choice in the context of perturbation theory. Clearly, this is more than the usual assignment of coordinate labels to points of a single spacetime [@sachs]. Furthermore, the correspondence established by relations such as (\[metricpert\]) is not unique, but rather (\[metricpert\]) typically defines a set of gauges, unless certain specific restrictions are satisfied by the fields involved (e.g., some metric components vanish). Leaving this problem aside, i.e. supposing that the gauge has been somehow completely fixed, let us look more precisely at the implications of (\[metricpert\]), adopting the geometrical description illustrated in Figure \[fig1\].

If we call ${\mbox{\boldmath $X$}}$ the chart on $\cM_0$ and ${\mbox{\boldmath $Z$}}$ the chart on $\cM$ we see that if we choose $z(o)=x(q)$, i.e. the correspondence between points of $\{g^{\rm b},\cM_{0}\}$ and $\{g,\cM\}$ implicit in (\[metricpert\]), we are implicitly defining a map $\varphi$ between $\cM_0$ and $\cM$, such that $\varphi={\mbox{\boldmath $Z$}}^{-1}\circ{\mbox{\boldmath $X$}}$. Thus from the geometrical point of view a gauge choice is an identification of points of $\cM_0$ and $\cM$. Therefore, we could as well start directly assigning the point identification map $\varphi$ first, calling $\varphi$ itself a gauge, and defining coordinates adapted to it later. This turns out to be a simpler way of proceeding 