We show that, given an arbitrary shift, the lapse N can be chosen so that
the extrinsic curvature K of the space slices with metric g​ in
arbitrary coordinates of a solution of Einstein's equations satisfies a
quasi-linear wave equation. We give a geometric first order symmetric
hyperbolic system verified in vacuum by g​, K and N. We show
that one can also obtain a quasi-linear wave equation for K by requiring N
to satisfy at each time an elliptic equation which fixes the value of the mean
extrinsic curvature of the space slices.Comment: 13 pages, latex, no figure