  By using the York canonical basis of ADM tetrad gravity, in a formulation
using radar 4-coordinates for the parametrization of the 3+1 splitting of the
space-time, it is possible to write the 4-Riemann tensor of a globally
hyperbolic, asymptotically Minkowskian space-time as a Hamiltonian tensor,
whose components are 4-scalars with respect to the ordinary world
4-coordinates, plus terms vanishing due to Einstein's equations. Therefore
"on-shell" we find the expression of the Hamiltonian 4-Riemann tensor.
Moreover, the 3+1 splitting of the space-time used to define the phase space
allows us to introduce a Hamiltonian set of null tetrads and to find the
Hamiltonian expression of the 4-Ricci scalars of the Newman-Penrose formalism.
  This material will be used in the second paper to study the 4-Weyl tensor,
the 4-Weyl scalars and the four Weyl eigenvalues and to clarify the notions of
Dirac and Bergmann observables.
