  After a study of the Hamiltonian group of gauge transformations of canonical
tetrad gravity on globally hyperbolic, asymptotically flat at spatial infinity,
spacetimes with Cauchy hypersurfaces $\Sigma_{\tau}$ diffeomorphic to $R^3, we
find the dependence of the cotriads on $\Sigma_{\tau}$ and of their momenta on
the six parameters associated with rotations and space diffeomorphisms. The
choice of 3-coordinates on $\Sigma_{\tau}$ is equivalent to the parametrization
of the cotriads with the last three degrees of freedom individuating the
3-geometries. The Shanmugadhasan canonical transformation, corresponding to the
choice of 3-orthogonal coordinates on $\Sigma_{\tau}$ and adapted to 13 of the
14 first class constraints, and the interpretation of the gauge transformations
are given. The gauge interpretation of tetrad gravity based on constraint
theory implies that a "Hamiltonian kinematical gravitational field" is an
equivalence class of pseudo-Riemannian spacetimes modulo the Hamiltonian group
of gauge transformations: it includes a conformal 3-geometry and all the
different 4-geometries (standard definition of a kinematical gravitational
field, $Riem M^4/Diff M^4$) connected to it by the gauge transformations. A
"Hamiltonian Einstein or dynamical gravitational field" is a kinematical one
which satisfies the Hamilton-Dirac equations generated by the ADM energy: it
coincides with the standard Einstein or dynamical gravitational field, namely a
4-geometry solution of Einstein's equations, since the Hilbert and ADM actions
both generate Einstein's equations so that the kinematical Hamiltonian gauge
transformations are dynamically restricted to the spacetime diffeomorphisms of
the solutions of Einstein's equations.
