A gravitational lens produces an odd number of images

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Abstract
Rigorous results are given to the effect that a transparent gravitational lens produces an odd number of images. Suppose that p is an event and T the history of a light source in a globally hyperbolic space‐time (M,g). Uhlenbeck’s Morse theory of null geodesics is used to show under quite general conditions that if there are at most a finite number n of future‐directed null geodesics from T to p, then M is contractible to a point. Moreover, n is odd and 1/2 (n−1) of the images of the source seen by an observer at p have the opposite orientation to the source. An analogous result is noted for Riemannian manifolds with positive definite metric.