The odd number theorem is a theorem in differential topology about strong gravitational lensing, which states that the number of multiple images produced by a bounded transparent lens must be odd.

Argument

. If we use direction cosines describing the bent light rays, we can write a vector field.

. 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, i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.

References