In this paper the authors analyze the structure of the Cauchy problem for general relativity by applying the theory of first-order symmetric hyperbolic systems. \par They show that (vacuum) Einstein equations can be split into an (elliptic) system of constraints and an evolutionary system. The evolutionary part is not symmetric hyperbolic in general. However, one can split the evolutionary part into a symmetric hyperbolic equation and a further constraint, which is noncovariant with respect to change of coordinates on the spatial manifold $S$. Hence, one can find spatial coordinates for which the antisymmetric part vanishes and, in those coordinates, solve the symmetric evolutionary part of the equations. For any initial condition that satisfies the elliptic constraints one can find a spatial metric $\gamma_{ij}(t, x) $ which together with a choice of the lapse $N$ and shift fields $\vec{N}$ defines a global Lorentzian metric $g$ that solves the original Einstein equations. \par This article may be recommended to researchers in the field of Cauchy problems in general relativity.