Variational principle for gravity in the extended phase-space

Variational formalism in the extended phase space for fields is applied to gravity. It is shown that the requirement of invariance under arbitrary local inertial frames implies a coupling of torsion to a 3form of matter fields on the one hand and to a 3-form (related to the Einstein tensor) on the other. Gravitational dynamics is restricted to torsion zero surface in the extended phase space for Einstein-Hilbert action.