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A gravitational lens produces an odd number of images. (English) Zbl 0569.53043
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.

##### MSC:
 53Z05 Applications of differential geometry to physics 53C80 Applications of global differential geometry to the sciences 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 83C99 General relativity
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