Geometrization of gravito-electromagnetic interactions from boundary conditions in the variational principle

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.