means by “perturbations of order $(k,k')$”.

Gauge invariance and gauge transformations {#gaugeiandt}
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Let us now suppose that two gauges $\varphi$ and $\psi$, described by pairs of vector fields $(^{\varphi}\eta,\,^{\varphi}\zeta)$ and $(^{\psi}\eta,\,^{\psi}\zeta)$ respectively, are defined on $\cN$, such that in the chart discussed above[^6]

[lclcl]{} \^\^[m]{} & = & \^\^m & = & 1 ,\
\^\^[m]{}& = & \^\^[m]{} & = & 0 ,\
\^\^[m+1]{} & = & \^\^[m+1]{} & = & 0 ,\
\^\^[m+1]{} & = & \^\^[m+1]{} & = & 1 . \[1010\]

Correspondingly, the integral curves of $(^{\varphi}\eta,\,^{\varphi}\zeta)$ and $(^{\psi}\eta,\,^{\psi}\zeta)$ define two two–parameter groups of diffeomorphisms $\varphi$ and $\psi$ on $\cN$, that connect any two leaves of the foliation. Thus, $(^{\varphi}\eta,\,^{\varphi}\zeta)$ and $(^{\psi}\eta,\,^{\psi}\zeta)$ are everywhere transverse to $\cM_{\lambda,\Omega}$ and points lying on the same integral surface of either of the two are to be regarded [*as the same point*]{} within the respective gauge: $\varphi$ and $\psi$ are both point identification maps, i.e. two different gauge choices.

The pairs of vector fields $(^{\varphi}\eta,\,^{\varphi}\zeta)$ and $(^{\psi}\eta,\,^{\psi}\zeta)$ can both be used to pull back a generic tensor $T$ and therefore to construct two other tensor fields $\varphi^*_{\lambda,\Omega}T$ and $\psi^*_{\lambda,\Omega}T$, for any given value of $(\lambda,\Omega)$. In particular, on $\cM_0$ we 