  A well-known result asserts that any isometric immersion with flat normal
bundle of a Riemannian manifold with constant sectional curvature into a space
form is (at least locally) holonomic. In this note, we show that this
conclusion remains valid for the larger class of Einstein manifolds. As an
application, when assuming that the index of relative nullity of the immersion
is a positive constant we conclude that the submanifold has the structure of a
generalized cylinder over a submanifold with flat normal bundle.
