RESUMEN

            Let f: M→N be an immersion where, M, N are Riemannian manifolds with metrics respectively g and h. A natural problem is to study relations between g and h* = f*h, the induced metric on M via f. For instance we can try to determine if, under some assumptions on f and the manifolds (M,g), (N,h), the two metrics h* and g are conformally related or even more if h* = g, that is, if f is an isometry. In this note we give some sufficient conditions to positively answer the first problem and show that a slight strengthening of them provides a necessary and sufficient criterion to solve the second.