GEOMETRICAL WELL-POSED SYSTEMS FOR THE EINSTEIN EQUATIONS

We show that, given an arbitrary shift, the lapse N cap be chosen so t hat the extrinsic curvature K of the space slices with metric ($) over bar g of a solution of Einstein's equations satisfies a quasilinear w ave equation. We give a geometrical first order symmetric hyperbolic s ystem verified in vacuum by ($) over bar g, K and N. We show that one can also obtain a quasi-linear wave equation for K by requiring N to s atisfy at each time an elliptic equation which fixes the value of the mean extrinsic curvature of the space slices.