This fact should be plausible, and can be proved by a straightforward calculation as follows.

Parametrize the 2-sphere S2 of radius 1 by (longitude,co-latitude). Then the usual metric is

Note that the curves constant are great circles.

Parametrize C= the cone on S2 by where r is the distance from the cone point. Then the usual metric on C (with the cone point at distance k from the sphere) is

A curve will have coordinates for t in [0,1]. If the curve is differentiable, its length is

Suppose the curve joins two points in the cone over the great circle , i.e. that . Then the curve given by , which lies entirely in the cone over the great circle , will have same endpoints as and will be shorter, since all the terms in the length integral were positive, and the term will now be zero.

The path of a light ray is the curve of shortest length between its endpoints. So if is the path of a light ray between two points in the cone over a great circle, then must lie entirely in that cone.

Back to Gravitational Lensing and Geometric Lensing.

Tony Phillips
1999-01-23