# 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.

### Sectional curvature

**curvaturecurvature tensorsmanifolds with constant sectional curvature**

### Index of radio propagation articles

**List of radio propagation topics**

conjugate 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.

### Conjugation

**conjugateConjugation (disambiguation)**

### 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.

### List of mathematical properties of points

**Points in mathematics**

### Steiner ellipse

**Steiner circumellipse**

### 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.