold, and \gamma is a geodesic that connects p and q.

Spherical geometry

sphericalspherecurved surface
For example, on a sphere, the north-pole and south-pole are connected by any meridian.

Meridian (geography)

meridianmeridiansmeridian line
For example, on a sphere, the north-pole and south-pole are connected by any meridian.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
Suppose p and q are points on a Riemannian manifold, and \gamma is a geodesic that connects p and q.

Antipodal point

antipodeantipodalantipodal points

Glossary of Riemannian and metric geometry

injectivity radiusgeodesic metric spaceproper
Conjugate points two points p and q on a geodesic \gamma are called conjugate if there is a Jacobi field on \gamma which has a zero at p and q.

Geodesics on an ellipsoid

geodesicsEarth geodesicsgeodesic
conjugate to the starting point.

Envelope (mathematics)

envelopeenvelope conditionenvelope function
In Riemannian geometry, if a smooth family of geodesics through a point P in a Riemannian manifold has an envelope, then P has a conjugate point where any geodesic of the family intersects the envelope.

Rauch comparison theorem

Let be Riemannian manifolds, let and be unit speed geodesic segments such that has no conjugate points along, and let be normal Jacobi fields along \gamma and such that and.

Principle of least action

principle of stationary actionleast action principleleast action
For example, Morse showed that the number of conjugate points in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian.

Clifton–Pohl torus

The Clifton–Pohl tori are also remarkable by the fact that they are the only non-flat Lorentzian tori with no conjugate points that are known.

Finsler manifold

Finsler geometryFinsler spaceFinsler metric
If F 2 is strongly convex, geodesics γ : [0, b] → M are length-minimizing among nearby curves until the first point γ(s) conjugate to γ(0) along γ, and for t > s there always exist shorter curves from γ(0) to γ(t) near γ, as in the Riemannian case.