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 :
.
Argument
If we use direction cosines describing the bent light rays, we can write a vector field on plane .
However, only in some specific directions , will the bent light rays reach the observer, i.e., the images only form where . Then we can directly apply the Poincaré–Hopf theorem .
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 and the number of negative indices . For the far field case, there is only one image, i.e., . So the total number of images is , i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.
References
Perlick V., Gravitational lensing from a geometric viewpoint, in B. Schmidt (ed.) "Einstein's field equations and their physical interpretations" Selected Essays in Honour of Jürgen Ehlers, Springer, Heidelberg (2000) pp. 373–425