  We consider $n+1$ dimensional smooth Riemannian and Lorentzian spaces
satisfying Einstein's equations. The base manifold is assumed to be smoothly
foliated by a one-parameter family of hypersurfaces. In both cases---likewise
it is usually done in the Lorentzian case---Einstein's equations may be split
into `Hamiltonian' and `momentum' constraints and a `reduced' set of field
equations. It is shown that regardless whether the primary space is Riemannian
or Lorentzian whenever the foliating hypersurfaces are Riemannian the
`Hamiltonian' and `momentum' type expressions are subject to a subsidiary first
order symmetric hyperbolic system. Since this subsidiary system is linear and
homogeneous in the `Hamiltonian' and `momentum' type expressions the
hyperbolicity of the system implies that in both cases the solutions to the
`reduced' set of field equations are also solutions to the full set of
equations provided that the constraints hold on one of the hypersurfaces
foliating the base manifold.
