?*]{}

Given two sets of Cauchy data one can ask the analogous question whether the corresponding maximal, globally hyperbolic developments in time admit an isometry which maps the respective emdedded data hypersurfaces onto each other. This may be quite difficult to decide in a concrete case but in principle there exists a clear criterion which only involves the two Cauchy data sets: they must be isometric. Unfortunately, such a straightforward answer seems not to be available in the general situation considered above.

Both the functions $\chi$ and $\chi'$ represent the mean extrinsic curvature of the hypersurface ${\cal T}$. They must be given, however, in the form $\chi = \chi(x^{\alpha})$ respectively $\chi' = \chi'(x^{\alpha'})$ with coordinates $x^{\alpha}$ and $x^{\alpha'}$, $\alpha, \alpha'  = 0, 1, 2$, which are different from each other on ${\cal T} \setminus \Sigma$ because $e_0 \neq e'_0$ there. To see that $\chi$ and $\chi'$ represent the same object the coordinate transformation $x^{\alpha} = x^{\alpha}(x^{\beta'})$ needs to be known. One might think of deriving this transformation by using the data $\Gamma$, $\Gamma'$ in the last two equations of (\[e0-eA-transport\]). But the boundary data do not contain any information on the metric induced on ${\cal T}$. The operators $D$ and 