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
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 ,
Then the curve
which lies entirely in the cone
over the great circle
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.