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
)
↦
(
u
′
,
v
′
)
. 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
=
δ
V
=
0
|
(
s
0
,
w
0
)
. Then we can directly apply the Poincaré–Hopf theorem
χ
=
∑
index
D
=
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.,
χ
=
n
+
−
n
−
=
1
. So the total number of images is
N
=
n
+
+
n
−
=
2
n
−
+
1
, i.e., odd. The strict proof needs Uhlenbeck’s Morse theory of null geodesics.
