Convexity and the finiteness of the number of geodesics. Applications to the multiple-image effect

We discuss the relations between the notion of convexity in a Riemannian fr amework and finiteness of the number of geodesics between two fixed points. The results obtained can be applied to the study of lightlike geodesics jo ining a point and a timelike curve in a conformally static Lorentzian manif old. These geodesics are interpreted as light rays between a light source a nd an event of a relativistic spacetime; thus our results may be used for a mathematical description of the gravitational lensing effect.