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In differential geometry, conjugate points or focal points[1][2] 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. All geodesics are locally 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) globally length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.[3]
- ^ Bishop, Richard L. and Crittenden, Richard J. Geometry of Manifolds. AMS Chelsea Publishing, 2001, pp.224-225.
- ^ Hawking, Stephen; Ellis, George (1973). The large scale structure of space-time. Cambridge university press.
- ^ Cheeger, Ebin. Comparison Theorems in Riemannian Geometry. North-Holland Publishing Company, 1975, pp. 17-18.