  We study the conditions of integrability when the boundary terms are
considered in the variation of the geometric contribution of the
Einstein-Hilbert action. We explore the emergent physical dynamics that is
obtained when we make a displacement from a background Riemann manifold to an
extended one, on which the non-metricity is nonzero. Under these circumstances,
a classical description of electrodynamics and non-perturbative gravitational
waves are considered in the extended manifold, when we variate the action.
