Uniformly convex space

The odd number theorem is a theorem in strong 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 $M:(u,v)\mapsto (u^{\prime },v^{\prime })$. 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 −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.

References

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