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 : formula_1. Argument. If we use direction cosines describing the bent light rays, we can write a vector field on formula_2 plane formula_3. However, only in some specific directions formula_4, will the bent light rays reach the observer, i.e., the images  only form where formula_5. Then we can directly apply the Poincaré–Hopf theorem formula_6. 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 formula_7 and the number of negative indices formula_8. For the far field case, there is only one image, i.e., formula_9. So the total number of images is formula_10, i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.