@BS]), a standard approach in which these issues do not appear. More sofisticated structures of the extended manifold $\cN$ have been considered in [@MAIA], in an attempt to give the background metric the status of a gauge invariant quantity.

Coming back to our construction, if a tensor $T_{\lambda,\Omega}$ is given on each $\cM_{\lambda,\Omega}$, we have that a tensor field $T$ is automatically defined on $\cN$ by the relation $T(p,\lambda,\Omega):= T_{\lambda,\Omega}(p)$, with $p\in\cM_{\lambda,\Omega}$.[^5] In particular, on each $\cM_{\lambda,\Omega}$ one has a metric $g_{\lambda,\Omega}$ and a set of matter fields $\tau_{\lambda,\Omega}$, satisfying the set of field equations \[g\_[,]{},\_[,]{}\]=0. Correspondingly, the fields $g$ and $\tau$ are defined on $\cN$.

We now want to define the perturbation in any tensor $T$, therefore we must find a way to compare $T_{\lambda,\Omega}$ with $T_0$: this requires a prescription for identifying points of $\cM_{\lambda,\Omega}$ with those of $\cM_0$. This is easily accomplished by assigning a diffeomorphism $\varphi_{\lambda,\Omega}:\cN\rightarrow\cN$ such that $\left.\varphi_{\lambda,\Omega}\right|_{\cM_0}:\cM_0
\rightarrow\cM_{\lambda,\Omega}$. Clearly, $\varphi_{\lambda,\Omega}$ can be regarded as the member of a two–parameter group of diffeomorphisms $\varphi$ on $\cN$, corresponding to the values of $\lambda,\Omega$ of the group parameter. Therefore, we could equally well give the vector fields $^{\varphi}\eta,\,^{\varphi}\zeta$ that generate $\varphi$. In the chart introduced above, $^{\varphi}\eta^m=1$, 