Odd number theorem

The odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology.

The theorem states that the number of multiple images produced by a bounded transparent lens must be odd.

Formulation
The gravitational lensing is a thought to mapped from what's known as image plane to source plane following the formula :

$$M: (u,v) \mapsto (u',v')$$.

Argument
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 $$ D=\delta V=0|_{(s_0,w_0)}$$. Then we can directly apply the Poincaré–Hopf theorem $$\chi=\sum \text{index}_D = \text{constant}$$.

The index of sources and sinks is +1, and that of saddle points is &minus;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., $$ \chi=n_{+}-n_{-}=1$$. So the total number of images is $$ N=n_{+}+n_{-}=2n_{-}+1 $$, i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.