The functions $g_{0 \nu}$ and $\partial_t g_{0 \nu}$ are used to remove the gauge freedom and to satisfy the gauge conditions. The remaining metric coefficients can be identified with the metric $h_{ab}$ and the remaining time derivatives are related to the second fundamental form $\kappa_{ab} = \kappa^*_{ab} + \frac{1}{3}\,\kappa\,h_{ab}$ with $h$-trace free part $\kappa^*_{ab}$ and mean extrinsic curvature $\kappa$. Three of the six functions $h_{ab}$ can locally be disposed of by removing the freedom to prescribe the coordinates $x^a$ and a further condition is imposed by the Hamiltonian constraint. Of the six functions $\kappa_{ab}$ the mean extrinsic curvature can be thought of as fixing the shape of the embedding of ${\cal S}$ into the solution space-time while the trace free part $\kappa^*_{ab}$ is restricted by the three momentum constraints. In both cases there remains a freedom of two functions which corresponds to the two degrees of freedom of the gravitational field.

While these considerations gives some insight into the structure and the role of the data it should be said that they are in fact very crude. For instance, it is known that a prescribed mean extrinsic curvature (and possibly a space-like surface $\Sigma$) can be used to determine an 