in order to derive the gauge transformations in the following section.

Let us follow this idea in the specific case of two parameters introducing, in the spirit of [@stewart; @BMMS; @SB; @BS], an $(m+2)$–dimensional manifold $\cN$, foliated by $m$–dimensional submanifolds diffeomorphic to $\cM$, so that $\cN=\cM\times\RR^2$. We shall label each copy of $\cM$ by the corresponding value of the parameters $\lambda,\Omega$. The manifold $\cN$ has a natural differentiable structure which is the direct product of those of $\cM$ and $\RR^2$. We can then choose charts on $\cN$ in which $x^{\mu}$ ($\mu=0,1,\dots m-1$) are coordinates of each leaf $\cM_{\lambda,\Omega}$ and $x^m=\lambda,\,x^{m+1}=\Omega$.

In this construction we are assuming that the perturbed spacetimes have the same manifold as the background one. With this we are not allowing the possibility of addressing questions like how can perturbations affect the differential structure of the background spacetime. These issues would require the use of a much more complicated mathematical apparatus, in particular the notion of limits of spacetimes introduced by Geroch [@GEROCH] in order to define the background manifold as a limit $\lambda,\Omega\rightarrow 0$ of a family of [*perturbed*]{} manifolds $\cM_{\lambda,\Omega}$. Instead, we consider perturbations as fields [*living*]{} on the background (as in [@stewart; @BMMS; @SB; 