The odd number theorem is a theorem in differential topology about strong gravitational lensing, which states that the number of multiple images produced by a bounded transparent lens must be odd.
Argument The gravitational lensing is a mapping from the image plane to the source plane: M : ( u , v ) ↦ ( u ′ , v ′ ) {\displaystyle M:(u,v)\mapsto (u',v')} . If we use direction cosines describing the bent light rays, we can write a vector field on ( u , v ) {\displaystyle (u,v)} plane V : ( s , w ) {\displaystyle V:(s,w)} . However, only in some specific directions V 0 : ( s 0 , w 0 ) {\displaystyle V_{0}:(s_{0},w_{0})} , will the bent light rays reach the observer, i.e., the images only form where D = δ V = 0 | ( s 0 , w 0 ) {\displaystyle D=\delta V=0|_{(s_{0},w_{0})}} . Then we can directly apply the Poincaré–Hopf theorem χ = ∑ index D = constant {\displaystyle \chi =\sum {\text{index}}_{D}={\text{constant}}} . The index of sources and sinks is +1, and that of saddle points is −1. So the Euler characteristic equals the difference between the number of positive indices n + {\displaystyle n_{+}} and the number of negative indices n − {\displaystyle n_{-}} . For the far field case, there is only one image, i.e., χ = n + − n − = 1 {\displaystyle \chi =n_{+}-n_{-}=1} . So the total number of images is N = n + + n − = 2 n − + 1 {\displaystyle N=n_{+}+n_{-}=2n_{-}+1} , i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.
References
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