The odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology. It says that the number of multiple images produced by a bounded transparent lens must be odd.
In fact, the gravitational lensing is a mapping from image plane to source plane M : ( u , v ) ↦ ( u ′ , v ′ ) . If we use direction cosines describing the bent light rays, we can write a vector field on ( u , v ) plane V : ( s , w ) . However, only in some specific directions 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 ) . Then we can directly apply the Poincaré–Hopf theorem χ = ∑ index D = 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 + and the number of negative indices n − . For the far field case, there is only one image, i.e., χ = n + − n − = 1 . So the total number of images is N = n + + n − = 2 n − + 1 , i.e., odd. The strict proof needs Uhlenbeck’s Morse theory of null geodesics.
