On local conditions for multiple imaging by bounded, smooth gravitational lenses

The conditions on the local value of the projected surface density, Σ, of a smooth and bounded gravitational lens, which determine whether such a lens is capable of producing multiple images are derived. It is shown that if Σ exceeds a critical value Σ_c at some point on such a lens then the lens can produce multiple images. The value of Σ_c depends only on fundamental constants and the distance scale of the lensing. The necessary conditions for lenses which have spherical symmetry or elliptical symmetry are given. It is also shown that there is no general lower bound on the value of Σ necessary for multiple imaging.