Gravitational lenses: odd or even images?

In this paper we first recall the Morse relations for light rays in vacuum. They can be obtained, under quite general assumptions, by the relativistic Fermat principle stated by Kovner and correctly proved by Perlick. They can be used for a global mathematical description of the gravitational lens effect. Afterwards we consider the well known thin lens model and we observe that the light rays having a geometrical index equal to 2 have a strong constraint about the surface density where they pass the deflector. Finally, we note that in situations similar, for example, to the Einstein cross (whose models predict the existence of one light ray having its geometrical index equal to 2), the constraint on the surface density suggests that the absorption may be so high that the related image is quite difficult to detect.