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Odd number theorem
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The Odd number theorem is a theorem in 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 
. If we use direction cosines describing the bended light rays, we can write a vector field on 
plane 
. However, only in some specific directions 
, the bended light rays will reach the observer, i.e., the images only forms at where 
. Then we can directly apply 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 indice 
and the number of negative indice 
. 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.
However, Gottlieb (1994) argues that the conditions under which the theorem can be applied to gravitational lensing are very restrictive and he gives examples with an even number of images.
- Burke W.L., 1981, "Multiple gravitational imaging by distributed masses", Astrophysical Journal 244, L1.
- McKenzie R.H., 1985, "A gravitational lens produced an odd number of images", Journal of Mathematical Physics 26, 1592.
- Gottlieb D.H., 1994. "A gravitational lens need not produce an odd number of images", Journal of Mathematical Physics 35, 5507–5510.
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