Two features of the local embedding problem of relativistic manifolds are discussed. It is first shown how preferred co-ordinates can be found, and a set of « potentials » introduced, such that the curved metric becomes unambiguously decomposed in a flat metric plus a genuine gravitational part. This decomposition of the metric holds locally, at least, over that extended space-time region corresponding to the co-ordinate patch of the local embedding. Next, the transformation group which preserves the form invariance of this decomposition of the metric is presented. It corresponds to a well-known enlarged Poincare group, with the characteristic property, however, that the « potentials » are transformed conjointly with the preferred co-ordinates.