Conjugate points
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In differential geometry, conjugate points are points that can be connected with geodesics in more than one way. For example, on a sphere, the north-pole and south-pole are connected by any meridian.
Formal definition
Suppose p and q are points on a Riemannian manifold, and c is a geodesic that connects p and q. Then p and q are conjugate points if there is a non-zero Jacobi field on c that vanishes on p and q.
Let us recall that any Jacobi field can be written as the derivative of a geodesic variation. Therefore one can construct a family of geodesics that connect conjugate points.
Examples
- On the sphere, any two points are conjugate.
- On , there are no conjugate points.
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