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Conjugate points

释义

Definition
Examples
See also
References

In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. Geodesics are locally length-minimizing, but, for example, on a sphere, any geodesic from the north-pole fail to be length-minimizing if it passes through the south-pole.^[1]

Definition

Suppose p and q are points on a Riemannian manifold, and is a geodesic that connects p and q. Then p and q are conjugate points along if there exists a non-zero Jacobi field along 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 fields). Therefore, if p and q are conjugate along , one can construct a family of geodesics that start at p and almost end at q. In particular,

if is the family of geodesics whose derivative in s at generates the Jacobi field J, then the end point

of the variation, namely , 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.

Examples

On the sphere , antipodal points are conjugate.
On , there are no conjugate points.
On Riemannian manifolds with non-positive sectional curvature, there are no conjugate points.

References

1. ^Cheeger, Ebin. Comparison Theorems in Riemannian Geometry. North-Holland Publishing Company, 1975, pp. 17-18.

1 : Riemannian geometry

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Definition

Examples

See also

References