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We derive the generalized Gauss-Codazzi-Mainardi (GCM) equation for a generalaffine connection with torsion and non-metricity. Moreover, we show that themetric compatibility and torsionless condition of a connection on a manifoldare inherited to the connection of its hypersurface. As a physical applicationto these results, we derive the (3+1)-Einstein Field Equation (EFE) for aspecial case of Metric-Affine f(R)-gravity when f(R)=R, the Metric-AffineGeneral Relativity (MAGR). Motivated by the concept of geometrodynamics, weintroduce additional variables on the hypersurface as a consequence ofnon-vanishing torsion and non-metricity. With these additional variables, weshow that for MAGR, the energy, momentum, and the stress-energy part of the EFEare dynamical, i.e., all of them contain the derivative of a quantity withrespect to the time coordinate. For the Levi-Civita connection, one couldrecover the Hamiltonian and the momentum (diffeomorphism) constraint, andobtain the standard dynamics of GR.

(3+1)-Formulation for Gravity with Torsion and Non-Metricity: The Stress-Energy-Momentum Equation