Trace of the extrinsic-curvature tensor and dynamics of Einstein equations

Spaces of the general theory of relativity such that families of hypersurfa ces are characterized by zero trace of the extrinsic-curvature tensor and t hat these hypersurfaces are closed or asymptotically flat are proven to be static. Owing to this circumstance, a Hamiltonian reduction based on the ex plicit solution of constraints with respect to the trace of the extrinsic-c urvature tensor can be used in the theory of gravity.