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.