It is well known that the Einstein equation is a system of nonlinear wave equations. The recently detected gravitational waves by LIGO are a direct consequence of the characteristic. In this lecture, we will see a formulation of the Einstein spacetimes as solutions to an elliptic variational problem, rather than a hyperbolic one, under the additional hypothesis of time-symmetry. We will consider spacetimes in dimensions four and five, as the latter provides a richer range of geometry.