  We obtain a system for the spatial metric and extrinsic curvature of a
spacelike slice that is hyperbolic non-strict in the sense of Leray and Ohya
and is equivalent to the Einstein equations. Its characteristics are the light
cone and the normal to the slice for any choice of lapse and shift functions,
and it admits a well-posed causal Cauchy problem in a Gevrey class of index
$\alpha=2$. The system becomes quasidiagonal hyperbolic if we posit a certain
wave equation for the lapse function, and we can then relate the results to our
previously obtained first order symmetric hyperbolic system for general
relativity.
