Odd number theorem

Argument

If we use direction cosines describing the bent light rays, we can write a vector field on ${\displaystyle (u,v)}$ plane ${\displaystyle V:(s,w)}$.

However, only in some specific directions ${\displaystyle V_0:(s_0,w_0)}$, will the bent light rays reach the observer, i.e., the images only form where ${\displaystyle D=\delta V=0{|}_{(s_0,w_0)}}$. Then we can directly apply the Poincaré–Hopf theorem ${\displaystyle \chi =\sum {\text{index}}_D=\text{constant}}$.

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 ${\displaystyle n_{+}}$ and the number of negative indices ${\displaystyle n_{-}}$. For the far field case, there is only one image, i.e., ${\displaystyle \chi =n_{+}-n_{-}=1}$. So the total number of images is ${\displaystyle N=n_{+}+n_{-}=2n_{-}+1}$, i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.