  In this work we prove the fact that, for a short time, it is possible to
construct a smooth parametrized family of isometric embeddings of an arbitrary
smooth parametrized family of Riemannian metrics on a smooth closed manifold
into an Euclidean space. In order to prove this statement we work out stability
estimates within Matthias G\"unther's local perturbation method to derive a
time-dependent local perturbation method around a free isometric embedding.
Iteratively, we use this time-dependent local perturbation method to construct
the desired family of isometric embeddings.
