A rigidity theorem for conformal immersions

The work of Elie Cartan suggests that results on isometric immersions might extend to conformal immersions once appropriate assumptions are made. This turns out to be the case for a result on isometric compositions. Given a hypersurface of a conformally flat submanifold, we provide sufficient conditions for a conformal immersion of the same manifold in any other conformally flat manifold to be locally a conformal composition. This means that the second immersion can be locally obtained composing the first one with a conformal immersion from the first ambient space into the second.